4>EPA
United States
Environmental Protection
Agency
ProUCL Version 5.1
Technical Guide
Statistical Software for Environmental Applications
for Data Sets with and without Nondetect
Observations
RESEARCH AND DEVELOPMENT
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EPA/600/R-07/041
October 2015
www.epa.gov
ProUCL Version 5.1
Technical
Statistical Software for Environmental Applications
for Data Sets with and without Nondetect
Observations
Prepared for:
Felicia Barnett, Director
ORD Site Characterization and Monitoring Technical Support Center (SCMTSC)
Superfund and Technology Liaison, Region 4
U.S. Environmental Protection Agency
61 Forsyth Street SW, Atlanta, GA 30303
Prepared by:
Anita Singh, Ph.D.1 and Ashok K. Singh, Ph.D.2
1 Lockheed Martin/SERAS
IS&GS-CIVIL
2890 Woodbridge Ave
Edison NJ 08837
2Professor of Statistics, Department of Hotel Management
University of Nevada Las Vegas
Las Vegas, NV89154
U.S. Environmental Protection Agency
Office of Research and Development
Washington, DC 20460
Notice: Although this work was reviewed by EPA and approved for publication, it may not necessarily reflect official
Agency policy. Mention of trade names and commercial products does not constitute endorsement or
recommendation for use.
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NOTICE
The United States Environmental Protection Agency (U.S. EPA) through its Office of Research and
Development (ORD) funded and managed the research described in this ProUCL Technical Guide. It has
been peer reviewed by the U.S. EPA and approved for publication. Mention of trade names or
commercial products does not constitute endorsement or recommendation by the U.S. EPA for
use.
• All versions of the ProUCL software including the current version ProUCL 5.1 have been
developed by Lockheed Martin, IS&GS - CIVIL under the Scientific, Engineering, Response and
Analytical Services contract with the U.S. EPA and is made available through the U.S. EPA
Technical Support Center (TSC) in Atlanta, Georgia (GA).
• Use of any portion of ProUCL that does not comply with the ProUCL Technical Guide is not
recommended.
• ProUCL contains embedded licensed software. Any modification of the ProUCL source code
may violate the embedded licensed software agreements and is expressly forbidden.
• ProUCL software provided by the EPA was scanned with McAfee VirusScan version 4.5.1 SP1
and is certified free of viruses.
With respect to ProUCL distributed software and documentation, neither the U.S. EPA nor any of their
employees, assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of
any information, apparatus, product, or process disclosed. Furthermore, software and documentation are
supplied "as-is" without guarantee or warranty, expressed or implied, including without limitation, any
warranty of merchantability or fitness for a specific purpose.
ProUCL software is a statistical software package providing statistical methods described in various U.S.
EPA guidance documents. ProUCL does not describe U.S. EPA policies and should not be considered to
represent U.S. EPA policies.
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Minimum Hardware Requirements
ProUCL 5.1 will function but will run slowly and page a lot.
• Intel Pentium 1.0 gigahertz (GHz)
• 45 MB of hard drive space
• 512 MB of memory (RAM)
• CD-ROM drive or internet connection
• Windows XP (with SP3), Vista (with SP1 or later), or Windows 7.
ProUCL 5.1 will function but some titles and some Graphical User Interfaces (GUIs) will need to be
scrolled. Definition without color will be marginal.
• 800 by 600 Pixels
• Basic Color is preferred
Preferred Hardware Requirements
• 1 GHz or faster Processor.
• 1 gigabyte (GB) of memory (RAM)
• 1024 by 768 Pixels or greater color display
Software Requirements
ProUCL 5.1 has been developed in the Microsoft .NET Framework 4.0 using the C# programming
language. To properly run ProUCL 5.1 software, the computer using the program must have the .NET
Framework 4.0 pre-installed. The downloadable .NET Framework 4.0 files can be obtained from one of
the following websites:
• http://msdn.microsoft.com/netframework/downloads/updates/default.aspx
http://www.microsoft.com/en-us/download/details.aspx?id= 17851
Quicker site for 32 Bit Operating systems
• http://www.microsoft.com/en-us/download/details.aspx?id=24872
Use this site if you have a 64 Bit operating system
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Installation Instructions when Downloading ProUCL 5.1 from the EPA Web Site
• Download the file SETUP.EXE from the EPA Web site and save to a temporary location.
• Run the SETUP.EXE program. This will create a ProUCL directory and two folders:
1) The TECHNICAL GUIDE (this document), and 2) DATA (example data sets).
• To run the program, use Windows Explorer to locate the ProUCL application file, and
Double click on it, or use the RUN command from the start menu to locate the
ProUCL.exe file, and run ProUCL.exe.
• To uninstall the program, use Windows Explorer to locate and delete the ProUCL folder.
Caution: If you have previous versions of the ProUCL, which were installed on your computer, you
should remove or rename the directory in which earlier ProUCL versions are currently located.
Installation Instructions when Copying ProUCL 5.1 from a CD
• Create a folder named ProUCL 5.1 on a local hard drive of the machine you wish to
install ProUCL 5.1.
• Extract the zipped file ProUCL.zip to the folder you have just created.
• Run ProUCL.exe
Note: If you have extension turned off, the program will show with the name ProUCL in your directory
and have an Icon with the label ProUCL.
Creating a Shortcut for ProUCL 5.1 on Desktop
• To create a shortcut of the ProUCL program on your desktop, go to your ProUCL
directory and right click on the executable program and send it to desktop. A ProUCL
icon will be displayed on your desktop. This shortcut will point to the ProUCL directory
consisting of all files required to execute ProUCL 5.1.
Caution: Because all files in your ProUCL directory are needed to execute the ProUCL software, one
needs to generate a shortcut using the process described above. Simply dragging the ProUCL executable
file from Window Explorer onto your desktop will not work successfully (an error message will appear)
as all files needed to run the software are not available on your desktop. Your shortcut should point to the
directory path with all required ProUCL files. All ProUCL files should reside in one directory on your
computer (and not on your Network System) and your shortcut should point to that directory.
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ProUCL 5.1
Software ProUCL version 5.1 (ProUCL 5.1), its earlier versions: ProUCL version 3.00.01, 4.00.02,
4.00.04, 4.00.05, 4.1.00, 4.1.01, and ProUCL 5.0.00, associated Facts Sheet, User Guides and Technical
Guides (e.g., EPA 2010a, 2010b, 2013a, 2013b) can be downloaded from the following EPA website:
http://www.epa.gov/osp/hstl/tsc/software.htm
http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm
Material for ProUCL webinars offered in March 2011, and relevant literature used in the development of
various ProUCL versions can also be downloaded from the above EPA website.
Contact Information for all Versions of ProUCL
Since 1999, the ProUCL software has been developed under the direction of the Technical Support Center
(TSC). As of November 2007, the direction of the TSC is transferred from Brian Schumacher to Felicia
Barnett. Therefore, any comments or questions concerning all versions of ProUCL software should be
addressed to:
Felicia Barnett, Director
ORD Site Characterization and Monitoring Technical Support Center (SCMTSC)
Superfund and Technology Liaison, Region 4
U.S. Environmental Protection Agency
61 Forsyth Street SW, Atlanta, GA 30303-8960
barnett.fericia@epa.gov
(404)562-8659
Fax: (404) 562-8439
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EXECUTIVE SUMMARY
The main objective of the ProUCL software funded by the United States Environmental Protection
Agency (EPA) is to compute rigorous statistics to help decision makers and project teams in making good
decisions at a polluted site which are cost-effective, and protective of human health and the environment.
The ProUCL software is based upon the philosophy that rigorous statistical methods can be used to
compute reliable estimates of population parameters and decision making statistics including: the upper
confidence limit (UCL) of the mean, the upper tolerance limit (UTL), and the upper prediction limit
(UPL) to help decision makers and project teams in making correct decisions. A few commonly used text
book type methods (e.g., Central Limit Theorem [CLT], Student's t-UCL) alone cannot address all
scenarios and situations occurring in environmental studies. Since many environmental decisions are
based upon a 95 percent (%) UCL (UCL95) of the population mean, it is important to compute UCLs of
practical merit. The use and applicability of a statistical method (e.g., student's t-UCL, CLT-UCL,
adjusted gamma-UCL, Chebyshev UCL, bootstrap-t UCL) depend upon data size, data skewness, and
data distribution. ProUCL computes decision statistics using several parametric and nonparametric
methods covering a wide-range of data variability, distribution, skewness, and sample size. It is
anticipated that the availability of the statistical methods in the ProUCL software covering a wide range
of environmental data sets will help the decision makers in making more informative and correct
decisions at Superfund and Resource Conservation and Recovery Act (RCRA) sites.
It is noted that for moderately skewed to highly skewed environmental data sets, UCLs based on the CLT
and the Student's t-statistic fail to provide the desired coverage (e.g., 0.95) to the population mean even
when the sample sizes are as large as 100 or more. The sample size requirements associated with the CLT
increases with skewness. It would be incorrect to state that a CLT or Student's statistic based UCLs are
adequate to estimate Exposure Point Concentrations (EPC) terms based upon skewed data sets. These
facts have been described in the published documents (Singh, Singh, and Engelhardt [1997, 1999]; Singh,
Singh, and laci 2002; and Singh et al. 2006) summarizing simulation experiments conducted on
positively skewed data sets to evaluate the performances of the various UCL computation methods. The
use of a parametric lognormal distribution on a lognormally distributed data set yields unstable
impractically large UCLs values, especially when the standard deviation (sd) of the log-transformed data
becomes greater than 1.0 and the data set is of small size less than (<) 30-50. Many environmental data
sets can be modeled by a gamma as well as a lognormal distribution. The use of a gamma distribution on
gamma distributed data sets tends to yield UCL values of practical merit. Therefore, the use of gamma
distribution based decision statistics such as UCLs, UPLs, and UTLs should not be dismissed by stating
that it is easier to use a lognormal model to compute these upper limits.
The suggestions made in ProUCL are based upon the extensive experience of the developers in
environmental statistical methods, published environmental literature, and procedures described in many
EPA guidance documents. These suggestions are made to help the users in selecting the most appropriate
UCL to estimate the EPC term which is routinely used in exposure assessment and risk management
studies of the USEPA. The suggestions are based upon the findings of many simulation studies described
in Singh, Singh, and Engelhardt (1997, 1999); Singh, Singh, and laci (2002); and Singh et al. (2006). It
should be pointed out that a typical simulation study does not (cannot) cover all real world data sets of
various sizes and skewness from all distributions. When deemed necessary, the user may want to consult
a statistician to select an appropriate upper limit to estimate the EPC term and other environmental
parameters of interest. For an analyte (data set) with skewness (sd of logged data) near the end points of
the skewness intervals presented in decision tables of Chapter 2 (e.g., Tables 2-9 through 2-11), the user
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may select the most appropriate UCL based upon the site conceptual site model (CSM), expert site
knowledge, toxicity of the analyte, and exposure risks associated with that analyte.
The inclusion of outliers in the computation of the various decision statistics tends to yield inflated values
of those decision statistics, which can lead to poor decisions. Often statistics that are computed for a data
set which includes a few outliers tend to be inflated and represent those outliers rather than representing
the main dominant population of interest (e.g., reference area). Identification of outliers, observations
coming from population(s) other than the main dominant population is suggested, before computing the
decision statistics needed to address project objectives. The project team may want to perform the
statistical evaluations twice, once with outliers and once without outliers. This exercise will help the
project team in computing reliable and defensible decision statistics which are needed to make cleanup
and remediation decisions at polluted sites.
The initial development during 1999-2000 and all subsequent upgrades and enhancements of the ProUCL
software have been funded by U.S. EPA through its Office of Research and Development (ORD).
Initially ProUCL was developed as a research tool for U.S. EPA scientists and researchers of the
Technical Support Center (TSC) and ORD- National Exposure Research Laboratory (NERL), Las Vegas.
Background evaluations, groundwater (GW) monitoring, exposure and risk management and cleanup
decisions in support of the Comprehensive Environmental Recovery, Compensation, and Liability Act
(CERCLA) and RCRA site projects of the U.S. EPA are often derived based upon test statistics such as
the Shapiro-Wilk (S-W) test, t-test, Wilcoxon-Mann-Whitney (WMW) test, analysis of variance
(ANOVA), and Mann-Kendall (MK) test and decision statistics including UCLs of the mean, UPLs, and
UTLs. To address the statistical needs of the environmental projects of the USEPA, over the years
ProUCL software has been upgraded and enhanced to include many graphical tools and statistical
methods described in many EPA guidance documents including: EPA 1989a, 1989b, 1991, 1992a, 1992b,
2000 Multi-Agency Radiation Survey and Site Investigation Manual (MARSSIM), 2002a, 2002b, 2002c,
2006a, 2006b, and 2009. Several statistically rigorous methods (e.g., for data sets with nondetects [NDs])
not easily available in the existing guidance documents and in the environmental literature are also
available in ProUCL 5.0/ProUCL 5.1.
ProUCL 5.1/ProUCL 5.0 has graphical, estimation, and hypotheses testing methods for uncensored-full
data sets and for left-censored data sets including ND observations with multiple detection limits (DLs) or
reporting limits (RLs). In addition to computing general statistics, ProUCL 5.1 has goodness-of-fit (GOF)
tests for normal, lognormal and gamma distributions, and parametric and nonparametric methods
including bootstrap methods for skewed data sets for computation of decision making statistics such as
UCLs of the mean (EPA 2002a), percentiles, UPLs for a pre-specified number of future observations
(e.g., k with k=\, 2, 3,...), UPLs for mean of future k (>1) observations, and UTLs (e.g., EPA 1992b,
2002b, and 2009). Many positively skewed environmental data sets can be modeled by a lognormal as
well as a gamma model. It is well-known that for moderately skewed to highly skewed data sets, the use
of a lognormal distribution tends to yield inflated and unrealistically large values of the decision statistics
especially when the sample size is small (e.g., <20-30). For gamma distributed skewed uncensored and
left-censored data sets, ProUCL software computes decision statistics including UCLs, percentiles, UPLs
for future k (>1) observations, UTLs, and upper simultaneous limits (USLs).
For data sets with NDs, ProUCL has several estimation methods including the Kaplan-Meier (KM)
method, regression on order statistics (ROS) methods and substitution methods (e.g., replacing NDs by
DL, DL/2). ProUCL 5.1 can be used to compute upper limits which adjust for data skewness;
specifically, for skewed data sets, ProUCL computes upper limits using KM estimates in gamma
(lognormal) UCL and UTL equations provided the detected observations in the left-censored data set
follow a gamma (lognormal) distribution. Some poor performing commonly used and cited methods such
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as the DL/2 substitution method and H-statistic based UCL computation method have been retained in
ProUCL 5.1 for historical reasons, and research and comparison purposes.
The Sample Sizes module of ProUCL can be used to develop data quality objectives (DQOs) based
sampling designs and to perform power evaluations needed to address statistical issues associated with a
variety of site projects. ProUCL provides user-friendly options to enter the desired values for the decision
parameters such as Type I and Type II error rates, and other DQOs used to determine the minimum
sample sizes needed to address project objectives. The Sample Sizes module can compute DQO-based
minimum sample sizes needed: to estimate the population mean; to perform single and two-sample
hypotheses testing approaches; and in acceptance sampling to accept or reject a batch of discrete items
such as a lot of drums containing hazardous waste. Both parametric (e.g., t-test) and nonparametric (e.g.,
Sign test, WMW test, test for proportions) sample size determination methods are available in ProUCL.
ProUCL has exploratory graphical methods for both uncensored data sets and for left-censored data sets
consisting of ND observations. Graphical methods in ProUCL include histograms, multiple quantile-
quantile (Q-Q) plots, and side-by-side box plots. The use of graphical displays provides additional insight
about the information contained in a data set that may not otherwise be revealed by the use of estimates
(e.g., 95% upper limits) and test statistics (e.g., two-sample t-test, WMW test). In addition to providing
information about the data distributions (e.g., normal or gamma), Q-Q plots are also useful in identifying
outliers and the presence of mixture populations (e.g., data from several populations) potentially present
in a data set. Side-by-side box plots and multiple Q-Q plots are useful to visually compare two or more
data sets, such as: site-versus-background concentrations, surface-versus-subsurface concentrations, and
constituent concentrations of several GW monitoring wells (MWs). ProUCL also has a couple of classical
outlier test procedures, such as the Dixon test and the Rosner test which can be used on uncensored data
sets as well as on left-censored data sets containing ND observations.
ProUCL has parametric and nonparametric single-sample and two-sample hypotheses testing approaches
for uncensored as well as left-censored data sets. Single-sample hypotheses tests: Student's t-test, Sign
test, Wilcoxon Signed Rank test, and the Proportion test are used to compare site mean/median
concentrations (or some other threshold such as an upper percentile) with some average cleanup standard,
Cs (or a not-to-exceed compliance limit, Ao) to verify the attainment of cleanup levels (EPA 1989a;
MARSSIM/EPA 2000; EPA 2006a) at remediated site areas of concern. Single-sample tests such as the
Sign test and Proportion test, and upper limits including UTLs and UPLs are also used to perform intra-
well comparisons. Several two-sample hypotheses tests as described in EPA guidance documents (e.g.,
2002b, 2006b, 2009) are also available in the ProUCL software. The two-sample hypotheses testing
approaches in ProUCL include: Student's t-test, WMW test, Gehan test and Tarone-Ware (T-W) test. The
two-sample tests are used to compare concentrations of two populations such as site versus background,
surface versus subsurface soils, and upgradient versus downgradient wells.
The Oneway ANOVA module in ProUCL has both classical and nonparametric Kruskal-Wallis (K-W)
tests. Oneway ANOVA is used to compare means (or medians) of multiple groups such as comparing
mean concentrations of areas of concern and to perform inter-well comparisons. In GW monitoring
applications, the ordinary least squares (OLS) regression model, trend tests, and time series plots are used
to identify upwards or downwards trends potentially present in constituent concentrations identified in
wells over a certain period of time. The Trend Analysis module performs the M-K trend test and Theil-
Sen (T-S) trend test on data sets with missing values; and generates trend graphs displaying a parametric
OLS regression line and nonparametric T-S trend line. The Time Series Plots option can be used to
compare multiple time-series data sets.
The use of the incremental sampling methodology (ISM) has been recommended by the Interstate
Technology and Regulatory Council (ITRC 2012) for collecting ISM soil samples to compute mean
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concentrations of the decision units (DUs) and sampling units (SUs) requiring characterization and
remediation activities. At many polluted sites, a large amount of discrete onsite and/or offsite background
data are already available which cannot be directly compared with newly collected ISM data. In order to
provide a tool to compare the existing discrete background data with actual field onsite or background
ISM data, a Monte Carlo Background Incremental Sample Simulator (BISS) module was incorporated in
ProUCL 5.0 and retained in ProUCL 5.1 (currently blocked from general use) which may be used on a
large existing discrete background data set. The BISS module simulates incremental sampling
methodology based equivalent background incremental samples. The availability of a large discrete
background data set collected from areas with geological conditions comparable to the DU(s) of interest
is a pre-requisite for successful application of this module. For now, the BISS module has been blocked
for use as this module is awaiting adequate guidance and instructions for its intended use on discrete
background data sets.
ProUCL software is a user-friendly freeware package providing statistical and graphical tools needed to
address statistical issues described in many U.S. EPA guidance documents. ProUCL 5.0/ProUCL 5.1 can
process many constituents (variables) simultaneously to: perform statistical tests (e.g., ANOVA and trend
test statistics) and compute decision statistics including UCLs of mean, UPLs, and UTLs - a capability
not available in several commercial software packages such as Minitab 16 and NADA for R (Helsel
2013). ProUCL also has the capability of processing data by group variables. Significant efforts have
been made to make the software as user friendly as possible. For example, on the various GOF graphical
displays, output sheets for GOF tests, OLS and ANOVA, in addition to critical values and/or ^-values, the
conclusion derived based upon those values is also displayed. ProUCL is easy to use and does not require
any programming skills as needed when using commercial software packages and programs written in R.
Methods incorporated in ProUCL have been tested and verified extensively by the developers,
researchers, scientists, and users. The results obtained by ProUCL are in agreement with the results
obtained by using other software packages including Minitab, SAS®, and programs written in R Script.
ProUCL 5.0/ProUCL 5.1 computes decision statistics (e.g., UPL, UTL) based upon the KM method in a
straight forward manner without flipping the data and re-flipping the computed statistics for left-censored
data sets; these operations are not easy for a typical user to understand and perform. This can become
unnecessarily tedious when computing decision statistics for multiple variables/analytes. Moreover,
unlike survival analysis, it is important to compute an accurate estimate of the sd which is needed to
compute decision making statistics including UPLs and UTLs. For left-censored data sets, ProUCL
computes a KM estimate of sd directly. These issues are elaborated by examples discussed in this User
Guide and in the accompanying ProUCL 5.1 Technical Guide.
ProUCL does not represent a policy software of the government. ProUCL has been developed on limited
resources, and it does provide many statistical methods often used in environmental applications. The
objective of the freely available user-friendly software, ProUCL is to provide statistical and graphical
tools to address environmental issues of environmental site projects for all users including those users
who cannot or may not want to program and/or do not have access to commercial software packages.
Some users have criticized ProUCL and pointed out some deficiencies such as: it does not have
geostatistical methods; it does not perform simulations; and does not offer programming interface for
automation. Due to the limited scope of ProUCL, advanced methods have not been incorporated in
ProUCL. For methods not available in ProUCL, users can use other statistical software packages such as
SAS® (available to EPA personnel) and R script to address their computational needs. Contributions from
scientists and researchers to enhance methods incorporated in ProUCL will be very much appreciated.
Just like other government documents (e.g., U.S. EPA 2009), various versions of ProUCL (2007, 2009,
2011, 2013, 2016) also make some rule-of thumb type suggestions (e.g., minimum sample size
requirement of 8-10) based upon professional judgment and experience of the developers. It is
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recommended that the users/project team/agencies make their own determinations about the rule-of-
thumb type suggestions made in ProUCL before applying a statistical method.
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ACRONYMS and ABBREVIATIONS
ACL
A-D, AD
AL
AOC
ANOVA
Ao
BC
BCA
BD
BISS
BTV
CC,cc
CERCLA
CL
CLT
COPC
Cs
CSM
Df
DL
DL/2 (t)
DL/2 Estimates
DOE
DQOs
DU
EA
EOF
EM
EPA
EPC
Alternative compliance or concentration limit
Anderson-Darling test
Action limit
Area(s) of concern
Analysis of variance
Not to exceed compliance limit or specified action level
Box-Cox transformation
Bias-corrected accelerated bootstrap method
Binomial distribution
Background Incremental Sample Simulator
Background threshold value
Confidence coefficient
Comprehensive Environmental Recovery, Compensation, and Liability Act
Compliance limit
Central Limit Theorem
Contaminant/constituent of potential concern
Cleanup standards
Conceptual site model
Degrees of freedom
Detection limit
UCL based upon DL/2 method using Student's t-distribution cutoff value
Estimates based upon data set with NDs replaced by 1/2 of the respective detection
limits
Department of Energy
Data quality objectives
Decision unit
Exposure area
Empirical distribution function
Expectation maximization
United States Environmental Protection Agency
Exposure point concentration
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GA
GB
GHz
GROS
GOF, G.O.F.
GUI
GW
Hi
H0
H-UCL
i.i.d.
ISM
ITRC
k, K
K
K,k
khat
k star
KM (%)
KM (Chebyshev)
KM(t)
KM(z)
K-M, KM
K-S, KS
K-W
LCL
LN,/w
LCL
LPL
LROS
LTL
Georgia
Gigabyte
Gigahertz
Gamma ROS
Goodness-of-fit
Graphical user interface
Groundwater
Alternative hypothesis
Null hypothesis
UCL based upon Land's H-statistic
Independently and identically distributed
Incremental sampling methodology
Interstate Technology & Regulatory Council
Positive integer representing future or next k observations
Shape parameter of a gamma distribution
Number of nondetects in a data set
MLE of the shape parameter of a gamma distribution
Biased corrected MLE of the shape parameter of a gamma distribution
UCL based upon Kaplan-Meier estimates using the percentile bootstrap method
UCL based upon Kaplan-Meier estimates using the Chebyshev inequality
UCL based upon Kaplan-Meier estimates using the Student's t-distribution critical
value
UCL based upon Kaplan-Meier estimates using critical value of a standard normal
distribution
Kaplan-Meier
Kolmogorov-Smirnov
Kruskal Wallis
Lower confidence limit
Lognormal distribution
Lower confidence limit of mean
Lower prediction limit
LogROS; robust ROS
Lower tolerance limit
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LSL
M,m
MARS SIM
MCL
MOD
MDL
MK, M-K
ML
MLE
TV
N
MVUE
MW
NARPM
ND, nd, Nd
NERL
NRC
OKG
OLS
ORD
OSRTI
OU
PCA
PDF, pdf
.pdf
PRO
PROP
p-values
QA
QC
Q-Q
R,r
Lower simultaneous limit
Applied to incremental sampling: number in increments in an ISM sample
Multi-Agency Radiation Survey and Site Investigation Manual
Maximum concentration limit, maximum compliance limit
Minimum detectable difference
Method detection limit
Mann-Kendall
Maximum likelihood
Maximum likelihood estimate
Number of observations/measurements in a sample
Number of observations/measurements in a population
Minimum variance unbiased estimate
Monitoring well
National Association of Remedial Project Managers
Nondetect
National Exposure Research Laboratory
Nuclear Regulatory Commission
Orthogonalized Kettenring Gnanadesikan
Ordinary least squares
Office of Research and Development
Office of Superfund Remediation and Technology Innovation
Operating unit
Principal component analysis
Probability density function
Files in Portable Document Format
Preliminary remediation goals
Proposed influence function
Probability-values
Quality assurance
Quality
Quantile-quantile
Applied to incremental sampling: number of replicates of ISM samples
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RAGS
RCRA
RL
RMLE
ROS
RPM
RSD
RV
S
SCMTSC
SD, Sd, sd
SND
SNV
SE
SSL
SQL
su
s-w, sw
T-S
TSC
TW, T-W
UCL
UCL95
UPL
UPL95
U.S. EPA, EPA
UTL
UTL95-95
USGS
USL
vs.
WMW
Risk Assessment Guidance for Superfund
Resource Conservation and Recovery Act
Reporting limit
Restricted maximum likelihood estimate
Regression on order statistics
Remedial Project Manager
Relative standard deviation
Random variable
Substantial difference
Site Characterization and Monitoring Technical Support Center
Standard deviation
Standard Normal Distribution
Standard Normal Variate
Standard error
Soil screening levels
Sample quantitation limit
Sampling unit
Shapiro-Wilk
Theil-Sen
Technical Support Center
Tarone-Ware
Upper confidence limit
95% upper confidence limit
Upper prediction limit
95% upper prediction limit
United States Environmental Protection Agency
Upper tolerance limit
95% upper tolerance limit with 95% coverage
U.S. Geological Survey
Upper simultaneous limit
Versus
Wilcoxon-Mann-Whitney
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WRS Wilcoxon Rank Sum
WSR Wilcoxon Signed Rank
Xp pth percentile of a distribution
< Less than
> Greater than
> Greater than or equal to
< Less than or equal to
A Greek letter denoting the width of the gray region associated with hypothesis testing
Z Greek letter representing the summation of several mathematical quantities, numbers
% Percent
a Type I error rate
(3 Type II error rate
0 Scale parameter of the gamma distribution
Z Standard deviation of the log-transformed data
A carat sign over a parameter, indicates that it represents a statistic/estimate computed
using the sampled data
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GLOSSARY
Anderson-Darling (A-D) test: The Anderson-Darling test assesses whether known data come from a
specified distribution. In ProUCL the A-D test is used to test the null hypothesis that a sample data set, x\,
..., xn came from a gamma distributed population.
Background Measurements: Measurements that are not site-related or impacted by site activities.
Background sources can be naturally occurring or anthropogenic (man-made).
Bias: The systematic or persistent distortion of a measured value from its true value (this can occur
during sampling design, the sampling process, or laboratory analysis).
Bootstrap Method: The bootstrap method is a computer-based method for assigning measures of
accuracy to sample estimates. This technique allows estimation of the sample distribution of almost any
statistic using only very simple methods. Bootstrap methods are generally superior to ANOVA for small
data sets or where sample distributions are non-normal.
Central Limit Theorem (CLT): The central limit theorem states that given a distribution with a mean, u,
and variance, a2, the sampling distribution of the mean approaches a normal distribution with a mean (u)
and a variance o2/N as N, the sample size, increases.
Censored Data Sets: Data sets that contain one or more observations which are nondetects.
Coefficient of Variation (CV): A dimensionless quantity used to measure the spread of data relative to
the size of the numbers. For a normal distribution, the coefficient of variation is given by s/xBar. It is also
known as the relative standard deviation (RSD).
Confidence Coefficient (CC): The confidence coefficient (a number in the closed interval [0, 1])
associated with a confidence interval for a population parameter is the probability that the random interval
constructed from a random sample (data set) contains the true value of the parameter. The confidence
coefficient is related to the significance level of an associated hypothesis test by the equality: level of
significance = 1 - confidence coefficient.
Confidence Interval: Based upon the sampled data set, a confidence interval for a parameter is a random
interval within which the unknown population parameter, such as the mean, or a future observation, x0,
falls.
Confidence Limit: The lower or an upper boundary of a confidence interval. For example, the 95% upper
confidence limit (UCL) is given by the upper bound of the associated confidence interval.
Coverage, Coverage Probability: The coverage probability (e.g., = 0.95) of an upper confidence limit
(UCL) of the population mean represents the confidence coefficient associated with the UCL.
Critical Value: The critical value for a hypothesis test is a threshold to which the value of the test
statistic is compared to determine whether or not the null hypothesis is rejected. The critical value for any
hypothesis test depends on the sample size, the significance level, a at which the test is carried out, and
whether the test is one-sided or two-sided.
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Data Quality Objectives (DQOs): Qualitative and quantitative statements derived from the DQO
process that clarify study technical and quality objectives, define the appropriate type of data, and specify
tolerable levels of potential decision errors that will be used as the basis for establishing the quality and
quantity of data needed to support decisions.
Detection Limit: A measure of the capability of an analytical method to distinguish samples that do not
contain a specific analyte from samples that contain low concentrations of the analyte. It is the lowest
concentration or amount of the target analyte that can be determined to be different from zero by a single
measurement at a stated level of probability. Detection limits are analyte and matrix-specific and may be
laboratory-dependent.
Empirical Distribution Function (EOF): In statistics, an empirical distribution function is a cumulative
probability distribution function that concentrates probability \ln at each of the n numbers in a sample.
Estimate: A numerical value computed using a random data set (sample), and is used to guess (estimate)
the population parameter of interest (e.g., mean). For example, a sample mean represents an estimate of
the unknown population mean.
Expectation Maximization (EM): The EM algorithm is used to approximate a probability density
function (PDF). EM is typically used to compute maximum likelihood estimates given incomplete
samples.
Exposure Point Concentration (EPC): The constituent concentration within an exposure unit to which
the receptors are exposed. Estimates of the EPC represent the concentration term used in exposure
assessment.
Extreme Values: Values that are well-separated from the majority of the data set coming from the
far/extreme tails of the data distribution.
Goodness-of-Fit (GOF): In general, the level of agreement between an observed set of values and a set
wholly or partly derived from a model of the data.
Gray Region: A range of values of the population parameter of interest (such as mean constituent
concentration) within which the consequences of making a decision error are relatively minor. The gray
region is bounded on one side by the action level. The width of the gray region is denoted by the Greek
letter delta, A, in this guidance.
H-Statistic: Land's statistic used to compute UCL of mean of a lognormal population
H-UCL: UCL based on Land's H-Statistic.
Hypothesis: Hypothesis is a statement about the population parameter(s) that may be supported or
rejected by examining the data set collected for this purpose. There are two hypotheses: a null hypothesis,
(Ho), representing a testable presumption (often set up to be rejected based upon the sampled data), and an
alternative hypothesis (HA), representing the logical opposite of the null hypothesis.
Jackknife Method: A statistical procedure in which, in its simplest form, estimates are formed of a
parameter based on a set of N observations by deleting each observation in turn to obtain, in addition to
the usual estimate based on N observations, N estimates each based on N-l observations.
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Kolmogorov-Smirnov (KS) test: The Kolmogorov-Smirnov test is used to decide if a data set comes
from a population with a specific distribution. The Kolmogorov-Smirnov test is based on the empirical
distribution function (EDF). ProUCL uses the KS test to test the null hypothesis if a data set follows a
gamma distribution.
Left-censored Data Set: An observation is left-censored when it is below a certain value (detection limit)
but it is unknown by how much; left-censored observations are also called nondetect (ND) observations.
A data set consisting of left-censored observations is called a left-censored data set. In environmental
applications trace concentrations of chemicals may indeed be present in an environmental sample (e.g.,
groundwater, soil, sediment) but cannot be detected and are reported as less than the detection limit of the
analytical instrument or laboratory method used.
Level of Significance (a): The error probability (also known as false positive error rate) tolerated of
falsely rejecting the null hypothesis and accepting the alternative hypothesis.
Lilliefors test: A goodness-of-fit test that tests for normality of large data sets when population mean and
variance are unknown.
Maximum Likelihood Estimates (MLE): MLE is a popular statistical method used to make inferences
about parameters of the underlying probability distribution of a given data set.
Mean: The sum of all the values of a set of measurements divided by the number of values in the set; a
measure of central tendency.
Median: The middle value for an ordered set of n values. It is represented by the central value when n is
odd or by the average of the two most central values when n is even. The median is the 50th percentile.
Minimum Detectable Difference (MOD): The MDD is the smallest difference in means that the
statistical test can resolve. The MDD depends on sample-to-sample variability, the number of samples,
and the power of the statistical test.
Minimum Variance Unbiased Estimates (MVUE): A minimum variance unbiased estimator (MVUE or
MVU estimator) is an unbiased estimator of parameters, whose variance is minimized for all values of the
parameters. If an estimator is unbiased, then its mean squared error is equal to its variance.
Nondetect (ND) values: Censored data values. Typically, in environmental applications, concentrations
or measurements that are less than the analytical/instrument method detection limit or reporting limit.
Nonparametric: A term describing statistical methods that do not assume a particular population
probability distribution, and are therefore valid for data from any population with any probability
distribution, which can remain unknown.
Optimum: An interval is optimum if it possesses optimal properties as defined in the statistical literature.
This may mean that it is the shortest interval providing the specified coverage (e.g., 0.95) to the
population mean. For example, for normally distributed data sets, the UCL of the population mean based
upon Student's t distribution is optimum.
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Outlier: Measurements (usually larger or smaller than the majority of the data values in a sample) that
are not representative of the population from which they were drawn. The presence of outliers distorts
most statistics if used in any calculations.
Probability - Values (p-value): In statistical hypothesis testing, the p-value associated with an observed
value, ^observed of some random variable T used as a test statistic is the probability that, given that the null
hypothesis is true, T will assume a value as or more unfavorable to the null hypothesis as the observed
value Observed. The null hypothesis is rejected for all levels of significance, a greater than or equal to the p-
value.
Parameter: A parameter is an unknown or known constant associated with the distribution used to model
the population.
Parametric: A term describing statistical methods that assume a probability distribution such as a
normal, lognormal, or a gamma distribution.
Population: The total collection of N objects, media, or people to be studied and from which a sample is
to be drawn. It is the totality of items or units under consideration.
Prediction Interval: The interval (based upon historical data, background data) within which a newly
and independently obtained (often labeled as a future observation) site observation (e.g., onsite,
compliance well) of the predicted variable (e.g., lead) falls with a given probability (or confidence
coefficient).
Probability of Type II (2) Error (P): The probability, referred to as (3 (beta), that the null hypothesis will
not be rejected when in fact it is false (false negative).
Probability of Type I (1) Error = Level of Significance (a): The probability, referred to as a (alpha),
that the null hypothesis will be rejected when in fact it is true (false positive).
pth Percentile or pth Quantile: The specific value, Xp of a distribution that partitions a data set of
measurements in such a way that the p percent (a number between 0 and 100) of the measurements fall at
or below this value, and (100-p) percent of the measurements exceed this value, Xp.
Quality Assurance (QA): An integrated system of management activities involving planning,
implementation, assessment, reporting, and quality improvement to ensure that a process, item, or service
is of the type and quality needed and expected by the client.
Quality Assurance Project Plan: A formal document describing, in comprehensive detail, the necessary
QA, quality control (QC), and other technical activities that must be implemented to ensure that the
results of the work performed will satisfy the stated performance criteria.
Quantile Plot: A graph that displays the entire distribution of a data set, ranging from the lowest to the
highest value. The vertical axis represents the measured concentrations, and the horizontal axis is used to
plot the percentiles/quantiles of the distribution.
Range: The numerical difference between the minimum and maximum of a set of values.
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Regression on Order Statistics (ROS): A regression line is fit to the normal scores of the order statistics
for the uncensored observations and is used to fill in values imputed from the straight line for the
observations below the detection limit.
Resampling: The repeated process of obtaining representative samples and/or measurements of a
population of interest.
Reliable UCL: see Stable UCL.
Robustness: Robustness is used to compare statistical tests. A robust test is the one with good
performance (that is not unduly affected by outliers and underlying assumptions) for a wide variety of
data distributions.
Resistant Estimate: A test/estimate which is not affected by outliers is called a resistant test/estimate
Sample: Represents a random sample (data set) obtained from the population of interest (e.g., a site area,
a reference area, or a monitoring well). The sample is supposed to be a representative sample of the
population under study. The sample is used to draw inferences about the population parameter(s).
Shapiro-Wilk (SW) test: Shapiro-Wilk test is a goodness-of-fit test that tests the null hypothesis that a
sample data set, x\,..., x» came from a normally distributed population.
Skewness: A measure of asymmetry of the distribution of the parameter under study (e.g., lead
concentrations). It can also be measured in terms of the standard deviation of log-transformed data. The
greater the standard deviation, the greater is the skewness.
Stable UCL: The UCL of a population mean is a stable UCL if it represents a number of practical merit
(e.g., a realistic value which can occur at a site), which also has some physical meaning. That is, a stable
UCL represents a realistic number (e.g., constituent concentration) that can occur in practice. Also, a
stable UCL provides the specified (at least approximately, as much as possible, as close as possible to the
specified value) coverage (e.g., -0.95) to the population mean.
Standard Deviation (sd, sd, SD): A measure of variation (or spread) from an average value of the
sample data values.
Standard Error (SE): A measure of an estimate's variability (or precision). The greater the standard
error in relation to the size of the estimate, the less reliable is the estimate. Standard errors are needed to
construct confidence intervals for the parameters of interests such as the population mean and population
percentiles.
Substitution Method: The substitution method is a method for handling NDs in a data set, where the ND
is replaced by a defined value such as 0, DL/2 or DL prior to statistical calculations or graphical analyses.
This method has been included in ProUCL 5.1 for historical comparative purposes but is not
recommended for use. The bias introduced by applying the substitution method cannot be quantified
with any certainty. ProUCL 5.1 will provide a warning when this option is chosen.
Uncensored Data Set: A data set without any censored (nondetects) observations.
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Unreliable UCL, Unstable UCL, Unrealistic UCL: The UCL of a population mean is unstable,
unrealistic, or unreliable if it is orders of magnitude higher than the other UCLs of a population mean. It
represents an unpractically large value that cannot be achieved in practice. For example, the use of Land's
H-statistic often results in an impractically large inflated UCL value. Some other UCLs, such as the
bootstrap-t UCL and Hall's UCL, can be inflated by outliers resulting in an impractically large and
unstable value. All such impractically large UCL values are called unstable, unrealistic, unreliable, or
inflated UCLs.
Upper Confidence Limit (UCL): The upper boundary (or limit) of a confidence interval of a parameter
of interest such as the population mean.
Upper Prediction Limit (UPL): The upper boundary of a prediction interval for an independently
obtained observation (or an independent future observation).
Upper Tolerance Limit (UTL): A confidence limit on a percentile of the population rather than a
confidence limit on the mean. For example, a 95% one-sided UTL for 95% coverage represents the value
below which 95% of the population values are expected to fall with 95 % confidence. In other words, a
95% UTL with coverage coefficient 95% represents a 95% UCL for the 95th percentile.
Upper Simultaneous Limit (USL): The upper boundary of the largest value.
xBar: arithmetic average of computed using the sampled data values
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ACKNOWLEDGEMENTS
We wish to express our gratitude and thanks to our friends and colleagues who have contributed during
the development of past versions of ProUCL and to all of the many people who reviewed, tested, and
gave helpful suggestions throughout the development of the ProUCL software package. We wish to
especially acknowledge EPA scientists including Deana Crumbling, Nancy Rios-Jafolla, Tim Frederick,
Dr. Maliha Nash, Kira Lynch, and Marc Stiffleman; James Durant of ATSDR, Dr. Steve Roberts of
University of Florida, Dr. Elise A. Striz of the National Regulatory Commission (NRC), and Drs. Phillip
Goodrum and John Samuelian of Integral Consulting Inc. for testing and reviewing ProUCL 5.0 and its
associated guidance documents, and for providing helpful comments and suggestions. We also wish to
thank Dr. D. Beal of Leidos for reviewing ProUCL 5.0.
Special thanks go to Ms. Donna Getty and Mr. Richard Leuser of Lockheed Martin for providing a
thorough technical and editorial review of ProUCL 5.1 and also ProUCL 5.0 User Guide and Technical
Guide. A special note of thanks is due to Ms. Felicia Barnett of EPA ORD Site Characterization and
Monitoring Technical Support Center (SCMTSC), without whose assistance the development of the
ProUCL 5.1 software and associated guidance documents would not have been possible.
Finally, we wish to dedicate the ProUCL 5.1 (and ProUCL 5.0) software package to our friend and
colleague, John M. Nocerino who had contributed significantly in the development of ProUCL and Scout
software packages.
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of
NOTICE ii
Software Requirements iii
Installation Instructions when Downloading ProUCL 5.1 from the EPA Web Site iv
ProUCL5.1 v
Contact Information for all Versions of ProUCL v
EXECUTIVE SUM MARY vi
ACRONYMS and ABBREVIATIONS xii
GLOSSARY xvii
ACKNOWLEDGEMENTS xxiii
INTRODUCTION OVERVIEW OF ProUCL VERSION 5.1 SOFTWARE 1
The Need for ProUCL Software 6
ProUCL 5.1 Capabilities 9
ProUCL 5.1 User Guide 16
CHAPTER 1 Guidance on the Use of Statistical Methods in ProUCL Software 17
1.1 Background Data Sets 17
1.2 Site Data Sets 18
1.3 Discrete Samples or Composite Samples? 19
1.4 Upper Limits and Their Use 20
1.5 Point-by-Point Comparison of Site Observations with BTVs, Compliance Limits and
Other Threshold Values 22
1.6 Hypothesis Testing Approaches and Their Use 23
1.6.1 Single Sample Hypotheses (Pre-established BTVs and Not-to-Exceed Values are
Known) 23
1.6.2 Two-Sample Hypotheses (BTVs and Not-to-Exceed Values are Unknown) 24
1.7 Minimum Sample Size Requirements and Power Evaluations 25
1.7.1 Why a data set of minimum size, n = 8 throughlO? 26
1.7.2 Sample Sizes for Bootstrap Methods 27
1.8 Statistical Analyses by a Group ID 28
1.9 Statistical Analyses for Many Constituents/Variables 28
1.10 Use of Maximum Detected Value as Estimates of Upper Limits 28
1.10.1 Use of Maximum Detected Value to Estimate BTVs and Not-to-Exceed Values
29
1.10.2 Use of Maximum Detected Value to Estimate EPC Terms 29
1.10.2. IChebyshev Inequality Based UCL95 30
1.11 Samples with Nondetect Observations 30
1.11.1 Avoid the Use of the DL/2 Substitution Method to Compute UCL95 30
1.11.2 ProUCL Does Not Distinguish between Detection Limits, Reporting limits, or
Method Detection Limits 31
1.12 Samples with Low Frequency of Detection 31
1.13 Some Other Applications of Methods in ProUCL 5.1 32
1.13.1 Identification of COPCs 32
1.13.2 Identification of Non-Compliance Monitoring Wells 32
1.13.3 Verification of the Attainment of Cleanup Standards, Cs 33
1.13.4 Using BTVs (Upper Limits) to Identify Hot Spots 33
1.14 Some General Issues, Suggestions and Recommendations made by ProUCL 33
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1.14.1 Handling of Field Duplicates 33
1.14.2 ProUCL Recommendation about ROS Method and Substitution (DL/2) Method
33
1.14.3 Unhandled Exceptions and Crashes in ProUCL 34
1.15 The Unofficial User Guide to ProUCL4 (Helsel and Gilroy 2012) 34
1.16 Box and Whisker Plots 41
CHAPTER 2 Goodness-of-Fit Tests and Methods to Compute Upper Confidence Limit of
Mean for Uncensored Data Sets without Nondetect Observations 45
2.1 Introduction 45
2.2 Goodness-of-Fit (GOF) Tests 47
2.2.1 Test Normality and Lognormality of a Data Set 48
2.2.7.7 Normal Quantile-quantile (Q-Q) Plot 48
2.2.7.2 Shapiro-Wilk (S-W) Test 49
2.2.1.3 Lilliefors Test 49
2.2.2 Gamma Distribution 50
2.2.2.7 Quantile-Quantile (Q-Q) Plot for a Gamma Distribution 57
2.2.2.2 Empirical Distribution Function (EDF)-Based Goodness-ofFit Tests. 51
2.3 Estimation of Parameters of the Three Distributions Incorporated in ProUCL 53
2.3.1 Normal Distribution 53
2.3.2 Lognormal Distribution 54
2.3.2.1 MLEs of the Parameters of a Lognormal Distribution 54
2.3.2.2 Relationship between Skewness and Standard Deviation, a 54
2.3.2.3 MLEs of the Quantiles of a Lognormal Distribution 56
2.3.2.4 MVUEs of Parameters of a Lognormal Distribution 57
2.3.3 Estimation of the Parameters of a Gamma Distribution 57
2.4 Methods for Computing aUCL of the Unknown Population Mean 60
2.4.1 (1-a)* 100 UCL of the Mean Based upon Student's t-Statistic 61
2.4.2 Computation of the UCL of the Mean of a Gamma, G (k, 9), Distribution 61
2.4.3 (1- a)* 100 UCL of the Mean Based Upon H-Statistic (H-UCL) 64
2.4.4 (1 - a)* 100 UCL of the Mean Based upon Modified-t-Statistic for Asymmetrical
Populations 72
2.4.5 (1 - a)* 100 UCL of the Mean Based upon the Central Limit Theorem 73
2.4.6 (1 - a)* 100 UCL of the Mean Based upon the Adjusted Central Limit Theorem
(Adjusted-CLT) 74
2.4.7 Chebyshev (1 - a)* 100 UCL of the Mean Using Sample Mean and Sample sd. 74
2.4.8 Chebyshev (1 - a)* 100 UCL of the Mean of a Lognormal Population Using the
MVUE of the Mean and its Standard Error 76
2.4.9 (1 - a)* 100 UCL of the Mean Using the Jackknife and Bootstrap Methods 77
2.4.9.1 (I- a) *1 00 UCL of the Mean Based upon the Jackknife Method 77
2.4.9.2 (1 - a) *100 UCL of the Mean Based upon the Standard Bootstrap
Method 78
2.4.9.3 (1 - a) *100 UCL of the Mean Based upon the Simple Percentile
Bootstrap Method 80
2.4.9.4 (1 - a) *100 UCL of the Mean Based upon the Bias-Corrected
Accelerated (BCA) Percentile Bootstrap Method 80
2.4.9.5 (1- a) *100 UCL of the Mean Based upon the Bootstrap-t Method 81
2.4.9.6 (1- a) *7 00 UCL of the Mean Based upon Hall's Bootstrap Method ...82
2.5 Suggestions and Summary 89
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2.5.1 Suggestions for Computing a 95% UCL of the Unknown Population Mean, (ii,
Using Symmetric and Positively Skewed Data Sets 90
2.5.1.1 Normally or Approximately Normally Distributed Data Sets 90
2.5.1.2 Gamma or Approximately Gamma Distributed Data Sets 91
2.5.1.3 Lognormally or Approximately Lognormally Distributed Skewed Data
Sets 92
2.5.1.4 Nonparametric Skewed Data Sets without a Discernible Distribution... 94
2.5.2 Summary of the Procedure to Compute a 95% UCL of the Unknown Population
Mean, \a\t Based upon Full Uncensored Data Sets without Nondetect
Observations 96
CHAPTER 3 Computing Upper Limits to Estimate Background Threshold Values Based
Upon Uncensored Data Sets without Nondetect Observations 98
3.1 Introduction 98
3.1.1 Description and Interpretation of Upper Limits used to Estimate BTVs 101
3.1.2 Confidence Coefficient (CC) and Sample Size 103
3.2 Treatment of Outliers 104
3.3 Upper p* 100% Percentiles as Estimates of BTVs 105
3.3.1 Nonparametric p*l00% Percentile 105
3.3.2 Normal p* 100% Percentile 106
3.3.3 Lognormalp* 100% Percentile 106
3.3.4 Gamma p* 100% Percentile 106
3.4 Upper Tolerance Limits 107
3.4.1 Normal Upper Tolerance Limits 107
3.4.2 Lognormal Upper Tolerance Limits 108
3.4.3 Gamma Distribution Upper Tolerance Limits 108
3.4.4 Nonparametric Upper Tolerance Limits 109
3.4.4.1 Determining the Order, r, of the Statistic, X(r), to Compute UTLp,(l-a)
110
3.4.4.2 Determining the Achieved Confidence Coefficient, CC achieve, Associated
WithX(r) 110
3.4.4.3 Determining the Sample Size 110
3.4.4.4 Nonparametric UTL Based upon the Percentile Bootstrap Method. ...Ill
3.4.4.5 Nonparametric UTL Based upon the Bias-Corrected Accelerated (BCA)
Percentile Bootstrap Method 777
3.5 Upper Prediction Limits 112
3.5.1 Normal Upper Prediction Limit 112
3.5.2 Lognormal Upper Prediction Limit 112
3.5.3 Gamma Upper Prediction Limit 113
3.5.4 Nonparametric Upper Prediction Limit 113
3.5.4.1 Upper Prediction Limit Based upon the Chebyshev Inequality 114
3.5.5 Normal, Lognormal, and Gamma Distribution based Upper Prediction Limits for
k Future Comparisons 114
3.5.6 Proper Use of Upper Prediction Limits 115
3.6 Upper Simultaneous Limits 115
3.6.1 Upper Simultaneous Limits for Normal, Lognormal and Gamma Distributions
116
CHAPTER 4 Computing Upper Confidence Limit of the Population Mean Based upon
Left-Censored Data Sets Containing Nondetect Observations 125
4.1 Introduction 125
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4.2 Pre-processing a Data Set and Handling of Outliers 127
4.2.1 Assessing the Influence of Outliers and Disposition of Outliers 127
4.2.2 Avoid Data Transformation 127
4.2.3 Do Not Use DL/2(t) UCL Method 128
4.2.4 Minimum Data Requirement 128
4.3 Goodness-of-Fit (GOF) Tests and Skewness for Left-Censored Data Sets 128
4.4 Nonparametric Kaplan-Meier (KM) Estimation Method 129
4.5 Regression on Order Statistics (ROS) Methods 131
4.5.1 Computation of the Plotting Positions (Percentiles) and Quantiles 132
4.5.2 Computing OLS Regression Line to Impute NDs 133
4.5.2.1 Influence of Outliers on Regression Estimates and Imputed NDs 134
4.5.3 ROS Method for Lognormal Distribution 134
4.5.3.1 Fully Parametric Log ROS Method 134
4.5.3.2 Robust ROS Method on Log-Transformed Data 135
4.5.3.3 Gamma ROS Method 138
4.6 A Hybrid KM Estimates and Distribution of Detected Observations Based Approach to
Compute Upper Limits for Skewed Data Sets - New in ProUCL 5.0/ ProUCL 5.1 141
4.6.1 Detected Data Set Follows aNormal Distribution 142
4.6.2 Detected Data Set Follows a Gamma Distribution 142
4.6.3 Detected Data Set Follows a Lognormal Distribution 143
4.6.3.1 Issues Associated with the Use of Lognormal distribution to Compute a
UCL of Mean for Data Sets with Nondetects 148
4.6.3.1.1 Impact of Using DL and DL/2 for Nondetects on UCL95 Computations 149
4.6.3.1.2 Impact of Outlier, 16.1 ppb on UCL95 Computations 150
4.7 Bootstrap UCL Computation Methods for Left-Censored Data Sets 151
4.7.1 Bootstrapping Data Sets with Nondetect Observations 152
4.7.1.1 UCL of Mean Based upon Standard Bootstrap Method 153
4.7.1.2 UCL of Mean Based upon Bootstrap-t Method 154
4.7.1.3 Percentile Bootstrap Method 154
4.7.1.4 Bias-Corrected Accelerated (BCA) Percentile Bootstrap Procedure.. 154
4.8 (l-a)*100% UCL Based upon Chebyshev Inequality 155
4.9 Saving Imputed NDs Using Stats/Sample Sizes Module of ProUCL 158
4.10 Parametric Methods to Compute UCLs Based upon Left-Censored Data Sets 158
4.11 Summary and Suggestions 158
CHAPTER 5 Computing Upper Limits to Estimate Background Threshold Values Based
upon Data Sets Consisting of Nondetect (ND) Observations 163
5.1 Introduction 163
5.2 Treatment of Outliers in Background Data Sets with NDs 163
5.3 Estimating BTVs Based upon Left-Censored Data Sets 164
5.3.1 Computing Upper Prediction Limits (UPLs) for Left-Censored Data Sets 164
5.3.1.1 UPLs Based upon Normal Distribution of Detected Observations and
KM Estimates 164
5.3.1.2 UPL Based upon the Chebyshev Inequality 165
5.3.1.3 UPLs Based upon ROS Methods 165
5.3.1.4 UPLs when Detected Data are Gamma Distributed. 165
5.3.1.5 UPLs when Detected Data are Lognormally Distributed. 166
5.3.2 Computing Upper p* 100% Percentiles for Left-Censored Data Sets 166
5.3.2.1 Upper Percentiles Based upon Standard Normal Z-Scores 166
5.3.2.2 Upper Percentiles when Detected Data are Lognormally Distributed 167
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5.3.2.3 Upper Percentiles when Detected Data are Gamma Distributed. 167
5.3.2.4 Upper Percentiles Based upon ROSMethods 167
5.3.3 Computing Upper Tolerance Limits (UTLs) for Left-Censored Data Sets 168
5.3.3.1 UTLs Based on KM Estimates when Detected Data are Normally
Distributed 168
5.3.3.2 UTLs Based on KM Estimates when Detected Data are Lognormally
Distributed 168
5.3.3.3 UTLs Based on KM Estimates when Detected Data are Gamma
Distributed 168
5.3.3.4 UTLs Based upon ROS Methods 169
5.3.4 Computing Upper Simultaneous Limits (USLs) for Left-Censored Data Sets.. 169
5.3.4.1 USLs Based upon Normal Distribution of Detected Observations and
KM Estimates 169
5.3.4.2 USLs Based upon Lognormal Distribution of Detected Observations and
KM Estimates 169
5.3.4.3 USLs Based upon Gamma Distribution of Detected Observations and
KM Estimates 169
5.3.4.4 USLs Based upon ROS Methods 770
5.4 Computing Nonparametric Upper Limits Based upon Higher Order Statistics 181
CHAPTER 6 Single and Two-sample Hypotheses Testing Approaches 182
6.1 When to Use Single Sample Hypotheses Approaches 182
6.2 When to Use Two-Sample Hypotheses Testing Approaches 183
6.3 Statistical Terminology Used in Hypotheses Testing Approaches 184
6.3.1 Test Form 1 184
6.3.2 Test Form 2 185
6.3.3 Selecting a Test Form 185
6.3.4 Errors Rates and Confidence Levels 185
6.4 Parametric Hypotheses Tests 187
6.5 Nonparametric Hypotheses Tests 187
6.6 Single Sample Hypotheses Testing Approaches 188
6.6.1 The One-Sample t-Test for Mean 188
6.6.1.1 Limitations and Robustness of One-Sample t-Test 188
6.6.1.2 Directions for the One-Sample t-Test 189
6.6.1.3 P-values 189
6.6.1.4 Relation between One-Sample Tests and Confidence Limits of the Mean
or Median 190
6.6.2 The One-Sample Test for Proportions 191
6.6.2.1 Limitations and Robustness 797
6.6.2.2 Directions for the One-Sample Test for Proportions 797
6.6.2.3 Use of the Exact Binomial Distribution for Smaller Samples 7 93
6.6.3 The Sign Test 194
6.6.3.1 Limitations and Robustness 194
6.6.3.2 Sign Test in the Presence ofNondetects 194
6.6.3.3 Directions for the Sign Test 194
6.6.4 The Wilcoxon Signed Rank Test 196
6.6.4.1 Limitations and Robustness 196
6.6.4.2 Wilcoxon Signed Rank (WSR) Test in the Presence ofNondetects 797
6.6.4.3 Directions for the Wilcoxon Signed Rank Test 797
6.7 Two-sample Hypotheses Testing Approaches 202
XXX
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6.7.1 Student's Two-sample t-Test (Equal Variances) 202
6.7.1.1 Assumptions and their Verification 202
6.7.1.2 Limitations and Robustness 203
6.7.1.3 Guidance on Implementing the Student's Two-sample t-Test 203
6.7.1.4 Directions for the Student's Two-sample t-Test. 203
6.7.2 The Satterthwaite Two-sample t-Test (Unequal Variances) 204
6.7.2.1 Limitations and Robustness 205
6.7.2.2 Directions for the Satterthwaite Two-sample t-Test 205
6.8 Tests for Equality of Dispersions 206
6.8.1 The F-Test forthe Equality of Two-Variances 206
6.8.1.1 Directions for the F-Test. 206
6.9 Nonparametric Tests 209
6.9.1 The Wilcoxon-Mann-Whitney (WMW) Test 209
6.9.1.1 Advantages and Disadvantages 209
6.9.1.2 WMW Test in the Presence of Nondetects 209
6.9.1.3 WMW Test Assumptions and Their Verification 270
6.9.1.4 Directions for the WMWTestwhen the Number of Site and Background
Measurements is small (n <20 or m <20) 270
6.9.1.5 Directions for the WMW Test when the Number of Site and Background
Measurements is Large (n > 20 and m > 20) 272
6.9.2 GehanTest 216
6.9.2.1 Limitations and Robustness 216
6.9.2.2 Directions for the Gehan Testwhen m>10 andn > 70 216
6.9.3 Tarone-Ware (T-W) Test 218
6.9.3.1 Limitations and Robustness 218
6.9.3.2 Directions for the Tarone-Ware Testwhen m > 70 andn >70 218
CHAPTER 7 Outlier Tests for Data Sets with and without Nondetect Values 223
7.1 Outliers in Environmental Data Sets 224
7.2 Outliers and Normality 225
7.3 Outlier Tests for Data Sets without Nondetect Observations 225
7.3.1 Dixon'sTest 225
7.3.1.1 Directions for the Dixon's Test 226
7.3.2 Rosner's Test 227
7.3.2.1 Directions for the Rosner's Test 227
7.4 Outlier Tests for Data Sets with Nondetect Observations 228
CHAPTER 8 Determining Minimum Sample Sizes for User Specified Decision Parameters
and Power Assessment 233
8.1.1 Sample Size Formula to Estimate Mean without Considering Type II ((3) Error
Rate 235
8.1.2 Sample Size Formula to Estimate Mean with Consideration to Both Type I (a)
and Type II ((3) Error Rates 236
8.2 Sample Sizes for Single-Sample Tests 237
8.2.1 Sample Size for Single-Sample t-test (Assuming Normality) 237
8.2.1.1 Case I (Right-Sided Alternative Hypothesis, Form 1) 238
8.2.1.2 Case II (Left-Sided Alternative Hypothesis, Form 2) 238
8.2.1.3 Case III (Two-Sided Alternative Hypothesis) 239
8.2.2 Single Sample Proportion Test 240
8.2.2.1 Case I (Right-Sided Alternative Hypothesis, Form 1) 241
8.2.2.2 Case II (Left-Sided Alternative Hypothesis, Form 2) 241
XXXI
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8.2.2.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 241
8.2.3 Nonparametric Single-sample Sign Test (does not require normality) .............. 243
8.2.3.1 Case I (Right-Sided Alternative Hypothesis) ....................................... 243
8.2.3.2 Case II (Left-Sided Alternative Hypothesis) ........................................ 243
8.2.3.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 243
8.2.4 Nonparametric Single Sample Wilcoxon Sign Rank (WSR) Test ..................... 244
8.2.4.1 Case I (Right-Sided Alternative Hypothesis) ....................................... 244
8.2.4.2 Case II (Left-Sided Alternative Hypothesis) ........................................ 245
8.2.4.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 245
8.3 Sample Sizes for Two-Sample Tests for Independent Sample ....................................... 246
8.3.1 Parametric Two-sample t-test (Assuming Normality) ....................................... 246
8.3.1.1 Case I (Right-Sided Alternative Hypothesis) ....................................... 246
8.3. 1.2 Case II (Left-Sided Alternative Hypothesis) ................................... 247
8.3.1.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 247
8.3.2 Wilcoxon-Mann-Whitney (WMW) Test (Nonparametric Test) ........................ 248
8.3.2.1 Case I (Right-Sided Alternative Hypothesis) ....................................... 248
8.3.2.2 Case II (Left-Sided Alternative Hypothesis) ........................................ 248
8.3.2.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 249
8.3.3 Sample Size forWMW Test Suggested by Noether(1987) ................................ 250
8.4 Acceptance Sampling for Discrete Objects [[[ 251
8.4.1 Acceptance Sampling Based upon Chi-square Distribution .............................. 252
8.4.2 Acceptance Sampling Based upon Binomial/Beta Distribution ........................ 252
CHAPTER 9 Oneway Analysis of Variance Module [[[ 254
9.1 Oneway Analysis of Variance (ANOVA) [[[ 254
9.1.1 General Oneway ANOVA Terminology [[[ 254
9.2 Classical Oneway ANOVA Model [[[ 255
9.3 Nonparametric Oneway ANOVA (Kruskal-Wallis Test) ............................................... 256
CHAPTER 10 Ordinary Least Squares Regression and Trend Analysis .......................... 260
10.1 Ordinary Least Squares Regression [[[ 260
70.7.7 Regression ANOVA Table [[[ 263
10.1.2 Confidence Interval and Prediction Interval around the Regression Line ......... 264
10.2 Trend Analysis [[[ 266
10.2.1 Mann-Kendall Test [[[ 267
10.2.1.1Large Sample Approximation for M-K Test. ......................................... 268
10.2.1. 2Step-by-Step Procedure to perform the Mann-Kendall Test. ................ 269
10.2.2 Theil - Sen Line Test [[[ 272
10. 2. 2. IStep-by-Step Procedure to Compute Theil-Sen Slope ........................... 273
273
1 0. 2. 2. 2Large Sample Inference for Theil - Sen Test Based upon Normal
Approximation [[[ 273
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REFERENCES 300
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INTRODUCTION
OVERVIEW OF ProUCL VERSION 5.1 SOFTWARE
The main objective of the ProUCL software funded by the U.S.EPA is to compute rigorous decision
statistics to help the decision makers in making reliable decisions which are cost-effective, and protective
of human health and the environment. The development of ProUCL software is based upon the
philosophy that rigorous statistical methods can be used to compute representative estimates of population
parameters (e.g., site mean, background percentiles) and accurate decision making statistics (including the
upper confidence limit [UCL] of the mean, upper tolerance limit [UTL], and upper prediction limit
[UPL]) which will assist decision makers and project teams in making sound decisions. The use and
applicability of a statistical method (e.g., student's t-UCL, Central Limit Theorem (CLT)-UCL, adjusted
gamma-UCL, Chebyshev UCL, bootstrap-t UCL) depend upon data size, data variability, data skewness,
and data distribution. ProUCL computes decision statistics using several parametric and nonparametric
methods covering a wide-range of data variability, skewness, and sample size. A couple of text book
methods described in most of the statistical text books (e.g., Hogg and Craig, 1995) based upon the
Student's t-statistic and the CLT alone cannot address all scenarios and situations commonly occurring in
environmental studies. It is incorrect to assume that Student's t-statistic and/or CLT based UCLs of mean
will provide the desired coverage (e.g., 0.95) to the population mean irrespective of the skewness of the
data set/population under consideration. These issues have been discussed in detail in Chapters 2 and 4 of
this Technical Guide. Several examples have been discussed throughout this guidance document and also
in the accompanying ProUCL 5.1 User Guide to elaborate on these issues.
The use of a parametric lognormal distribution on a lognormally distributed data set tends to yield
unstable impractically large UCL values, especially when the standard deviation of the log-transformed
data is greater than 1.0 and the data set is of small size such as less than 30-50 (Hardin and Gilbert 1993;
Singh, Singh, and Engelhardt 1997). Many environmental data sets can be modeled by a gamma as well
as a lognormal distribution. Generally, the use of a gamma distribution on gamma distributed data sets
yields UCL values of practical merit (Singh, Singh, and laci 2002). Therefore, the use of gamma
distribution based decision statistics such as UCLs, UPL, and UTLs cannot be dismissed just because it is
easier to use a lognormal model to compute these upper limits. The two distributions do not behave in a
similar manner. The advantages of computing the gamma distribution-based decision statistics are
discussed in Chapters 2 through 5 of this guidance document.
Since many environmental decisions are made based upon a 95% UCL of the population mean, it is
important to compute reliable UCLs and other decision making statistics of practical merit. In an effort to
compute stable UCLs of the population mean and other decision making statistics, in addition to
computing the Student's t statistic and the CLT based statistics (e.g., UCLs, UPLs), significant effort has
been made to incorporate rigorous statistical methods for computing UCLs (and other limits) in the
ProUCL software, covering a wide-range of data skewness and sample sizes (e.g., Singh, Singh, and
Engelhardt, 1997; Singh, Singh, and laci, 2002; and Singh, Singh, 2003). It is anticipated that the
availability of the statistical methods in the ProUCL software, which can be applied to a wide range of
environmental data sets, will help decision makers in making more informative, practical and sound
decisions.
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It is noted that even for skewed data sets, practitioners tend to use the CLT or Student's t-statistic based
UCLs of mean for "large" sample sizes of 25-30 (rule-of-thumb to use CLT). However, this rule-of-
thumb does not apply for moderately to highly skewed data sets, specifically when a (standard deviation
of the log-transformed data) starts exceeding 1. The large sample size requirement associated with the use
of the CLT depends upon the skewness of the data distribution under consideration. The large sample
requirement associated with CLT for the sample mean to follow an approximate normal distribution
increases with the data skewness; and for highly skewed data sets, even samples of size greater than
(>)100 may not be large enough for the sample mean to follow an approximate normal distribution. For
moderately skewed to highly skewed environmental data sets, as expected, UCLs based on the CLT and
the Student's t-statistic fail to provide the desired coverage of the population mean even when the sample
sizes are as large as 100 or more. These facts have been verified in the published simulation experiments
conducted on positively skewed data sets (e.g., Singh, Singh, and Engelhardt, 1997; Singh, Singh, and
laci, 2002); some graphs showing the simulation results are provided in Appendix B.
The initial development and all subsequent upgrades and enhancements of the ProUCL software have
been funded by the U.S. EPA through its Office of Research and Development (ORD). Initially ProUCL
was developed as a research tool for scientists and researchers of the Technical Support Center and ORD-
NERL, Las Vegas. During 1999-2001, the initial intent and objectives of developing the ProUCL
software (Version 1.0 and Version 2.0) were to provide a statistical research tool for EPA scientists which
can be used to compute theoretically sound 95% upper confidence limits (UCL95s) of the mean routinely
used in exposure assessment, risk management and cleanup decisions made at various CERCLA and
RCRA sites (EPA 1992a, 2002a). During 2002, the peer-reviewed ProUCL version 2.1 (with Chebyshev
inequality based UCLs) was released for public use. Several researchers have developed rigorous
parametric and nonparametric statistical methods (e.g., Johnson 1978; Grice and Bain 1980; Efron [1981,
1982]; Efron and Tibshirani 1993; Hall [1988, 1992]; Sutton 1993; Chen 1995; Singh, Singh, and
Engelhardt 1997; Singh, Singh, and laci 2002] to compute upper limits (e.g., UCLs) which adjust for data
skewness. Since Student's t-UCL, CLT-UCL, and percentile bootstrap UCL fail to provide the desired
coverage to the population mean of skewed distributions, several parametric (e.g., gamma distribution
based) and nonparametric (e.g., bias-corrected accelerated [BCA] bootstrap and bootstrap-t, Chebyshev
UCL) UCL computation methods which adjust for data skewness were incorporated in ProUCL versions
3.0 and 3.00.02 during 2003-2004. ProUCL version 3.00.02 also had graphical Q-Q plots and GOF tests
for normal, lognormal, and gamma distributions; capabilities to statistically analyze multiple variables
simultaneously were also incorporated in ProUCL 3.00.02 (EPA 2004).
It is important to compute decision statistics (e.g., UCLs, UTLs) which are cost-effective and protective
of human health and the environment (balancing between Type I and Type II errors), therefore, one
cannot dismiss the use of the better [better than t-UCL, CLT-UCL, ROS and KM percentile bootstrap
UCL, KM-UCL (t)] performing UCL computation methods including gamma UCLs and the various
bootstrap UCLs which adjust for data skewness. During 2004-2007, ProUCL was upgraded to versions
4.00.02, and 4.00.04. These upgrades included exploratory graphical (e.g., Q-Q plots, box plots) and
statistical (e.g., maximum likelihood estimation [MLE], KM, and ROS) methods for left-censored data
sets consisting of nondetect (NDs) observations with multiple DLs or RLs. For uncensored and left-
censored data sets, these upgrades provide statistical methods to compute upper limits: percentiles, UPLs
and UTLs needed to estimate site-specific background level constituent concentrations or background
threshold values (BTVs). To address statistical needs of background evaluation projects (e.g.,
MARSSIM/EPA 2000, EPA 2002b), several single-sample and two-sample hypotheses testing
approaches were also included in these ProUCL upgrades.
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During 2008-2010, ProUCL was upgraded to ProUCL 4.00.05. The upgraded ProUCL was enhanced by
including methods to compute gamma distribution based UPLs and UTLs (Krishnamoorthy, Mathew, and
Mukherjee 2008). The Sample Size module to compute DQOs-based minimum sample sizes, needed to
address statistical issues associated with environmental projects (e.g., EPA 2000, 2002c, 2006a, 2006b),
was also incorporated in ProUCL 4.00.05.
During 2009-2011, ProUCL 4.00.05 was upgraded to ProUCL 4.1 and 4.1.01. ProUCL 4.1 (2010) and
4.1.01 (2011) retain all capabilities of the previous versions of ProUCL software. Two new modules:
Oneway ANOVA and Trend Analysis were included in ProUCL 4.1. The Oneway ANOVA module has
both parametric and nonparametric ANOVA tests to perform inter-well comparisons. The Trend
Analysis module can be used to determine potential upward or downward trends present in constituent
concentrations identified in GW monitoring wells (MWs). The Trend Analysis module can compute
Mann-Kendall (MK) and Theil-Sen (T-S) trend statistics to determine upward or downward trends
potentially present in analyte concentrations. ProUCL 4.1 also has the OLS Regression module. In
ProUCL 4.1, some modifications were made in decision tables which are used to make suggestions
regarding the use of UCL95 for estimating EPCs. Specifically, based upon experience, developers of
ProUCL re-iterated that the use of a lognormal distribution for estimating EPCs and BTVs should be
avoided, as the use of the lognormal distribution tends to yield unrealistic and unstable values of decision
making statistics including UCLs, UPLs, and UTLs. This is especially true when the sample size is <20-
30 and the data set is moderately to highly skewed. During March 2011, webinars were presented
describing the capabilities and use of the methods available in ProUCL 4.1, which can be downloaded
from the EPA ProUCL website.
ProUCL version 5.0.00 (EPA 2013, 2014) represents an upgrade of ProUCL 4.1.01 (EPA June 2011)
which represents an upgrade of ProUCL 4.1.00 (EPA 2010). For uncensored and left-censored data sets,
ProUCL 5.0.00 (ProUCL 5.0) contains all statistical and graphical methods that were available in the
previous versions of the ProUCL software package except for some poor performing and restricted (e.g.,
can be used only when a single detection limit is present) estimation methods such as the MLE and
winsorization methods for left-censored data sets. ProUCL has GOF tests for normal, lognormal, and
gamma distributions for uncensored and left-censored data sets with NDs. ProUCL 5.0 has the extended
version of the Shapiro-Wilk (S-W) test to perform normal and lognormal GOF tests for data sets of sizes
up to 2000 (Royston [1982, 1982a]). In addition to normal and lognormal distribution- based decision
statistics, ProUCL software computes UCLs, UPLs, and UTLs based upon the gamma distribution.
Several enhancements were made in the UCLs/EPCs and Upper Limits/BTVs modules of the ProUCL
5.0 software. A new statistic, an upper simultaneous limit (USL) (Singh and Nocerino 2002; Wilks 1963)
has been incorporated in the Upper Limits/BTVs module of ProUCL 5.0 for data sets consisting of NDs
with multiple DLs. A two-sample hypothesis test, the Tarone-Ware (T-W; Tarone and Ware, 1978) test
has also been incorporated in ProUCL 5.0. Nonparametric tolerance limits have been enhanced, and for
specific values of confidence coefficients, coverage probability, and sample size, ProUCL 5.0 outputs the
confidence coefficient (CC) actually achieved by a UTL. The Trend Analysis and OLS Regression
modules can handle missing events when computing trend test statistics and generating trend graphs.
Some new methods using KM estimates in gamma (and lognormal) distribution-based UCL, UPL, and
UTL equations have been incorporated to compute the decision statistics for data sets consisting of
nondetect observations. To facilitate the computation of UCLs from ISM based samples (ITRC 2012); the
minimum sample size requirement has been lowered to 3, so that one can compute the UCL95 based upon
ISM data sets of sizes >3.
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All known bugs, typographical errors, and discrepancies found by the developers and users of the
ProUCL software package were addressed in ProUCL version 5.0.00. Specifically, a discrepancy found
in the estimate of mean based upon the KM method was fixed in ProUCL 5.0. Some changes were made
in the decision logic used in the Goodness of Fit and UCLs/EPCs modules. In practice, based upon a
given data set, it is well known that the two statistical tests (e.g., T-S and OLS trend tests) can lead to
different conclusions. To streamline the decision logic associated with the computation of the various
UCLs, the decision tables in ProUCL 5.0 were updated. Specifically, for each distribution if at least one
of the two GOF tests (e.g., Shapiro-Wilk or Lilliefors test for normality) determines that the hypothesized
distribution holds, then ProUCL concludes that the data set follows the hypothesized distribution, and
decision statistics are computed accordingly. Additionally, for gamma distributed data sets, ProUCL 5.0
suggests the use of the: adjusted gamma UCL for samples of sizes < 50 (instead of 40 suggested in
previous versions); and approximate gamma UCL for samples of sizes >50.
Also, for samples of larger sizes (e.g., with n > 100) and small values of the gamma shape parameter, k
(e.g., k < 0.1), significant discrepancies were found in the critical values of the two gamma GOF test
statistics (Anderson-Darling [A-D] and Kolmogorov Smirnov [K-S] tests) obtained using the two gamma
deviate generation algorithms: Whitaker (1974) and Marsaglia and Tsang (2000). For values of k < 0.2,
the critical values of the two gamma GOF tests: A-D and K-S tests have been updated using the currently
available more accurate gamma deviate generation algorithm due to Marsaglia and Tsang's (2000); more
details about the implementation of their algorithm can be found in Kroese, Taimre, and Botev (2011).
For values of the shape parameter, k=0.025, 0.05, 0.1, and 0.2, the critical value tables for these two tests
were updated by incorporating the newly generated critical values for the three significance levels: 0.05,
0.1, and 0.01. The updated tables are provided in Appendix A of the ProUCL 5.0/ProUCL 5.1 Technical
Guide. It should be noted that for k=0.2, the older and the newly generated critical values are in general
agreement; therefore, critical values for k=0.2 were not replaced in tables summarized in Appendix A.
ProUCL 5.0 also has a new Background Incremental Sample Simulator (BISS) module (blocked for
general public use) which can be used on a large existing discrete background data set to simulate
background incremental samples. The availability of a large discrete data set collected from areas with
geological formations and conditions comparable to the DUs (background or onsite) of interest is a
requirement for successful application of this module. The simulated BISS data can be compared with the
actual field ISM (ITRC 2012) data collected from the various DUs using other modules of ProUCL 5.0.
The values of the BISS data are not directly available to users; however, the simulated BISS data can be
accessed by the various modules of ProUCL 5.0 to perform desired statistical evaluations. For example,
the simulated background BISS data can be merged with the actual field ISM data after comparing the
two data sets using a two-sample t-test; the simulated BISS or the merged data can be used to compute a
UCL of the mean or a UTL.
Note: The ISM methodology used to develop the BISS module is a relatively new approach; methods
incorporated in this BISS module requires further investigation. For now, the BISS module has been
blocked for use in ProUCL 5.0/ProUCL 5.1 as this module is awaiting adequate guidance and instructions
for its intended use on discrete background data sets.
ProUCL 5.0 is a user-friendly freeware package providing statistical and graphical tools needed to
address statistical issues described in several EPA guidance documents. Considerable effort was made to
provide a detailed technical guide to help practitioners understand the statistical methods needed to
address the statistical needs of their environmental projects. ProUCL generates detailed output sheets and
graphical displays for each method which can be used to educate students learning environmental
statistical methods. Like previous versions, ProUCL 5.0 can process many variables simultaneously to
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compute various tests (e.g., ANOVA and trend test statistics) and decision statistics including UCL of the
mean, UPLs, and UTLs, a capability not available in other software packages such as Minitab 16 and
NADA for R (Helsel 2013). Without the availability of this option, the user has to compute decision and
test statistics for one variable at a time which becomes cumbersome when dealing with a large number of
variables. ProUCL 5.0 also has the capability of processing data by groups. ProUCL 5.0 is easy to use; it
does not require any programming skills as needed when using programs written in R Script.
Deficiencies Identified in ProUCL 5.0: For ProUCL to be compatible with Microsoft Office 8 and
provide Excel-compatible Spreadsheet functionality (e.g., ability to input/output *.xlsx files), ProUCL 5.0
used FarPoint Spread 5 for .NET; and for graphics, ProUCL 5.0 used the development software package,
ChartFx 7. The look and feel of ProUCL 5.0 is quite different from its previous versions; all main menu
options were re-arranged. However, the use of upgraded development softwares resulted in some
problems. Specifically, it takes an unacceptably long time to save large ProUCL 5.0 generated output files
using FarPoint Spread 5. Also the use of ChartFx 7 caused some problems in properly labeling axes for
histograms. Additionally, some unhandled exceptions and crashes were noted by users. The unhandled
exceptions were mainly noted for "bad" data sets including data sets not following ProUCL input format;
data sets with not enough observations; and data sets with not enough detects.
ProUCL 5.1: ProUCL 5.1 represents an upgrade of ProUCL 5.0 to address deficiencies identified in
ProUCL 5.0. ProUCL 5.1 retains all capabilities of ProUCL 5.0 as described above. All modules in
ProUCL 5.1, and their look and feel is the same as in ProUCL 5.0. In this document, any statement made
about the capabilities of ProUCL 5.0 also apply to ProUCL version 5.1; and to save time, not all screen
shots used in ProUCL 5.0 manuals have been replaced in the ProUCL 5.1 User Guide and Technical
Guide. Upgrades in ProUCL 5.1 (not available in earlier versions) have been labeled as New in ProUCL
5.1 in this document.
All known bugs, crashes, and unhandled exceptions (e.g., on bad data sets) found in ProUCL 5.0 have
been addressed in ProUCL 5.1. In ProUCL 5.1, some enhancements have been made in the Trend
Analysis option of the Statistical Test module of ProUCL 5.1. ProUCL 5.1 computes and outputs
residuals for the non-parametric T-S trend line which may be helpful to compute a prediction band around
the T-S trend line. In addition to generating Q-Q plots based upon detected observations, the Goodness of
Fit Tests option of the Statistical Tests module of ProUCL 5.1 generates censored probability plots for
data sets with NDs. Some changes have been made in the decision table used to make suggestions for
UCL selection based upon a gamma distribution. New licensing agreements were obtained for the
development softwares: FarPoint and ChartFx. Due to deficiencies present in the development software,
ProUCL 5.1 generated large output files still take a long time to be saved. However, there is a quick work
around to this problem, instead of saving the output sheet using ProUCL, one can copy the output
spreadsheet and save the copied output sheet using Excel. This operation can be carried out instantly.
Note about Histograms: ChartFx 7.0 has some inherent deficiencies, as a result labeling of bins along the
x-axis on a histogram is still not as desirable as one would like it to be. The x-axis display will start from
zero instead of the proper lowest histogram value. Occurrences are rare but they can occur. Some tools
have been added in ProUCL 5.1, and relevant statistics (e.g., start point, midpoint, and end point) of a
histogram bar can be displayed by hovering the cursor on that bar.
Software ProUCL version 5.1, its earlier versions: ProUCL version 3.00.02, 4.00.02, 4.00.04, 4.1.00,
4.1.01 and ProUCL 5.0, associated Facts Sheet, User Guides and Technical Guides (e.g., EPA [2004,
2007, 2009a, 2009b, 2010a, 2010b, 2013a, 2013b]) can be downloaded from the EPA website:
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http://www.epa.gov/osp/hstl/tsc/software.htm
http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm
The Need for ProUCL Software
EPA guidance documents (e.g., EPA [1989a, 1989b, 1992a, 1992b, 1994, 1996, 2000, 2002a, 2002b,
2002c, 2006a, 2006b, 2009a, and 2009b]) describe statistical methods including: DQOs-based sample
size determination procedures, methods to compute decision statistics: UCL95, UPL, and UTLs,
parametric and nonparametric hypotheses testing approaches, Oneway ANOVA, OLS regression, and
trend determination approaches. Specifically, EPA guidance documents (2000, 2002c, 2006a, 2006b)
describe DQOs-based parametric and nonparametric minimum sample size determination procedures
needed: to compute decision statistics (e.g., UCL95); to perform site versus background comparisons
(e.g., t-test, proportion test, WMW test); and to determine the number of discrete items (e.g., drums filled
with hazardous material) that need to be sampled to meet the DQOs (e.g., specified proportion, po of
defective items, allowable error margin in an estimate of mean). Statistical methods are used to compute
test statistics (e.g., S-W test, t-test, WMW test, T-S trend statistic) and decision statistics (e.g., 95% UCL,
95% UPL, UTL95-95) needed to address statistical issues associated with CERCLA and RCRA site
projects. For example, exposure and risk management and cleanup decisions in support of EPA projects
are often made based upon the mean concentrations of the contaminants/constituents of potential concern
(COPCs). Site-specific BTVs are used in site versus background evaluation studies. A UCL95 is used to
estimate the EPC terms (EPA 1992a, 2002a); and upper limits such as upper percentiles, UPLs, or UTLs
are used to estimate BTVs or not-to-exceed values (EPA 1992b, 2002b, and 2009). The estimated BTVs
are used to address several objectives: to identify the COPCs; to identify the site areas of concern
(AOCs); to perform intra-well comparisons to identify MWs not meeting specified standards; and to
compare onsite constituent concentrations with site-specific background level constituent concentrations.
Oneway ANOVA is used to perform inter-well comparisons and OLS regression and trend tests are often
used to determine potential trends present in constituent concentrations identified in GW monitoring wells
(MWs). Most of the methods described in this paragraph are available in the ProUCL 5.1 (ProUCL 5.0)
software package.
It is noted that not much guidance is available in the guidance documents cited above to compute rigorous
UCLs, UPLs, and UTLs for moderately to highly skewed uncensored and left-censored data sets
containing NDs with multiple DLs, a common occurrence in environmental data sets. Several parametric
and nonparametric methods are available in the statistical literature (Singh, Singh, and Engelhardt 1997;
Singh, Singh, and laci 2002; Krishnamoorthy et al. 2008; Singh, Maichle, and Lee, 2006) to compute
UCLs and other upper limits which adjust for data skewness. During the years, as new methods became
available to address statistical issues related to environmental projects, those methods were incorporated
in ProUCL software so that environmental scientists and decision makers can make more accurate and
informed decisions. Until 2006, not much guidance was provided on how to compute UCL95s of the
mean and other upper limits (e.g., UPLs and UTLs) based upon data sets containing NDs with multiple
DLs. For data sets with NDs, Singh, Maichle, and Lee (2006) conducted an extensive simulation study to
compare the performances of the various estimation methods (in terms of bias in the mean estimate) and
UCL computation methods (in terms of coverage provided by a UCL). They demonstrated that the
nonparametric KM method performs well in terms of bias in estimates of mean. They also concluded that
UCLs computed using the Student's t-statistic and percentile bootstrap method using the KM estimates do
not provide the desired coverage to the population mean of skewed data sets. They demonstrated that
depending upon sample size and data skewness, UCLs computed using KM estimates, the BCA bootstrap
method (mildly skewed data sets), the bootstrap-t method, and the Chebyshev inequality (moderately to
highly skewed data sets) provide better coverage (closer to the specified 95% coverage) to the population
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mean than other UCL computation methods. Based upon their findings, during 2006-2007, several UCL
and other upper limits computation methods based upon KM and ROS estimates were incorporated in the
ProUCL 4.0 software. It is noted that since the inclusion of the KM method in ProUCL 4.0 (2007), the
use of the KM method based upper limits has become popular in many environmental applications to
estimate EPC terms and BTVs. The KM method is also described in the latest version of the unified
RCRA guidance document (U.S. EPA 2009).
It is not easy to justify distributional assumptions of data sets consisting of both detects and NDs with
multiple DLs. Therefore, based upon the published literature and experience, parametric UCL (and other
upper limits) computation methods such as the MLE method (Cohen 1991) and the expectation
maximization (EM) method (Gleit 1985) for normal and lognormal distributions were not included
ProUCL 5.0 (and ProUCL 5.1) even though these methods were available in earlier versions of ProUCL.
Additionally, the winsorization method (Gilbert 1987) available in an earlier version of ProUCL has also
been excluded from ProUCL 5.0 (ProUCL 5.1) due to its poor performance. During 2015, some
researchers (e.g., from New Mexico State University, Las Cruces, NM) suggested that the EM method
performs better than some of the methods available in ProUCL 5.0, especially the gamma ROS (GROS)
method; a method which can be used on left-censored data sets with multiple DLs. The literature has
articles dealing with MLE and EM methods for data sets with a single censoring point (DL). Further
research needs to be conducted on methods for computing reliable estimates of the mean, sd, and upper
limits based upon parametric MLE and EM methods for data sets with NDs and multiple DLs. As always,
it is the desire of the developers of ProUCL to incorporate the best available methods in ProUCL. The
developers of ProUCL welcome/encourage other researchers to share their findings about the EM method
showing that EM method performs better than methods already available in ProUCL 5.0/ProUCL 5.1 for
data sets with single/multiple censoring points. The developers of ProUCL have been enhancing the
ProUCL software with better performing methods as those methods become available. Efforts will be
made to incorporate contributed code (with acknowledgement) for superior methods in future versions of
ProUCL. ProUCL software is also used for teaching environmental statistics courses therefore, in addition
to statistical and graphical methods routinely used to address statistical needs of environmental projects,
some poor performing methods such as the substitution DL/2 method and Land's (1975) H-statistic based
UCL computation method have been retained in ProUCL version 5.1 for research and comparison
purposes.
Methods incorporated in ProUCL 5.1 and in its earlier versions have been tested and verified extensively
by the developers, researchers, scientists, and users. Specifically, the results obtained by ProUCL 5.1 are
in agreement with the results obtained by using other software packages including Minitab, SAS®, and
programs available in R-Script (not all methods are available in these software packages). Additionally,
like ProUCL 5.0, ProUCL 5.1 outputs several intermediate results (e.g., khat and biased corrected kstar
estimates of the gamma shape parameter, k, and critical values (e.g., tolerance factor, K, used to compute
UTLs; critical value, d2max, used to compute USL) needed to compute decision statistics of interest,
which may help interested users to verify statistical results computed by the ProUCL software. Whenever
applicable, ProUCL provides warning messages and based upon professional experience and findings of
simulation studies, makes suggestions to help a typical user in selecting the most appropriate decision
statistic (e.g., UCL).
Note: The availability of intermediate results and critical values can be used to compute lower limits and
two-sided intervals which are not as yet available in the ProUCL software.
For left-censored data sets, ProUCL 5.1 computes decision statistics (e.g., UCL, UPL, and UTL) based
upon KM estimates computed in a straight forward manner without flipping the data and re-flipping the
decision statistics; these operations are not easy for a typical user to understand and perform and can
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become quite tedious when multiple analytes need to be processed. Moreover, in environmental
applications it is important to compute accurate estimates of sd which are needed to compute decision
making statistics including UPLs and UTLs. Decision statistics (UPL, UTL) based upon a KM estimate
of the of sd and computed using indirect methods can be different from the statistics computed using an
estimate of sd obtained using the KM method directly, especially when one is dealing with a skewed data
set or when using a log-transformation. These issues are elaborated by examples discussed in the
accompanying ProUCL 5.1 Technical Guide.
For uncensored data sets, researchers (e.g., Johnson 1978; Chen 1995; Efron and Tibshirani 1993; Hall
[1988, 1992], and additional references found in Chapters 2 and 3) developed parametric (e.g., gamma
distribution based) and nonparametric (bootstrap-t and Hall's bootstrap method, modified-t) methods for
computation of decision statistics which adjust for data skewness. For uncensored positively skewed data
sets, Singh, Singh, and laci (2002) performed simulation experiments to compare the performances (in
terms of coverage probabilities) of the various UCL computation methods described in the literature.
They demonstrated that for skewed data sets, UCLs based upon Student's t statistic, central limit theorem
(CLT), and percentile bootstrap method tend to underestimate the population mean (EPC). It is reasonable
to state that the findings of the simulation studies performed on uncensored skewed data sets comparing
the performances of the various UCL computation methods can be extended to skewed left-censored data
sets. Based upon the findings of those studies performed on uncensored data sets and also using the
findings summarized in Singh, Maichle, and Lee (2006), it was concluded that t-statistic, CLT, and the
percentile bootstrap method based UCLs computed using KM estimates (and also ROS estimates)
underestimate the population mean of moderately skewed to highly skewed data sets. Interested users
may want to verify these statements by performing simulation experiments or other forms of rigorous
testing. Like uncensored skewed data sets, for left-censored data sets, ProUCL 5.1 offers several
parametric and nonparametric methods for computing UCLs and other limits which adjust for data
skewness.
Due to the lack of research and methods, in earlier versions of the ProUCL software (e.g., ProUCL
4.00.02, ProUCL 4.0), KM estimates were used in the normal distribution based equations for computing
the various upper limits for left-censored data sets. However, normal distribution based upper limits (e.g.,
t-UCL) using KM estimates (or any other estimates such as ROS estimates) fail to provide the specified
coverage (e.g., 0.95) of the parameters (e.g., mean, percentiles) of populations with skewed distributions
(Singh, Singh, laci 2002; Johnson 1978; Chen 1995). For skewed data sets, ProUCL 5.0/ProUCL 5.1
computes UCLs applying KM estimates in UCL equations for skewed data sets (e.g., gamma and
lognormal); therefore, some changes have been made in the decision tables of ProUCL 5.0/ProUCL 5.1
for computing UCL95s. Also, the nonparametric UCL computation methods (e.g., percentile bootstrap)
do not provide the desired coverage to the population means of skewed distributions (e.g., Hall [1988,
1992], Efron and Tibshirani, 1993). For example, the use of t-UCL or the percentile bootstrap UCL
method on robust ROS estimates or on KM estimates underestimates the population mean for moderately
skewed to highly skewed data sets. Chapters 3 and 5 of the ProUCL Technical Guide describe parametric
and nonparametric KM methods for computing upper limits (and available in ProUCL 5.0/ ProUCL 5.1)
which adjust for data skewness.
The KM method yields good estimates of the population mean and std (Singh, Maichle, and Lee2006);
however upper limits computed using the KM or ROS estimates in normal equations or in the percentile
bootstrap method do not account for skewness present in the data set. Appropriate UCL computation
methods which account for data skewness should be used on KM or ROS estimates. For left-censored
data sets, ProUCL 5.0/ProUCL 5.1 compute upper limits using KM estimates in gamma (lognormal)
UCL, UPL, and UTL equations (e.g., also suggested in U.S. EPA 2009) provided the detected
observations in the left-censored data set follow a gamma (lognormal) distribution.
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Recently, the use of the ISM methodology has been recommended (ITRC 2012) for collecting soil
samples with the purpose of estimating mean concentrations of DUs requiring analysis of human and
ecological risk and exposure. ProUCL can be used to compute UCLs based upon ISM data as described
and recommended in the ITRC ISM Technical and Regulatory Guide (2012). At many sites, large
amounts of discrete background data are already available which are not directly comparable to the actual
field ISM data (onsite or background). To compare the existing discrete background data with field ISM
data, the BISS module (blocked for general use in ProUCL version 5.1 awaiting guidance and instructions
for its intended use) of ProUCL 5.1 can be used on a large (e.g., consisting of at least 30 observations)
existing discrete background data set. The BISS module simulates the incremental sampling methodology
based equivalent incremental background samples; and each simulated BISS sample represents an
estimate of the mean of the population represented by the discrete background data set. The availability of
a large discrete background data set collected from areas with geological conditions comparable to the
DU(s) of interest (onsite DUs) is a requirement for successful application of this module. The user cannot
see the simulated BISS data; however, the simulated BISS data can be accessed by other modules of
ProUCL 5.0 (ProUCL 5.1) for performing desired statistical evaluations. For example, the simulated
BISS data can be merged with the actual field ISM background data after comparing the two data sets
using a two-sample t-test. The actual field ISM or the merged ISM and BISS data can be accessed by
modules of ProUCL to compute a UCL of the mean or a UTL.
ProUCL 5.1 Capabilities
Assumptions: Like most statistical methods, statistical methods for computing upper limits (e.g., UCLs,
UPLs, UTLs) are also based upon certain assumptions including the availability of a randomly collected
data set consisting of independently and identically distributed (i.i.d) observations representing the
population (e.g., site area, reference area) under investigation. A UCL of the mean (of a population) and
BTV estimates (UPL, UTL) should be computed using a randomly collected (simple random or
systematic random) data set representing a single statistical population (e.g., site population or
background population). When multiple populations (e.g., background and site data mixed together) are
present in a data set, the recommendation is to separate them first by using the population partitioning
techniques (e.g., Singh, Singh, and Flatman 1994) prior to computing the appropriate decision statistics
(e.g., 95% UCLs). Regardless of how the populations are separated, decision statistics should be
computed separately for each identified population. The topic of population partitioning and the
extraction of a valid site-specific background data set from a broader mixture data set potentially
consisting of both onsite and offsite data are beyond the scope of ProUCL 5.0/ProUCL 5.1. Parametric
estimation and hypotheses testing methods (e.g., t-test, UCLs, UTLs) are based upon distributional (e.g.,
normal distribution, gamma) assumptions. ProUCL includes GOF tests for determining if a data set
follows a normal, a gamma, or a lognormal distribution.
Multiple Constituents/Variables: Environmental scientists need to evaluate many constituents in their
decision making processes including exposure and risk assessment, background evaluations, and site
versus background comparisons. ProUCL can process multiple constituents/variables simultaneously in a
user-friendly manner; an option not available in other freeware or commercial software packages such as
NADA for R (Helsel 2013). This option is very useful when one has to process many variables/analytes
and compute decision statistics (e.g., UCLs, UPLs, and UTLs) and/or test statistics (e.g., ANOVA test,
trend test) for those variables/analytes.
Analysis by a Group Variable: ProUCL also has the capability of processing data by groups. A valid
group column should be included in the data file. The analyses of data categorized by a group ID variable
such as: 1) Surface versus (vs.) Subsurface; 2) AOC1 vs. AOC2; 3) Site vs. Background; and 4)
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Upgradient vs. Downgradient MWs are common in many environmental applications. ProUCL offers this
option for data sets with and without nondetects. The Group option provides a way to perform statistical
tests and methods including graphical displays separately for each of the group (samples from different
populations) that may be present in a data set. For example, the same data set may consist of analytical
data from multiple groups or populations representing site, background, two or more AOCs, surface soil,
subsurface soil, and GW. By using this option, the graphical displays (e.g., box plots, Q-Q plots,
histograms) and statistics (including computation of background statistics, UCLs, ANOVA test, trend test
and OLS regression statistics) can be easily computed separately for each group in the data set.
Exploratory Graphical Displays for Uncensored and Left-Censored Data Sets: Graphical methods
included in the Graphs module of ProUCL include: Q-Q plots (data in same column), multiple Q-Q plots
(data in different columns), box plots, multiple box plots (data in different columns), and histograms.
These graphs can also be generated for data sets containing ND observations. Additionally, the OLS
Regression and Trend Analysis module can be used to generate graphs displaying parametric OLS
regression lines with confidence and prediction intervals around the regression and nonparametric Theil-
Sen trend lines. The Trend Analysis module can generate trend graphs for data sets without a sampling
event variable, and also generates time series graphs for data sets with a sampling event (time) variable.
Like ProUCL 5.0, ProUCL 5.1 accepts only numerical values for the event variable. Graphical displays of
a data set are useful for gaining added insight regarding a data set that may not otherwise be clear by
looking at test statistics such as T-S test or MK statistics. Unlike test statistics (e.g., t-test, MK test, AD
test) and decision statistics (e.g., UCL, UTL), graphical displays do not get influenced by outliers and ND
observations. It is suggested that the final decisions be made based upon statistical results as well as
graphical displays.
Side-by-side box plots or multiple Q-Q plots are useful to graphically compare concentrations of two or
more groups (e.g., several monitoring wells). The GOF module of ProUCL generates Q-Q plots for
normal, gamma, and lognormal distributions based upon uncensored as well as left-censored data sets
with NDs. All relevant information such as the test statistics, critical values and probability-values (p-
values), when available are also displayed on the GOF Q-Q plots. In addition to providing information
about the data distribution, a normal Q-Q plot in the original raw scale also helps to identify outliers and
multiple populations that may be present in a data set. On a Q-Q plot, observations well-separated from
the majority of the data may represent potential outliers coming from a population different from the main
dominant population (e.g., background population). In a Q-Q plot, jumps and breaks of significant
magnitude suggest the presence of observations coming from multiple populations (onsite and offsite
areas). ProUCL can also be used to display box plots with horizontal lines displayed/superimposed at
pre-specified compliance limits (CLs) or computed upper limits (e.g., UPL, UTL). This kind of graph
provides a visual comparison of site data with compliance limits and/or BTV estimates.
Outlier Tests: ProUCL also provides a couple of classical outlier test procedures (EPA 2006b, 2009), the
Dixon test and the Rosner test. The details of these outlier tests are described in Chapter 7. These outlier
tests often suffer from "masking effects" in the presence of multiple outliers. It is suggested that the
classical outlier procedures should always be accompanied by graphical displays including box plots and
Q-Q plots. Description and use of the robust and resistant outlier procedures (Rousseeuw and Leroy 1987;
Singh and Nocerino 1995) are beyond the scope of ProUCL 5.1. Interested users are encouraged to try
the Scout 2008 software package (EPA 2009d) for robust outlier identification methods especially when
dealing with multivariate data sets consisting of observations for several variables/analytes/constituents.
Outliers represent observations coming from populations different from the main dominant population
represented by the majority of the data set. Outliers distort most statistics (e.g., mean, UCLs, UPLs, test
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statistics) of interest. Therefore, it is desirable to compute decisions statistics based upon data sets
representing the main population and not to compute distorted statistics by accommodating a few low
probability outliers (e.g., by using a lognormal distribution). Moreover, it should be noted that even
though outliers might have minimal influence on hypotheses testing statistics based upon ranks (e.g.,
WMW test), outliers do distort several nonparametric statistics including bootstrap methods such as
bootstrap-t and Hall's bootstrap UCLs and other nonparametric UPLs and UTLs computed using higher
order statistics.
Goodness-of-Fit Tests: In addition to computing simple summary statistics for data sets with and without
NDs, ProUCL 5.1 includes GOF tests for normal, lognormal and gamma distributions. To test for
normality (lognormality) of a data set, ProUCL includes the Lilliefors test and the extended S-W test for
samples of sizes up to 2000 (Royston 1982, 1982a). For the gamma distribution, two GOF tests: the A-D
test (Anderson and Darling 1954) and K-S test (Schneider 1976, 1978) are available in ProUCL. For
samples of larger sizes (e.g., with n > 100) and small values of the gamma shape parameter, k (e.g., k <
0.1), significant discrepancies were found in the critical values of the two gamma GOF test statistics (A-D
and K-S tests) obtained using the two gamma deviate generation algorithms: Whitaker (1974) and
Marsaglia and Tsang (2000). In ProUCL 5.0 (and ProUCL 5.1), for values of k < 0.2, the critical values of
the two gamma GOF tests: A-D and K-S tests have been updated using the currently available more
efficient gamma deviate generation algorithm due to Marsaglia and Tsang's (2000); more details about the
implementation of their algorithm can be found in Kroese, Taimre, and Botev (2011). For these two GOF
and values of the shape parameter, k=0.025, 0.05, 0.1, and 0.2, critical value tables have been updated by
incorporating the newly generated critical values for three levels of significance: 0.05, 0.1, and 0.01. The
updated tables are provided in Appendix A of the ProUCL Technical Guide. It was noted that for k=0.2,
the older (generated in 2002) and the newly generated critical values are in general agreement; therefore,
critical values for k=0.2 were not replaced in tables summarized in Appendix A.
ProUCL also generates GOF Q-Q plots for normal, lognormal, and gamma distributions displaying all
relevant statistics including GOF test statistics. GOF tests for data sets with and without NDs are
described in Chapters 2 and 3 of the ProUCL Technical Guide. For data sets containing NDs, it is not
easy to verify the distributional assumptions correctly, especially when the data set consists of a large
percentage of NDs with multiple DLs and NDs exceeding some detected values. Historically, decisions
about distributions of data sets with NDs are based upon GOF test statistics computed using the data
obtained: without NDs; replacing NDs by 0, DL, or DL/2; using imputed NDs based upon a ROS (e.g.,
lognormal ROS) method. For data sets with NDs, ProUCL 5.1 can perform GOF tests using the methods
listed above. ProUCL 5.1 can also generate censored probability plots (Q-Q plots) which are very similar
to Q-Q plots generated using detected data. Using the Imputed NDs using ROS Methods option of the
Stats/Sample Sizes module of ProUCL 5.0, additional columns can be generated for storing imputed
(estimated) values for NDs based upon normal ROS, gamma ROS, and lognormal ROS (also known as
robust ROS) methods.
Sample Size Determination and Power Evaluation: The Sample Sizes module in ProUCL can be used to
develop DQO-based sampling designs needed to address statistical issues associated with environmental
projects. ProUCL 5.1 provides user-friendly options for entering the desired/pre-specified values for
decision parameters (e.g., Type I and Type II error rates) and other DQOs used to determine minimum
sample sizes for statistical applications including: estimation of the mean, single and two-sample
hypothesis testing approaches, and acceptance sampling for discrete items (e.g., drums containing
hazardous waste). Both parametric (e.g., t-test) and nonparametric (e.g., Sign test, WRS test) sample size
determination methods as described in EPA (2000, 2002c, 2006a, 2006b) guidance documents are
available in ProUCL 5.1. ProUCL also has the sample size determination option for acceptance sampling
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of lots of discrete objects such as a lot (batch, set) of drums containing hazardous waste (e.g., RCRA
applications, EPA 2002c). When the sample size for an application (e.g., verification of cleanup level) is
not computed using the DQOs-based sampling design process, the Sample Size module can be used to
assess the power of the test statistic used in retrospect. The mathematical details of the Sample Sizes
module are given in Chapter 8 of the ProUCL Technical Guide.
Bootstrap Methods: Bootstrap methods are computer intensive nonparametric methods which can be used
to compute decision statistics of interest when a data set does not follow a known distribution, or when it
is difficult to analytically derive the distributions of statistics of interest. It is well-known that for
moderately skewed to highly skewed data sets, UCLs based upon standard bootstrap and the percentile
bootstrap methods do not perform well (e.g., Efron [1981, 1982]; Efron and Tibshirani 1993; Hall
[1988,1992]; Singh, Singh, and laci 2002; Singh, Maichle and Lee 2006) as the interval estimates based
upon these bootstrap methods fail to provide the specified coverage to the population mean (e.g., UCL95
does not provide adequate 95% coverage of population mean). For skewed data sets, Efron and Tibshirani
(1993) and Hall (1988, 1992) considered other bootstrap methods such as the BCA, bootstrap-t and Hall's
bootstrap methods. For skewed data sets, bootstrap-t and Hall's bootstrap (meant to adjust for skewness)
methods perform better (e.g., in terms of coverage for the population mean) than the other bootstrap
methods. However, it has been noted (e.g., Efron and Tibshirani 1993, Singh, Singh, and laci 2002) that
these two bootstrap methods tend to yield erratic and inflated UCL values (orders of magnitude higher
than other UCLs) in the presence of outliers. Similar behavior of the bootstrap-t UCL and Hall's bootstrap
UCL methods is observed for data sets consisting of NDs and outliers. For nonparametric uncensored and
left-censored data sets with NDs, depending upon data variability and skewness, ProUCL recommends
the use of BCA bootstrap, bootstrap-t, or Chebyshev inequality based methods for computing decision
statistics. Due to the reasons described above, whenever applicable, ProUCL 5.0/ProUCL 5.1 provides
cautionary notes and warning messages regarding the use of bootstrap-t and Halls bootstrap UCL
methods.
Hypotheses Testing Approaches: ProUCL software has both single-sample (e.g., Student's t-test, sign
test, proportion test, WSR test) and two-sample (Student's t-test, WMW test, Gehan test, and T-W test)
parametric and nonparametric hypotheses testing approaches. Hypotheses testing approaches in ProUCL
can handle both full-uncensored data sets and left-censored data sets with NDs. Most of the hypotheses
tests also report associated /"-values. For some hypotheses tests (e.g., WMW test, WSR test, proportion
test), large sample /"-values based upon the normal approximation are computed using continuity
correction factors. The mathematical details of the various single-sample and two-sample hypotheses
testing approaches are described in Chapter 6 the ProUCL Technical Guide.
• Single-Sample Tests: Parametric (Student's t-test) and nonparametric (Sign test, WSR test, tests for
proportions and percentiles) hypotheses testing approaches are available in ProUCL. Single-sample
hypotheses tests are used when environmental parameters such as the cleanup standard, action level,
or compliance limits are known, and the objective is to compare site concentrations with those known
threshold values. A t-test (or a sign test) may be used to verify the attainment of cleanup levels in an
AOC after a remediation activity has taken place or a test for proportion may be used to verify if the
proportion of exceedances of an action level (Ao or a CL) by sample observations collected from an
AOC (or a MW) exceeds a certain specified proportion (e.g., 1%, 5%, 10%).
The differences between these tests should be noted and understood. A t-test or a Wilcoxon Signed
Rank (WSR) test are used to compare the measures of location and central tendencies (e.g., mean,
median) of a site area (e.g., AOC) to a cleanup standard, Cs, or action level also representing a
measure of central tendency (e.g., mean, median); whereas, a proportion test determines if the
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proportion of site observations from an AOC exceeding a compliance limit (CL) exceeds a specified
proportion, P0 (e.g., 5%, 10%). The percentile test compares a specified percentile (e.g., 95th) of the
site data to a pre-specified upper threshold (e.g., action level).
• Two-Sample Tests: Hypotheses tests (Student's t-test, WMW test, Gehan test, T-W test) are used to
perform site versus background comparisons, compare concentrations of two or more AOCs, or to
compare concentrations of GW collected from MWs. As cited in the literature, some of the
hypotheses testing approaches (e.g., nonparametric two-sample WMW) deal with a single detection
limit scenario. When using the WMW test on a data set with multiple detection limits, all
observations (detects and NDs) below the largest detection limit need to be considered as NDs
(Gilbert 1987). This in turn tends to reduce the power and increase uncertainty associated with test.
As mentioned before, it is always desirable to supplement the test statistics and conclusions with
graphical displays such as multiple Q-Q plots and side-by-side box plots. The Gehan test or T-W test
(new in ProUCL 5.1) should be used in cases where multiple detection limits are present.
Note about Ouantile Test: For smaller data sets, the Quantile test as described in U.S. EPA documents
(U.S. EPA [1994, 2006b]; Hollander and Wolfe, 1999) is available in ProUCL 4.1(see ProUCL 4.1
Technical Guide). In the past, some users incorrectly used this test for larger data sets. Due to lack of
resources, this test has not been expanded for data sets of all sizes. Therefore, to avoid confusion and its
misuse for larger data sets, the Quantile test was not included in ProUCL 5.0 and ProUCL 5.1.
Computation of Upper Limits including UCLs. UPLs. UTLs. and USLs: ProUCL software has parametric
and nonparametric methods including bootstrap and Chebyshev inequality based methods to compute
decision making statistics such as UCLs of the mean (EPA 2002a), percentiles, UPLs for future k (>1)
observations, UTLs (U.S. EPA [1992b and 2009]) and upper simultaneous limits (USLs) (Singh and
Nocerino [1995, 2002]) based upon uncensored full data sets and left-censored data sets containing NDs
with multiple DLs. Methods incorporated in ProUCL cover a wide range of skewed data distributions
with and without NDs. In addition to normal and lognormal distributions based upper limits, ProUCL 5.0
can compute parametric UCLs, percentiles, UPLs for future k (>1) observations, UTLs, and USLs based
upon gamma distributed data sets. For data sets with NDs, ProUCL has several estimation methods
including the Kaplan-Meier (KM) method (1958), ROS methods (Helsel 2005) and substitution methods
such as replacing NDs with the DL or DL/2 (Gilbert 1987; U.S. EPA 2006b). Substitution method and
other poor performing methods (e.g., H-UCL for lognormal distribution) have been retained, as requested
by U.S. EPA scientists, in ProUCL 5.0/ProUCL 5.1 for research and comparison purposes. One may not
interpret the availability of these poor performing methods in ProUCL as recommended methods by
ProUCL or by the U.SEP A for computing decision statistics.
Computation of UCLs Based upon Uncensored Data Sets without NDs: Parametric UCL computation
methods in ProUCL for uncensored data sets include: Student's t-UCL, Approximate gamma UCL (using
chi-square approximation), Adjusted gamma UCL (adjusted for level significance), Land's H-UCL, and
Chebyshev inequality-based UCL (using minimum variance unbiased estimates (MVUEs) of parameters
of a lognormal distribution). Nonparametric UCL computation methods for data sets without NDs
include: CLT-based UCL, Modified-t-statistic-based UCL (adjusted for skewness), Adjusted-CLT-based
UCL (adjusted for skewness), Chebyshev inequality-based UCL (using sample mean and standard
deviation), Jackknife method-based UCL, UCL based upon standard bootstrap, UCL based upon
percentile bootstrap, UCL based upon BCA bootstrap, UCL based upon bootstrap-t, and UCL based upon
Hall's bootstrap method. The details of UCL computation methods for uncensored data sets are
summarized in Chapter 2 of the ProUCL Technical Guide.
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Computations of UPLs. UTLs. and USLs Based upon Uncensored Data Sets without NDs: For
uncensored data sets without NDs, ProUCL can compute parametric percentiles, UPLs for k (k>l) future
observations, UPLs for mean of k (>1) future observations, UTLs, and USLs based upon the normal,
gamma, and lognormal distributions. Nonparametric upper limits are typically based upon order statistics
of a data set. Depending upon the size of the data set, the higher order statistics (maximum, second
largest, third largest, and so on) are used to compute these upper limits (e.g., UTLs). Depending upon the
sample size, specified CC and coverage probability, ProUCL 5.1 outputs the actual CC achieved by a
nonparametric UTL. The details of the parametric and nonparametric computation methods for UPLs,
UTLs, and USLs are described in Chapter 3 of the ProUCL Technical Guide.
Computation of UCLs. UPLs. UTLs. and USLs Based upon Left-Censored Data Sets with NDs: For data
sets with NDs, ProUCL computes UCLs, UPLs, UTLs, and USLs based upon the mean and sd computed
using lognormal ROS (LROS, robust ROS), Gamma ROS (GROS), KM, and DL/2 substitution methods.
To adjust for skewness in non-normally distributed data sets, ProUCL uses bootstrap methods and
Chebyshev inequality when computing UCLs and other limits using estimates of the mean and sd
obtained using the methods (details in Chapters 4 and 5) listed above. ProUCL 5.1 (new in ProUCL 5.0)
uses parametric methods on KM (and ROS) estimates, provided detected observations in the left-censored
data set follow a parametric distribution. For example, if the detected data follow a gamma distribution,
ProUCL uses KM estimates in gamma distribution-based equations when computing UCLs, UTLs, and
other upper limits. When detected data do not follow a discernible distribution, depending upon size and
skewness of detected data, ProUCL recommends the use of Kaplan-Meier (1958) estimates in bootstrap
methods and the Chebyshev inequality for computing nonparametric decision statistics (e.g., UCL95,
UPL, UTL) of interest. ProUCL computes KM estimates directly using left-censored data sets without
flipping data and requiring re-flipping of decision statistics. The KM method incorporated in ProUCL
computes both sd and standard error (SE) of the mean. As mentioned earlier, for historical reasons and
for comparison and research purposes, the DL/2 substitution method and H-UCL based upon LROS
method have been retained in ProUCL 5.0/ProUCL 5.1. The inclusion of the substitution and LROS
methods in ProUCL should not be inferred as an endorsement of those methods by ProUCL software and
its developers. The details of the UCL computation methods for data sets with NDs are given in Chapter 4
and the detail description of the various other upper limits: UPLs, UTLs, and USLs for data sets with NDs
are given in Chapter 5 of the ProUCL Technical Guide.
Oneway ANOVA. OLS Regression and Trend Analysis: The Oneway ANOVA module has both
classical and nonparametric K-W ANOVA tests as described in EPA guidance documents (e.g., EPA
[2006b, 2009]). Oneway ANOVA is used to compare means (or medians) of multiple groups such as
comparing mean concentrations of several areas of concern or performing inter-well comparisons of
COPC concentrations at several MWs. The OLS Regression option computes the classical OLS
regression line and generates graphs displaying the OLS line, confidence bands and prediction bands
around the regression line. All statistics of interest including slope, intercept, and correlation coefficient
are displayed on the OLS line graph. The Trend Analysis module has two nonparametric trend tests: the
M-K trend test and T-S trend test. Using this option, one can generate trend graphs and time-series graphs
displaying a T-S trend line and all other statistics of interest with associated />-values. In addition to slope
and intercept, the T-S test in ProUCL 5.1 computes and outputs residuals based upon the computed
nonparametric T-S line.
In GW monitoring applications, OLS regression, trend tests, and time series plots are often used to
identify trends (e.g., upwards, downwards) in constituent concentrations of GW monitoring wells over a
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certain period of time (U.S. EPA 2009). The details of Oneway ANOVA are given in Chapter 9 and OLS
regression line and Trend tests methods are described in Chapter 10 of the ProUCL Technical Guide.
BISS Module: At many sites, a large amount of discrete onsite and background data are already available
which are not directly comparable to actual field ISM data. In order to provide a tool to compare the
existing discrete data with ISM data, the BISS module of ProUCL 5.0 may be used on a large existing
discrete data set. The ISM methodology used to develop the BISS module is a relatively new approach;
methods incorporated in this BISS module require further investigation. For now, the BISS module has
been blocked for use in ProUCL 5.0/ProUCL 5.1 as this module is awaiting adequate guidance for its
intended use on discrete background data sets.
Note: It is pointed out that in this document, all statements made about the capabilities of ProUCL 5.0
also apply to ProUCL version 5.1; and to save time, many screen shots used in ProUCL 5.0 manuals have
been used in ProUCL 5.1 manuals (User Guide and Technical Guide). All upgrades in ProUCL 5.1 (not
available in earlier versions) have been identified as new in ProUCL 5.1 in this document.
Recommendations and Suggestions in ProUCL: Until 2006, not much guidance was available on how to
compute a UCL95 of the mean and other upper limits (e.g., UPLs and UTLs) for skewed left-censored
data sets containing NDs with multiple DLs, a common occurrence in environmental data sets. For
uncensored positively skewed data sets, Singh, Singh, and laci (2002) summarize some simulation results
comparing the performances (in terms of coverage probabilities) of several UCL computation methods
described in the statistical and environmental literature. They noted that the optimal choice of a decision
statistic (e.g., UCL95) depends upon the sample size, data distribution and data skewness. They
incorporated the results of their findings in ProUCL 3.1 and higher versions to select the most appropriate
UCL to estimate the EPC term.
For data sets with NDs, Singh, Maichle, and Lee (2006) conducted a similar simulation study to compare
the performances of the various estimation methods (in terms of bias in the mean estimate); and some
UCL computation methods (in terms of coverage provided by a UCL). They demonstrated that the KM
estimation method performs well in terms of bias in estimates of the mean; and for skewed data sets, the t-
statistic, CLT, and the percentile bootstrap method based UCLs computed using KM estimates (and ROS
estimates) underestimate the population mean. From these findings summarized in Singh, Singh, and laci
(2002) and Singh, Maichle, and Lee (2006), it is natural to state and assume the findings of the simulation
studies performed on uncensored skewed data sets comparing performances of the various UCL
computation methods can be extended to skewed left-censored data sets.
Like uncensored data sets without NDs, for data sets with NDs, there is no one single best UCL (and
other upper limits such as UTL, UPL) which can be used to estimate an EPC (and background threshold
values) for all data sets of varying sizes, distribution, and skewness. The optimal choice of a decision
statistic depends upon the size, distribution, and skewness of detected observations.
For data sets with and without NDs, ProUCL computes decision statistics including UCLs, UPLs, and
UTLs using several parametric and nonparametric methods covering a wide-range of sample size, data
variability and skewness. Using the results and findings summarized in the literature cited above, and
based upon the sample size, data distribution, and data skewness, modules of ProUCL make suggestions
about using the most appropriate decision statistic(s) to estimate population parameter(s) of interest (e.g.,
EPC). The suggestions made in ProUCL are based upon the extensive professional applied and theoretical
experience of the developers in environmental statistical methods, published literature, results of
simulation studies conducted by the developers of ProUCL and procedures described in many U.S. EPA
15
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guidance documents. These suggestions are made to help the users in selecting the most appropriate UCL
to estimate an EPC which is routinely used in exposure assessment and risk management studies of the
U.S. EPA. It should be pointed out that a typical simulation study cannot cover all data sets of various
sizes and skewness from all types of distributions. For an analyte (data set) with skewness (sd of logged
data) near the end points of the skewness intervals described in decision tables of Chapter 2 (e.g., Tables
2-9 through 2-11) of the ProUCL Technical Guide, the user/project team may select the most appropriate
UCL based upon the site CSM, expert site knowledge, toxicity of the analyte, and exposure risks
associated with that analyte. The project team should make the final decision regarding using or not using
the suggestions/recommendations made by ProUCL. If deemed necessary, the project team may want to
consult a statistician.
Even though, ProUCL software has been developed using limited government funding, ProUCL 5.1
provides many statistical and graphical methods described in U.S. EPA documents for data sets with and
without NDs. However, one may not compare the availability of methods in ProUCL 5.1 with methods
available in the commercial software packages such as SAS® and Minitab 16. For example, trend tests
correcting for seasonal/spatial variations and geostatistical methods are not available in the ProUCL
software. For those methods, the user is referred to commercial software packages such as SAS®. As
mentioned earlier, is the developers of ProUCL recommended supplementing test results (e.g., two-
sample test) with graphical displays (e.g., Q-Q plots, side-by-side box plots) especially when data sets
contain NDs and outliers. With the inclusion of the BISS, Oneway ANOVA, OLS Regression Trend
and the user-friendly DQOs based Sample Size modules, ProUCL represents a comprehensive software
package equipped with statistical methods and graphical tools needed to address many environmental
sampling and statistical needs as described in the various CERCLA (U.S. EPA 1989a, 1992a, 2002a,
2002b, 2006a, 2006b), MARSSIM (U.S. EPA 2000), and RCRA (U.S. EPA 1989b, 1992b, 2002c, 2009)
guidance documents.
Finally, the users of ProUCL are cautioned about the use of methods and suggestions described in some
recent environmental literature. For example, many decision statistics (e.g., UCLs, UPLs, UTLs,)
computed using the methods (e.g., percentile bootstrap, statistics using KM estimates and t-critical
values) described in Helsel (2005, 2012) will fail to provide the desired coverage for environmental
parameters of interest (mean, upper percentile) of moderately skewed to highly skewed populations and
conclusions derived based upon those decisions statistics may lead to incorrect conclusions which may
not be cost-effective or protective of human health and the environment.
Note: The look and feel of ProUCL 5.1 is similar to that of ProUCL 5.0; and they share the same names
for the various modules and drop-down menus. For modules where no changes have been made in
ProUCL since 2010 (e.g., Sample Sizes), screen shots as used in ProUCL 5.0 documents have been used
in ProUCL 5.1 documents. Some of the screen shots generated using ProUCL 5.1 might have ProUCL 5.0
in their titles as those screen shots have not been re-generated and replaced.
ProUCL 5.1 User Guide
In addition to this Technical Guide, a User Guide also accompanies the ProUCL 5.1 software, providing
details of using the statistical and graphical methods incorporated in ProUCL 5.1. The User Guide
provides details about the input and output operations that can be performed using ProUCL 5.1. The User
guide also provides details about saving edited input files, output Excel-type spreadsheets and graphical
displays generated by ProUCL 5.1.
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CHAPTER 1
Guidance on the Use of Statistical Methods in ProUCL
Software
Decisions based upon statistics computed using discrete data sets of small sizes (e.g., < 6) cannot be
considered reliable enough to make decisions that affect human health and the environment. For example,
a background data set of size < 6 is not large enough to characterize a background population, compute
BTV estimates, or to perform background versus site comparisons. Several U.S. EPA guidance
documents (e.g., EPA 2000, 2006a, 2006b) detail DQOs and minimum sample size requirements needed
to address statistical issues associated with different environmental applications. In order to obtain
reliable statistical results, an adequate amount of data should be collected using project-specified DQOs
(i.e., CC, decision error rates). The Sample Sizes module of ProUCL computes minimum sample sizes
based on DQOs specified by the user and described in many guidance documents. In some cases, it may
not be possible (e.g., due to resource constraints) to collect the calculated number of samples needed to
meet the project-specific DQOs. Under these circumstances one can use the Sample Sizes module to
assess the power of the test statistic resulting from the reduced number of samples which were collected.
Based upon professional experience, the developers of ProUCL 4 software and its later versions have
been making some rule-of-thumb suggestions regarding minimum sample size requirements needed to
perform statistical evaluations such as: estimation of environmental parameters of interest (i.e., EPCs and
BTVs), comparing site data with background data or with some pre-established screening levels (e.g.,
action levels [ALs], compliance limits [CLs]). Those rule-of thumb suggestions are described later in
Section 1.7 of this chapter. It is noted that those minimum sample requirements have been adopted by
some other guidance documents including the RCRA Guidance Document (EPA 2009).
This chapter also describes the differences between the various statistical upper limits including upper
confidence limits (UCLs) of the mean, upper prediction limits (UPLs) for future observations, and upper
tolerance intervals (UTLs) often used to estimate the environmental parameters of interest including EPC
terms and BTVs. The use of a statistical method depends upon the environmental parameter(s) being
estimated or compared. The measures of central tendency (e.g., means, medians, or their UCLs) are used
to compare site mean concentrations with a cleanup standard, Cs, also representing some central tendency
measure of a reference area or some other known threshold representing a measure of central tendency.
The upper threshold values, such as the CLs, alternative concentration limits (ACL), or not-to-exceed
values, are used when individual point-by-point observations are compared with those threshold values.
Depending upon whether the environmental parameters (e.g., BTVs, not-to-exceed value, or EPC term)
are known or unknown, different statistical methods with different data requirements are needed to
compare site concentrations with pre-established (known) or estimated (unknown) standards and BTVs.
Several upper limits, and single and two sample hypotheses testing approaches, for both full-uncensored
and left-censored data sets are available in the ProUCL software package for performing the comparisons
described above.
1.1 Background Data Sets
Based upon the CSM and regional and expert knowledge about the site, the project team selects
background or reference areas. Depending upon the site activities and the pollutants, the background area
can be site-specific or a general reference area with conditions comparable to the site before
contamination due to site related activities. An appropriate random sample of independent observations
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(i.i.d) should be collected from the background area. A defensible background data set represents a
"single" environmental population possibly without any outliers. In a background data set, in addition to
reporting and/or laboratory errors, statistical outliers may also be present. A few elevated statistical
outliers present in a background data set may actually represent potentially contaminated locations
belonging to an impacted site area and/or possibly from other sources; those elevated outliers may not be
coming from the background population under evaluation. Since the presence of outliers in a data set
tends to yield distorted (poor and misleading) values of the decision making statistics (e.g., UCLs, UPLs
and UTLs), elevated outliers should not be included in background data sets and estimation of BTVs.
The objective here is to compute background statistics based upon a data set which represents the main
background population, and does not accommodate the few low probability high outliers (e.g., coming
from extreme tails of the data distribution) that may also be present in the sampled data. The occurrence
of elevated outliers is common when background samples are collected from various onsite areas (e.g.,
large Federal Facilities). The proper disposition of outliers, to include or not include them in statistical
computations, should be decided by the project team. The project team may want to compute decision
statistics with and without the outliers to evaluate the influence of outliers on the decision making
statistics.
A couple of classical outlier tests (Dixon and Rosner tests) are available in ProUCL. Since both of these
classical tests suffer from masking effects (e.g., some extreme outliers may mask the occurrence of other
intermediate outliers), it is suggested that these classical outlier tests be supplemented with graphical
displays such as a box plot and a Q-Q plot on a raw scale. The use of exploratory graphical displays helps
in determining the number of outliers potentially present in a data set. The use of graphical displays also
helps in identifying extreme high outliers as well as intermediate and mild outliers. The use of robust and
masking-resistant outlier identification procedures (Singh and Nocerino, 1995, Rousseeuw and Leroy,
1987) is recommended when multiple outliers are present in a data set. Those methods are beyond the
scope of ProUCL 5.1. However, several robust outlier identification methods are available in the Scout
2008 version 1.0 software package (EPA 2009d) available at
An appropriate background data set of a reasonable size (preferably computed using the DQOs processes)
is needed for the data set to be representative of background conditions and to compute upper limits (e.g.,
estimates of BTVs) and compare site and background data sets using hypotheses testing approaches. A
background data set should have a minimum of 10 observations, however more observations is preferable.
1.2 Site Data Sets
A data set collected from a site population (e.g., AOC, exposure area [EA], DU, group of MWs) should
be representative of the population under investigation. Depending upon the areas under investigation,
different soil depths and soil types may be considered as representing different statistical populations. In
such cases, background versus site comparisons may have to be conducted separately for each of those
sub-populations (e.g., surface and sub-surface layers of an AOC, clay and sandy site areas). These issues,
such as comparing depths and soil types, should also be considered in the planning stages when
developing sampling designs. Specifically, the availability of an adequate amount of representative data is
required from each of those site sub-populations/strata defined by sample depths, soil types, and other
characteristics.
Site data collection requirements depend upon the objective(s) of the study. Specifically, in background
versus site comparisons, site data are needed to perform:
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point-by-point onsite comparisons with pre-established ALs or estimated BTVs. Typically, this
approach is used when only a small number (e.g., < 6) of onsite observations are compared with a
BTV or some other not-to-exceed value. If many onsite values need to be compared with a BTV,
the recommended upper limit to use is the UTL or upper simultaneous limit (USL) to control the
false positive error rate (Type I Error Rate). More details can be found in Chapter 3 of this
guidance document. Alternatively, one can use hypothesis testing approaches (Chapter 6)
provided enough observations (at least 10, more are preferred) are available.
single-sample hypotheses tests to compare site data with a pre-established cleanup standards, Cs
(e.g., representing a measure of central tendency); proportion test to compare site proportion of
exceedances of an AL with a pre-specified allowable proportion, Po. These hypotheses testing
approaches are used on site data when enough site observations are available. Specifically, when
at least 10 (more are desirable) site observations are available; it is preferable to use hypotheses
testing approaches to compare site observations with specified threshold values. The use of
hypotheses testing approaches can control both types of error rates (Type 1 and Type 2) more
efficiently than the point-by-point individual observation comparisons. This is especially true as
the number of point-by-point comparisons increases. This issue is illustrated by the following
table summarizing the probabilities of exceedances (false positive error rate) of a BTV (e.g., 95th
percentile) by onsite observations, even when the site and background populations have
comparable distributions. The probabilities of these chance exceedances increase as the site
sample size increases.
Sample Size
1
2
5
8
10
12
64
Probability of
Exceedance
0.05
0.10
0.23
0.34
0.40
0.46
0.96
• two-sample hypotheses tests to compare site data distribution with background data distribution
to determine if the site concentrations are comparable to background concentrations. An adequate
amount of data needs to be made available from the site as well as the background populations. It
is preferable to collect at least 10 observations from each population under comparison.
Notes: From a mathematical point of view, one can perform hypothesis tests on data sets consisting of
only 3-4 data values; however, the reliability of the test statistics (and the conclusions derived) thus
obtained is questionable. In these situations it is suggested to supplement the test statistics decisions with
graphical displays.
1.3 Discrete Samples or Composite Samples?
ProUCL can be used for discrete sample data sets, as well as on composite sample data sets. However, in
a data set (background or site), samples should be either all discrete or all composite. In general, both
discrete and composite site samples may be used for individual point-by-point site comparisons with a
threshold value, and for single and two-sample hypotheses testing applications.
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• When using a single-sample hypothesis testing approach, site data can be obtained by collecting
all discrete or all composite samples. The hypothesis testing approach is used when many (> 10)
site observations are available. Details of the single-sample hypothesis approaches are widely
available in EPA guidance documents (MARSSIM 2000, EPA 1989a, 2006b). Several single-
sample hypotheses testing procedures available in ProUCL are described in Chapter 6 of this
document.
• If a two-sample hypothesis testing approach is used to perform site versus background
comparisons, then samples from both of the populations should be either all discrete samples, or
all composite samples. The two-sample hypothesis testing approaches are used when many (e.g.,
at least 10) site, as well as background, observations are available. For better results with higher
statistical power, the availability of more observations perhaps based upon an appropriate DQOs
process (EPA 2006a) is desirable. Several two-sample hypotheses tests available in ProUCL 5.1
are described in Chapter 6 of this document.
1.4 Upper Limits and Their Use
The computation and use of statistical limits depend upon their applications and the parameters (e.g., EPC
term, BTVs) they are supposed to be estimating. Depending upon the objective of the study, a pre-
specified cleanup standard, Cs, can be viewed as representing: 1) an average (or median) constituent
concentration, //o; or 2) a not-to-exceed upper threshold concentration value, Ao. These two threshold
values, /no, and Ao, represent two significantly different parameters, and different statistical methods and
limits are used to compare the site data with these two very different threshold values. Statistical limits,
such as a UCL of the population mean, a UPL for an independently obtained "single" observation, or
independently obtained "&" observations (also called future k observations, next k observations, or k
different observations), upper percentiles, and UTLs are often used to estimate the environmental
parameters: EPC (/no) and a BTV (Ao). A new upper limit, USL was included in ProUCL 5.0 which may
be used to estimate a BTV based upon a well-established background data set representing a single
statistical population without any outliers.
It is important to understand and note the differences between the uses and numerical values of these
statistical limits so that they can be properly used. The differences between UCLs and UPLs (or upper
percentiles), and UCLs and UTLs should be clearly understood. A UCL with a 95% confidence limit
(UCL95) of the mean represents an estimate of the population mean (measure of the central tendency),
whereas a UPL95, a UTL95%-95% (UTL95-95), and an upper 95th percentile represent estimates of a
threshold from the upper tail of the population distribution such as the 95th percentile. Here, UPL95
represents a 95% upper prediction limit, and UTL95-95 represents a 95% confidence limit of the 95th
percentile. For mildly skewed to moderately skewed data sets, the numerical values of these limits tend to
follow the order given as follows.
Sample Mean < UCL95 of Mean < Upper 95th Percentile < UPL95 of a Single Observation < UTL95-95
Example 1-1. Consider a real data set collected from a Superfund site. The data set has several inorganic
COPCs, including aluminum (Al), arsenic (As), chromium (Cr), iron (Fe), lead (Pb), manganese (Mn),
thallium (Tl) and vanadium (V). Iron concentrations follow a normal distribution. This data set has been
used in several examples throughout the two ProUCL guidance documents (Technical Guide and User
Guide), therefore it is provided as follows.
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Aluminum
6280
3830
3900
5130
9310
15300
9730
7840
10400
16200
6350
10700
15400
12500
2850
9040
2700
1710
3430
6790
11600
4110
7230
4610
Arsenic
1.3
1.2
2
1.2
3.2
5.9
2.3
1.9
2.9
3.7
1.8
2.3
2.4
2.2
1.1
3.7
1.1
1
1.5
2.6
2.4
1.1
2.1
0.66
Chromium
8.7
8.1
11
5.1
12
20
12
11
13
20
9.8
14
17
15
8.4
14
4.5
3
4
11
16.4
7.6
35.5
6.1
Iron
4600
4330
13000
4300
11300
18700
10000
8900
12400
18200
7340
10900
14400
11800
4090
15300
6030
3060
4470
9230
Lead
16
6.4
4.9
8.3
18
14
12
8.7
11
12
14
14
19
21
16
25
20
11
6.3
13
98.5
53.3
109
8.3
Manganese
39
30
10
92
530
140
440
130
120
70
60
110
340
85
41
66
21
8.6
19
140
72.5
27.2
118
22.5
Thallium
0.0835
0.068
0.155
0.0665
0.071
0.427
0.352
0.228
0.068
0.456
0.067
0.0695
0.07
0.214
0.0665
0.4355
0.0675
0.066
0.067
0.068
0.13
0.068
0.095
0.07
Vanadium
12
8.4
11
9
22
32
19
17
21
32
15
21
28
25
8
24
11
7.2
8.1
16
Several upper limits for iron are summarized as follows, and it be seen that they follow the order (in
magnitude) as described above.
Table 1-1. Computation of Upper Limits for Iron (Normally Distributed)
Mean
9618
Median
9615
Min
3060
Max
18700
UCL95
11478
UPL95 for a
Single
Observation
18145
UPL95 for 4
Observations
21618
UTL95-95
21149
95%
Upper
Percentile
17534
For highly skewed data sets, these limits may not follow the order described above. This is especially true
when the upper limits are computed based upon a lognormal distribution (Singh, Singh, and Engelhardt
1997). It is well known that a lognormal distribution based H-UCL95 (Land's UCL95) often yields
unstable and impractically large UCL values. An H-UCL95 often becomes larger than UPL95 and even
larger than a UTL 95%-95% and the largest sample value. This is especially true when dealing with
skewed data sets of smaller sizes. Moreover, it should also be noted that in some cases, a H-UCL95
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becomes smaller than the sample mean, especially when the data are mildly skewed and the sample size is
large (e.g., > 50, 100).
There is a great deal of confusion about the appropriate use of these upper limits. A brief discussion about
the differences between the applications and uses of the statistical limits described above is provided as
follows.
• A UCL represents an average value that is compared with a threshold value also representing an
average value (pre-established or estimated), such as a mean Cs. For example, a site 95% UCL
exceeding a Cs, may lead to the conclusion that the cleanup standard, Cs has not been attained by the
average site area concentration. It should also be noted that UCLs of means are typically computed
from the site data set.
• A UCL represents a "collective" measure of central tendency, and it is not appropriate to compare
individual site observations with a UCL. Depending upon data availability, single or two-sample
hypotheses testing approaches are used to compare a site average or a site median with a specified or
pre-established cleanup standard (single-sample hypothesis), or with the background population
average or median (two-sample hypothesis).
• A UPL, an upper percentile, or a UTL represents an upper limit to be used for point-by-point
individual site observation comparisons. UPLs and UTLs are computed based upon background data
sets, and point-by-point onsite observations are compared with those limits. A site observation
exceeding a background UTL may lead to the conclusion that the constituent is present at the site at
levels greater than the background concentrations level.
• When enough (e.g., at least 10) site observations are available, it is preferable to use hypotheses
testing approaches. Specifically, single-sample hypotheses testing (comparing site to a specified
threshold) approaches should be used to perform site versus a known threshold comparison; and two-
sample hypotheses testing (provided enough background data are also available) approaches should
be used to perform site versus background comparison. Several parametric and nonparametric single
and two-sample hypotheses testing approaches are available in ProUCL 5.0/ProUCL 5.1.
It is re-emphasized that only averages should be compared with averages or UCLs, and individual site
observations should be compared with UPLs, upper percentiles, UTLs, or USLs. For example, the
comparison of a 95% UCL of one population (e.g., site) with a 90% or 95% upper percentile of another
population (e.g., background) cannot be considered fair and reasonable as these limits (e.g., UCL and
UPL) estimate and represent different parameters.
1.5 Point-by-Point Comparison of Site Observations with BTVs, Compliance
Limits and Other Threshold Values
The point-by-point observation comparison method is used when a small number (e.g., < 6) of site
observations are compared with pre-established or estimated BTVs, screening levels, or preliminary
remediation goals (PRGs). Typically, a single exceedance of the BTV by an onsite (or a monitoring well)
observation may be considered an indication of the presence of contamination at the site area under
investigation. The conclusion of an exceedance by a site value is sometimes confirmed by re-sampling
(taking a few more collocated samples) at the site location (or a monitoring well) exhibiting constituent
concentrations in excess of the BTV. If all collocated sample observations (or all sample observations
collected during the same time period) from the same site location (or well) exceed the BTV or PRO, then
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it may be concluded that the location (well) requires further investigation (e.g., continuing treatment and
monitoring) and possibly cleanup.
When BTV constituent concentrations are not known or pre-established, one has to collect or extract a
background data set of an appropriate size that can be considered representative of the site background.
Statistical upper limits are computed using the background data set thus obtained, which are used as
estimates of BTVs. To compute reasonably reliable estimates of BTVs, a minimum of 10 background
observations should be collected, perhaps using an appropriate DQOs process as described in EPA (2000,
2006a). Several statistical limits listed above are used to estimate BTVs based upon a defensible (free of
outliers, representing the background population) background data set of an adequate size.
The point-by-point comparison method is also useful when quick turnaround comparisons are required in
real time. Specifically, when decisions have to be made in real time by a sampling/screening crew, or
when only a few site samples are available, then individual point-by-point site concentrations are
compared either with pre-established cleanup goals or with estimated BTVs. The sampling crew can use
these comparisons to: 1) screen and identify the COPCs, 2) identify the potentially polluted site AOCs, or
3) continue or stop remediation or excavation at an onsite area of concern.
If a larger number of samples (e.g., >10) are available from the AOC, then the use of hypotheses testing
approaches (both single-sample and a two-sample) is preferred. The use of hypothesis testing approaches
tends to control the error rates more tightly and efficiently than the individual point-by-point site
comparisons.
1.6 Hypothesis Testing Approaches and Their Use
Both single-sample and two-sample hypotheses testing approaches are used to make cleanup decisions at
polluted sites, and also to compare constituent concentrations of two (e.g., site versus background) or
more populations (e.g., MWs).
1.6.1 Single Sample Hypotheses (Pre-established BTVs and Not-to-Exceed Values are Known)
When pre-established BTVs are used such as the U.S. Geological Survey (USGS) background values
(Shacklette and Boerngen 1984), or thresholds obtained from similar sites, there is no need to extract,
establish, or collect a background data set. When the BTVs and cleanup standards are known, one-sample
hypotheses are used to compare site data (provided enough site data are available) with known and pre-
established threshold values. It is suggested that the project team determine (e.g., using DQOs) or decide
(depending upon resources) the number of site observations that should be collected and compared with
the "pre-established" standards before coming to a conclusion about the status (clean or polluted) of the
site AOCs. As mentioned earlier, when the number of available site samples is < 6, one might perform
point-by-point site observation comparisons with a BTV; and when enough site observations (at least 10)
are available, it is desirable to use single-sample hypothesis testing approaches. Depending upon the
parameter (fio, Ao), represented by the known threshold value, one can use single-sample hypotheses tests
for population mean or median (t-test, sign test), or use single-sample tests for proportions and
percentiles. The details of the single-sample hypotheses testing approaches can be found in EPA (2006b)
guidance document and in Chapter 6 of this document.
One-Sample t-Test: This test is used to compare the site mean, fi, with some specified cleanup standard,
Cs, where the Cs represents an average threshold value, /no. The Student's t-test (or a UCL of the mean) is
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used (assuming normality of site data set or when sample size is large, such as larger than 30, 50) to
verify the attainment of cleanup levels at a polluted site after some remediation activities.
One-Sample Sign Test or Wilcoxon Signed Rank (WSR) Test: These tests are nonparametric tests and can
also handle ND observations, provided the detection limits of all NDs fall below the specified threshold
value, Cs. These tests are used to compare the site location (e.g., median, mean) with some specified Cs
representing a similar location measure.
One-Sample Proportion Test or Percentile Test: When a specified cleanup standard, Ao, such as a PRG or
a BTV represents an upper threshold value of a constituent concentration distribution rather than the mean
threshold value, juo, then a test for proportion or a test for percentile (equivalently UTL 95-95 UTL 95-90)
may be used to compare site proportion (or site percentile) with the specified threshold or action level, Ao.
1.6.2 Two-Sample Hypotheses (BTVs and Not-to-Exceed Values are Unknown)
When BTVs, not-to-exceed values, and other cleanup standards are not available, then site data are
compared directly with the background data. In such cases, two-sample hypothesis testing approaches are
used to perform site versus background comparisons. Note that this approach can be used to compare
concentrations of any two populations including two different site areas or two different monitoring wells
(MWs). In order to use and perform a two-sample hypothesis testing approach, enough data should be
available from each of the two populations. Site and background data requirements (e.g., based upon
DQOs) for performing two-sample hypothesis test approaches are described in EPA (2000, 2002b, 2006a,
2006b) and also in Chapter 6 of this Technical Guide. While collecting site and background data, for
better representation of populations under investigation, one may also want to account for the size of the
background area (and site area for site samples) in sample size determination. That is, a larger number
(>15-20) of representative background (and site) samples should be collected from larger background
(and site) areas; every effort should be made to collect as many samples as determined by the DQOs-
based sample sizes.
The two-sample (or more) hypotheses approaches are used when the site parameters (e.g., mean, shape,
distribution) are being compared with the background parameters (e.g., mean, shape, distribution). The
two-sample hypotheses testing approach is also used when the cleanup standards or screening levels are
not known a priori. Specifically, in environmental applications, two-sample hypotheses testing
approaches are used to compare average or median constituent concentrations of two or more populations.
To derive reliable conclusions with higher statistical power based upon hypothesis testing approaches, an
adequate amount of data (e.g., minimum of 10 samples) should be collected from all of the populations
under investigation.
The two-sample hypotheses testing approaches incorporated in ProUCL 5.1 are listed as follows:
• Student t-test (with equal and unequal variances) - Parametric test assumes normality
• Wilcoxon-Mann-Whitney (WMW) test - Nonparametric test handles data with NDs with one DL
- assumes two populations have comparable shapes and variability
• Gehan test - Nonparametric test handles data sets with NDs and multiple DLs - assumes
comparable shapes and variability
• Tarone-Ware (T-W) test - Nonparametric test handles data sets with NDs and multiple DLs -
assumes comparable shapes and variability
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The Gehan and T-W tests are meant to be used on left-censored data sets with multiple DLs. For best
results, the samples collected from the two (or more) populations should all be of the same type obtained
using similar analytical methods and apparatus; the collected site and background samples should all be
discrete or all composite (obtained using the same design and pattern), and be collected from the same
medium (soil) at similar depths (e.g., all surface samples or all subsurface samples) and time (e.g., during
the same quarter in groundwater applications) using comparable (preferably same) analytical methods.
Good sample collection methods and sampling strategies are given in EPA (1996, 2003) guidance
documents.
Note: ProUCL 5.1 (and previous versions) has been developed using limited government funding.
ProUCL 5.1 is equipped with statistical and graphical methods needed to address many environmental
sampling and statistical issues as described in the various CERCLA, MARSSIM, and RCRA documents
cited earlier. However, one may not compare the availability of methods in ProUCL 5.1 with methods
incorporated in commercial software packages such as SAS® and Minitab 16. Not all methods available in
the statistical literature are available in ProUCL.
1.7 Minimum Sample Size Requirements and Power Evaluations
Due to resource limitations, it is not be possible (nor needed) to sample the entire population (e.g.,
background area, site area, AOCs, EAs) under study. Statistics is used to draw inference(s) about the
populations (clean, dirty) and their known or unknown statistical parameters (e.g., mean, variance, upper
threshold values) based upon much smaller data sets (samples) collected from those populations. To
determine and establish BTVs and site specific screening levels, defensible data set(s) of appropriate
size(s) representing the background population (e.g., site-specific, general reference area, or historical
data) need to be collected. The project team and site experts should decide what represents a site
population and what represents a background population. The project team should determine the
population area and boundaries based upon all current and intended future uses, and the objectives of data
collection. Using the collected site and background data sets, statistical methods supplemented with
graphical displays are used to perform site versus background comparisons. The test results and statistics
obtained by performing such site versus background comparisons are used to determine if the site and
background level constituent concentrations are comparable; or if the site concentrations exceed the
background threshold concentration level; or if an adequate amount of remediation approaching the BTV
or some cleanup level has been performed at polluted site AOCs.
To perform statistical tests and compute upper limits, determine the number of samples that need to be
collected from the populations (e.g., site and background) under investigation using appropriate DQOs
processes (EPA 2000, 2006a, 2006b). ProUCL has the Sample Sizes module which can be used to
develop DQOs based sampling designs needed to address statistical issues associated with polluted sites
projects. ProUCL provides user-friendly options to enter the desired/pre-specified values of decision
parameters (e.g., Type I and Type II error rates) to determine minimum sample sizes for the selected
statistical applications including: estimation of mean, single and two-sample hypothesis testing
approaches, and acceptance sampling. Sample size determination methods are available for the sampling
of continuous characteristics (e.g., lead or Radium 226), as well as for attributes (e.g., proportion of
occurrences exceeding a specified threshold). Both parametric (e.g., t-tests) and nonparametric (e.g., Sign
test, test for proportions, WRS test) sample size determination methods are available in ProUCL 5.1 and
in its earlier versions (e.g., ProUCL 4.1). ProUCL also has sample size determination methods for
acceptance sampling of lots of discrete objects such as a batch of drums containing hazardous waste (e.g.,
RCRA applications, U.S. EPA 2002c).
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However, due to budgetary or logistical constraints, it may not be possible to collect the same number of
samples as determined by applying a DQO process. For example, the data might have already been
collected (as often is the case) without using a DQO process, or due to resource constraints, it may not
have been possible to collect as many samples as determined by using a DQO-based sample size formula.
In practice, the project team and the decision makers tend not to collect enough background samples. It is
suggested to collect at least 10 background observations before using statistical methods to perform
background evaluations based upon data collected using discrete samples. The minimum sample size
recommendations described here are useful when resources are limited, and it may not be possible to
collect as many background and site samples as computed using DQOs based sample size determination
formulae. In case data are collected without using a DQO process, the Sample Sizes module can be used
to assess the power of the test statistic in retrospect. Specifically, one can use the standard deviation of the
computed test statistic (EPA 2006b) and compute the sample size needed to meet the desired DQOs. If the
computed sample size is greater than the size of the data set used, the project team may want to collect
additional samples to meet the desired DQOs.
Note: From a mathematical point of view, the statistical methods incorporated in ProUCL and described
in this guidance document for estimating EPC terms and BTVs, and comparing site versus background
concentrations can be performed on small site and background data sets (e.g., of sizes as small as 3).
However, those statistics may not be considered representative and reliable enough to make important
cleanup and remediation decisions which will potentially impact human health and the environment.
ProUCL provides messages when the number of detects is <4-5, and suggests collecting at least 8-10
observations. Based upon professional judgment, as a rule-of-thumb, ProUCL guidance documents
recommend collecting a minimum of 10 observations when data sets of a size determined by a DQOs
process (EPA 2006) cannot be collected. This however, should not be interpreted as the general
recommendation and every effort should be made to collect DQOs based number of samples. Some recent
guidance documents (e.g., EPA 2009) have also adopted this rule-of-thumb and suggest collecting a
minimum of about 8-10 samples in the circumstance that data cannot be collected using a DQO-based
process. However, the project team needs to make these determinations based upon their comfort level
and knowledge of site conditions.
• To allow users to compute decision statistics using data from ISM (ITRC, 2012) samples,
ProUCL 5.1 will compute decision statistics (e.g., UCLs, UPLs, UTLs) based upon samples of
sizes as small as 3. The user is referred to the ITRC ISM Technical Regulatory Guide (2012) to
determine which UCL (e.g., Student's t-UCL or Chebyshev UCL) should be used to estimate the
EPC term.
1.7.1 Why a data set of minimum size, n = 8 through 10?
Typically, the computation of parametric upper limits (UPL, UTL, UCL) depends upon three values: the
sample mean, sample variability (standard deviation) and a critical value. A critical value depends upon
sample size, data distribution, and confidence level. For samples of small size (< 8-10), the critical values
are large and unstable, and upper limits (e.g., UTLs, UCLs) based upon a data set with fewer than 8-10
observations are mainly driven by those critical values. The differences in the corresponding critical
values tend to stabilize when the sample size becomes larger than 8-10 (see tables below, where degrees
of freedom \df\ = sample size - 1). This is one of the reasons ProUCL guidance documents suggest a
minimum data set size of 10 when the number of observations determined from sample-size calculations
based upon EPA DQO process exceed the logistical/financial/temporal/constraints of a project. For
samples of sizes 2-11, 95% critical values used to compute upper limits (UCLs, UPLs, UTLs, and USLs)
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based upon a normal distribution are summarized in the subsequent tables. In general, a similar pattern is
followed for critical values used in the computation of upper limits based upon other distributions.
For the normal distribution, Student's t-critical values are used to compute UCLs and UPLs which are
summarized as follows.
Table of Critical Values of t-Statistic
df= sample size-l= (n-1)
Upper-tail probability p
df
1
2
3
4
5
t,
7
&
9
10
.10
3.07 S
.886
.633
.533
.476
.440
.415
.397
.383
.372
.05
6.314
2.920
2.353
2.132
2.015
.943
.895
.860
.833
.812
.025
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
.112
15.89
4.849
3.482
2.999
2.757
2.612
2.517
2.449
2.398
2.359
.(.11
31.S2
6.965
4.541
3.747
3.365
3.143
2.99S
2.896
1.821
2.764
One can see that once the sample size starts exceeding 9-10 (df= 8, 9), the difference between the critical
values starts stabilizing. For example, for upper tail probability (= level of significance) of 0.05, the
difference between critical values for df= 9 and df=\0 is only 0.021, where as the difference between
critical values for df= 4 and 5 is 0.117; similar patterns are noted for other levels of significance. For the
normal distribution, critical values used to compute UTL90-95, UTL95-95, USL90, and USL95 are
described as follows. One can see that once the sample size starts exceeding 9-10, the difference between
the critical values starts decreasing significantly.
UTL90-95 UTL95-95
USL90
USL95
3
4
5
6
7
8
9
10
11
6.155
4.162
3.407
3.006
2.755
2.582
2.454
2.355
2.275
7.656
5.144
4.203
3.708
3.399
3.187
3.031
2.911
2.815
1.148
1.425
1.602
1.729
1.828
1.909
1.977
2.036
2.088
1.153
1.462
1.671
1.822
1.938
2.032
2.11
2.176
2.234
Note: Nonparametric upper limits (UPLs, UTLs, and USLs) are computed using higher order statistics of
a data set. To achieve the desired confidence coefficient, samples of sizes much greater than 10 are
required. For details, refer to Chapter 3. It should be noted that critical values of USLs are significantly
lower than critical values for UTLs. Critical values associated with UTLs decrease as the sample size
increases. Since, as the sample size increases the maximum of the data set also increases, and critical
values associated with USLs increase with the sample size.
1.7.2 Sample Sizes for Bootstrap Methods
Several nonparametric methods including bootstrap methods for computing UCL, UTL, and other limits
for both full-uncensored data sets and left-censored data sets with NDs are available in ProUCL 5.1.
Bootstrap resampling methods are useful when not too few (e.g., < 15-20) and not too many (e.g., > 500-
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1000) observations are available. For bootstrap methods (e.g., percentile method, BCA bootstrap method,
bootstrap-t method), a large number (e.g., 1000, 2000) of bootstrap resamples are drawn with replacement
from the same data set. Therefore, to obtain bootstrap resamples with at least some distinct values (so that
statistics can be computed from each resample), it is suggested that a bootstrap method should not be used
when dealing with small data sets of sizes less than 15-20. Also, it is not necessary to bootstrap a large
data set of size greater than 500 or 1000; that is when a data set of a large size (e.g., > 500) is available,
there is no need to obtain bootstrap resamples to compute statistics of interest (e.g., UCLs). One can
simply use a statistical method on the original large data set.
Note: Rules-of-thumb about minimum sample size requirements described in this section are based upon
professional experience of the developers. ProUCL software is not a policy software. It is recommended
that the users/project teams/agencies make determinations about the minimum number of observations
and minimum number of detects that should be present in a data set before using a statistical method.
1.8 Statistical Analyses by a Group ID
The analyses of data categorized by a group ID variable such as: 1) Surface vs. Subsurface; 2) AOC1 vs.
AOC2; 3) Site vs. Background; and 4) Upgradient vs. Downgradient monitoring wells are common in
environmental applications. ProUCL 5.1 offers this option for data sets with and without NDs. The
Group Option provides a tool for performing separate statistical tests and for generating separate
graphical displays for each member/category of the group (samples from different populations) that may
be present in a data set. The graphical displays (e.g., box plots, quantile-quantile plots) and statistics (e.g.,
background statistics, UCLs, hypotheses tests) of interest can be computed separately for each group by
using this option. Moreover, using the Group Option, graphical methods can display multiple graphs
(e.g., Q-Q plots) on the same graph providing graphical comparison of multiple groups.
It should be pointed out that it is the user's responsibility to provide an adequate amount of data to
perform the group operations. For example, if the user desires to produce a graphical Q-Q plot (e.g., using
only detected data) with regression lines displayed, then there should be at least two detected data values
(to compute slope, intercept, sd) in the data set. Similarly if the graphs are desired for each group
specified by the group ID variable, there should be at least two observations in each group specified by
the group variable. When ProUCL data requirements are not met, ProUCL does not perform any
computations, and generates a warning message (colored orange) in the lower Log Panel of the output
screen of ProUCL 5.1.
1.9 Statistical Analyses for Many Constituents/Variables
ProUCL software can process multiple analytes/variables simultaneously in a user-friendly manner This
option is useful when one has to process multiple variables and compute decision statistics (e.g., UCLs,
UPLs, and UTLs) and test statistics (e.g., ANOVA test, trend test) for multiple variables. It is the user's
responsibility to make sure that each selected variable has an adequate amount of data so that ProUCL
can perform the selected statistical method correctly. ProUCL displays warning messages when a selected
variable does not have enough data needed to perform the selected statistical method.
1.10 Use of Maximum Detected Value as Estimates of Upper Limits
Some practitioners use the maximum detected value as an estimate of the EPC term. This is especially
true when the sample size is small such as < 5, or when a UCL95 exceeds the maximum detected values
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(EPA 1992a). Also, many times in practice, the BTVs and not-to-exceed values are estimated by the
maximum detected value (e.g., nonparametric UTLs, USLs).
1.10.1 Use of Maximum Detected Value to Estimate BTVs and Not-to-Exceed Values
BTVs and not-to-exceed values represent upper threshold values from the upper tail of a data distribution;
therefore, depending upon the data distribution and sample size, the BTVs and other not-to-exceed values
may be estimated by the largest or the second largest detected value. A nonparametric UPL, UTL, and
USL are often estimated by higher order statistics such as the maximum value or the second largest value
(EPA 1992b, 2009, Hahn and Meeker 1991). The use of higher order statistics to estimate the UTLs
depends upon the sample size. For data sets of size: 1) 59 to 92 observations, a nonparametric UTL95-95
is given by the maximum detected value; 2) 93 to 123 observations, a nonparametric UTL95-95 is given
by the second largest maximum detected value; and 3) 124 to 152 observations, a UTL95-95 is given by
the third largest detected value in the sample, and so on.
1.10.2 Use of Maximum Detected Value to Estimate EPC Terms
Some practitioners tend to use the maximum detected value as an estimate of the EPC term. This is
especially true when the sample size is small such as < 5, or when a UCL95 exceeds the maximum
detected value. Specifically, the EPA (1992a) document suggests the use of the maximum detected value
as a default value to estimate the EPC term when a 95% UCL (e.g., the H-UCL) exceeds the maximum
value in a data set. ProUCL computes 95% UCLs of the mean using several methods based upon normal,
gamma, lognormal, and non-discernible distributions. In the past, a lognormal distribution was used as the
default distribution to model positively skewed environmental data sets. Additionally, only two methods
were used to estimate the EPC term based upon: 1) normal distribution and Student's t-statistic, and 2)
lognormal distribution and Land's H-statistic (Land 1971, 1975). The use of the H-statistic often yields
unstable and impractically large UCL95 of the mean (Singh, Singh, and Engelhardt 1997; Singh, Singh,
and laci 2002). For highly skewed data sets of smaller sizes (< 30, < 50), H-UCL often exceeds the
maximum detected value. Since the use of a lognormal distribution has been quite common (suggested as
a default model in the risk assessment guidance for Superfund [RAGS] document [EPA 1992a]), the
exceedance of the maximum value by an H-UCL95 is frequent for many skewed data sets of smaller sizes
(e.g., < 30, < 50). These occurrences result in the possibility of using the maximum detected value as an
estimate of the EPC term.
It should be pointed out that in some cases, the maximum observed value actually might represent an
impacted location. Obviously, it is not desirable to use an observation potentially representing an
impacted location to estimate the EPC for an AOC. The EPC term represents the average exposure
contracted by an individual over an EA during a long period of time; the EPC term should be estimated
by using an average value (such as an appropriate 95% UCL of the mean) and not by the maximum
observed concentration. One needs to compute an average exposure and not the maximum exposure. As
can be seen in figures described in Appendix B, for data sets of small sizes (e.g., < 10-20), the Max Test
(U.S. EPA 1996)does not provide the specified 95% coverage to the population mean, and for larger data
sets it overestimates the EPC term, which may lead to unnecessary further remediation.
Several methods, some of which are described in EPA (2002a) and other EPA documents, are available in
versions of ProUCL (i.e., ProUCL 3.00.02 [EPA 2004], ProUCL 4.0 [U.S. EPA 2007], ProUCL 4.00.05
[EPA 2009, 2010], ProUCL 4.1 [EPA 2011]) for estimating the EPC terms. For data sets with NDs,
ProUCL 5.0 (and ProUCL 5.1) has some new UCL (and other limits) computation methods which were
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not available in earlier versions of ProUCL. It is unlikely that the UCLs based upon those methods will
exceed the maximum detected value, unless some outliers are present in the data set.
1.10.2.1 Chebyshev Inequality Based UCL95
ProUCL 5.1 (and its earlier versions) displays a warning message when the suggested 95% UCL (e.g.,
Hall's or bootstrap-t UCL with outliers) of the mean exceeds the detected maximum concentration. When
a 95% UCL does exceed the maximum observed value, ProUCL suggests the use of an alternative UCL
computation method based upon the Chebyshev inequality. One may use a 97.5% or 99% Chebyshev
UCL to estimate the mean of a highly skewed population. The use of the Chebyshev inequality to
compute UCLs tends to yield more conservative (but stable) UCLs than other methods available in
ProUCL software. In such cases, when the sample size is large (and other UCL methods such as the
bootstrap-t method yield unrealistically high values due to presence of outliers), one may want to use a
95% Chebyshev UCL or a Chebyshev UCL with a lower confidence coefficient such as 90% as an
estimate of the population mean, especially when the sample size is large (e.g., >100, 150). The details (as
functions of sample size and skewness) for the use of those UCLs are summarized in various versions of
ProUCL Technical Guides (EPA 2004, 2007, 2009, 2010d, 2011, 2013a).
Notes: Using the maximum observed value to estimate the EPC term representing the average exposure
contracted by an individual over an EA is not recommended. For the sake of interested users, ProUCL
displays a warning message when the recommended 95% UCL (e.g., Hall's bootstrap UCL) of the mean
exceeds the observed maximum concentration. For such scenarios (when a 95% UCL does exceed the
maximum observed value), an alternative UCL computation method based upon Chebyshev inequality is
suggested by the ProUCL software.
1.11 Samples with Nondetect Observations
ND observations are inevitable in most environmental data sets. Singh, Maichle, and Lee (2006) studied
the performances (in terms of coverages) of the various UCL95 computation methods including the
simple substitution methods (such as the DL/2 and DL methods) for data sets with ND observations. They
concluded that the UCLs obtained using the substitution methods, including the replacement of NDs by
DL/2; do not perform well even when the percentage of ND observations is low, such as less than 5% to
10%. They recommended avoiding the use of substitution methods for computing UCL95 based upon
data sets with ND observations.
7.77.7 Avoid the Use of the DL/2 Substitution Method to Compute UCL95
Based upon the results of the report by Singh, Maichle, and Lee (2006), it is recommended to avoid the
use of the DL/2 substitution method when performing a GOF test, and when computing the summary
statistics and various other limits (e.g., UCL, UPL, UTLs) often used to estimate the EPC terms and
BTVs. Until recently, the substitution method has been the most commonly used method for computing
various statistics of interest for data sets which include NDs. The main reason for this has been the lack of
the availability of the other rigorous methods and associated software programs that can be used to
estimate the various environmental parameters of interest. Today, several methods (e.g., using KM
estimates) with better performance, including the Chebyshev inequality and bootstrap methods, are
available for computing the upper limits of interest. Several of those parametric and nonparametric
methods are available in ProUCL 4.0 and higher versions. The DL/2 method is included in ProUCL for
historical reasons as it had been the most commonly used and recommended method until recently (EPA
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2006b). EPA scientists and several reviewers of the ProUCL software had suggested and requested the
inclusion of the DL/2 substitution method in ProUCL for comparison and research purposes.
Notes: Even though the DL/2 substitution method has been incorporated in ProUCL, its use is not
recommended due to its poor performance. The DL/2 substitution method has been retained in ProUCL
5.1 for historical and comparison purposes. NERL-EPA, Las Vegas strongly recommends avoiding the
use of this method even when the percentage of NDs is as low as 5% to 10%.
1.11.2 ProUCL Does Not Distinguish between Detection Limits, Reporting limits, or Method
Detection Limits
ProUCL 5.1 (and all previous versions) does not make distinctions between method detection limits
(MDLs), adjusted MDLs, sample quantitation limits (SQLs), reporting limits (RLs), or DLs. Multiple
DLs (or RLs) in ProUCL mean different values of the detection limits. It is user's responsibility to
understand the differences between these limits and use appropriate values (e.g., DLs) for nondetect
values below which the laboratory cannot reliably detect/measure the presence of the analyte in collected
samples (e.g., soil samples). A data set consisting of values less than the DLs (or MDLs, RLs) is
considered a left-censored data set. ProUCL uses statistical methods available in the statistical literature
for left-censored data sets for computing statistics of interest including mean, sd, UCL, and estimates of
BTVs.
The user determines which qualifiers (e.g., J, U, UJ) will be considered as nondetects. Typically, all
values with U or UJ qualifiers are considered as nondetect values. It is the user's responsibility to enter a
value which can be used to represent a ND value. For NDs, the user enters the associated DLs or RLs
(and not zeros or half of the detection limits). An indicator column/variable, D_x taking a value, 0, for all
nondetects and a value, 1, for all detects is assigned to each variable, x, with NDs. It is the user's
responsibility to supply the numerical values for NDs (should be entered as reported DLs) not qualifiers
(e.g., J, U, B, UJ). For example, for thallium with nondetect values, the user creates an associated column
labeled as D_thallium to tell the software that the data set will have nondetect values. This column,
D_thallium consists of only zeros (0) and ones (1); zeros are used for all values reported as NDs and ones
are used for all values reported as detects.
1.12 Samples with Low Frequency of Detection
When all of the sampled values are reported as NDs, the EPC term and other statistical limits should also
be reported as a ND value, perhaps by the maximum RL or the maximum RL/2. The project team will
need to make this determination. Statistics (e.g., UCL95) based upon only a few detected values (e.g., <
4) cannot be considered reliable enough to estimate EPCs which can have a potential impact on human
health and the environment. When the number of detected values is small, it is preferable to use ad hoc
methods rather than using statistical methods to compute EPCs and other upper limits. Specifically, for
data sets consisting of < 4 detects and for small data sets (e.g., size < 10) with low detection frequency
(e.g., < 10%), the project team and the decision makers should decide, on a site-specific basis, how to
estimate the average exposure (EPC) for the constituent and area under consideration. For data sets with
low detection frequencies, other measures such as the median or mode represent better estimates (with
lesser uncertainty) of the population measure of central tendency.
Additionally, when most (e.g., > 95%) of the observations for a constituent lie below the DLs, the sample
median or the sample mode (rather than the sample average) may be used as an estimate of the EPC. Note
that when the majority of the data are NDs, the median and the mode may also be represented by a ND
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value. The uncertainty associated with such estimates will be high. The statistical properties, such as the
bias, accuracy, and precision of such estimates, would remain unknown. In order to be able to compute
defensible estimates, it is always desirable to collect more samples.
1.13 Some Other Applications of Methods in ProUCL 5.1
In addition to performing background versus site comparisons for CERCLA and RCRA sites, performing
trend evaluations based upon time-series data sets, and estimating EPCs in exposure and risk evaluation
studies, the statistical methods in ProUCL can be used to address other issues dealing with environmental
investigations that are conducted at Superfund or RCRA sites.
1.13.1 Identification of COPCs
Risk assessors and remedial project managers (RPMs) often use screening levels or BTVs to identify
COPCs during the screening phase of a cleanup project at a contaminated site. The screening for COPCs
is performed prior to any characterization and remediation activities that are conducted at the site. This
comparison is performed to screen out those constituents that may be present in the site medium of
interest at low levels (e.g., at or below the background levels or some pre-established screening levels)
and may not pose any threat and concern to human health and the environment. Those constituents may
be eliminated from all future site investigations, and risk assessment and risk management studies.
To identify the COPCs, point-by-point site observations are compared with some pre-established soil
screening levels (SSL) or estimated BTVs. This is especially true when the comparisons of site
concentrations with screening levels or BTVs are conducted in real time by the sampling or cleanup crew
onsite. The project team should decide the type of site samples (discrete or composite) and the number of
site observations that should be collected and compared with the screening levels or the BTVs. In case
BTVs or screening levels are not known, the availability of a defensible site-specific background or
reference data set of reasonable size (e.g., at least 10) is required for computing reliable and
representative estimates of BTVs and screening levels. The constituents with concentrations exceeding
the respective screening values or BTVs may be considered COPCs, whereas constituents with
concentrations (e.g., in all collected samples) lower than the screening values or BTVs may be omitted
from all future evaluations.
1.13.2 Identification of Non-Compliance Monitoring Wells
In MW compliance assessment applications, individual (often discrete) constituent concentrations from a
MW are compared with some pre-established limits such as an ACL or a maximum concentration limit
(MCL). An exceedance of the MCL or the BTV (e.g., estimated by a UTL95-95 or a UPL95) by a MW
concentration may be considered an indication of contamination in that MW. For individual concentration
comparisons, the presence of contamination (determined by an exceedance) may have to be confirmed by
re-sampling from that MW. If concentrations of constituents in the original sample and re-sample(s)
exceed the MCL or BTV, then that MW may require further scrutiny, perhaps triggering remediation
activates. If the concentration data from a MW for 4 to 5 continuous quarters (or some other designated
time period determined by the project team) are below the MCL or BTV level, then that MW may be
considered as complying with (achieving) the pre-established or estimated standards.
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1.13.3 Verification of the Attainment of Cleanup Standards, Cs
Hypothesis testing approaches are used to verify the attainment of the cleanup standard, Cs, at site AOCs
after conducting remediation and cleanup at those site AOCs (EPA 1989a, 1994). In order to assess the
attainment of cleanup levels, a representative data set of adequate size perhaps obtained using the DQO
process (or a minimum of 10 observations should be collected) needs to be made available from the
remediated/excavated areas of the site under investigation. The sample size should also account for the
size of the remediated site areas: meaning that larger site areas should be sampled more (with more
observations) to obtain a representative sample of the remediated areas under investigation. Typically, the
null hypothesis of interest is H0: Site Mean, /& > Cs versus the alternative hypothesis, Hi: Site Mean, /4 <
Cs, where the cleanup standard, Cs, is known a priori.
1.13.4 Using BTVs (Upper Limits) to Identify Hot Spots
The use of upper limits (e.g., UTLs) to identify hot spot(s) has also been mentioned in the Guidance for
Comparing Background and Chemical Concentrations in Soil for CERCLA Sites (EPA 2002b). Point-by-
point site observations are compared with a pre-established or estimated BTV. Exceedances of the BTV
by site observations may represent impacted locations with elevated concentrations (hot spots).
1.14 Some General Issues, Suggestions and Recommendations made by
ProUCL
Some general issues regarding the handling of multiple DLs by ProUCL and recommendations made
about various substitution and ROS methods for data sets with NDs are described in the following
sections.
1.14.1 Handling of Field Duplicates
ProUCL does not pre-process field duplicates. The project team determines how field duplicates will be
handled and pre-processes the data accordingly. For an example, if the project team decides to use
average values for field duplicates, then averages need to be computed and field duplicates need to be
replaced by their respective average values. It is the user's responsibility to feed in appropriate values
(e.g., averages, maximum) for field duplicates. The user is advised to refer to the appropriate EPA
guidance documents related to collection and use of field duplicates for more information.
1.14.2 ProUCL Recommendation about ROS Method and Substitution (DL/2) Method
For data sets with NDs, ProUCL can compute point estimates of population mean and standard deviation
using the KM and ROS methods (and also using the DL/2 substitution method). The substitution method
has been retained in ProUCL for historical and research purposes. ProUCL uses Chebyshev inequality,
bootstrap methods, and normal, gamma, and lognormal distribution based equations on KM (or ROS)
estimates to compute upper limits (e.g., UCLs, UTLs). The simulation study conducted by Singh,
Maichle and Lee (2006) demonstrated that the KM method yields accurate estimates of the population
mean. They also demonstrated that for moderately skewed to highly skewed data sets, UCLs based upon
KM estimates and BCA bootstrap (mild skewness), KM estimates and Chebyshev inequality (moderate to
high skewness), and KM estimates and bootstrap-t method (moderate to high skewness) yield better (in
terms of coverage probability) estimates of EPCs than other UCL methods based upon the Student's t-
statistic on KM estimates, percentile bootstrap method on KM or ROS estimates.
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1.14.3 UnhandledExceptions and Crashes in ProUCL
A typical statistical software, especially developed under limited resources may not be able to
accommodate data sets with all kinds of deficiencies such as all missing values for a variable, or all
nondetect values for a variable. An inappropriate/insufficient data set can occur in various forms and not
all of them can be addressed in a scientific program like ProUCL. Specifically, from a programming point
of view, it can be quite burdensome on the programmer to address all potential deficiencies that can occur
in a data set. ProUCL 5.1 addresses many data deficiencies and produces warming messages. All data
deficiencies causing unhandled exceptions which were identified by users have been addressed in
ProUCL 5.1. However, when ProUCL yields an unhandled exception or crashes, it is highly likely that
there is something wrong with the data set; the user is advised to review the input data set to make sure
that the data set follows ProUCL data and formatting requirements.
1.15 The Unofficial User Guide to ProUCL4 (Helsel and Gilroy 2012)
Several ProUCL 4.1 users sent inquiries about the validity of the comments made about the ProUCL
software in the Unofficial User Guide to ProUCL4 (Helsel and Gilroy, 2012) and in the Practical Stats
webinar, "ProUCL v4: The Unofficial User Guide," presented by Dr. Helsel on October 15, 2012 (Helsel
2012a). Their inquiries led us to review comments made about the ProUCL4 software and its associated
guidance documents (EPA 2007, 2009a, 2009b, 2010c, 2010d, and 2011) in the "The Unofficial Users
Guide to ProUCL4" and in the webinar, "ProUCL v4: The Unofficial User Guide". These two documents
collectively are referred to as the Unofficial ProUCLv4 User Guide in this ProUCL document. The pdf
document describing the material presented in the Practical Stats Webinar (Helsel 2012a) was
downloaded from the http://www.practicalstats.com website.
In the "ProUCL v4: The Unofficial User Guide", comments have been made about the software and its
guidance documents, therefore, it is appropriate to address those comments in the present ProUCL
guidance document. It is necessary to provide the detailed response to assure that: 1) rigorous statistical
methods are used to compute decision making statistics; and 2) the methods incorporated in ProUCL
software are not misrepresented and misinterpreted. Some general responses and comments about the
material presented in the webinar and in the Unofficial User Guide to ProUCLv4 are described as follows.
Specific comments and responses are also considered in the respective chapters of ProUCL 5.1 (and
ProUCL 5.0) guidance documents.
Note: It is noted that the Kindle version of "ProUCL v4: Unofficial User Guide" is no longer available on
Amazon. Several incorrect theoretical statements and statements misrepresenting ProUCL 4 were made in
that Unofficial User Guide; therefore, a brief response to some of those statements has been retained in
ProUCL 5.1 guidance documents.
ProUCL is a freeware software package which has been developed under limited government funding to
address statistical issues associated with various environmental site projects. Not all statistical methods
(e.g., Levene test) described in the statistical literature have been incorporated in ProUCL. One should not
compare ProUCL with commercial software packages which are expensive and not as user-friendly as the
ProUCL software when addressing environmental statistical issues. The existing and some new statistical
methods based upon the research conducted by ORD-NERL, EPA Las Vegas during the last couple of
decades have been incorporated in ProUCL to address the statistical needs of various environmental site
projects and research studies. Some of those new methods may not be available in text books, in the
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library of programs written in R-script, and in commercial software packages. However, those methods
are described in detail in the cited published literature and also in the ProUCL Technical Guides (e.g.,
EPA 2007, 2009a, 2009b, 2010c, 2010d, and 2011). Even though for uncensored data sets, programs
which compute gamma distribution based UCLs and UPLs are available in R Script, programs which
compute a 95% UCL of mean based upon a gamma distribution on KM estimates are not as easily
available.
• In the Unofficial ProUCL v4 User Guide, several statements have been made about percentiles. There
are several ways to compute percentiles. Percentiles computed by ProUCL may or may not be
identical (don't have to be) to percentiles computed by NADA for R (Helsel 2013) or described in
Helsel and Gilroy (2012). To address users' requests, ProUCL 4.1 (2011) and its higher versions
compute percentiles that are comparable to the percentiles computed by Excel 2003 and higher
versions.
• The literature search suggests that there are a total of nine (9) known types of percentiles, i.e., 9
different methods of calculating percentiles in statistics literature (Hyndman and Fan, 1996). The R
programming language (R Core Team 2012) computes percentiles using those 9 methods using the
following statement in R
Quantile (x, p, type=k) where p = percentile, k = integer between 1-9
ProUCL computes percentiles using Type 7; Minitab 16 and SPSS compute percentiles using Type 6.
It is simply a matter of choice, as there is no 'best' type to use. Many software packages use one type
for calculating a percentile, and another for generating a box plot (Hyndman and Fan 1996).
• An incorrect statement "By definition, the sample mean has a 50% chance of being below the true
population mean" has been made in Helsel and Gilroy (2012) and also in Helsel (2012a). The above
statement is not correct for means of skewed distributions (e.g., lognormal or gamma) commonly
occurring in environmental applications. Since Helsel (2012) prefers to use a lognormal distribution,
the incorrectness of the above statement has been illustrated using a lognormal distribution. The
mean and median of a lognormal distribution (details in Section 2.3.2 of Chapter 2 of this Technical
Guide) are given by:
mean =jul = exp(ju + 0.5a2); and median =M = exp(//)
From the above equations, it is clear that the mean of a lognormal distribution is always greater than
the median for all positive values of a (sd of log-transformed variable). Actually the mean is greater
than the pth percentile when a >2zp. For example, when p = 0.80, zp = 0.845, and mean of a
lognormal distribution, ju\ exceeds xo.so, the 80th percentile when a > 1.69. In other words, when a >
1.69 the lognormal mean will exceed the 80th percentile of a lognormal distribution. Here zp
represents thepth percentile of the standard normal distribution (SND) with mean 0 and variance 1.
To demonstrate the incorrectness of the above statement, a small simulation study was conducted.
The distribution of sample means based upon samples of size 100 were generated from lognormal
distributions with ju =4, and varying skewness. The experiment was performed 10,000 times to
generate the distributions of sample means. The probabilities of sample means less than the
population means were computed. The following results are noted.
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Table 1-2. Probabilities p(~x < ^) Computed for Lognormal Distributions with u=4 and Varying Values of a
Results are based upon 10000 Simulation Runs for Each Lognormal Distribution Considered
Parameter
/>(*
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impute NDs) as the objective is not to impute NDs. To impute NDs, ProUCL uses ROS methods
(Gamma ROS and log ROS) requiring place holders; and ProUCL computes plotting positions for all
detects and NDs to generate a proper regression model which is used to impute NDs. Also for
comparison purposes, ProUCL can be used to generate Q-Q plots on data sets obtained by replacing
NDs by their respective DLs or DL/2s. In these cases, no NDs are imputed, and there is no need to
retain placeholders for NDs. On these Q-Q plots, ProUCL displays some relevant statistics which are
computed based upon the data displayed on those graphs.
Helsel (2012a) states that the Summary Statistics module does not display KM estimates and that
statistics based upon logged data are useless. Typically, estimates computed after processing the data
do not represent summary statistics. Therefore, KM and ROS estimates are not displayed in the
Summary Statistics module. These statistics are available in several other modules including the
UCL and BTV modules. At the request of several users, summary statistics are computed based upon
logged data. It is believed that the mean, median, or standard deviation of logged data do provide
useful information about data skewness and data variability.
To test for the equality of variances, the F-test, as incorporated in ProUCL, performs fairly well and
the inclusion of the Levene's (1960) test will not add any new capability to the ProUCL software.
Therefore, taking budget constraints into consideration, Levene's test has not been incorporated in the
ProUCL software.
o Although it makes sense to first determine if the two variances are equal or unequal, this is
not a requirement to perform a t-test. The t-distribution based confidence interval or test for
fii - fi2 based on the pooled sample variance does not perform better than the approximate
confidence intervals based upon Satterthwaite's test. Hence testing for the equality of
variances is not required to perform a two-sample t-test. The use of Welch-Satterthwaite's or
Cochran's method is recommended in all situations (see Hayes 2005).
Helsel (2012a) suggests that imputed NDs should not be made available to the users. The developers
of ProUCL and other researchers like to have access to imputed NDs. As a researcher, for exploratory
purposes only, one may want to have access to imputed NDs to be used in exploratory advanced
methods such as multivariate methods including data mining, cluster and principal component
analyses. It is noted that one cannot easily perform exploratory methods on multivariate data sets with
NDs. The availability of imputed NDs makes it possible for researchers and scientists to identify
potential patterns present in complex multivariate data by using data mining exploratory methods on
those multivariate data sets with NDs. Additional discussion on this topic is considered in Chapter 4
of this Technical Guide.
o The statements summarized above should not be misinterpreted. One may not use parametric
hypothesis tests such as a t-test or a classical ANOVA on data sets consisting of imputed
NDs. These methods require further investigation as the decision errors associated with such
methods remain unquantified. There are other methods such as the Gehan and T-W tests in
ProUCL 5.0/ProUCL 5.1 which are better suited to perform two-sample hypothesis tests
using data sets with multiple detection limits.
Outliers: Helsel (2012a) and Helsel and Gilroy (2012) make several comments about outliers. The
philosophy (with input from EPA scientists) of the developers of ProUCL about the outliers in
environmental applications is that those outliers (unless they represent typographical errors) may
potentially represent impacted (site related or otherwise) locations or monitoring wells; and therefore
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may require further investigation. Moreover, decision statistics such as a UCL95 based upon a data
set with outliers gets inflated and tends to represent those outliers instead of representing the
population average. Therefore, a few low probability outliers coming from the tails of the data
distribution may not be included in the computation of the decision making upper limits (UCLs,
UTLs), as those upper limits get distorted by outliers and tend not to represent the parameters they are
supposed to estimate.
o The presence of outliers in a data set tends to destroy the normality of the data set. In other
words, a data set with outliers can seldom (may be when outliers are mild, lying around the
border of the central and tail parts of a normal distribution) follow a normal distribution.
There are modern robust and resistant outlier identification methods (e.g., Rousseeuw and
Leroy 1987; Singh and Nocerino 1995) which are better suited to identify outliers present in
a data set; several of those robust outlier identification methods are available in the Scout
2008 version 1.0 (EPA 2009) software package.
o For both Rosner and Dixon tests, it is the data set (also called the main body of the data set)
obtained after removing the outliers (and not the data set with outliers) that needs to follow a
normal distribution (Barnett and Lewis 1994). Outliers are not known in advance. ProUCL
has normal Q-Q plots which can be used to get an idea about potential outliers (or mixture
populations) present in a data set. However, since a lognormal model tends to accommodate
outliers, a data set with outliers can follow a lognormal distribution; this does not imply that
the outlier which may actually represent an impacted/unusual location does not exist! In
environmental applications, outlier tests should be performed on raw data sets, as the cleanup
decisions need to be made based upon values in the raw scale and not in log-scale or some
other transformed space. More discussion about outliers can be found in Chapter 7 of this
Technical Guide.
In Helsel (2012a), it is stated, "Mathematically, the lognormal is simpler and easier to interpret than
the gamma (opinion)." We do agree with the opinion that the lognormal is simpler and easier to use
but the log-transformation is often misunderstood and hence incorrectly used and interpreted.
Numerous examples (e.g., Example 2-1 and 2-2, Chapter 2) are provided in the ProUCL guidance
documents illustrating the advantages of the using a gamma distribution.
It is further stated in Helsel (2012a) that ProUCL prefers the gamma distribution because it
downplays outliers as compared to the lognormal. This argument can be turned around - in other
words, one can say that the lognormal is preferred by practitioners who want to inflate the effect of
the outlier. Setting this argument aside, we prefer the gamma distribution as it does not transform the
variable so the results are in the same scale as the collected data set. As mentioned earlier, log-
transformation does appear to be simpler but problems arise when practitioners are not aware of the
pitfalls (e.g., Singh and Ananda 2002; Singh, Singh, and laci 2002) associated with the use of
lognormal distribution.
Helsel (2012a) and Helsel and Gilroy (2012) state that "lognormal and gamma are similar, so usually
if one is considered possible, so is the other." This is another incorrect and misleading statement;
there are significant differences in the two distributions and in their mathematical properties. Based
upon the extensive experience in environmental statistics and published literature, for skewed data
sets that follow both lognormal and gamma distributions, the developers favor the use of the gamma
distribution over the lognormal distribution. The use of the gamma distribution based decision
statistics is preferred to estimate the environmental parameters (mean, upper percentile). A lognormal
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model tends to hide contamination by accommodating outliers and multiple populations whereas a
gamma distribution adjusts for skewness but tends not to accommodate contamination (elevated
values) as can be seen in Examples 2-1 and 2-2 of Chapter 2 of this Technical Guide. The use of the
lognormal distribution on a data set with outliers tends to yield inflated and distorted estimates which
may not be protective of human health and the environment; this is especially true for skewed data
sets of small of sizes <20-30; the sample size requirement increases with skewness.
o In the context of computing a UCL95 of mean, Helsel and Gilroy (2012) and Helsel (2012a) state
that GROS and LROS methods are probably never better than the KM method. It should be
noted that these three estimation methods compute estimates of mean and standard deviation and
not the upper limits used to estimate EPCs and BTVs. The use of the KM method does yield good
estimates of the mean and standard deviation as noted by Singh, Maichle, and Lee (2006). The
problem of estimating mean and standard deviation for data sets with nondetects has been studied
by many researchers as described in Chapter 4 of this document. Computing good estimates of
mean and sd based upon left-censored data sets addresses only half of the problem. The main
issue is to compute decision statistics (UCL, UPL, UTL) which account for uncertainty and data
skewness inherently present in environmental data sets.
o Realizing that for skewed data sets, Student's t-UCL, CLT-UCL, and standard and percentile
bootstrap UCLs do not provide the specified coverage to the population mean for uncensored data
sets, many researchers (e.g., Johnson 1978; Chen 1995; Efron and Tibshirani 1993; Hall [1988,
1992]; Grice and Bain 1980; Singh, Singh, and Engelhardt 1997; Singh, Singh, and laci 2002)
developed parametric (e.g., gamma) and nonparametric (e.g., bootstrap-t and Hall's bootstrap
method, modified-t, and Chebyshev inequality) methods for computing confidence intervals and
upper limits which adjust for data skewness. One cannot ignore the work and findings of the
researchers cited above, and assume that Student's t-statistic based upper limits or percentile
bootstrap method based upper limits can be used for all data sets with varying skewness and
sample sizes.
o Analytically, it is not feasible to compare the various estimation and UCL computation methods
for skewed data sets containing ND observations. Instead, researchers use simulation
experiments to learn about the distributions and performances of the various statistics (e.g., KM-t-
UCL, KM-percentile bootstrap UCL, KM-bootstrap-t UCL, KM-Gamma UCL). Based upon the
suggestions made in published literature and findings summarized in Singh, Maichle, and Lee
(2006), it is reasonable to state and assume that the findings of the simulation studies performed
on uncensored skewed data sets comparing the performances of the various UCL computation
methods can be extended to skewed left-censored data sets.
o Like uncensored skewed data sets, for left-censored data sets, ProUCL 5.0/ProUCL 5.1 has
several parametric and nonparametric methods to compute UCLs and other limits which adjust
for data skewness. Specifically, ProUCL uses KM estimates in gamma equations; in the
bootstrap-t method, and in the Chebyshev inequality to compute upper limits for left-censored
skewed data sets.
Helsel (2012a) states that ProUCL 4 is based upon presuppositions. It is emphasized that ProUCL
does not make any suppositions in advance. Due to the poor performance of a lognormal model, as
demonstrated in the literature and illustrated via examples throughout ProUCL guidance documents,
the use of a gamma distribution is preferred when a data set can be modeled by a lognormal model
and a gamma model. To provide the desired coverage (as close as possible) for the population mean,
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in earlier versions of ProUCL (version 3.0), in lieu of H-UCL, the use of Chebyshev UCL was
suggested for moderately and highly skewed data sets. In later (3.00.02 and higher) versions of
ProUCL, depending upon skewness and sample size, for gamma distributed data sets, the use of the
gamma distribution was suggested for computing the UCL of the mean.
Upper limits (e.g., UCLs, UPLs, UTLs) computed using the Student's t statistic andpercentile bootstrap
method (Helsel 2012, NADAfor R, 2013) often fail to provide the desired coverage (e.g., 95% confidence
coefficient) to the parameters (mean, percentile) of most of the skewed environmental populations. It is
suggested that the practitioners compute the decision making statistics (e.g., UCLs, UTLs) by taking: data
distribution; data set size; and data skewness into consideration. For uncensored and left-censored data
sets, several such upper limits computation methods are available in ProUCL 5.1 and its earlier versions.
Contrary to the statements made in Helsel and Gilroy (2012), ProUCL software does not favor statistics
which yield higher (e.g., nonparametric Chebyshev UCL) or lower (e.g., preferring the use of a gamma
distribution to using a lognormal distribution) estimates of the environmental parameters (e.g., EPC and
BTVs). The main objectives of the ProUCL software funded by the U.S. EPA is to compute rigorous
decision statistics to help the decision makers and project teams in making sound decisions which are
cost-effective and protective of human health and the environment.
Cautionary Note: Practitioners and scientists are cautioned about: 1) the suggestions made about the
computations of upper limits described in some recent environmental literature such as the NADA books
(Helsel [2005, 2012]); and 2) the misleading comments made about the ProUCL software in the training
courses offered by Practical Stats during 2012 and 2013. Unfortunately, comments about ProUCL made
by Practical Stats during their training courses lack professionalism and theoretical accuracy. It is noted
that NADA packages in R and Minitab (2013) developed by Practical Stats do not offer methods which
can be used to compute reliable or accurate decision statistics for skewed data sets. Decision statistics
(e.g., UCLs, UTLs, UPLs) computed using the methods (e.g., UCLs computed using percentile bootstrap,
and KM and LROS estimates and t-critical values) described in the NADA books and incorporated in
NADA packages do not take data distribution and data skewness into consideration. The use of statistics
suggested in NADA books and in Practical Stats training sessions often fail to provide the desired
specified coverage to environmental parameters of interest for moderately skewed to highly skewed
populations. Conclusions derived based upon those statistics may lead to incorrect conclusions which
may not be cost-effective or protective of human health and the environment.
Page 75 (Helsel [2012]): One of the reviewers of the ProUCL 5.0 software drew our attention to the
following incorrect statement made on page 75 of Helsel (2012):
"If there is only 1 reporting limit, the result is that the mean is identical to a substitution of the reporting
limit for censored observations."
An example of a left-censored data set containing ND observations with one reporting limit of 20 which
illustrates this issue is described as follows.
Y D_y
20 0
20 0
20 0
7 1
58 1
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92 1
100 1
72 1
11 1
27 1
The mean and standard deviation based upon the KM and two substitution methods: DL/2 and DL are
summarized as follows:
Kaplan-Meier (KM) Statistics
Mean 39.4
SD 35.56
DL Substitution method (replacing censored values by the reporting limit)
Mean 42.7
SD 34.77
DL/2 Substitution method (replacing NDs by the reporting limit)
Mean 39.7
SD 37.19
The above example illustrates that the KM mean (when only 1 detection limit is present) is not actually
identical to the mean estimate obtained using the substitution, DL (RL) method. The statement made in
Helsel's text (and also incorrectly made in his presentations such as the one made at the U.S. EPA 2007
National Association of Regional Project Managers (NARPM) Annual Conference:
http ://www.ttemidev. com/narpm2007 Admin/conference/)
holds only when all observations reported as detects are greater than the single reporting limit, which is
not always true for environmental data sets consisting of analytical concentrations.
1.16 Box and Whisker Plots
At the request of ProUCL users, a brief description of box plots (also known as box and whisker plots) as
developed by Tukey (Hoaglin, Mosteller and Tukey 1991) is provided in this section. A box and
whiskers plot represents a useful and convenient exploratory tool and provides a quick five point
summary of a data set. In statistical literature, one can find several ways to generate box plots. The
practitioners may have their own preferences to use one method over the other. Box plots are well
documented in the statistical literature and description of box plots can be easily obtained by surfing the
net. Therefore, the detailed description about the generation of box plots is not provided in ProUCL
guidance documents. ProUCL also generates box plots for data set with NDs. Since box plots are used
for exploratory purposes to identify outliers and also to compare concentrations of two or more groups, it
does not really matter how NDs are displayed on those box plots. ProUCL generates box plots using
detection limits and draws a horizontal line at the highest detection limit. Users can draw up to four
horizontal lines at other levels (e.g., a screening level, a BTV, or an average) of their choice.
All box plot methods, including the one in ProUCL, represent five-point summary graphs including: the
lowest and the highest data values, median (50th percentile=second quartile, Q2), 25th percentile (lower
quartile, Ql), and 75th percentile (upper quartile, Q3). A box and whisker plot also provides information
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about the degree of dispersion (interquartile range (IQR) = Q3-Ql=length/height of the box in a box plot),
the degree of skewness (suggested by the length of the whiskers) and unusual data values known as
outliers. Specifically, ProUCL (and other software packages) use the following to generate a box and
whisker plot.
• Q1= 25th percentile, Q2= 50th (median), and Q3 = 75th percentile
• Interquartile range= IQR = Q3-Q1 (the length/height of the box in a box plot)
• Lower whisker starts at Q1 and the upper whisker starts at Q3.
• Lower whisker extends up to the lowest observation or (Ql - 1.5 * IQR) whichever is higher
• Upper whisker extends up to the highest observation or (Q3 + 1.5* IQR) whichever is lower
• Horizontal bars (also known as fences) are drawn at the end of whiskers
• Guidance in statistical literature suggests that observations lying outside the fences (above the
upper bar and below the lower bar) are considered potential outliers
An example box plot generated by ProUCL is shown in the following graph.
Box Plot for Lead
Box Plot with Fences and Outlier
It should be pointed out that the use of box plots in different scales (e.g., raw-scale and log-scale) may
lead to different conclusions about outliers. Below is an example illustrating this issue.
Example 1-2. Consider an actual data set consisting of copper concentrations collected a Superfund Site.
The data set is: 0.83, 0.87, 0.9, 1, 1, 2, 2, 2.18, 2.73, 5, 7, 15, 22, 46, 87.6, 92.2, 740, and 2960. Box plots
using data in the raw-scale and log-scale are shown in Figures 1-1 and 1-2.
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Box Plot for Copper
Figure 1-1. Box Plot of Raw Data in Original Scale
Based upon the last bullet point of the description of box plots described above, from Figure 1-1, it is
concluded that two observations 740 and 2960 in the raw scale represent outliers.
Box Plot for In(copper)
Figure 1-2. Box Plot of Data in Log-Scale
However, based upon the last bullet point about box plots, from Figure 1-2, it is concluded that two
observations 740 and 2960 in the log-scale do not represent outliers. The log-transformation has
accommodated the two outliers. This is one of the reasons ProUCL guidance suggests avoiding the use of
log-transformed data. The use of a log-transformation tends to hide/accommodate outliers/contamination.
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Note: ProUCL uses a couple of development tools such as FarPoint spread (for Excel type input and
output operations) and ChartFx (for graphical displays). ProUCL generates box plots using the built-in
box plot feature in ChartFx. The programmer has no control over computing various statistics (e.g., Ql,
Q2, Q3, IQR) using ChartFx. So box plots generated by ProUCL can differ slightly from box plots
generated by other programs (e.g., Excel). However, for all practical and exploratory purposes, box plots
in ProUCL are equally good (if not better) as available in the various commercial software packages for
investigating data distribution (skewed or symmetric), identifying outliers, and comparing multiple
groups (main objectives of box plots).
Precision in Box Plots: Box plots generated using ChartFx round values to the nearest integer. For
increased precision of graphical displays (all graphical displays generated by ProUCL), the user can use
the process described as follows.
Position your cursor on the graph and right-click, a popup menu will appear. Position the cursor on
Properties and right-click; a windows form labeled Properties will appear. There are three choices at
the top: General, Series and Y-Axis. Position the e cursor over the Y-Axis choice and left-click. You
can change the number of decimals to increase the precision, change the step to increase or decrease the
number Y-Axis values displayed and/or change the direction of the label. To show values on the plot
itself, position your cursor on the graph and right-click; a pop-up menu will appear. Position the cursor on
Point Labels and right-click. There are other options available in this pop-up menu including changing
font sizes.
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CHAPTER 2
Goodness-of-Fit Tests and Methods to Compute Upper
Confidence Limit of Mean for Uncensored Data Sets
without Nondetect Observations
2.1 Introduction
Many environmental decisions including exposure and risk assessment and management, and cleanup
decisions are made based upon the mean concentrations of the contaminants/constituents of potential
COPCs. To address the uncertainty associated with the sample mean, a UCL95 is used to estimate the
unknown population mean, fj,\. A UCL95 is routinely used to estimate the EPC) term (EPA 1992a; EPA
2002a). A UCL95 of the mean represents that limit such that one can be 95% confident that the
population mean, JJL\, will be less than that limit. From a risk point of view, a UCL95 of the mean
represents a number that is considered health protective when used to compute risk and health hazards.
Since, many environmental decisions are made based upon a UCL95, it is important to compute a
reliable, defensible (from human health point of view) and cost-effective estimate of the EPC. To
compute reliable estimates of practical merit, ProUCL software provides several parametric and
nonparametric UCL computation methods covering a wide-range of skewed distributions (e.g.,
symmetric, mildly skewed to highly skewed) for data sets of various sizes. Based upon simulation results
summarized in the literature (Singh, Singh, and Engelhardt [1997], Singh, Singh and laci [2002]), data
distribution, data set size, and skewness, ProUCL makes suggestions on how to select an appropriate
UCL95 of the mean to estimate the EPC. It should be noted that a simulation study cannot cover all
possible real world data sets of various sizes and skewness following different probability distributions.
This ProUCL Technical Guide provides sufficient guidance to help a user select the most appropriate
UCL as an estimate of the EPC. The ProUCL software makes suggestions to help a typical user select an
appropriate UCL from all the UCLs incorporated in ProUCL and those available in the statistical
literature. UCL values, other than those suggested by ProUCL, may be selected based upon project
personnel's experiences and project needs. The user may want to consult a statistician before selecting an
appropriate UCL95.
The ITRC (2012) regulatory document recommends the use of a Student's t-UCL95 and Chebyshev
inequality based UCL95 to estimate EPCs for ISM based soil samples collected from DUs. In order to
facilitate the computation of ISM data based estimates of the EPC, ProUCLS.l (and ProUCL 5.0) can
compute a UCL95 of the mean based upon data sets of sizes as small as 3. Additionally, the UCL module
of ProUCL can be used on ISM-based data sets with NDs.
However, it is advised that the users do not compute decision making statistics (e.g., UCLs, upper
prediction limits [UPLs], upper tolerance limits [UTLs])from discrete data sets consisting of less than 8-
10 observations.
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For uncensored data sets without ND observations, theoretical details of the Student's t- and percentile
bootstrap UCL computation methods, as well as the more complicated bootstrap-t and gamma distribution
methods, are described in this Chapter. One should not ignore the use of gamma distribution based UCLs
(and other upper limits) just because it is easier to use a lognormal distribution. Typically, environmental
data sets are positively skewed, and a default lognormal distribution (EPA 1992a) is used to model such
data distributions. Additionally, an H-statistic based Land's (Land, 1971, 1975) H-UCL is then typically
used to estimate the EPC. Hardin and Gilbert (1993), Singh, Singh, and Engelhardt (1997, 1999), Schultz
and Griffin (1999), and Singh, Singh, and laci (2002) pointed out several problems associated with the
use of the lognormal distribution and the H-statistic to compute UCL of the mean. For lognormal data sets
with high standard deviation (sd), a, of the natural log-transformed data (e.g., a exceeding 1.0 to 1.5), the
H-UCL becomes unacceptably large, exceeding the 95% and 99% data quantiles, and even the maximum
observed concentration, by orders of magnitude (Singh, Singh, and Engelhardt 1997). The H-UCL is also
very sensitive to a few low or a few high values. For example, the addition of a single low measurement
can cause the H-UCL to increase by a large amount (Singh, Singh, and laci, 2002) by increasing
variability. Realizing that the use of the H-statistic can result in an unreasonably large UCL, it has been
recommended (EPA 1992a) that the maximum value be used as an estimate of the EPC in cases when the
H-UCL exceeds the largest value in the data set. For uncensored data sets without any NDs, ProUCL
makes suggestions/recommendations on how to compute an appropriate UCL95 based upon data set size,
data skewness and distribution.
In practice, many skewed data sets follow a lognormal as well as a gamma distribution. Singh, Singh, and
laci (2002) observed that UCLs based upon a gamma distribution yield reliable and stable values of
practical merit. It is, therefore, desirable to test whether an environmental data set follows a gamma
distribution. A gamma distribution based UCL95 of the mean provides approximately 95% coverage to
the population mean, ju\ = kO of a gamma distribution, G (k, 9) with k and 9 respectively representing the
shape and scale parameters. For data sets following a gamma distribution with shape parameter, k > 1, the
EPC should be estimated using an adjusted gamma (when «<50) or approximate gamma (when «>50)
UCL95 of the mean. For highly skewed gamma distributed data sets with values of the shape parameter, k
< 1.0, a 95% UCL may be computed using the bootstrap-t-method or Hall's bootstrap method when the
sample size, n, is smaller, such as <15 to 20. For larger sample sizes with n> 20, a UCL of the mean may
be computed using the adjusted or approximate gamma UCL (Singh, Singh, and laci 2002) computation
method. Based upon professional judgment and practical experience of the authors, some of these
suggestions have been modified. Specifically, in earlier versions ProUCL, the cutoff value for the shape
parameter, k was 0.1 which has been changed to 1.0 in this version.
Unlike the percentile bootstrap and bias-corrected accelerated bootstrap (BCA) methods, bootstrap-t and
Hall's bootstrap methods (Efron and Tibshirani, 1993) account for data skewness and their use is
recommended on skewed data sets when computing UCLs of the mean. However, the bootstrap-t and
Hall's bootstrap methods sometimes result in erratic, inflated, and unstable UCL values, especially in the
presence of outliers (Efron and Tibshirani 1993). Therefore, these two methods should be used with
caution. The user should examine the various UCL results and determine if the UCLs based upon the
bootstrap-t and Hall's bootstrap methods represent reasonable and reliable UCL values. If the results of
these two methods are much higher than the rest of the UCL computation methods, it could be an
indication of erratic behavior of these two bootstrap UCL computation methods. ProUCL prints out a
warning message whenever the use of these two bootstrap methods is recommended. In cases where these
two bootstrap methods yield erratic and inflated UCLs, the UCL of the mean may be computed using the
Chebyshev inequality.
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ProUCL has graphical (e.g., quantile-quantile [Q-Q] plots) and formal goodness-of-fit (GOF) tests for
normal, lognormal, and gamma distributions. These GOF tests are available for data sets with and without
NDs. The critical values of the Anderson-Darling (A-D) test statistic and the Kolmogorov-Smirnov (K-S)
test statistic to test for gamma distributions were generated using Monte Carlo simulation experiments
(Singh, Singh, and laci 2002). Those critical values have been incorporated in ProUCL software and are
tabulated in Appendix A for various levels of significance.
ProUCL computes summary statistics for raw, as well as, log-transformed data sets with and without ND
observations. In this Technical Guide and in ProUCL software, log -transformation (log) stands for the
natural logarithm (In, LN) or log to the base e. For uncensored data sets, mathematical algorithms and
formulae used in ProUCL to compute the various UCLs are summarized in this chapter. ProUCL also
computes the maximum likelihood estimates (MLEsj and the minimum variance unbiased estimates
(MVUEs) of the population parameters of normal, lognormal, and gamma distributions. Nonparametric
UCL computation methods in ProUCL include: Jackknife, central limit theorem (CLT), adjusted-CLT,
modified Student's t (adjusts for skewness) Chebyshev inequality, and bootstrap methods. It is well
known that the Jackknife method (with sample mean as an estimator) and Student's t-method yield
identical UCL values. Moreover, it is noted that UCLs based upon the standard bootstrap and the
percentile bootstrap methods do not perform well by not providing the specified coverage of the mean for
skewed data sets.
Note on Computing Lower Confidence Limits (LCLs) of Mean: For some environmental projects an LCL
of the unknown population mean is needed to achieve the project DQOs. At present, ProUCL does not
directly compute LCLs of mean. However, for data sets with and without nondetects, excluding the
bootstrap methods, gamma distribution, and H-statistic based LCLs of mean, the same critical value (e.g.,
normal z value, Chebyshev critical value, or t-critical value) can be used to compute a LCL of mean as
used in the computation of the UCL of the mean. Specifically, to compute a LCL, the '+' sign used in the
computation of the corresponding UCL needs to be replaced by the '-' sign in the equation used to
compute that UCL (excluding gamma, lognormal H-statistic, and bootstrap methods). For specific details,
the user may want to consult a statistician. For data sets without nondetect observations, the user may
want to use the Scout 2008 software package (EPA 2009d, 2010) to directly compute the various
parametric and nonparametric LCLs of mean.
2.2 Goodness-of-Fit (GOF) Tests
Let x\, X2, ..., xn be a representative random sample (e.g., representing lead concentrations) from the
underlying population (e.g., site areas under investigation) with unknown mean, [j.\, and variance, oi1. Let
fj. and a represent the population mean and the population standard deviation (sd) of the log -transformed
(natural log to the base e) data. Let y and sy (a ) be the sample mean and sample sd, respectively, of the
log -transformed data, yt = log (xi); 1= 1,2, ... ,n. Specifically, let
n
(2-2)
Similarly, let x and sx be the sample mean and sd of the raw data, xi , X2 , .. , xn, obtained by replacing y
by x in equations (2-1) and (2-2), respectively. In this chapter, irrespective of the underlying distribution,
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ju\, and a\2 represent the mean and variance of the random variable X (in original units), whereas // and a1
represent the mean and variance of Y = loge(X).
Three data distributions have been considered in ProUCL 5.1 (and in older versions). These include the
normal, lognormal, and the gamma distributions. Shapiro-Wilk, for n <2000, and Lilliefors (1967) test
statistics are used to test for normality or lognormality of a data set. Lilliefors test (along with graphical
Q-Q plot) seems to perform fairly well for samples of size 50 and higher. In ProUCL 5.1, updated critical
values of Lilliefors test developed by Moling and Abdi (2007) and provided in the Encyclopedia of
Measurement and Statistics have been used. The empirical distribution function (EDF) based methods,
the K-S and A-D tests, are used to test for a gamma distribution. Extensive critical values for these two
test statistics have been obtained via Monte Carlo simulation experiments (Singh, Singh, and laci 2002).
For interested users, those critical values are given in Appendix A for various levels of significance. In
addition to these formal tests, the informal histogram and Q-Q plots (also called probability plots) are also
available for visual inspection of the data distributions (Looney and Gulledge 1985). Q-Q plots also
provide useful information about the presence of potential outliers and multiple populations in a data set.
A brief description of the GOF tests follows.
No matter which normality test is used, it may fail to detect the actual non-normality of the population
distribution if the sample size is small, «<20 and with large sample sizes, «>50 or so, a small deviation
from normality will lead to rejection of the normality hypothesis. The modified K-S test known as
Lilliefors test is suggested for checking the normality assumption when the mean and sd of population
distribution is not known.
2.2.1 Test Normality and Lognormality of a Data Set
ProUCL tests for normality and lognormality of a data set using three different methods described below.
The program tests normality or lognormality at three different levels of significance, 0.01, 0.05, and 0.1
(or confidence levels: 0.99, 0.95, and 0.90). For normal distributions, ProUCL outputs approximate
probability values (p-values) for the S-W GOF test. The details of those methods can be found in the cited
references.
2.2.1.1 Normal Quantile-quantile (Q-Q) Plot
A normal Q-Q represents a graphical method to test for approximate normality or lognormality of a data
set (Hoaglin, Mosteller, and Tukey 1983; Singh 1993; Looney and Gulledge, 1985). A linear pattern
displayed by the majority of the data suggests approximate normality or lognormality (when performed
on log-transformed data) of the data set. For example, a high value, 0.95 or greater, of the correlation
coefficient of the linear pattern may suggest approximate normality (or lognormality) of the data set under
study. However, on this graphical display, observations well-separated from the linear pattern displayed
by the majority of data may represent outlying observations not belonging to the dominant population,
whose distribution one is assessing based upon a data set. Apparent jumps and breaks in the Q-Q plot
suggest the presence of multiple populations. The correlation of the Q-Q plot for such a data set may still
be high but that does not signify that the data set follows a normal distribution.
Notes: Graphical displays provide added insight into a data set which might not be apparent based upon
statistics such as S-W statistic or a correlation coefficient. The correlation coefficient of a Q-Q plot with
curves, jumps and breaks can be high, which does not necessarily imply that the data follow a normal or
lognormal distribution. AGOF test of a data set should always be judged based upon a formal (e.g., S-W
statistic) as well as informal graphical displays. The normal Q-Q plot is used as an exploratory tool to
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identify outliers or to identify multiple populations. On all Q-Q plots, ProUCL displays relevant statistics
including: mean, sd, GOF test statistic, associated critical value, /"-value (when available), and the
correlation coefficient.
There is no substitute for graphical displays of data sets. Graphical displays provide added insight about
data sets and do not get distorted by outliers and/or mixture populations. The final conclusion regarding
the data distribution should be based upon the formal GOF tests as wells as Q-Q plots. This statement is
true for all GOF tests: normal, lognormal, and gamma distributions.
2.2.1.2 Shapiro-Wilk (S-W) Test
The S-W test is a powerful test used to test the normality or lognormality of a data set. ProUCL performs
this test for samples of size up to 2000 (Royston 1982, 1982a). For sample sizes < 50, in addition to a test
statistic and critical value, an approximate /"-value associated with S-W test is also displayed. For samples
of size >50, only approximate />-values are displayed. Based upon the selected level of significance and
the computed test statistic, ProUCL informs the user if the data set is normally (or lognormally)
distributed. This information should be used to compute an appropriate UCL of the mean.
2.2.1.3 Lilliefors Test
This test is useful for data sets of larger size (Lilliefors 1967; Dudewicz and Misra 1988; Conover 1999).
This test is a slight modification of the Kolmogorov-Smirnov (K-S) test and is more powerful than a one-
sample K-S (with the estimated population mean and sd). In version 5.1 of ProUCL, critical values of
Lilliefors test developed by Moling and Abdi and provided in the Encyclopedia of Measurement and
Statistics (Salkind, N. Editor 2007) have been used and incorporated in the program. The critical values
as described in Salkind (2007) are used for n up to 50, and for values of «>50 approximate critical values
are computed using the following formula:
Critical Values = Factor If (n); where /(«) = —= 0.01.
V«
The Factor used in the above equation depends upon the level of significance, a; Factor values are 0.741,
0.819, 0.895, and 1.035 for a = 0.20, 0.1, 0.05, and 0.01 respectively.
Based upon the selected level of significance and the computed test statistic, ProUCL informs the user if
the data set is normally or lognormally distributed. This information should be used to compute an
appropriate UCL of the mean. The program outputs the relevant statistics on the Q-Q plot of data.
• For convenience, normality, lognormality, or gamma distribution test results for a built-in level of
significance of 0.05 are displayed on the UCL and background statistics output sheets. This helps
the user in selecting the most appropriate UCL to estimate the EPC. It should be pointed out that
sometimes, the two GOF tests may lead to different conclusions. In such situations, ProUCL
displays a message that data are approximately normally or lognormally distributed. It is
suggested that the user makes a decision based upon the information provided by the associated
Q-Q plot and the values of the GOF test statistics.
New in ProUCL 5.1: Based upon the author's professional experience and in an effort to streamline the
decision process for computing upper limits (e.g., UCL95), some changes were made in the decision logic
applied in ProUCL for suggesting/recommending UCL values. Specifically, ProUCL 5.1 makes
decisions about the data distribution based upon both the Lilliefors and S-W GOF test statistics for
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normal and lognormal distributions and both the A-D and K-S GOF test statistics for the gamma
distribution. When a data set passes one of the two GOF tests for a distribution, ProUCL outputs a
statement that the data set follows that approximate distribution and suggests using appropriate decision
statistic(s). Specifically, when only one of the two GOF statistic leads to the conclusion that data are
normal, lognormal or gamma, ProUCL outputs the conclusion that the data set follows that approximate
distribution and all suggestions provided by ProUCL regarding the use of parametric or nonparametric
decision statistics are made based upon this conclusion. As a result, UCLs suggested by ProUCL 5.1 can
differ from the UCLs suggested by earlier versions of ProUCL.
Note: When dealing with a small data set, n <50, and Lilliefors test suggests that data are normal and the
S-W test suggests that data are not normal, ProUCL will suggest that the data set follows an approximate
normal distribution. However, for smaller data sets, Lilliefors test results may not be reliable, therefore
the user is advised to review GOF tests for other distributions and proceed accordingly. It is emphasized,
when a data set follows a distribution (e.g., distribution A) using all GOF tests, and also follows an
approximate distribution (e.g., distribution B) using one of the available GOF tests, it is preferable to use
distribution A over distribution B. However, when distribution A is a highly skewed (e.g., sd of logged
data >1.0) lognormal distribution, use the guidance provided on the ProUCL generated output.
In practice, depending upon the power associated with statistical tests, two tests (e.g., two sample t-test
vs. WMW test; S-W test vs. Lilliefors test) used to address the same statistical issue (comparing two
groups, assessing data distribution) can lead to different conclusions (e.g., GOF tests for normality in
Example 2-4); this is especially true when dealing with data sets of smaller sizes. The power of a test can
be increased by collecting more data. If this is not feasible due to resource constraints, the collective
project team should determine which conclusion to use in the decision making process. It may, in these
cases, be appropriate to consult a statistician.
2.2.2 Gamma Distribution
A continuous random variable, X (e.g., concentration of an analyte), is said to follow a gamma
distribution, G(k, 6) with parameters k > 0 (shape parameter) and 6 > 0 (scale parameter), if its probability
density function is given by the following equation:
f(x; k, 9} = —i— x^e-*16; x> 0
•s \ ? ? / /)/tT—< /- 7 \ ? /^->\
(2-3)
= 0; otherwise
Many positively skewed data sets follow a lognormal as well as a gamma distribution. The use of a
gamma distribution tends to yield reliable and stable 95% UCL values of practical merit. It is therefore
desirable to test if an environmental data set follows a gamma distribution. If a skewed data set does
follow a gamma model, then a 95% UCL of the population mean should be computed using a gamma
distribution. For data sets which follow a gamma distribution, the adjusted 95% UCL of the mean based
upon a gamma distribution is optimal (Bain and Engelhardt 1991) and approximately provides the
specified 95% coverage of the population mean, jj.\ = W (Singh, Singh, and laci 2002).
The GOF test statistics for a gamma distribution are based upon the EDF. The two EDF tests incorporated
in ProUCL are the K-S test and the A-D test, which are described in D'Agostino and Stephens (1986) and
Stephens (1970). The graphical Q-Q plot for a gamma distribution has also been incorporated in ProUCL.
The critical values for the two EDF tests are not available, especially when the shape parameter, k, is
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small (k < 1). Therefore, the associated critical values have been computed via extensive Monte Carlo
simulation experiments (Singh, Singh, and laci 2002). The critical values for the two test statistics are
given in Appendix A. The 1%, 5%, and 10% critical values of these two test statistics have been
incorporated in ProUCL. The GOF tests for a gamma distribution depend upon the MLEs of the gamma
parameters, k and 6, which should be computed before performing the GOF tests. Information about
estimation of gamma parameters, gamma GOF tests, and construction of gamma Q-Q plots is not readily
available in statistical textbooks. Therefore, a detailed description of the methods for a gamma
distribution is provided as follows.
2. 2. 2. 1 Quantile-Quantile (Q-Q) Plot for a Gamma Distribution
Let xi, Jt2, ... , xn be a random sample from the gamma distribution, G(k,9); and let X(i) < X(2) < ... < X(n)
represent the ordered sample. Let k and 9 represent the maximum likelihood estimates (MLEs) of k and
9, respectively; details of the computation of the MLEs of k and 9 can be found in Singh, Singh, and laci
(2002). The Q-Q plot for a gamma distribution is obtained by plotting the scatter plot of pairs,
(x0i,x(i)) i := 1, 2, ..., n. The gamma quantiles, Jtoi, are given by the equation, x0i = z0i0/2; i := 1, 2, ...,
n, where the quantiles zoi (already ordered) are obtained by using the inverse chi-square distribution and
are given as follows:
; /:=1,2,...,« (2-4)
9 *
In (2-4), x2k represents a chi-square random variable with 2k degrees of freedom (df). The program,
PPCHI2 (Algorithm AS91) described in Best and Roberts (1975) has been used to compute the inverse
chi-square percentage points given by equation (2-4). All relevant statistics including the MLE of k are
also displayed on a gamma Q-Q plot.
Like a normal Q-Q plot, a linear pattern displayed by the majority of the data on a gamma Q-Q plot
suggests that the data set follows an approximate gamma distribution. For example, a high value (e.g.,
0.95 or greater) of the correlation coefficient of the linear pattern may suggest an approximate gamma
distribution of the data set under study. However, on this Q-Q plot, points well-separated from the bulk of
data may represent outliers. Apparent breaks and jumps in the gamma Q-Q plot suggest the presence of
multiple populations. The correlation coefficient of a Q-Q plot with outliers and jumps can still be high
which does not signify that the data follow a gamma distribution. Therefore, a graphical Q-Q plot and
other formal GOF tests, the A-D test or K-S test, should be used on the same data set to determine the
distribution of a data set.
2.2.2.2 Empirical Distribution Function (EDF) -Based Goodness-of Fit Tests
Let F(x) be the cumulative distribution function (CDF) of a gamma distributed random variable, X. Let Z
= F(X), then Z represents a uniform U(0,l) random variable (Hogg and Craig 1995). For each x,, compute
z; by using the incomplete gamma function given by the equation z; = F (x;); /':=!, 2, ..., n. The
algorithm (Algorithm AS 239, Shea 1988) as given in the book Numerical Recipes in C, the Art of
Scientific Computing (Press et al. 1990) has been used to compute the incomplete gamma function.
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Arrange the re suiting z; in ascending order as Z(i) < zp) <...< Z(n). Le t z = zt / « be the mean of the
\t=i )
n, zi; i := 1, 2, ..., n.
Compute the following two statistics:
D+ = max . {1 / n - z(i) } , and D = max . {z(i) - (i -!)/«} (2-5)
The K-S test statistic is given byD = max(D+,D~) ; and the A-D test statistic is given as follows:
A2 =-n- (l/«) {(2/ - l)[log z(!) + log(l - z(B+1_0)]} (2-6)
i=l
As mentioned before, the critical values for these two statistics, D and A2, are not readily available. For
the A-D test, only the asymptotic critical values are available in the statistical literature (D'Agostino and
Stephens 1986). Some raw critical values for the K-S test are given in Schneider (1978), and Schneider
and Clickner (1976). Critical values of these test statistics are computed via Monte Carlo experiments
(Singh, Singh, and laci 2002). It is noted that the distributions of the K-S test statistic, D, and the A-D test
statistic, A2, do not depend upon the scale parameter, 6; therefore, the scale parameter, 6, has been set
equal to 1 in all simulation experiments. In order to generate critical values, random samples from gamma
distributions were generated using the algorithm as given in Whittaker (1974). It is observed that the
simulated critical values are in close agreement with all available published critical values.
The critical values simulated by Singh, Singh, and laci (2002) for the two test statistics have been
incorporated in the ProUCL software for three levels of significance, 0.1, 0.05, and 0.01. For each of the
two tests, if the test statistic exceeds the corresponding critical value, then the hypothesis that the data set
follows a gamma distribution is rejected. ProUCL computes the GOF test statistics and displays them on
the gamma Q-Q plot and also on the UCL and background statistics output sheets generated by ProUCL.
Like all other tests, in practice these two GOF test may lead to different conclusions. In such situations,
ProUCL outputs a message that the data follow an approximate gamma distribution. The user should
make a decision based upon the information provided by the associated gamma Q-Q plot and the values
of the GOF test statistics.
Computation of the Gamma Distribution Based Decision Statistics and Critical Values: When computing
the various decision statistics (e.g., UCL and BTVs), ProUCL uses biased corrected estimates, kstar, k
and theta star, 9* (described in Section 2.3.3) of the shape, k, and scale, 9 , parameters of the gamma
distribution. It is noted that the critical values for the two gamma GOF tests reported in the literature
(D'Agostino and Stephens 1986; Schneider and Clickner 1976; and Schneider 1978) are computed using
the MLE estimates, k and 9 of the two gamma parameters, k and$ . Therefore, the critical values of A-
D and K-S tests incorporated in ProUCL have also been computed using the MLE estimates: khat, k and
theta hat, 9 of the two gamma parameters, k and 9 .
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Updated Critical Values of Gamma GOF Test Statistics (New in ProUCL 5.0): For values of the gamma
distribution shape parameter, k < 0.1, critical values of the two gamma GOF tests, A-D and K-S tests,
have been updated in ProUCL 5.0 and higher versions. Critical values incorporated in earlier versions
were simulated using the gamma deviate generation algorithm (Whittaker 1974) available at the time and
with the source code described in the book Numerical Recipes in C, the Art of Scientific Computing (Press
et al. 1990). Th gamma deviate generation algorithm available at the time was not very efficient
especially for smaller values of the shape parameter, k < 0.1. For values of the shape parameter, £< 0.1,
significant discrepancies were found in the critical values of the two gamma GOF test statistics obtained
using the two gamma deviate generation algorithms: Whitaker (1974) and Marsaglia and Tsang (2000).
Therefore, for values of k < 0.2, critical values for the two gamma GOF tests have been re-generated and
tables of critical values of the two gamma GOF tests have been updated in Appendix A. Specifically, for
values of the shape parameter, k < 0.1, critical values of the two gamma GOF tests have been generated
using the more efficient gamma deviate generation algorithm as described in Marsaglia and Tsang's
(2000) and Best (1983). Detailed description about the implementation of Marsaglia and Tsang's
algorithm to generate gamma deviates can be found in Kroese, Taimre, and Botev (2011). From a
practical point of view, for values of k greater than 0.1, the simulated critical values obtained using
Marsaglia and Tsang's algorithm (2000) are in general agreement with the critical values of the two GOF
test statistics incorporated in ProUCL 4.1 for the various values of the sample size considered. Therefore,
those critical values for values of k > 0.1 do not have to be updated.
Note: In March 2015 minor discrepancies were identified in critical values of the gamma GOF A-D tests,
as summarized in Tables A1-A6 of ProUCL 5.0 Technical Guide. For example, for a specified sample
size and level of significance, a, the critical values for GOF tests are expected to decrease as k increases.
Due to inherent random variability in the simulated gamma data sets, critical values do not follow
(deviations are minor occurring in 2nd or 3rd decimal places) this trend in a few cases. However, from a
practical and decision making point of view those differences are minor (see below). These discrepancies
can be eliminated by performing simulation experiments using more iterations. In ProUCL 5.1, these
discrepancies in the critical values of gamma GOF tests have been fixed via interpolation.
For example, in Table A-3, for the A-D test, with significance level a= 0.05 and «=7, critical values for
£=10, 20, and 50 are 0.708, 0.707, and 0.708. Also, in Table A-4 for «=200 and £=0.025, the critical
value is 0.070489, and for «=200, £=0.05, the critical value is 0.07466. Due to a lack of resources and
time, the critical values have not been re-simulated; however, this value has been replaced by an
interpolated value using simulated values for £=0.025 and £=0.1.
2.3 Estimation of Parameters of the Three Distributions Incorporated in
ProUCL
Let jui and a\- represent the mean and variance of the random variable, X, and ju and o2 represent the mean
and variance of the random variable Y = log(X). Also, a represents the standard deviation of the log-
transformed data. For both lognormal and gamma distributions, the associated random variable can take
only positive values. It is typical of environmental data sets to consist of only positive concentrations.
2.3.7 Normal Distribution
Let X be a continuous random variable (e.g., lead concentrations in surface soils of a site), which follows
a normal distribution, N (MI, a\2) with mean, ju\, and variance, a\2. The probability density function of a
normal distribution is given by the following equation:
53
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-GO
-------
For positively skewed data sets, the various levels of skewness can be defined in terms a or its MLE
estimate, sy. These levels are described as follows in Table 2-1. ProUCL software uses the sample sizes
and skewness levels defined below to make suggestions/recommendations to select an appropriate UCL
as an estimate of the EPC.
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Table 2-1. Skewness as a Function of a (or its MLE, sy = 6), sd of log(X)
Standard Deviation
of Logged Data Skewness
a < 0.5 Symmetric to mild skewness
0.5 < a < 1.0 Mild skewness to moderate skewness
1.0 < a < 1.5 Moderate skewness to high skewness
1.5 < a < 2.0 High skewness
9 0 < < 1 0 Very high skewness (moderate probability of
~ ' outliers and/or multiple populations)
- „ Extremely high skewness (high probability of
~ ' outliers and/or multiple populations)
Note: When data are mildly skewed with a < 0.5, the three distributions considered in ProUCL tend to
yield comparable upper limits irrespective of the data distribution.
2.3.2.3 MLEs of the Quantiles of a Lognormal Distribution
For highly skewed (a > 1.5) lognormally distributed populations, the population mean, jj.\, often exceeds
the higher quantiles (80%, 90%, 95%) of the distribution. Therefore, the estimation of these quantiles is
also of interest. This is especially true when one may want to use MLEs of the higher order quantiles such
as 95%, 97.5%, etc. as estimates of the EPC. The formulae to compute these quantiles are described here.
The />th quantile (or 100 />th percentile), xp, of the distribution of a random variable, X, is defined by the
probability statement, P(X < xp) = p. If zp is the pth quantile of the standard normal random variable, Z,
with P(Z < zp) = p, then the pth quantile of a lognormal distribution is given by xp = exp(« + zpa). Thus
the MLE of the pih quantile is given by:
xp = exp(/} + zpa) (2-13)
It is expected that 95% of the observations coming from a lognormal LN(w, o2) distribution would lie at or
below exp(« + 1.650). The 0.5th quantile of the standard normal distribution is ZQ.S = 0, and the 0.5th
quantile (or median) of a lognormal distribution is M= exp(«), which is obviously smaller than the mean,
[j.\, as given by equation (2-8).
Notes: The mean, ju\, is greater than xp if and only if a > 2zp. For example, when/? = 0.80, zp = 0.845, ju\
exceeds xo.so, the 80th percentile if and only if a > 1.69, and, similarly, the mean, jj.\, will exceed the 95th
percentile if and only if a > 3.29 (extremely highly skewed). ProUCL computes the MLEs of the 50%
(median), 90%, 95%, and 99% percentiles of lognormally distributed data sets.
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2.3.2.4 MVUEs of Parameters of a Lognormal Distribution
Even though the sample mean x is an unbiased estimator of the population mean, /j.\, it does not possess
the minimum variance (MV). The MVUEs of ju\ and a\2 of a lognormal distribution are given as follows:
(2-14)
(2-15)
The series expansion of the function g»(x) is given in Bradu and Mundlak (1970), and Aitchison and
Brown (1969). Tabulations of this function are also provided by Gilbert (1987). Bradu and Mundlak
(1970) computed the MVUE of the variance of the estimate, fa,
-gn((n-2)s2yl(n-m (2-16)
The square root of the variance given by equation (2-16) is called the standard error (SE) of the
estimate, fa, given by equation (2-14). The MVUE of the median of a lognormal distribution is given by
M = exp(y)gn[-s2y l(2(n -1))] (2-17)
For a lognormally distributed data set, ProUCL also computes these MVUEs given by equations (2-14)
through (2-17).
2.3.3 Estimation of the Parameters of a Gamma Distribution
The population mean and variance of a two-parameter gamma distribution, G(k, 0), are functions of both
parameters, k and 6. In order to estimate the mean, one has to obtain estimates of k and 9. The
computation of the MLE of k is quite complex and requires the computation of Digamma and Trigamma
functions. Several researchers (Choi and Wette 1969; Bowman and Shenton 1988; Johnson, Kotz, and
Balakrishnan 1994) have studied the estimation of the shape and scale parameters of a gamma
distribution. The MLE method to estimate the shape and scale parameters of a gamma distribution is
described below.
As before, let x\, X2, ..., xn be a random sample (e.g., representing constituent concentrations) of size n
from a gamma distribution, G(k, 0), with unknown shape and scale parameters, k and 0, respectively. The
log- likelihood function (obtained using equation (2-3)) is given as follows:
~~ ' (2-18)
To find the MLEs of k and 0, one differentiates the log-likelihood function as given in (2-18) with respect
to k and 0, and sets the derivatives to zero. This results in the following two equations:
1 ~ " ' \and (2-19)
57
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1
k9 = -^_jxi=x (2-20)
Solving equation (2-20) for 9 , and substituting the result in (2-19), we get following equation:
^
(2-21)
There does not exist a closed form solution of equation (2-21). This equation needs to be solved
numerically fork , which requires the use of digamma and trigamma functions. An estimate of k can be
computed iterative ly by using the Newton-Raphson method (Press et al. 1990), leading to the following
iterative equation:
, ,, ,
k,=k,,- *~y - - (2-22)
The iterative process stops when k starts to converge. In practice, convergence is typically achieved in
fewer than 10 iterations. In equation (2-22),
Here *V(k) is the digamma function and ~*V'(k) is the trigamma function. Good approximate values for
these two functions (Choi and Wette 1969) can be obtained using the following two approximations. For
k > 8, these functions are approximated by:
log( £) - {l + [l - (1 / 1 0 - 1 1(2 Ik2 ))/ £2 }(6k)]l(2k) , and (2-23)
- (1/5-1 /(Ik2 ))/k2 ^(3k)}/(2k)}/k (2-24)
For k < 8, one can use the following recurrence relations to compute these functions:
+ 1) - 1 / k , and (2-25)
(2-26)
In ProUCL, equations (2-23) through (2-26) have been used to estimate k. The iterative process requires
an initial estimate of k. A good starting value for k in this iterative process is given by ko = 1 / (2M). Thorn
(1968) suggested the following approximation as an initial estimate of k:
4M V 3
1 1 + ,1 + -M (2-27)
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Bowman and Shenton (1988) suggest using k , given by (2-27) as a starting value of k for the iterative
procedure, calculating kl at the 7th iteration using the following formula:
( }
M
Both equations (2-22) and (2-28) have been used to compute the MLE of k. It is observed that the
estimate, k , based upon the Newton-Raphson method, as given by equation (2-22), is in close agreement
with the one obtained using equation (2-28) with Thorn's approximation as an initial estimate. Choi and
Wette (1969) further concluded that the MLE of k, k , is biased high. A bias-corrected (Johnson, Kotz,
and Balakrishnan 1994) estimate of k is given by:
k* = (n - 3)k I n + 2 /(3n) (2-29)
In (2-29), k is the MLE of k obtained using either (2-22) or (2-28). Substitution of equation (2-29) in
equation (2-20) yields an estimate of the scale parameter, 6, given as follows:
0* =xlk* (2-30)
ProUCL computes simple MLEs of £ and 0, and also bias-corrected estimates given by (2-29) and (2-30)
of k and 6. The bias -corrected estimate (called k star and theta star in ProUCL graphs and output sheets)
of k as given by (2-29) has been used in the computation of the UCLs (as given by equations (2-34) and
(2-35) below) of the mean of a gamma distribution.
Note on Bias Corrected Estimates, k and 6 : As mentioned above, Choi and Wette (1969) concluded
that the MLE, k , of k is biased high. They suggested the use of the bias-corrected (Johnson, Kotz, and
Balakrishnan 1994) estimate of £ given by (2-29) above. However, recently the developers performed a
simulation study to evaluate the bias in the MLE of the mean of a gamma distribution for various values
of the shape parameter, k and sample size, n. For smaller values of £ (e.g., <0.2), the bias in the mean
estimate (in absolute value) and mean square error (MSE) based upon the biased corrected MLE, k are
higher than those computed using the MLE estimate, k ; and for higher values of £ (e.g., >0.2), the bias in
the mean estimate and MSE computed using the biased corrected MLE, k are lower than those
computed using the MLE, k . For values of k around 0.2, the use of k and k yields comparable results
for all values of the sample size. The bias in the mean estimate obtained using the MLE, k , increases as k
increases, and as expected, bias and MSE decrease as the sample size increases. The results of this study
will be published elsewhere.
At present for uncensored and left-censored data sets, ProUCL computes all gamma UCLs and other
upper limits (Chapters 3, 4 and 5) using bias corrected estimates, k and 6 of k and 6. ProUCL
generated output sheets display many intermediate results including k and k* ; 6 and 9* . Interested
users may want to compute UCLs and other upper limits using MLE estimates, k and 6 ofk and 6 for
values of k described in the above paragraph.
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2.4 Methods for Computing a UCL of the Unknown Population Mean
ProUCL computes a (1 - a)*100 UCL of the population mean, ju\, using several parametric and
nonparametric methods. ProUCL can compute a (1 - a)*100 UCL (except for adjusted gamma UCL and
Land's H-UCL) of the mean for any user selected confidence coefficient, (1 - a), lying in the interval
[0.5, 1.0]. For the computation of the adjusted gamma UCL, three confidence levels, namely: 0.90, 0.95,
and 0.99 are supported by the ProUCL software. An approximate gamma UCL can be computed for any
level of significance in the interval [0.5, 1.0].
Parametric UCL Computation Methods in ProUCL include:
• Student's t-statistic (assumes normality or approximate normality) based UCL,
• Approximate gamma UCL (assumes approximate gamma distribution),
• Adjusted gamma UCL (assumes approximate gamma distribution),
• Land's H-Statistic UCL (assumes lognormality), and
• Chebyshev inequality based UCL: Chebyshev (MVUE) UCL obtained using MVUE of the
parameters (assumes lognormality).
Nonparametric UCL Computation Methods in ProUCL include:
• Modified-t-statistic (modified for skewness) UCL,
• Central Limit Theorem (CLT) UCL to be used for large samples,
• Adjusted Central Limit Theorem UCL: adjusted-CLT UCL (adjusted for skewness),
• Chebyshev UCL: Chebyshev (Mean, sd) obtained using classical sample mean and standard
deviation,
• Jackknife UCL (yields the same result as Student's t-statistic UCL),
• Standard bootstrap UCL,
• Percentile bootstrap UCL,
• BCA bootstrap UCL,
• Bootstrap-t UCL, and
• Hall's bootstrap UCL.
For skewed data sets, Modified-t and adjusted CLT methods adjust for skewness. However, this
adjustment is not adequate (Singh, Singh, and laci, 2002) for moderately skewed to highly skewed data
sets (levels of skewness described in Table 2-1). Even though some UCL methods (e.g., CLT, UCL
based upon Jackknife method, standard bootstrap, and percentile bootstrap methods) do not perform well
enough to provide the specified coverage to the population mean of skewed distributions. These methods
have been included in ProUCL for comparison, academic, and research purposes. These comparisons are
also necessary to demonstrate why the use of a Student's t-based UCL and Kaplan-Meier (KM) method
based UCLs using t-critical values as suggested in some environmental books should be avoided.
Additionally, the inclusion of these methods also helps the user to make better decisions. Based upon the
sample size, n, data skewness, a, and data distribution, ProUCL makes suggestions regarding the use of
one or more 95% UCL methods to estimate the EPC. For additional gudidance, the users may want to
consult a statistician to select the most appropriate UCL95 to estimate an EPC.
It is noted that in the environmental literature, recommendations about the use of UCLs have been made
without accounting for the skewness and sample size of the data set. Specifically, Helsel (2005, 2012)
suggests the use t-statistic and percentile bootstrap method on robust regression on order statistics (ROS)
and KM estimates to compute UCL95s without considering data skewness and sample size. For
60
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moderately skewed to highly skewed data sets, the use of such UCLs underestimates the population mean.
These issues are illustrated by examples discussed in the following sections and also in Chapters 4 and 5.
2. 4.1 (1 - a) *100 UCL of the Mean Based upon Student 's t-Statistic
The widely used Student's t-statistic is given by:
-
(2-31)
sxn
Where x and sx are, respectively, the sample mean and sample standard deviation obtained using the raw
data. For normally distributed data sets, the test statistic given by equation (2-31) follows the Student's t-
distribution with (n -1) df. Let ta,n-\ be the upper a* quantile of the Student's t-distribution with (n -1) df.
A (1 - a)*100 UCL of the population mean, ju\, is given by:
(2-32)
For a normally (when the skewness is approximately 0) distributed data sets, equation (2-32) provides the
best (optimal) way of computing a UCL of the mean. Equation (2-32) may also be used to compute a
UCL of the mean based upon symmetric or mildly skewed (|skewness|<0.5) data sets, where the skewness
is defined in Table 2-1. For moderately skewed data sets (e.g., whencr, the sd of log-transformed data,
starts approaching and exceeding 0.5), the UCL given by (2-32) fails to provide the desired coverage of
the population mean. This is especially true when the sample size is smaller than 20-25 (graphs
summarized in Appendix B). The situation gets worse (coverage much smaller) for higher values of the
sd, tj , or its MLE, sy.
Notes: ProUCL 5.0 and later versions make a decision about the data distribution based upon both of the
GOF test statistics: Lilliefors and Shapiro-Wilk GOF statistics for normal and lognormal distributions;
and A-D and K-S GOF test statistics for gamma distribution. Specifically, when only one of the two GOF
statistic lead to the conclusion that data are normal (lognormal or gamma), ProUCL outputs the
conclusion that the data set follows an approximate normal (lognormal, gamma) distribution; all decision
statistics (parametric or nonparametric) are computed based upon this conclusion. Due to these changes,
UCL(s) suggested by ProUCL 5.1 can differ from the UCL(s) suggested by ProUCL 4.1. Some examples
illustrating these differences have been considered later in this chapter and also in Chapter 4.0.
2. 4.2 Computation of the UCL of the Mean of a Gamma, G (k, 6), Distribution
It is well-known that the use of a lognormal distribution often yields unstable and unrealistic values of the
decision statistics including UCLs and UTLs for moderately skewed to highly skewed lognormally
distributed data sets; especially when the data set is of a small size (e.g., <30, 50, ...). Even though
methods exist to compute 95% UCLs of the mean, UPLs and UTLs based upon gamma distributed data
sets (Grice and Bain 1980; Wong 1993; Krishnamoorthy et al. 2008), those methods have not become
popular due to their computational complexity and/or the lack of their availability in commercial software
packages (e.g., Minitab 16). Despite the better performance (in terms of coverage and stability) of the
decision making statistics based upon a gamma distribution, some practitioners tend to dismiss the use of
gamma distribution based decision statistics by not acknowledging them (EPA 2009; Helsel 2012) and/or
61
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stating that the use of a lognormal distribution is easier to compute the various upper limits. Throughout
this document, several examples have been used to illustrate these issues.
For gamma distributions, ProUCL software has both approximate (used for n>50) and adjusted (when
«<50) UCL computation methods. Critical values of the chi-square distribution and an estimate of the
gamma shape parameter, k along with the sample mean are used to compute gamma UCLs. As seen
above, computation of an MLE of k is quite involved, and this works as a deterrent to the use of a gamma
distribution-based UCL of the mean. However, the computation of a gamma UCL currently should not be
a problem due to the easy availability of statistical software to compute these estimates. It is noted that
some of the gamma distribution based methods incorporated in ProUCL (e.g., prediction limits, tolerance
limits) are also available in the R Script library.
Update in ProUCL 5.0 and Higher Versions: For gamma distributed data sets, all versions of ProUCL
compute both adjusted and approximate gamma UCLs. However, in earlier versions of ProUCL, an
adjusted gamma UCL was recommended for data sets of sizes <40 (instead of 50 as in ProUCL 5.1), and
an approximate gamma UCL was recommended for data sets of sizes>40, whereas ProUCL 5.1 suggests
using approximate gamma UCL for sample sizes >50.
Given a random sample, x\, x2, ... , xn, of size n from a gamma, G(k, 6), distribution, it can be shown that
1m, 19 follows a chi-square distribution, j^nk with v = Ink degrees of freedom (df). When the shape
parameter, k, is known, a uniformly most powerful test of size of a of the null hypothesis, H0: [j.\ > Cs,
against the alternative hypothesis, HA: n\ < Cs, is to reject H0 if x/Cs < x^nlc(a)/2nk . The corresponding
(1 - a) 100% uniformly most accurate UCL for the mean, ju\, is then given by the probability statement.
l-a (2-33)
Where, %2u(a) denotes the cumulative percentage point of the chi-square distribution (e.g., a is the area
in the left tail) with v (=2nk) df. That is, if Y follows xl, thenP(7 < xl(.aJ) = a • m practice, k is not
known and needs to be estimated from data. A reasonable method is to replace k by its bias-corrected
estimate,A: , as given by equation (2-29). This yields the following approximate (1 - a)*100 UCL of the
mean,//i.
Approximate - UCL = 2nk*x/xlnp («) (2-34)
It should be pointed out that the UCL given by equation (2-34) is an approximate UCL without guarantee
that the confidence level of (1 - a) will be achieved by this UCL. Simulation results summarized in
Singh, Singh, and laci (2002) suggest that an approximate gamma UCL given by (2-34) does provide the
specified coverage (95%) for values of k > 0.5. Therefore, for values of k> 0.5, one should use the
approximate gamma UCL given by equation (2-34) to estimate the EPC.
For smaller sample sizes, Grice and Bain (1980) computed an adjusted probability level, ft (adjusted level
of significance), which can be used in (2-34) to achieve the specified confidence level of (7 - a). For a =
0.05 (confidence coefficient of 0.95), a = 0.1, and a = 0.01, these probability levels are given below in
Table 2-2 for some values of the sample size n. One can use interpolation to obtain an adjusted ft for
values of n not covered in Table 2-2. The adjusted (1 - a)*100 UCL of the gamma mean, ju\ = kO, is given
by the following equation:
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Adjusted- UCL = 2nk*x/x^- Off) (2-35)
Where ft is given in equation (2-2) for a = 0.05, 0.1, and 0.01. Note that as the sample size, n, becomes
large, the adjusted probability level, /?, approaches the specified level of significance, a. Except for the
computation of the MLE of k, equations (2-34) and (2-35) provide simple chi-square-distribution-based
UCLs of the mean of a gamma distribution. It should also be noted that the UCLs given by (2-34) and (2-
35) only depend upon the estimate of the shape parameter, k, and are independent of the scale parameter,
6, and its ML estimate. Consequently, coverage probabilities for the mean associated with these UCLs do
not depend upon the values of the scale parameter, 6.
Table 2-2. Adjusted Level of Significance, /?
« = 0.05 « = 0.1 « = 0.01
N probability level, /? probability level, /? probability level, /?
5
10
20
40
—
0.0086
0.0267
0.0380
0.0440
0.0500
0.0432
0.0724
0.0866
0.0934
0.1000
0.0000
0.0015
0.0046
0.0070
0.0100
For gamma distributed data sets, Singh, Singh, and laci (2002) noted that the coverage probabilities
provided by the 95% UCLs based upon bootstrap-t and Hall's bootstrap methods (discussed below) are in
close agreement. For larger samples, these two bootstrap methods approximately provide the specified
95% coverage and for smaller data sets (from a gamma distribution), the coverage provided by these two
methods is slightly lower than the specified level of 0.95.
Notes
Note 1: Gamma UCLs do not depend upon the standard deviation of the data set which gets distorted by
the presence of outliers. Thus, unlike the lognormal distribution, outliers have reduced influence on the
computation of the gamma distribution based upon decision statistics including the UCL of the mean - a
fact generally not known to a typical user.
Note 2: For all gamma distributed data sets for all values of k and n, all modules and all versions of
ProUCL compute the various upper limits based upon the mean and standard deviation obtained using the
bias-corrected estimate, k*. As noted earlier, the estimate k* does yield better estimates (reduced bias)
for all values of k >0.2. For values of k <0.2, the differences between the various limits obtained using k
and A: are not that significant. However from a theoretical point of view, when k <0.2, it is desirable to
compute the mean, standard deviation, and the various upper limits using the MLE estimate, k . ProUCL
generated output sheets display many intermediate results including k andk ; 9 and 9 . Interested users
may want to compute UCLs and other upper limits using MLE estimates, k and 0,ofk and 6 for values
of £ described in the above paragraph.
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2.4.3 (1- a) *100 UCL of the Mean Based Upon H-Statistic (H-UCL)
The one-sided (1 - aj*100 UCL for the mean, ju\, of a lognormal distribution as derived by Land (1971,
1975) is given as follows:
UCL = exp(y + 0.5s* + SyH^/Jn-l) (2-36)
Tables of H-statistic critical values can be found in Land (1975). When the population is lognormal, Land
(1971) showed that theoretically the UCL given by equation (2-36) possesses optimal properties and is
the uniformly most accurate unbiased confidence limit. However, in practice, the H-statistic based UCL
can be quite disappointing and misleading, especially when the data set is skewed and/or consists of
outliers, or represents a mixture data set coming from two or more populations (Singh, Singh, and
Engelhardt 1997, 1999; Singh, Singh, and laci 2002). Even a minor increase in the sd, sy, drastically
inflates the MVUE of pi and the associated H-UCL. The presence of low as well as high data values
increases sy, which in turn inflates the H-UCL. Furthermore, it has been observed (Singh, Singh,
Engelhardt 1997, 1999) that for samples of sizes smaller than 20-30 (sample size requirement also
depends upon skewness), and for values of a approaching and exceeding 1.0 (moderately skewed to
highly skewed data), the use of the H-statistic results in impractical and unacceptably large UCL values.
Notes: In practice, many skewed data sets can be modeled by both gamma and lognormal distributions;
however, there are differences in the properties and behavior of these two distributions. Decision statistics
computed using the two distributions can differ significantly (see Example 2-2 below). It is noted that
some recent documents (Helsel and Gilroy, 2012) incorrectly state that the two distributions are similar.
Helsel (2012, 2012a) suggests the use a lognormal distribution due its computational ease. However, one
should not compromise the accuracy and defensibility of estimates and decision statistics by using easier
methods which may underestimate (e.g., using a percentile bootstrap UCL based upon a skewed data set)
or overestimate (e.g., H-UCL) the population mean. Computation of defensible estimates and decision
statistics taking the sample size and data skewness into consideration is always recommended. For
complicated and skewed data sets, several UCL computation methods (e.g., bootstrap-t, Chebyshev
inequality, and Gamma UCL) are available in ProUCL to compute appropriate decision statistics (UCLs,
UTLs) covering a wide-range of data skewness and sample sizes.
For lognormally distributed data sets, the coverage provided by the bootstrap-t 95% UCL is a little lower
than the coverage provided by the 95% UCL based upon Hall's bootstrap method (Appendix B).
However, it is noted that for lognormally distributed data sets, the coverage provided by these two
bootstrap methods is significantly lower than the specified 0.95 coverage for samples of all sizes. This is
especially true for moderately skewed to highly skewed (a >1.0) lognormally distributed data sets. For
such data sets, a Chebyshev inequality based UCL can be used to estimate the population mean. The H-
statistic often results in unstable values of the UCL95, especially when the sample size is small, «<20, as
shown in Examples 2-1 through 2-3.
Example 2-1. Consider the silver data set with n=56 (from NADA for R package [Helsel, 2013]). The
normal GOF test graph is shown in Figure 2-1. It can be seen that the data set has an extreme outlier (an
observation significantly different from the main body of the data set). The data set contains NDs, and
therefore is considered in Chapter 4 and 5 again. Here this data set is considered assuming that all
observations represent detected values. The data set does not follow a gamma distribution (Figure 2-3) but
can be modeled by a lognormal distribution as shown in Figure 2-2, accommodating the outlier 560. The
histogram shown in Figure 2-4 suggests that data are highly skewed. The sd of the logged data = 1.74.
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The various UCLs computed using ProUCL 5.0 are displayed in Table 2-3 (with outlier) and Table 2-4
(without outlier) following the Q-Q plots.
Normal Q-Q Plot for Silver
Silva
(i-BG
Mean = 1545
Sd = 75.19
Slope = 31.03
Intercept = 15.45
Conelalion. R = 0.406
Shapiro-WifcTesI
Appiox.Tesl Value = 11206
p Value = 0.000
• Best Fit Line
Theoretical Ouantiles (Standard Normal)
Figure 2-1. Normal Q-Q Plot of Raw Data in Original Scale
Lognormal Q-Q Plot for Silver
Mean = 0.6
Sd = 1.746
Slope-1.732
Intercept = 16
Correlation, R =0.975
Lilhflurj I eii
Test Statistic -0.117
Critical Value(0.05l = 0.11
Data Appeal Lognoimal
• Best Fit Line
Theoretical Ouantiles (Standard Normal)
Figure 2-2. Lognormal Q-Q plot with GOF Test Statistics
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Gamma Q-Q Plot for Silver
N=56
Mean = 15. 4482
k star = 0 3141
tbetasl
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Table 2-3. Lognormal and Nonparametric UCLs for Silver Data including the outlier 560
Silver
Total Number of Observations
General Statistics
56
Minimum 0.1
Maximum 560
SD 75.19
Coefficient of Variation 4.868
Number of Distinct Observations 22
Number of Missing Observations 0
Mean 15.45
Median 1.3
Std, Error of Mean 10.05
Skewness 7.174
Lognormal GOF Test
Shapiro Wilk Test Statistic 0.951 Shapiro Wilk Lognormal GOF Test
5% Shapiro Wilk P Value 0.0464 Data Not Lognormal at 5% Significance Level
Ulliefore Test Statistic 0117 LJIIiefors Lognormal GOF Test
5% Ljlliefore Critical Value 0.118 Data appear Lognormal at 5% Significance Level
Data appear Approximate Lognormal at 5% Significance Level
Lognormal Statistics
Minimum of Logged Data -2.303
Maximum of Logged Data 6.328
Mean of logged Data
SD of logged Data
0.6
1.746
Assuming Lognonnal Distribution
95%H-UCL 1S.54
95% Chebyshev (MVUE) UCL 19.12
99% Chebyshev (MVUE) UCL 33.59
Nonparametric Distribution Free UCLs
95%CLTUCL 31.98
95% Standard Bootstrap UCL 32.23
95% Hall's Bootstrap UCL 94.1
95% BCA Bootstrap UCL 52.45
90% Chebyshev(Mean. Sd} UCL 45.59
97.5% Chebyshev{Mean, Sd) UCL 78.2
90% Chebyshev (MVUE) UCL 15.61
97.5% Chebyshev (MVUE) UCL 24
95% Jackknife UCL 32.26
95% Bootstrap-t UCL 180.4
95% Percentile Bootstrap UCL 35.5
95% Chebyshev{Mean. Sd) UCL 59.25
99% Chebyshev(Mean, Sd) UCL 115.4
Suggested UCL to Use
95% H-UCL 13.54
The histogram without the outlier is shown in Figure 2-5. The data is positively skewed with skewness =
5.45. UCLs based upon the data set without the outlier are summarized in Table 2-4 as follows. A quick
comparison of the results presented in Tables 2-3 and 2-4 reveals how the presence of an outlier distorts
the various decision making statistics.
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Histogram for Silver without Outlier 560
Silva
Minimum
Maximum
SD
Ekewneii
Kurtosis
0 Medlar
0 Normal Distiibutior
Figure 2-5. Histogram of Silver Data Set Excluding Outlier 560
Table 2-4. Lognormal and Nonparametric UCLs Not Including the Outlier Observation 560
Silver
General Statistics
Total Number of Observations 55
Number of Distinct Observations 21
Number of Missing Observations 0
Minimum 0.1
Maximum 90
SD 12.95
Coefficient of Variation 2.334
Mean 5.547
Median 1.2
Std. Error of Mean 1.746
Skewness 5.45
Lognormal GOF Test
Shapiro Wilk Test Statistic 0.959
5% Shapiro Wilk P Value 0.114
Ulliefors Test Statistic 0.122
5% LJlliefors Critical Value 0.119
Shapiro Wilk Lognormal GOF Test
Data appear Lognormal at 5% Significance Level
I lilliefors Lognonnal GOF Test
Data Not Lognormal at 5% Significance Level
Data appear Approximate Lognormal at 5% Significance Level
Lognonnal Statistics
Minimum of Logged Data -2.303
Maximum of Logged Data 4.5
Mean of logged Data 0.496
SD of logged Data 1.577
Assuming Lognormal Distribution
95%H-UCL 11.11 90% Chebyshev (MVUE) UCL 10.13
95% Chebyshev (MVUE) UCL 12.26
97.5% Chebyshev (M VU E) UC L 1 5.22
99'. Chebyshev (MVUE) UCL 21 .04
Nonparametric Distribution Free UCLs
95SCLTUCL S.419 95% Jackknife UCL 8.469
95% Standard Bootstrap UCL S.371 95% Bootstrap UCL 12.12
95't Halls Bootstrap UCL 19.2
95% BCA Bootstrap UCL 10.47
90% Chebyshev(Hean, Sd) UCL 10.78
97.5*4 Chebyshev(Mean.Sd) UCL 16.45
95% Percentile Bootstrap UCL 8.642
95% Chebyshev(Mean, Sd) UCL 1 3.16
99% Chebyshev(Mean. Sd) UCL 22.92
Suggested UCL to Use
95%H-UCL 11.11
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Example 2-2: The positively skewed data set consisting of 25 observations, with values ranging from
0.35 to 170, follows a lognormal as well as a gamma distribution. The data set is: 0.3489, 0.8526, 2.5445,
2.5602, 3.3706, 4.8911, 5.0930, 5.6408, 7.0407, 14.1715, 15.2608, 17.6214, 18.7690, 23.6804, 25.0461,
31.7720, 60.7066, 67.0926, 72.6243, 78.8357, 80.0867, 113.0230, 117.0360, 164.3302, and 169.8303.
The mean of the data set is 44.09. The data set is positively skewed with sd of log-transformed data =
1.68. The normal GOF results are shown in the Q-Q plot of Figure 2-6, it is noted that the data do not
follow a normal distribution. The data set follows a lognormal as well as a gamma distribution as shown
in Figures 2-6a and 2-6b and also in Tables 2-5 and 2-6. The various lognormal and nonparametric
UCL95s (Table 2-5) and Gamma UCL95s (Table 2-6) are summarized in the following.
• The lognormal distribution based UCL95 is 229.2 which is unacceptably higher than all other UCLs
and an order of magnitude higher than the sample mean of 44.09. A more reasonable Gamma
distribution based UCL95 of the mean is 74.27 (recommended by ProUCL).
• The data set is highly skewed (Figure 2-6) with sd of the log-transformed data = 1.68; a Student's t-
UCL of 61.66 and a nonparametric percentile bootstrap UCL95 of 60.32 may represent
underestimates of the population mean.
• The intent of the ProUCL software is to provide users with methods which can be used to compute
reliable decision statistics required to make decisions which are cost-effective and protective of
human health and the environment.
Q-Q Plot for X
Theoretical Quantiles (Standard Normal)
Figure 2-6. Normal Q-Q Plot of X
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Gamma Q-Q Plot for X
Mean-44 0892
kstai-0.5924
thetastar = 74.4199
Slope = 0.9291
Intercept = 4.0114
Correlation, R =0.9685
Anderson-Darling Test
Test Statistic-0.374
Critical Value[0 05) = 0.794
Data appear Gamma Distributed
• Best Fit Line
Theoretical Quantiles of Gamma Distribution
Figure 2-6a. Gamma Q-Q Plot of X
Lognormal Q-Q Plot for X
Slope = 1696
Intercept-^. 335
Correlation, R = 0.97S
Shapiio-Wilk Test
Exact Test Statistic = 0948
DiticalValue(0.05] = 0.918
Data Appear Lognormal
Appiox. Test Value-0.949
p-Vabe = 0.247
• Best Fit Line
Theoretical Quantiles (Standard Normal)
Figure 2-6b. Lognormal Q-Q Plot of X
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Table 2-5. Nonparametric and Lognormal UCL95
General Statistics
Total Number of Observations 25
Minimum 0-349
Maximum 163.3
SD 5134
Coefficient of Variation 1-1 64
Number of Distinct Observations 25
Number of Missing Observations 0
Mean 44.09
Median 18.77
Std. Eirorof Mean 10.27
Skewness 1.234
Lognormal GOF Test
Shapiro Wilk Test Statistic 0.94B Shapiro Wilk Lognonrial GOF Test
5% Shapiro Wilk Critical Value 0.318 Data appear Lognormal at 5% Signtficance Level
Uliefors Test Statistic 0.135 Lilliefors Lognonnal GOF Test
5% Lilliefors Critical Value (3.177 Data appear Lognormai at 5% Signtficance Level
Data appear Lognormal at 5% Significance Level
Lognonnal Statistics
Minimum of Logged Data -1.053
Maximum of Logged Data 5.135
Assigning Lognonnal Distribution
35% H-LICL 223.2
95% Chebyshev (MVUE) UCL 176.3
99% Chebyshev (MVUE) UCL 323
Nonparametric Distribution Free UCLs
35% CLT UCL 60.93
95% Standard Bootstrap UCL 60.57
95% Halls Bootstrap UCL 62.55
95% BCA Bootstrap UCL 64.8
30%Chebyshev{Mean,Sd)UCL 74.83
97.5% ChebyshevfMean. Sd) UCL 108.2
Mean of logged Data
SD of logged Data
2.S35
1.68
90% Chebyshev (M VU E) DC L 140.6
97.5% Chebyshev (MVUE) UCL 225.8
95% Jackfcnife UCL 61.66
95% Bootstrap^ UCL 65.58
95% Percentile Bootstrap UCL 60.32
35% ChebyshevfMean, Sd} UCL 83.85
39% ChebyshevfMean. Sd} UCL 146.3
Notes: The use of H-UCL is not recommended for moderately skewed to highly skewed data sets of
smaller sizes (e.g., 30, 50, 70, etc.). ProUCL computes and outputs H-statistic based UCLs for historical
and academic reasons. This example further illustrates that there are significant differences between a
lognormal and a gamma model; for positively skewed data sets, it is recommended to test for a gamma
model first. If data follow a gamma distribution, then the UCL of the mean should be computed using a
gamma distribution. The use of nonparametric methods is preferred when computing a UCL95 for
skewed data sets which do not follow a gamma distribution.
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Table 2-6. Gamma UCL95
Total Number of Observations
General Statistics
25
Minimum 0.349
Maximum 163.8
SD 51.34
Coefficient of Variation 1.164
Number of Distinct Observations 25
Number of Missing Observations 0
Mean 44.09
Median 18.77
SD of logged Data 1.68
Skewness 1.294
Gamma GOF Test
A-D Test Statistic 0.374 Anderson-Darling Gamma GOF Test
5% A-D Critical Value 0.794 Data appear Gamma Distributed at 5% Significance Level
K-S Test Statistic 0113 Kolmogrov Smirnoff Gamma GOF Test
5% K-S Critical Value 0.1S3 Data appear Gamma Distributed at 5% Significance Level
Data appear Gamma Distributed at 5% Significance Level
Gamma Statistics
k hat (MLE) 0.643
Theta hat (MLE) 68.58
nu hat (MLE) 32.15
MLE Mean (bias corrected) 44.09
Adjusted Level of Significance 8.0395
Assuming Gamma Distribution
95% .Approximate Gamma UCL 71.77
Suggested UCL to Use
35% Adjusted Gamma UCL 7^.27
k star (bias corrected MLE) 0.592
Theta star {bias corrected MLE) 74.42
nu star {bias corrected) 29.62
MLE Sd (bias corrected) 57.28
Approximate Chi Square Value {0.05} 18.2
Adjusted Chi Square Value 17.59
35% .Adjusted Gamma UCL
2.4.4 (1 - a)* 100 UCL of the Mean Based upon Modified-t-Statistic for Asymmetrical
Populations
It is well known that percentile bootstrap, standard bootstrap, and Student's t-statistic based UCL of the
mean do not provide the desired coverage of a population mean (Johnson 1978, Sutton 1993, Chen 1995,
Efron and Tibshirani 1993) of skewed data distributions. Several researchers including: Chen (1995),
Johnson (1978), Kleijnen, Kloppenburg, and Meeuwsen (1986), and Sutton (1993) suggested the use of
the modified-t-statistic and skewness adjusted CLT for testing the mean of a positively skewed
distribution. The UCLs based upon the modified t-statistic and adjusted CLT methods were included in
earlier versions of ProUCL (e.g., versions 1.0 and 2.0) for research and comparison purposes prior to the
availability of Gamma distribution based UCLs in ProUCL 3.0 (2004). Singh, Singh, and laci (2002)
noted that these two skewness adjusted UCL computation methods work only for mildly skewed
distributions. These methods have been retained in later versions of ProUCL for academic reasons. The
(1 - a)*100 UCL of the mean based upon a modified t-statistic is given by:
(2-37)
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Where ju3, an unbiased moment estimate (Kleijnen, Kloppenburg, and Meeuwsen 1986) of the third
central moment is given as follows:
This modification for a skewed distribution does not perform well even for mildly to moderately skewed
data sets. Specifically, the UCZ given by equation (2-37) may not provide the desired coverage of the
population mean, ju\, when a starts approaching and exceeding 0.75 (Singh, Singh, and laci 2002). This is
especially true when the sample size is smaller than 20-25. This small sample size requirement increases
as a increases. For example, when a starts approaching and exceeding 1 to 1.5, the UCL given by
equation (2-37) does not provide the specified coverage (e.g., 95%), even for samples as large as 100.
2. 4.5 (1 - a) *100 UCL of the Mean Based upon the Central Limit Theorem
The CLT states that the asymptotic distribution, as n approaches infinity, of the sample mean, xn , is
normally distributed with mean, [j.\, and variance, o\2ln irrespective of the distribution of the population.
More precisely, the sequence of random variables given by:
a/ -\in
has a standard normal limiting distribution. For large sample sizes, n, the sample mean, X , has an
approximate normal distribution irrespective of the underlying distribution function (Hogg and Craig
1995). The large sample requirement depends upon the skewness of the underlying distribution function
of individual observations. The large sample requirement for the sample mean to follow a normal
distribution increases with skewness. Specifically, for highly skewed data sets, even samples of size 100
may not be large enough for the sample mean to follow a normal distribution. This issue is illustrated in
Appendix B. Since the CLT method requires no distributional assumptions, this is a nonparametric
method. As noted by Hogg and Craig (1995), if a\ is replaced by the sample standard deviation, sx, the
normal approximation for large n is still valid. This leads to the following approximate large sample (1 -
a)* 100 UCL of the mean:
UCL =x + zasjn (2-40)
An often cited and used rule of thumb for a sample size associated with a CLT based method is n > 30.
However, this may not be adequate if the population is skewed, specifically when a (sd of log-
transformed variable) starts exceeding 0.5 to 0.75 (Singh, Singh, laci 2002). In practice, for skewed data
sets, even a sample as large as 100 is not large enough to provide adequate coverage to the mean of
skewed populations. Noting these observations, Chen (1995) proposed a refinement of the CLT approach,
which makes a slight adjustment for skewness.
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2. 4.6 (1 - a)* 100 UCL of the Mean Based upon the Adjusted Central Limit Theorem
(Adjusted-CLT)
The "adjusted-CLT" UCL is obtained if the standard normal quantile, za, in the upper limit of equation
(2-40) is replaced by the following adjusted critical value (Chen 1995):
Thus, the adjusted- CLT (1 - a)* 100 UCL for the mean, ju\, is given by
UCL = x + [za + 4 (1 + 2zza )/(6 Vw) jyx /Vw (2-42)
Here k3 , the coefficient of skewness (raw data), is given by
Skewness (raw data) k3 = Ju3/s3x (2-43)
where, fi3, an unbiased estimate of the third moment, is given by equation (2-38). This is another large
sample approximation for the UCL of the mean of skewed distributions. This is a nonparametric method,
as it does not depend upon any of the distributional assumptions.
Just like the modified-t-UCL, it is observed that the adjusted-CLT UCL also does not provide the
specified coverage to the population mean when the population is moderately skewed, specifically when a
becomes larger than 0.75. This is especially true when the sample size is smaller than 20 to25. This large
sample size requirement increases as the skewness (or a) increases. For example, when a starts
approaching and exceeding 1.5, the UCL given by equation (2-42) does not provide the specified
coverage (e.g., 95%), even for samples as large as 100. It is noted that UCL given by (2-42) does not
provide adequate coverage to the mean of a gamma distribution, especially when the shape parameter (or
its estimate) k< 1.0 and the sample size is small.
Notes: UCLs based upon these skewness adjusted methods, such as the Johnson's modified-t and Chen's
adjusted-CLT, do not provide the specified coverage to the population mean even for mildly to
moderately skewed (e.g., a in [0.5, 1 .0]) data sets. The coverage of the population mean provided by these
UCLs becomes worse (much smaller than the specified coverage) for highly skewed data sets. These
methods have been retained in ProUCL 5.1 for academic and research purposes.
2. 4. 7 Chebyshev (1 - a) *100 UCL of the Mean Using Sample Mean and Sample sd
Several commonly used UCL95 computation methods (e.g., Student's t-UCL, percentile and BCA
bootstrap UCLs) fail to provide the specified coverage (e.g., 95%) to the population mean of skewed data
sets. The use of a lognormal distribution based H-UCL (EPA 2006a, EPA 2009) is still commonly used to
estimate EPCs based upon lognormally distributed skewed data sets. However, the use of Land's H-
statistic yields unrealistically large UCL95 values for moderately skewed to highly skewed data sets. On
the other hand, when the mean of a logged data set is negative, the H-statistic tends to yield an
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impractically low value of H-UCL (See Example 2-1 above) especially when the sample size is large
(e.g., > 30-50). To address some of these issues associated with lognormal H-UCLs, Singh, Singh, and
Engelhardt (1997) proposed the use of the Chebyshev inequality to compute a UCL of the mean of
skewed distributions. They noted that a Chebyshev UCL tends to yield stable, realistic, and conservative
estimates of the EPCs. The use of the Chebyshev UCL has been recently adopted by the ITRC (2012) to
compute UCLs of the mean based upon data sets obtained using the incremental sampling methodology
(ISM) approach.
For moderately skewed data sets, the Chebyshev inequality yields conservative but realistic UCL95 . For
highly skewed data sets, even a Chebyshev inequality fails to yield a UCL95 providing 95% coverage for
the population mean (Singh, Singh, and laci 2002; Appendix B). To address these issues, ProUCL
computes and displays 97.5% or 99% Chebyshev UCLs. The user may want to consult a statistician to
select the most appropriate UCL (e.g., 95% or 97.5% UCL) for highly skewed nonparametric data sets.
Since the use of the Chebyshev inequality tends to yield conservative UCL95s, especially for moderately
skewed data sets of large sizes (e.g., >50), ProUCL 5.1 also outputs a UCL90 based upon the Chebyshev
inequality.
The two-sided Chebyshev theorem (Hogg and Craig 1995) states that given a random variable, X, with
finite mean and standard deviation, /j.\ and o\, we have
P(-kal < x - ft < kxr1 ) > 1 - 1 / k2 (2-44)
This result can be applied to the sample mean, x (with mean, ju\ and variance, a\ /«), to compute a
conservative UCL for the population mean, /j.\. For example, if the right side of equation (2-44) is equated
to 0.95, then k = 4.47, and UCL = x + 4.47^ / ~Jn represents a conservative 95% upper confidence limit
for the population mean, ju\. Of course, this would require the user to know the value of a\. The obvious
modification would be to replace a\ with the sample standard deviation, sx, but since this is estimated
from data, the result is not guaranteed to be conservative. However, in practice, the use of the sample sd
does yield conservative values of the UCL95 unless the data set is highly skewed with sd of the log-
transformed data exceeding 2 to 2.5, and so forth. In general, the following equation can be used to obtain
a (1 - a)* 100 UCL of the population mean, ju\:
UCL =x + V(1/«K /Jn (2-45)
A slight refinement of equation (2-45) is given as follows:
UCL = x + ((l/a)-V)sx n (2-46)
All versions of ProUCL compute the Chebyshev (1 - a)* 100 UCL of the population mean using equation
(2-46). This UCL is labeled as Chebyshev (Mean, Sd) on the output sheets generated by ProUCL. Since
this Chebyshev method requires no distributional assumptions, it is a nonparametric method. This UCL
may be used to estimate the population mean, ju\, when the data are not normal, lognormal, or gamma
distributed, especially when sd, a (or its estimate, sy) becomes large such as > 1.5.
From simulation results summarized in Singh, Singh, and laci (2002) and graphical results presented in
Appendix B, it is observed that for highly skewed gamma distributed data sets (with shape parameter k <
0.5), the coverage provided by the Chebyshev 95% UCL (given by equation (2-46)) is smaller than the
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specified coverage of 0.95. This is especially true when the sample size is smaller than 10-20. As
expected, for larger samples sizes, the coverage provided by the 95% Chebyshev UCL is at least 95%. For
larger samples, the Chebyshev 95% UCL tends to result in a higher (but stable) UCL of the mean of
positively skewed gamma distributions.
Based upon the number of observations and data skewness, ProUCL suggests using a 95%, 97.5%, or a
99% Chebyshev UCL. If these limits appear to be higher than expected, collectively the project team
should make the decision regarding using an appropriate confidence coefficient (CC) to compute a
Chebyshev inequality based upper limit. ProUCL can compute upper limits (e.g., UCLs, UTLs) for any
user-specified level of confidence coefficient in the interval [0.5, 1]. For convenience, ProUCL 5.0 also
displays Chebyshev inequality based 90% UCL of the mean.
Note about Chebyshev Inequality based UCLs: The developers of ProUCL have made significant efforts
to make suggestions that allows the user to choose the most appropriate UCL95 to estimate the EPC.
However, suggestions made in ProUCL may not cover all real world data sets, especially smaller data sets
with higher variability. Based upon the results of the simulation studies and graphical displays presented
in Appendix B, the developers noted that for smaller data sets with high variability (e.g., sd of logged data
>1, 1.5, etc.) even a conservative Chebyshev UCL95 tends not to provide the desired 95% coverage to the
population mean. In these scenarios, ProUCL suggests the use of a Chebyshev UCL97.5 or a Chebyshev
UCL99 to provide the desired coverage (0.95) for the population mean. It is suggested that when data are
highly skewed and ProUCL is recommending the use of a Chebyshev inequality based UCL, the project
team collectively determines which UCL will be the most appropriate to address the project needs.
ProUCL can calculate UCLs for many levels including non-typical levels such as 98%, 96%, 92%.
2.4.8 Chebyshev (1 - a) *100 UCL of the Mean of a Lognormal Population Using the MVUE of
the Mean and its Standard Error
Earlier versions of ProUCL (when gamma UCLs were not available in ProUCL) used equation (2-44) on
the MVUEs of the lognormal mean and sd to compute a UCL (denoted by (1 - a)* 100 Chebyshev
(MVUE)) of the population mean of a lognormal population. In general, if ^i is an unknown mean, /^ is
an estimate, and al (/}j) is an estimate of the standard error of//j, then the following equation:
UCL = & +V((l/«)-l)
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For a confidence coefficient of 0.95, ProUCL UCLs/EPCs module makes suggestions which are based
upon the extensive experience of the developers of ProUCL with environmental statistical methods,
published literature (Singh, Singh, and Engelhardt 1997, Singh and Nocerino 2002, Singh, Singh, and laci
2002, and Singh, Maichle, and Lee 2006) and procedures described in the various guidance documents.
However, the project team is responsible for determining whether to use the suggestions made by
ProUCL. This determination should be based upon the conceptual site model (CSM), expert site and
regional knowledge. The project team may want to consult a statistician.
2.4.9 (1 - a) *100 UCL of the Mean Using the Jackknife and Bootstrap Methods
Bootstrap and jackknife methods (Efron 1981, 1982; Efron and Tibshirani 1993) are nonparametric
statistical resampling techniques which can be used to reduce the bias in point estimates and construct
approximate confidence intervals for parameters, such as the population mean, population percentiles.
These methods do not require any distributional assumptions and can be applied to a variety of situations.
The bootstrap methods incorporated in ProUCL for computing upper limits include: the standard
bootstrap method, percentile bootstrap method, BCA percentile bootstrap method, bootstrap-t method
(Efron, 1981, 1982; Hall 1988), and Hall's bootstrap method (Hall 1992; Manly 1997).
As before, let x\, x2, ... , x» represent a random sample of size n from a population with an unknown
parameter, 6, and let 6 be an estimate of 9, which is a function of all n observations. Here, the
parameter, #, could be the population mean and a reasonable choice for the estimate, 6, might be the
sample mean, x . Another choice for 6 is the MVUE of the mean of a lognormal population, especially
when dealing with lognormally distributed data sets.
2.4.9.1 (l-o.) *100 UCL of the Mean Based upon the Jackknife Method
For the jackknife method, n estimates of 9 are computed by deleting one observation at a time (Dudewicz
and Misra 1988). For each index, / (1=1,2,...n), denote by #(!), the estimate of 9(computed similarly as
6) omit the /'th observation from the original sample of size n and compute the arithmetic mean of these n
jackknifed estimates using:
0=-£0(0 (2-48)
n i=\
A quantity known as the /'th "pseudo-value" is given by:
J,=nO-(n-1)0(0 (2-49)
Using equations (2-48) and (2-49), compute the jackknife estimator of #as follows:
J(0) = - Y J. = n6 - (n -1)0 (2-50)
11
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If the original estimate 6 is biased, then under certain conditions, part of the bias is removed by the
/*,
jackknife method, and an estimate of the standard error (SE) of the jackknife estimate, J(0), is given by
(2-51)
//
Next, using the jackknife estimate, compute a t-type statistic given by
(2-52)
The t-type statistic given above follows an approximate Student's t- distribution with (n - 1) df, which
can be used to derive the following approximate (1-a)* 100% UCL for 9,
If the sample size, n, is large, then the upper a* Hpantile in the above equation can be replaced with the
corresponding upper a* standard normal quantile, za. Observe, also, that when 6 is the sample mean, x ,
then the jackknife estimate is the same as the sample mean, J(x) = x, the estimate of the standard error
given by equation (2-51) simplifies to sjn112, and the UCL in equation (2-53) reduces to the familiar t-
statistic based UCL given by equation (2-32). ProUCL uses the jackknife estimate as the sample mean,
that yields J(x) = X , which in turn translates equation (2-53) to Student's t- UCL given by equation (2-
32). This method has been included in ProUCL to satisfy the curiosity of those users who are unaware
that the jackknife method (with sample mean as the estimator) yields a UCL of the population mean
identical to the UCL based upon the Student's t- statistic as given by equation (2-32).
Notes: It is well known that the Jackknife method (with sample mean as an estimator) and Student's t-
method yield identical UCLs. However, some users may be unaware of this fact, and some researchers
may want to see these issues described and discussed in one place. Also, it has been suggested that a 95%
UCL based upon the Jackknife method on the full data set obtained using the robust ROS (LROS) method
may provide adequate coverage (Shumway, Kayhanian, and Azari 2002) to the population mean of
skewed distributions, which of course is not true since like Student's t-UCL, the Jackknife UCL does not
account for data skewness. Finally, users are cautioned to note that for large data sets («>10,000), the
Jackknife method may take a long time (several hours) to compute a UCL of the mean.
2.4.9.2 (1 - a)*100 UCL of the Mean Based upon the Standard Bootstrap Method
In bootstrap resampling methods, repeated samples of size n each are drawn with replacement from a
given data set of size n. The process is repeated a large number of times (e.g., 2000 times), and each time
an estimate, $., of 6 is computed. The estimates are used to compute an estimate of the SE of 9. A
description of the bootstrap methods, illustrated by application to the population mean, /j.\, and the sample
mean, x , is given as follows.
78
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Step 1. Let (Jen, Xi2, ... , xm) represent the /'th bootstrap sample of size n with replacement from the original
data set, (x\, x2, ..., x»); denote the sample mean using this bootstrap sample by xt.
Step 2. Repeat Step 1 independently TV times (e.g., 1000-2000), each time calculating a new estimate.
Denote these estimates (KM means, ROS means) by xl,x2,..., XN. The bootstrap estimate of the
population mean is the arithmetic mean, XB, of the TV estimates xf: /' := 1, 2, ..., N. The bootstrap
estimate of the SE of the estimate, x , is given by:
(2-54)
If some parameter, 6 (e.g., the population median), other than the mean is of concern with an associated
estimate (e.g., the sample median), then same steps described above are applied with the parameter and its
estimates used in place of ^i and x . Specifically, the estimate, Qi , would be computed, instead of xt , for
each of the TVbootstrap samples. The general bootstrap estimate, denoted by<9B , is the arithmetic mean of
those TV estimates. The difference,^ -0, provides an estimate of the bias in the estimate, 6 , and an
estimate of the SE of 9 is given by:
(2-55)
A (l-a)*100 standard bootstrap UCL for 6>is given by
UCL=9 + zaaB (2-56)
ProUCL computes the standard bootstrap UCL by using the population mean and sample mean, given by
//i and x . The UCL obtained using the standard bootstrap method is quite similar to the UCL obtained
using the Student's t-statistic given by equation (2-32), and, as such, does not adequately adjust for
skewness. For skewed data sets, the coverage provided by the standard bootstrap UCL is much lower than
the specified coverage (e.g., 0.95).
Notes: Typically, bootstrap methods are not recommended for small data sets consisting of less than 10-
15 distinct values. Also, it is not desirable to use bootstrap methods on larger (« > 500) data sets. For
small data sets, several bootstrap re-samples could be identical and/or all values in a bootstrap re-sample
could be identical; no statistical computations can be performed on data sets with all identical
observations. For larger data sets, there is no need to perform and use bootstrap methods as a large data
set is already representative of the population itself. Methods based upon normal approximations, applied
to data sets of larger sizes (n > 500), yield good estimates and results. Also, for larger data, bootstrap
methods and especially the Jackknife method can take a long time to compute statistics of interest.
79
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2.4.9.3 (1 - a) *100 UCL of the Mean Based upon the Simple Percentile Bootstrap
Method
Bootstrap resampling of the original data set of size n is used to generate the bootstrap distribution of the
unknown population mean. In this method, the TV bootstrapped means, xt, /:=1,2,...,7V, are arranged in
ascending order as^(1) < x(2} <•••< x(N). The (1 - a)* 100 UCL of the population mean, [j,\, is given by
the value that exceeds the (1 - a)* 100 of the generated mean values. The 95% UCL of the mean is the
95th percentile of the generated means and is given by:
95%Percentile UCL = 95th% jf.;;: = 1, 2, ..., N (2-57)
For example, when N= 1000, the bootstrap 95% percentile UCL is given by the 950th ordered mean value
given by jc . It is well-known that for skewed data sets, the UCL95 of the mean based upon the
percentile bootstrap method does not provide the desired coverage (95%) for the population mean. The
users of ProUCL and other software packages are cautioned about the suggested use of the percentile
bootstrap method for computing UCL95s of the mean based upon skewed data sets. Noting the
deficiencies associated with the upper limits (UCLs) computed using the percentile bootstrap method,
researchers (Efron 1981; Hall 1988, 1992; Efron and Tibshirani 1993) have developed and proposed the
use of skewness adjusted bootstrap methods. Simulations results and graphs presented in Appendix A
verify that for skewed data sets, the coverage provided by the percentile bootstrap UCL95 and standard
bootstrap UCL is much lower than the coverages provided by the UCL95s based upon the bootstrap-t and
the Hall's bootstrap methods. It is observed that for skewed (lognormal and gamma) data sets, the BCA
bootstrap method performs slightly better (in terms of coverage probability) than the percentile method.
2.4.9.4 (1 - a)*100 UCL of the Mean Basedupon the Bias-Corrected Accelerated (BCA)
Percentile Bootstrap Method
The BCA bootstrap method adjusts for bias in the estimate (Efron and Tibshirani 1993; and Manly 1997).
Results and graphs summarized in Appendix B suggest that the BCA method does provide a slight
improvement over the simple percentile and standard bootstrap methods. However, for skewed data sets
(parametric or nonparametric), the improvement is not adequate enough and yields UCLs with a coverage
probability much lower than the coverage provided by bootstrap-t and Hall's bootstrap methods. This is
especially true when the sample size is small. For skewed data sets, the BCA method also performs better
than the modified-t-UCL. Based upon gamma distributed data sets, the coverage provided by the BCA
95%UCL approaches 0.95 as the sample size increases. For lognormally distributed data sets, the
coverage provided by the BCA 95%UCL is much lower than the specified coverage of 0.95.
The BCA upper confidence limit of intended (1 - a)*100 coverage is given by the following equation:
BCA - UCL = Jc("2) (2-58)
80
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Here x("2) is the a2*100*percentile computed using TVbootstrap means xt; /': = 1, 2, ..., N. For example,
when N = 2000, x(a^ = (a2N)th ordered statistic of the TV bootstrapped means, xt; /': = 1, 2, ..., TV denoted
by x(a JV) represents a BCA-UCL; 0.2 is given by the following probability statement:
O
z0+z^
(2-59)
-------
From the simulation results summarized in Singh, Singh, and laci (2002) and in Appendix B, it is
observed that for skewed data sets, the bootstrap-t method tends to yield more conservative (higher) UCL
values than the other UCLs obtained using the Student's t, modified-t, adjusted-CZr, and other bootstrap
methods described above. It is noted that for highly skewed (k < 0.1 or a > 2) data sets of small sizes (n <
10 to 15), the bootstrap-t method performs better (in terms of coverage) than other (adjusted gamma UCL,
or Chebyshev inequality UCL) UCL computation methods.
2.4.9.6 (l-o.) *100 UCL of the Mean Based upon Hall's Bootstrap Method
Hall (1992) proposed a bootstrap method that adjusts for bias as well as skewness. This method has been
included in UCL guidance document for CERCLA sites (EPA 2002a). In this method, xt, sX:i , and&3.,
the sample mean, the sample standard deviation, and the sample skewness, respectively, are computed
from the /'th bootstrap re-sample (/' = 1, 2,..., N) of the original data. Let x be the sample mean, sxbe the
/*,
sample standard deviation, and k3 be the sample skewness (as given by equation (2-43)) computed using
the original data set of size n. The quantities, Wt and Qi, given below are computed for the TV bootstrap
samples:
W, = (3c;. - 3c)/sx, , and Q, (W, } = W,+ k3,W,213 + kffi 127 + k3i l(6n)
The quantities, Qi (Wi.) are arranged in ascending order. For a specified (1 - a) confidence coefficient,
compute the (aN)ih ordered value, qa, of the quantities, <2,(J^). Next, compute W(qa~) using the inverse
function, which is given as follows:
W(qa) = 3^[\ + k3(qa-k3/(6n))) -Ij/k3 (2-63)
In equation (2-63), k3 is computed using equation (2-43). Finally, the (1 - a)*100 UCL of the population
mean based upon Hall's bootstrap method is given as follows:
UCL=x-W(qJsx (2-64)
For both lognormal and gamma distributions, bootstrap-t and Hall's bootstrap methods perform better
than the other bootstrap methods, namely, the standard bootstrap method, simple percentile, and bootstrap
BCA percentile methods. For highly skewed lognormal data sets, the coverages based upon Hall's
method and bootstrap-t method are significantly lower than the specified coverage, 0.95. This is true even
for samples of larger sizes (n > 100). For lognormal data sets, the coverages provided by Hall's bootstrap
and bootstrap-t methods do not increase much with the sample size, n. For highly skewed (sd > 1.5, 2.0)
data sets of small sizes (n < 15), Hall's bootstrap method and the bootstrap-t method perform better than
the Chebyshev UCL, and for larger samples, the Chebyshev UCL performs better than Hall's and
bootstrap-t methods.
Notes: The bootstrap-t and Hall's bootstrap methods sometimes yield inflated and erratic values,
especially in the presence of outliers (Efron and Tibshirani 1993). Therefore, these two methods should
82
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be used with caution. If outliers are present in a data set and the project team decides to use them in UCL
computations, the use of alternative UCL computation methods (e.g., based upon the Chebyshev
inequality) is suggested. These issues are examined in Example 2-3.
Also, when a data set follows a normal distribution without outliers, these nonparametric bootstrap
methods, percentile bootstrap method, BCA bootstrap method and bootstrap-t method, will yield
comparable results to the Student's t-UCL and modified-t UCL.
Moreover, when a data set is mildly skewed sd of logged data <0.5), parametric methods and bootstrap
methods discussed in this chapter tend to yield comparable UCL values.
Example 2-3: Consider the pyrene data set with n = 56 selected from the literature (She 1997; Helsel
2005). The pyrene data set has been used in several chapters of this technical guide to illustrate the
various statistical methods incorporated in ProUCL. The pyrene data set contains several NDs and will be
considered again in Chapter 4. However, here, the data set is considered as an uncensored data set to
discuss the issues associated with skewed data sets containing outliers; and how outliers can distort UCLs
based upon bootstrap-t and Hall's bootstrap UCL computation methods. The Rosner outlier test (see
Chapter 7) and normal Q-Q plot displayed in Figure 2-7 below confirm that the observation, 2982.45, is
an extreme outlier. However, the lognormal distribution accommodated this outlier and the data set with
this outlier follows a lognormal distribution (Figure 2-8). Note that the data set including the outlier does
not follow a gamma distribution.
Q-Q Plot for Pyrene
Mean-1732
Sd - 391 4
Sbpe-195.9
lnte(ceDt = 173.2
Cotrelation, R ^0.432
Theoretical Quantiles (Standard Normal)
Figure 2-7. Normal Q-Q Plot of She's Pyrene Data Set
83
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Lognormal Q-Q Plot for Pyrene
Pyrene
Mean - 4 66
Srj, 0.787
Slope-0.762
lni«cep,-4.66
Ctmeiation,R= 0.951
Tesi Slaiiste = 0 099
Difc^V^iie[005)-011E
Data Appear Lggrormal
• Be;! Fit Line
Theoretical Quantiles (Standard Normal)
Figure 2-8. Lognormal Q-Q Plot of She's Pyrene Data Set
Several lognormal and nonparametric UCLs (with outlier) are summarized in Table 2-7 below.
Table 2-7. Nonparametric and Lognormal UCLs on Pyrene Data Set with Outlier 2982
Pyrene
General Statistics
Total Number of Observations 56
Minimum 28
Maximum 2982
SD 391.4
Coefficient of Variation 2.26
Number of Distinct Observations 44
Number of Missing Observations 0
Mean 173.2
Median 104
Std. Error of Mean 52.3
Skewness 6.967
Lognormal GOF Test
Shapiro Wilk Test Statistic 0.924 Shapiro Wilk Lognormal GOF Test
5% Shapiro Wilk P Value 0.00174 Data Not Lognormal at 5% Significance Level
LJIIiefors Test Statistic 0.0992 Lilliefors Lognormal GOF Test
5% Lilliefors Critical Value 0.118 Data appear Lognormal at 5% Significance Level
Data appear Approximate Lc
Locator
Minimum of Logged Data 3.3j
Maximum of Logged Data 8
Assuming Loc
95%H-IJCL 130.2
95% Chebyshev (MVUE) UCL 21 6.3
99% Chebyshev (MVUE) UCL 310.4
Nonparametnc I
95%CLTUCL 259.2
95'i Standard Bootstrap UCL 254.5
m Hall's Bootstrap UCL 588.5
95% BCA Bootstrap UCL 336.7
90% ChebysneviMean. Sd) UCL 330.1
97.5% Chebyshev(Mean. Sd) UCL 499.8
ignormal at 5% Significance Level
Tial Statistics
2 Mean of logged Data
SD of logged Data
normal Distribution
90% Chebyshev (MVUE) UCL
97.5% Chebyshev (MVUE) UCL
distribution Free UCLs
95%JackknifeUCL
95% Bootstrap^ UCL
95% Percentile Bootstrap UCL
95% ChebyshevfMean. Sd) UCL
99% ChebyshevfMean. Sd) UCL
4.66
0.787
193.5
248.1
26C.7
525.2
276.5
401.1
693.6
84
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Looking at the mean (173.2), standard deviation (391.4), and SE (52.3) in the original scale, the H-UCL
(180.2) appears to represent an underestimate of the population mean; a nonparametric UCL such as a
90% Chebyshev or a 95% Chebyshev UCL may be used to estimate the population mean. Since there is
an outlier present in the data set, both bootstrap-t (UCL=525.2) and Hall's bootstrap (UCL=588.5)
methods yield elevated values for the UCL95. A similar pattern was noted in Example 2-1 where the data
set included an extreme outlier.
Computations of UCLs without the Outlier 2982
The data set without the outlier follows both a gamma and lognormal distribution with sd of the log-
transformed data = 0.649 suggesting that the data are moderately skewed. The gamma GOF test results
are shown in Figure 2-9. The UCL output results for the pyrene data set without the outlier are
summarized in Table 2-8. Since the data set is moderately skewed and the sample size of 55 is fairly
large, all UCL methods (including bootstrap-t and Hall's bootstrap methods) yield comparable results.
ProUCL suggested the use of a gamma UCL95. This example illustrates how the inclusion of even a
single outlier distorts all statistics of interest. The decision statistics should be computed based upon a
data set representing the main dominant population.
Gamma Q-Q Plot for Pyrene-1
N-55
kstai- 24544
tlleta star - 49.7506
Steps-1.0899
llllelcepl-.1D.5e72
Cpmelatini. R - 0.9852
Anderson-Darling Test
Test Statistic = D 461
Critical Vakle(0 051-0.760
Data appear Gamma Distributed
Theoretical Quantiles of Gamma Distribution
Figure 2-9. Gamma GOF Test on Pyrene Data Set without the Outlier
85
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Table 2-8. Gamma, Nonparametric and Lognormal UCLs on Pyrene Data Set without
Outlier=2982
Gamma GOF Test
A-D Test Statistic 0.461 Andenson-Darting Gamma GOF Test
0.76 Detected data appear Gamma Distributed at 5% Significance Level
0.0316 Kolmogrov-Smirnoff Gamma GOF Test
0.121 Detected data appear Gamma Distributed at 55, Significance Level
5% A-D Critical Value
K-S Test Statistic
5% K-S Critical Value
Detected data appear Gamma Distributed at 5% Significance Level
k hat (MLE)
ThetahatfMLE)
nu hat (MLE)
MLE Mean (bias coirectedj
Gamma Statistics
2.5S3
47.27
234.2
1221
Adjusted Level of Significance 0.0456
k star (bias corrected MLE}
Theta star (bias corrected MLE)
nu star [bias corrected)
MLE Sd (bias corrected)
Approximate Chi Square Value (D.05)
Adjusted Chi Square Value
2454
49.75
270
77.94
232.9
232
Assuming Gamma Distribution
55% Approximate Gamma UCL (use when n;=5D) 141.5
95% Adjusted Gamma UCL (use when n;50) 142.1
Lognomal GOF Test
Shapiro Wilk Test Statistic D.976 Shapiro Wilk Lognomal GOF Test
5'i Shapiro Wilk P Value 0.552 Data appear Lognormal at 5% Significance Level
Lilliefore Test Statistic 0.0553 LJIIiefors Lognomal GOF Test
5% Ulliefors Critical Value 0.115 Data appear Lognormal at 5'i Significance Level
Data appear Lognormal at 5% Significance Level
Lognormal Statistics
Minimum of Logged Data 3.332
Maximum of Logged Data €.129
Assuming LognormaJ Distribution
95% H-UCL 146.2
95% Chebyshev (MVU E) UCL 172.6
99% Chebyshev (MVU E) UCL 237.3
Mean of logged Data
SD of logged Data
4.599
0.649
90% Chebyshev (MVUE) UCL 156.8
97.5% Chebyshev (MVU E) UCL 194.4
Table 2-8 (continued). Gamma, Nonparametric and Lognormal UCLs on Pyrene Data Set without
Outlier=2982
Nonparametric Distribution Free UCL Statistics
Data appear to follow a Discernible Distribution at 5% Significance Level
Nonparametric Distribution Free UCLs
95% CLT UCL
95% Standard Bootstrap UCL
95% Halls Bootstrap UCL
95% BCA Bootstrap UCL
90%Chebyshev(Mean, Sd) UCL
97.5%Chebyshev(Mean. Sd) UCL
141
141
145
145.1
156.6
193.8
95%JackknifeUCL 141.3
95% Bootstrap-t UCL 146.2
95% Percentile Bootstrap UCL 141.5
95% ChebyshevfMean, Sd) UCL 172.2
99% ChebyshevfMean, Sd) UCL 236.4
Suggested UCL to Use
95% Approximate Gamma UCL 141.5
86
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Example 2-4: Consider the chromium concentration data set of size 24 from a real polluted site to
illustrate the differences in UCL95 suggested by ProUCL 4.1 and ProUCL 5.0/ProUCL 5.1. The data set
is provided here in full as it has been also used in several examples in Chapter 3.
Aluminum Arsenic Chromium
6280
3830
3900
5130
9310
15300
9730
7840
10400
16200
6350
10700
15400
12500
2850
9040
2700
1710
3430
6790
11600
4110
7230
4610
1.3
1.2
2
1.2
3.2
5.9
2.3
1.9
2.9
3.7
1.8
2.3
2.4
2.2
1.1
3.7
1.1
1
1.5
2.6
2.4
1.1
2.1
0.66
8.7
8.1
11
5.1
12
20
12
11
13
20
9.8
14
17
15
8.4
14
4.5
3
4
11
16.4
7.6
35.5
6.1
Iron
4600
4330
13000
4300
11300
18700
10000
8900
12400
18200
7340
10900
14400
11800
4090
15300
6030
3060
4470
9230
Lead
16
6.4
4.9
8.3
18
14
12
8.7
11
12
14
14
19
21
16
25
20
11
6.3
13
98.5
53.3
109
8.3
Mn
39
30
10
92
530
140
440
130
120
70
60
110
340
85
41
66
21
8.6
19
140
72.5
27.2
118
22.5
Thallium Vanadium
0.0835
0.068
0.155
0.0665
0.071
0.427
0.352
0.228
0.068
0.456
0.067
0.0695
0.07
0.214
0.0665
0.4355
0.0675
0.066
0.067
0.068
0.13
0.068
0.095
0.07
12
8.4
11
9
22
32
19
17
21
32
15
21
28
25
8
24
11
7.2
8.1
16
The chromium concentrations follow an approximate normal distribution (determined using the two
normality tests) and also a gamma distribution. ProUCL 5.1 uses the conclusion based upon both
(Shapiro-Wilk and Lilliefors) normality tests and ProUCL 4.1 uses the conclusion based only upon the
Shapiro-Wilk test leading to the conclusion that the data set does not follow a normal distribution and
suggested the use of gamma UCLs. UCL results computed and suggested by ProUCL 5.1 and ProUCL
4.1 are summarized as follows. Data are mildly skewed (with sd of logged data = 0.57), therefore,
UCL95s obtained using normal and gamma distributions are comparable.
87
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UCLs Suggested by ProUCL 5.0/ProUCL 5.1
Chromium
Total Number of Observations
General Satisfies
24
Minimum 3
Maximum 35.5
SD 6.832
Coefficient of Variation G.576
Number of Distinct Observations 19
Number of Missing Observations 0
Mean 11.37
Median 11
3d. Error of Mean 1.407
Skewness 1.728
Shapiro Wilk Test Statistic
5% Shapiro Wilk Critical Value
Ljlliefors Test Statistic
5% Ljlliefors Critical Value
Normal GOF Test
8.87
0.316
0.134
8.181
Shapiro Wilk GOF Test
Data Not Normal at 5% Significance Level
Ljlliefors GOF Test
Data appear Normal at 5% Significance Level
Data appear Approximate Normal at 5% Significance Level
Assuming Normal Distribution
95!'. Normal UCL
35% Student s-t UCL
14.3-8
35% UCLs (Adjusted for Skewness)
35% Adjusted-C LT UCL fChen-1395} 14.81
95% Modified^ UCL (Johnson-1378) 14.46
Suggested UCL to Use
95% Students* UCL 14.38
UCLs Suggested by ProUCL 4.1
Gamma Distribution Test
k star (bias corrected) 3.128
Theta Star 3.825
MLEofMean 11.97
MLE of Standard Deviation 6.766
nustar 150.2
Approximate Chi Square Value (.05) 122.8
Adjusted Level of Significance 0.0332
Adjusted Chi Square Value 121.1
Anderson-Darling Test Statistic 0.208
Anderson-Darling 5% Critical Value 0.75
Kdmogorov-Smirnov Test Statistic 0.0325
Kolmogorov-Smirnov 5% Critical Value 0.179
Data appear Gamma Distribided at 5% Significance Levd
Assuming Gamma Distribution
95% Approximate Gamma UCL (Use when n >=40) 14.63
95% .Adjusted Garrrra UCL (Use when n < iO) Kgi
Potential UCLto Use
Data Distribution
Data appear Gamma Distributed at 5% Significance Levd
Nonparametric Statistics
95% CLT UCL 14.28
95% Jackknife UCL 14.38
95% Standard Bootstrap UCL 14.28
95% Bootstrap-t UCL 15.19
35% Hall's Bootstrap UCL 16.77
95% Percentile Bootstrap UCL 14.37
95% BCA Bootstrap UCL 14.95
95% Chebyshev(Mean. Sd) UCL 18.1
37.5% Cnebyshev(Mean. Sd) UCL 20.75
93% Chebyshev(Mean. Sd) UCL 25.96
Use 95% Adjusted Garrrra UCL 14.84
88
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Example 2-5: Consider another mildly skewed real-world data set consisting of lead (Pb) concentrations
from a polluted site Questions were raised regarding ProUCL suggesting that the data are approximate
normal and suggesting the use of the Student's t-UCL This example is included to illustrate that when data
are mildly skewed (sd of logged data <0.5), the differences between UCLs computed using different
distributions are not substantial from a practical point of view. The mildly skewed (with sd of logged data
=0.47), zinc (Zn) data set of size 11 is given by: 38.9, 45.4, 40.1, 101.4, 166.7, 53.9, 57. 35.7, 43.2, 72.9,
and 72.1. The Zn data set follows an approximate normal (using the Lilliefors test). As we know, the
Lilliefors test works well for data sets of size >50; so it is valid to question why ProUCL suggests the use
of a normal Student's t-UCL. This data set also follows a gamma (using both tests) and lognormal
distribution (using both tests). Student's t-UCL95 suggested by ProUCL (using approximate normality) =
87.26, Gamma UCL95 (adjusted) = 93.23, Gamma UCL95 (approximate) = 88.75, and a lognormal
UCL95 = 90.51. So all UCLs are comparable for this mildly skewed data set.
Note: When a data set follows all three distributions (when this happens, it is highly likely that data set is
mildly skewed), one may want to use a UCL for the distribution with the highest p-value. Also when
skewness in terms of sd of logged data is <0.5, all three distributions yield comparable UCLs.
New in ProUCL 5.0 and ProUCL 5.1: Some changes have been made in the decision tables which are
used to make suggestions for selecting a UCL to estimate EPCs. In earlier versions, data distribution
conclusions (internally) in the UCL and BTV modules were based upon only one GOF test statistic (e.g.,
Shapiro Wilk test for normality or lognormality). In ProUCL 5.0 and ProUCL 5.1, data distribution
conclusions are based upon both GOF statistics (e.g., both Shapiro -Wilk and Lilliefors tests for
normality) available in ProUCL. When only one of the GOF test passes, it is determined that the data set
follows an approximate distribution and ProUCL makes suggestions accordingly. However, when a data
set follows more than one distribution, the use of the distribution passing both GOF tests is preferred. For
data sets with NDs, ProUCL 5.0/5.1 offers more UCL computation methods than ProUCL 4.1. These
updates and additions have been incorporated in the decision tables of ProUCL 5.1. Due to these upgrades
and additions, suggestions regarding the use of a UCL made by ProUCL 4.1 and ProUCL 5.1 can differ
for some data sets.
Suggestions made by ProUCL are based upon simulations performed by the developers. A typical
simulation study does not (cannot) cover all data sets of various sizes and skewness from the various
distributions. The ProUCL Technical Guide provides sufficient guidance which can help a user select the
most appropriate UCL as an estimate of the EPC. ProUCL makes these UCL suggestions to help a
typical user select the appropriate UCL from the various available UCLs. Non-statisticians may want to
seek help from a qualified statistician.
2.5 Suggestions and Summary
The suggestions provided by ProUCL for selecting an appropriate UCL of the mean are summarized in
this section. These suggestions are made to help the users in selecting the most appropriate UCL to
estimate the EPC which is routinely used in exposure assessment and risk management studies of the
USEPA. The suggestions are based upon the findings of the simulation studies described in Singh, Singh,
and Engelhardt (1997, 1999); Singh, Singh, and laci (2002); Singh et al. (2006); and Appendix B. A
typical simulation study does not (cannot) cover all data sets of all sizes and skewness from all
distributions. For an analyte (data set) with skewness (sd of logged data) near the end points of the
skewness intervals described in decision tables, Table 2-9 through Table 2-11, the user may select the
most appropriate UCL based upon expert site knowledge, toxicity of the analyte, and exposure risk
associated with that analyte. ProUCL makes these UCL suggestions to help a typical user in selecting the
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appropriate UCL from the many available UCLs. Non-statisticians may want to seek help from a qualified
statistician.
UCL suggestions have been summarized for: 1) normally distributed data sets, 2) gamma distributed data
sets, 3) lognormally distributed data sets, and 4) nonparametric data sets (data sets not following any of
the three distributions available in ProUCL). For a given data set, an appropriate UCL can be computed
by using more than one method. Therefore, depending upon the data size, distribution, and skewness,
sometimes ProUCL may suggest more than one UCL. In such situations, the user may choose any of the
suggested UCLs. If needed, the user may consult a statistician for additional insight. When the use of a
Chebyshev inequality based UCL (e.g., UCL95) is suggested, the user may want to compare that UCL95
with other UCLs including the Chebyshev UCL90 (as Chebyshev inequality tends to yield conservative
UCLs), before deciding upon the use of an appropriate UCL to estimate the population (site) average.
2.5.7 Suggestions for Computing a 95% UCL of the Unknown Population Mean, pi, Using
Symmetric and Positively Skewed Data Sets
For mildly skewed data sets with a or a < 0.5, most of the parametric and nonparametric methods
(excluding Chebyshev inequality which is used on skewed data sets) tend to yield comparable UCL
values. Any UCL computation method may be used to estimate the EPC. However, for highly skewed
(<7>2.0) parametric and nonparametric data sets, there is no simple solution to compute a reliable 95%
UCL of the population mean, ju\. As mentioned earlier, the UCL95 based upon skewness adjusted
methods, such as Johnson's modified-t and Chen's adjusted-CLT, do not provide the specified coverage
to the population mean even for moderately skewed (a in the interval [0.5, 1.0]) data sets for samples of
sizes as large as 100. The coverage of the population mean by these skewness-adjusted UCLs gets poorer
(much smaller than the specified level) for highly skewed data sets, where skewness levels have been
defined in Table 2-1 as functions of (7 (standard deviation of logged data). Interested users may also want
to consult graphs provided in Appendix B for a better understanding of the summary and suggestions
described in this section.
2.5.1.1 Normally or Approximately Normally Distributed Data Sets
For normally distributed data sets, several methods such as: the Student's t-statistic, modified-t-statistic,
and bootstrap-t computation methods yield comparable UCL95s providing coverage probabilities close to
the nominal level, 0.95.
• For normally distributed data sets, a UCL based upon the Student's t-statistic, as given by
equation (2-32), provides the optimal UCL of the population mean. Therefore, for normally
distributed data sets, one should always use a 95% UCL based upon the Student's t-statistic.
• The 95% UCL of the mean given by equation (2-32) based upon the Student's t-statistic
(preferably modified-t) may also be used on non-normal data sets with sd, sy of the log-
transformed data less than 0.5, or when the data set follows an approximate normal distribution.
A data set is approximately normal when: 1) the normal Q-Q plot displays a linear pattern
(without outliers, breaks and jumps) and the resulting correlation coefficient is high (0.95 or
higher); and/or 2) one of the two GOF tests for a normal distribution incorporated in ProUCL
suggests that data are normally distributed.
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• Student's t-UCL may also be used to estimate the EPC when the data set is symmetric (but
possibly not normally distributed). A measure of symmetry (or skewness) is k3, which is given
/*,
by equation (2-43). A value of k3 close to zero (absolute value of skewness is roughly less than
0.2 or 0.3) suggests approximate symmetry. The approximate symmetry of a data distribution can
also be judged by looking at a box plot and/or a histogram.
Note: Use Student's t-UCL for normally distributed data sets. For approximately normally distributed data
sets, non-normal symmetric data sets (when skewness is less than 0.2-0.3), and mildly skewed data sets
with logged sd<0.5, one may use the modified t-UCL.
2.5.1.2 Gamma or Approximately Gamma Distributed Data Sets
In practice, many skewed data sets can be modeled both by a lognormal distribution and a gamma
distribution. Estimates of the unknown population mean based upon the two distributions can differ
significantly (see Example 2- 2 above). For data sets of small size (<20 and even <50) the 95% H-UCL of
the mean based upon a lognormal model often results in unjustifiably large and impractical 95% UCL
values. In such cases, a gamma model, G (k, 0), may be used to compute a 95% UCL provided the data
set follows a gamma distribution.
• One should always first check if a given skewed data set follows a gamma distribution. If a data
set does follow a gamma distribution or an approximate gamma distribution (suggested by
gamma Q-Q plots and gamma GOF tests), one should use a 95% UCL based upon a gamma
distribution to estimate the EPC. For gamma distributed data sets of sizes > 50 with shape
parameter, k>\, the use of the approximate gamma UCL95 is recommended to estimate the EPC.
• For gamma distributed data sets of sizes <50, with shape parameter, k >1, the use of the adjusted
gamma UCL95 is recommended.
• For highly skewed gamma distributed data sets of small sizes (e.g., <15 or <20) and small values
of the shape parameter, k (e.g., k < =1.0), a gamma UCL95 may fail to provide the specified 0.95
coverage for the population mean (Singh, Singh, and laci 2002); the use of a bootstrap-t UCL95
or Hall's bootstrap UCL95 is suggested for small highly skewed gamma distributed data sets to
estimate the EPC. The small sample size requirement increases as skewness increases. That is as
k decreases, the required sample size, n, increases. In the case Hall's bootstrap and bootstrap-t
methods yield inflated and erratic UCL results (e.g., when outliers are present), the 95% UCL of
the mean may be computed based upon the adjusted gamma 95% UCL.
• For highly skewed gamma distributed data sets of sizes > 15 and small values of the shape
parameter, k (k < 1.0), the adjusted gamma UCL95 (when available) may be used to estimate the
EPC, otherwise one may want to use the approximate gamma UCL.
• For highly skewed gamma distributed data sets of sizes > 50 and small values of the shape
parameter, k (k < 1.0), the approximate gamma UCL95 may be used to estimate the EPC.
• The use of an H-UCL should be avoided for highly skewed (a > 2.0) lognormally distributed
data sets. For such highly skewed lognormally distributed data sets that cannot be modeled by a
gamma or an approximate gamma distribution, the use of nonparametric UCL computation
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methods based upon the Chebyshev inequality (larger samples) or bootstrap-t and Hall's
bootstrap methods (smaller samples) is preferred.
Notes: Bootstrap-t and Hall's bootstrap methods should be used with caution as sometimes these methods
yield erratic, unreasonably inflated, and unstable UCL values, especially in the presence of outliers (Efron
and Tibshirani 1993). In the case Hall's bootstrap and bootstrap-t methods yield inflated and erratic UCL
results, the 95% UCL of the mean may be computed based upon the adjusted gamma 95% UCL. ProUCL
prints out a warning message associated with the recommended use of the UCLs based upon the
bootstrap-t method or Hall's bootstrap method.
Table 2-9. Summary Table for the Computation of a 95% UCL of the Unknown Mean, fii,
of a Gamma Distribution; Suggestions are made Based upon Biased Adjusted Estimates
k"(Skewness „ , „. „
Sample Size, n Suggestion
Bias Adjusted)
Approximate gamma 95% UCL (Gamma KM or
k* > 1.0 n>=50 GROS)
,% <-„ Adjusted gamma 95% UCL (Gamma KM or GROS)
K ^ l.U n^-ju
£* ,, 95% UCL based upon bootstrap-t
~ ' or Hall's bootstrap method*
Adjusted gamma 95% UCL (Gamma KM) if
k* <1.0 n>\5,n<50 available, otherwise use approximate gamma 95%
UCL(Gamma KM)
k* <1.0 n > 50 Approximate gamma 95% UCL (Gamma KM)
*In case the bootstrap-t or Hall's bootstrap method yields an erratic, inflated, and unstable UCL value, the
UCL of the mean should be computed using the adjusted gamma UCL method.
Note: Suggestions made in Table 2-9 are used for uncensored as well as left-censored data sets. This table
is not repeated in Chapter 4. All suggestions have been made based upon bias adjusted estimates, K of k.
When the data set is uncensored, use upper limits based upon the sample size and bias adjusted MLE
estimates; and when the data set is left-censored, use upper limits based upon the sample size and biased
adjusted estimates obtained using the KM method or GROS method provided k*>\. When k*>\, UCLs
based upon the GROS method and gamma UCLs computed using KM estimates tend to yield comparable
UCLs from a practical point of view.
2.5.1.3 Lognormally or Approximately Lognormally Distributed Skewed Data Sets
For lognormally, LN (ju, a1), distributed data sets, the H-statistic-based UCL provides the specified 0.95
coverage for the population mean for all values of a; however, the H-statistic often results in unjustifiably
large UCL values that do not occur in practice. This is especially true when skewness is high (a > 1.5-2.0)
and the data set is small («<20-50). For skewed (a or a > 0.5) lognormally distributed data sets, the
Student's t-UCL95, modified-t-UCL95, adjusted-CLT UCL95, standard bootstrap and percentile
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bootstrap UCL95 methods fail to provide the specified 0.95 coverage for the population mean for samples
of all sizes. Based upon the results of the research conducted to evaluate the appropriateness of the
applicability of a lognormal distribution based estimates of the EPC (Singh, Singh, and Engelhardt 1997;
Singh, Singh, and laci 2002), the developers of ProUCL suggest avoiding the use of the lognormal
distribution to estimate the EPC. Additionally, the use of the lognormal distribution based Chebyshev
(MVUE) UCL should also be avoided unless skewness is mild with the sd of log-transformed data <0.5-
0.75. The Chebyshev (MVUE) UCL has been retained in ProUCL software for historical and information
purposes. ProUCL 5.0 and higher versions do not suggest its use.
• ProUCL5.0 computes and outputs H-statistic based UCLs and Chebyshev (MVUE) UCLs for
historical, research, and comparison purposes as it is noted that some recent guidance documents
(EPA 2009) are recommending the use of lognormal distribution based decision statistics.
ProUCL can compute an H-UCL of the mean for samples of sizes up to 1000.
• It is suggested that all skewed data sets be first tested for a gamma distribution. For gamma
distributed data sets, decisions statistics should be computed using gamma distribution based
exact or approximate statistical methods as summarized in Section 2.5.1.2.
• For lognormally distributed data sets that cannot be modeled by a gamma distribution, methods as
summarized in Table 2-10 may be used to compute a UCL of the mean to estimate the EPC. For
highly skewed (e.g., sd >1.5) lognormally distributed data sets which do not follow a gamma
distribution, one may want to compute a UCL using nonparametric bootstrap methods (Efron and
Tibshirani 1993) and the Chebyshev (Mean, Sd) UCL.
Table 2-10. Summary Table for the Computation of a UCL of the Unknown Mean, [it, of a
Lognormal Population to Estimate the EPC
25
«<20
20<«<50
«>50
«<20
20<«<50
50<«<70
«>70
«<30
30<«<70
70<«< 100
Suggestions
Student's t, modified-t, or H-UCL
H-UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
97.5% or 99% Chebyshev (Mean, Sd) UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
99% Chebyshev (Mean, Sd) UCL
97.5% Chebyshev (Mean, Sd) UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
99% Chebyshev (Mean, Sd)
97.5% Chebyshev (Mean, Sd) UCL
95% Chebyshev (Mean, Sd) UCL
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3.5«
Sample Size, n
n> 100
n< 15
15<«<50
50<«< 100
100 150
For all n
Suggestions
H-UCL
Bootstrap-t or Hall's bootstrap method*
99% Chebyshev(Mea«, Sd)
97.5% Chebyshev (Mean, Sd) UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
Use nonparametric methods*
*In the case that the Hall's bootstrap or bootstrap-t methods yield an erratic unrealistically large UCL95
value, a UCL of the mean may be computed based upon the Chebyshev inequality: Chebyshev (Mean, Sd)
UCL
** For highly skewed data sets with a exceeding 3.0, 3.5, pre-process the data. It is very likely that the
data includes outliers and/or come from multiple populations. The population partitioning methods may
be used to identify mixture populations present in the data set.
2.5.1.4
Nonparametric Skewed Data Sets without a Discernible Distribution
For moderately and highly skewed data sets which are neither gamma nor lognormal, one can use a
nonparametric Chebyshev UCL, bootstrap-t, or Hall's bootstrap UCL (for small samples) of the mean to
estimate the EPC. For skewed nonparametric data sets with negative and zero values, use a 95%
Chebyshev (Mean, Sd) UCL for the population mean, ju\. For all other nonparametric data sets with only
positive values, the following procedure may be used to estimate the EPC. The suggestions described here
are based upon simulation experiments and may not cover all skewed data sets or various sizes originating
from the real world practical studies and applications.
• As noted earlier, for mildly skewed data sets with a (or 30) with a< 0.5 one can use the BCA
bootstrap method or the adjusted CLT to compute a 95% UCL of the mean, ju\.
• For nonparametric moderately skewed data sets (e.g., a or its estimate, a in the interval [0.5, 1]),
one may use a 95% Chebyshev (Mean, Sd) UCL of the population mean, jj.\. In practice, for
values of a closer to 0.5, a 95% Chebyshev (Mean, Sd) UCL may represent an over estimate of
the EPC. The user is advised to compare 95% and 90% Chebyshev (Mean, Sd) UCLs.
• For nonparametric moderately and highly skewed data sets (e.g., a in the interval [1.0, 2.0]),
depending upon the sample size, one may use a 97.5% Chebyshev (Mean, Sd) UCL or a 95%
Chebyshev (Mean, Sd) UCL to estimate the EPC.
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• For highly and extremely highly skewed data sets with a in the interval [2.0, 3.0], depending
upon the sample size, one may use Hall's UCL95 or the 99% Chebyshev (Mean, Sd) UCL or the
97.5% Chebyshev (Mean, Sd) UCL or the 95% Chebyshev (Mean, Sd) UCL to estimate the EPC.
For skewed data sets with(7>3, none of the methods considered in this chapter provide the specified 95%
coverage for the population mean, ju\. The coverages provided by the various methods decrease as a (a)
increases. For such data sets of sizes less than 30, a 95% UCL can be computed based upon Hall's
bootstrap method or bootstrap-t method. Hall's bootstrap method provides the highest coverage (but <
0.95) when the sample size is small; and the coverage for the population mean provided by Hall's method
(and the bootstrap-t method) does not increase much as the sample size, n, increases. However, as the
sample size increases, the coverage provided by the Chebyshev (Mean, Sd) UCL increases. Therefore, for
larger skewed data sets wither>3, the EPC may be estimated by the 99% Chebyshev (Mean, Sd) UCL.
The large sample size requirement increases as a increases. Suggestions are summarized in Table 2-11.
Table 2-11. Summary Table for the Computation of a 95% UCL of the Unknown Mean, [ti, Based
upon a Skewed Data Set (with All Positive Values) without a Discernible Distribution, Where 3.5**
Sample Size, n
For all n
For all n
For all n
n<20
20
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*If Hall's bootstrap method yields an erratic and unstable UCL value (e.g., happens when outliers are
present), a UCL of the population mean may be computed based upon the 99% Chebyshev (Mean, Sd)
method.
** For highly skewed data sets with a exceeding 3.0 to 3.5, pre-process the data. Data sets with such
high skewness are complex and it is very likely that the data includes outliers and/or come from multiple
populations. The population partitioning methods may be used to identify mixture populations present in
the data set.
2.5.2 Summary of the Procedure to Compute a 95% UCL of the Unknown Population Mean,
pi, Based upon Full UncensoredData Sets without Nondetect Observations
A summary of the process used to compute an appropriate UCL95 of the mean is summarized as follows.
• Formal GOF tests are performed first so that based on the determined data distribution, an
appropriate parametric or nonparametric UCL of the mean can be computed to estimate the EPC.
ProUCL generates formal GOF Q-Q plots to graphically evaluate the distribution (normal,
lognormal, or gamma) of the data set.
• For a normally or approximately normally distributed data set, the user is advised to use a
Student's t-distribution-based UCL of the mean. Student's t-UCL or modified-t-statistic based
UCL can be used to compute the EPC when the data set is symmetric (e.g.,
to 0.3) or mildly skewed, that is, when a or a is less than 0.5.
is smaller than 0.2
For mildly skewed data sets with a (sd of logged data) less than 0.5, all distributions available in
ProUCL tend to yield comparable UCLs. Also, when a data set follows all three distributions in
ProUCL, compute the upper limits based upon the distribution with highest /"-value.
For gamma or approximately gamma distributed data sets, the user is advised to: 1) use the
approximate gamma UCL when biased adjusted MLE, k* of k >1 and n > 50; 2) use the adjusted
gamma UCL when biased MLE, k* of k > 1 and n < 50; 3) use the bootstrap-t method or Hall's
bootstrap method when k < 1 and the sample size, n < 15 (or <20, sample size requirement
depends upon k); 4) use the adjusted gamma UCL (if available) for k* < 1 and sample size, 15 <
n < 50; and 5) use approximate gamma UCL when k* <\ but n >50. If the adjusted gamma UCL
is not available, then use the approximate gamma UCL as an estimate of the EPC. When the
bootstrap-t method or Hall's bootstrap method yields an erratic inflated UCL (when outliers are
present) result, the UCL may be computed using the adjusted gamma UCL (if available) or the
approximate gamma UCL.
For lognormally or approximately lognormally distributed data sets, ProUCL recommends a UCL
computation method based upon the sample size, n, and standard deviation of the log-transformed
data, a. These suggestions are summarized in Table 2-10.
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• For nonparametric data sets, which are not normally, lognormally, or gamma distributed, a
nonparametric UCL is used to estimate the EPC. Methods used to estimate EPCs based upon
nonparametric data sets are summarized in Table 2-11. For example, for mildly skewed
nonparametric data sets (sd of logged data <0.5) of smaller sizes (n <30), one may use a
modified-t UCL or BCA bootstrap UCL; and for larger mildly skewed data sets, one may use a
CLT-UCL, adjusted-CLT UCL, or BCA bootstrap UCL.
• For moderately skewed to highly skewed nonparametric data sets, the use of a Chebyshev (Mean,
Sd) UCL is suggested. For extremely skewed data sets (a > 3.0), even a Chebyshev inequality-
based 99% UCL of the mean fails to provide the desired coverage (e.g., 0.95) of the population
mean. It is likely that such high skewed data sets do not occur with high probability representing
a single statistical population.
• For highly skewed data sets with a exceeding 3.0, 3.5, it is suggested the user pre-processes the
data. It is very likely that the data contains outliers and/or come from multiple populations.
Population partitioning methods (available in Scout; EPA 2009d) may be used to identify mixture
populations present in the data set; and decision statistics, such as EPCs, may be computed
separately for each of the identified sub-population.
Notes: It should be pointed out that when dealing with a small data set (e.g., <50), and the Lilliefors test
suggests that data are normal and S-W test suggests that data are not normal, ProUCL will suggest that
the data set follows an approximate normal distribution. However, for smaller data sets, Lilliefors test
results may not be reliable, therefore the user is advised to review GOF tests for other distributions and
proceed accordingly. It is emphasized, when a data set follows a distribution (e.g., distribution A) using
all GOF tests, and also follows an approximate distribution (e.g., Distribution B) using one of the
available GOF tests, it is preferable to use distribution A over distribution B. However, when distribution
A is a highly skewed (sd of logged data >1.0) lognormal distribution, use the guidance provided on the
ProUCL generated output.
Once again, contrary to the common belief and practice, for moderately skewed to highly skewed data
sets, the CLT and t-statistic based UCLs of the mean cannot provide defensible estimates of EPCs.
Depending upon data skewness of a nonparametric data set, sample size as large as 50, 70, or 100 is not
large enough to apply the CLT and conclude that the sample mean approximately follows a normal
distribution. The sample size requirement increases with skewness. The use of nonparametric methods
such as bootstrap-t and Chebyshev inequality based upper limits is suggested for skewed data sets.
Finally, ProUCL makes suggestions about the use of one or more UCLs based upon the data distribution,
sample size, and data skewness. Most of the suggestions made in ProUCL are based upon the simulation
studies performed by the developers and their professional experience. However, simulations performed
do not cover all real world scenarios and data sets. The users may use UCLs values other than those
suggested by ProUCL based upon their own experiences and project needs.
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CHAPTER 3
Computing Upper Limits to Estimate Background
Threshold Values Based Upon Uncensored Data Sets
without Nondetect Observations
3.1 Introduction
In background evaluation studies, site-specific (e.g., soils, groundwater) background level constituent
concentrations are needed to compare site concentrations with background level concentrations also
known as background threshold values (BTVs). The BTVs are estimated, based upon sampled data
collected from reference areas and/or unimpacted site-specific background areas (e.g., upgradient wells)
as determined by the project team. The first step in establishing site-specific background level constituent
concentrations is to collect an appropriate number of samples from the designated background or
reference areas. The Stats/Sample Sizes module of ProUCL software can be used to compute DQOs-
based sample sizes. Once an adequate amount of data has been collected, the next step is to determine the
data distribution. This is typically done using exploratory graphical tools (e.g., Q-Q plots) and formal
GOF tests. Depending upon the data distribution, one will use a parametric or nonparametric methods to
estimate BTVs.
In this chapter and also in Chapter 5 of this document, a BTV is a parameter of the background population
representing an upper threshold (e.g., 95th upper percentile) of the background population. When one is
interested in comparing averages, a BTV may represent an average value of a background population
which can be estimated by a UCL95 (e.g., Chapter 21 of EPA 2009 RCRA Guidance). However, in
ProUCL guidance and in ProUCL software, a BTV represents an upper threshold of the background
population. The Upper Limits/BTVs module of ProUCL software computes upper limits which are often
used to estimate a BTV representing an upper threshold of the background population. With this
definition of a BTV, an onsite observation in exceedance of a BTV estimate may be considered as not
coming from the background population; such a site observation may be considered as exhibiting some
evidence of contamination due to site-related activities. Sometimes, locations exhibiting concentrations
higher than a BTV estimate are re-sampled to verify the possibility of contamination. Onsite values less
than BTVs represent unimpacted locations and can be considered part of the background (or comparable
to the background) population. This approach, comparing individual site or groundwater (GW)
monitoring well (MW) observations with BTVs, is particularly helpful to: 1) identify and screen
constituents/contaminants of concern (COCs); and 2) use after some remediation activities (e.g.,
installation of a GW treatment plant) have already taken place and the objective is to determine if the
remediated areas have been remediated close enough to the background level constituent concentrations.
Background versus site comparisons can also be performed using two-sample hypothesis tests (see
Chapter 6). However, BTV estimation methods described in this chapter are useful when not enough site
data are available to perform hypotheses tests such as the two-sample t-test or the nonparametric
Wilcoxon Rank Sum (WRS) test. When enough (more than 8 to 10 observations) site data are available,
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hypotheses testing approaches can be used to compare onsite and background data or onsite data with
some pre-established threshold or screening values. The single-sample hypothesis tests (e.g., t-test, WRS
test, proportion test) are used when screening levels or BTVs are known or pre-established. The two-
sample hypotheses tests are used when enough data (at least 8-10 observations from each population) are
available from background (e.g., upgradient wells) as well as site (e.g., monitoring wells) areas. This
chapter describes statistical limits that may be used to estimate the BTVs for full uncensored data sets
without any ND observations. Statistical limits for data sets consisting of NDs are discussed in Chapter 5.
It is implicitly assumed that the background data set used to estimate BTVs represents a single statistical
population. However, since outliers (well-separated from the main dominant data) are inevitable in most
environmental applications, some outliers such as the observations coming from populations other than
the background population may also be present in a background data set. Outliers, when present, distort
decision statistics of interest (e.g., upper prediction limits [UPLs], upper tolerance limits [UTLs]), which
in turn may lead to incorrect remediation decisions that may not be cost-effective or protective of human
health and the environment. The BTVs should be estimated by statistics representing the dominant
background population represented by the majority of the data set. Upper limits computed by including a
few low probability high outliers (e.g., coming from the far tails of data distribution) tend to represent
locations with those elevated concentrations rather than representing the main dominant background
population. It is suggested that all relevant statistics be computed using the data sets with and without low
probability occasional outliers. This extra step often helps the project team to see the potential influence
of outlier(s) on the decision making statistics (UCLs, UPLs, UTLs) and to make informative decisions
about the disposition of outliers. That is, the project team and experts familiar with the site should decide
which of the computed statistics (with outliers or without outliers) represent more accurate estimate(s) of
the population parameters (e.g., mean, EPC, BTV) under consideration. Since the treatment and handling
of outliers in environmental applications is a subjective and controversial topic, the project team
(including decision makers, site experts) may decide to treat outliers on a site-specific basis using all
existing knowledge about the site and reference areas under investigation. A couple of classical outlier
tests, incorporated in ProUCL, are discussed in Chapter 7.
Extracting a Site-Specific Background Data Set from a Broader Mixture Data Set: Typically, not many
background samples are collected due to resource constraints and difficulties in identifying suitable
background areas with anthropogenic activities and natural geological characteristics comparable to
onsite areas (e.g., at large Federal Facilities, mining sites). Under these conditions, due to confounding of
site related chemical releases with anthropogenic influences and natural geological variability, it becomes
challenging to:l) identify background/reference areas with comparable anthropogenic activities and
geological conditions/formations; and 2) collect an adequate amount of data needed to perform
meaningful and defensible site versus background comparisons for each geological stratum to determine
chemical releases only due to the site related operations and releases. Moreover, a large number of
background samples (not impacted by site related chemical releases) may need to be collected
representing the various soil types and anthropogenic activities present at the site; which may not be
feasible due to resource constraints and difficulties in identifying background areas with anthropogenic
activities and natural geological characteristics comparable to onsite areas. The lack of sufficient
background data makes it difficult to perform defensible background versus site comparisons and
compute reliable estimates of BTVs. A small background data set may not adequately represent the
background population; and due to uncertainty and larger variability, the use of a small data set tends to
yield non-representative estimates of BTVs.
Knowing the complexity of site conditions and that within all environmental site samples (data sets) exist
both background level concentrations and concentrations indicative of site-related releases, sometimes it
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is desirable to extract a site-specific background data set from a mixture data set consisting of all
available onsite and offsite concentrations. This is especially true for larger sites including Federal
Facilities and Mining Sites. Several researchers (Sinclair 1976; Holgresson and Jorner 1978;
Fleischhauer and Korte 1990) have used normal Q-Q/probability plots methods to delineate multiple
populations which can be present in a mixture data set collected from environmental, geological and
mineral exploration studies.
Therefore, when not enough observations are available from reference areas with geological and
anthropogenic influences comparable to onsite areas, the project team may want to use an iterative
population partitioning methods (Singh, Singh, and Flatman 1994; Fleischhauer and Korte 1990) on a
broader mixture data set to extract a site-specific background data set with geological conditions and
anthropogenic influences comparable to those of the various onsite areas. Using the information provided
by iteratively generated Q-Q plots, the project team then determines a background breakpoint (BP)
distinguishing between background level concentrations and onsite concentrations potentially
representing locations impacted by onsite releases. The background BP is determined based upon the
information provided by iterative Q-Q plots, site CSM, expert site knowledge, and toxicity of the
contaminant. The extracted background data set is used to compute upper limits (BTVs) which take data
(contaminant) variability into consideration. If all parties of a project team do not come to a consensus on
a background BP, then the best approach is to: identify comparable background areas and collect a
sufficient amount of background data representing all formations and potential anthropogenic influences
present at the site. The topics of population partitioning and the extraction of a site-specific background
data set from a mixture data set are beyond the scope of ProUCL software and this technical guidance
document. It requires the development of a separate chapter describing the iterative population
partitioning method including the identification and extraction of a defensible background data set from a
mixture data set consisting of all available data collected from background areas (if available), and
unimpacted and impacted onsite locations.
A review of the environmental literature reveals that one or more of the following statistical upper limits
are used to estimate BTVs:
• Upper percentiles
• Upper prediction limits (UPLs)
• Upper tolerance limits (UTLs)
• Upper Simultaneous Limits (USLs) - New in ProUCL 5.0/ProUCL 5.1
Note: The upper limits which are selected to estimate the BTV are dependent on the project objective
(e.g., comparing a single future observation, or comparing an unknown number of observations with a
BTV estimate). . ProUCL does not provide suggestions as to which estimate of a BTV is appropriate for a
project; the appropriate upper limit is determined by the project team. Once the project team has decided
on an upper limit (e.g., UTL95-95), a similar process used to select a UCL95 can be used for selecting a
UTL95-95 from among the UTLs computed by ProUCL. The differences between the various limits used
to estimate BTVs are not clear to many practitioners. Therefore, a detailed discussion about the use of the
different limits with their interpretation is provided in the following sections. Since 0.95 is the most
commonly used confidence coefficient (CC), these limits are described for a CC of 0.95 and coverage
probability of 0.95 associated with a UTL. ProUCL can compute these limits for any valid combination of
CC and coverage probabilities including some commonly used values of CC levels (0.80, 0.90, 0.95,
0.99) and coverage probabilities (0.80, 0.90, 0.95, 0.975).
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Caution: To provide a proper balance between false positives and false negatives, the upper limits
described above, especially a 95% USL (USL95), should be used only when the background data set
represents a single environmental population without outliers (observations not belonging to background).
Inclusion of multiple populations and/or outliers tends to yield elevated values of USLs (and also of
UPLs and UTLs) which can result in a high number (and not necessarily high percentage) of undesirable
false negatives, especially for data sets of larger sizes (n > 30).
Note on Computing Lower Limits: In many environmental applications (e.g., in GW monitoring), one
needs to compute lower limits including: lower confidence limits (LCL) of the mean, lower prediction
limits (LPLs), lower tolerance limits (LTLs), or lower simultaneous limit (LSLs). At present, ProUCL
does not directly compute a LCL, LPL, LTL, or a LSL. For data sets with and without NDs, ProUCL
outputs several intermediate results and critical values (e.g., khat, nuhat, tolerance factor K for UTLs,
d2max for USLs) needed to compute the interval estimates and lower limits. For data sets with and
without NDs, except for the bootstrap methods, the same critical value (e.g., normal z value, Chebyshev
critical value, or t-critical value) can be used to compute a parametric LPL, LSL, or a LTL (for samples of
size >30 to be able to use Natrella's approximation in LTL) as used in the computation of a UPL, USL, or
a UTL (for samples of size >30). Specifically, to compute a LPL, LSL, and LTL («>30) the '+' sign used
in the computation of the corresponding UPL, USL, and UTL («>30) needs to be replaced by the '-' sign
in the equations used to compute UPL, USL, and UTL («>30). For specific details, the user may want to
consult a statistician. For data sets without ND observations, the Scout 2008 software package (EPA
2009d) can compute the various parametric and nonparametric LPLs, LTLs (all sample sizes), and LSLs.
3.1.1 Description and Interpretation of Upper Limits used to Estimate BTVs
Based upon a background data set, upper limits such as a 95% upper confidence limit of the 95th
percentile (UTL95-95) are used to estimate upper threshold value(s) of the background population. It is
expected that observations coming from the background population will lie below that BTV estimate with
a specified CC. BTVs should be estimated based upon an "established" data set representing the
background population under consideration.
Established Background Data Set: This data set represents background conditions free of outliers which
potentially represent locations impacted by the site and/or other activities. An established background
data set should be representative of the environmental background population. This can be determined by
using a normal Q-Q plot on a background data set. If there are no jumps and breaks in the normal Q-Q
plot, the data set may be considered representative of a single environmental population. A single
environmental background population here means that the background (and also the site) can be
represented by a single geological formation, or by single soil type, or by a single GW aquifer etc.
Outliers, when present in a data set, result in inflated values of many decision statistics including UPLs,
UTLs, and USLs. The use of inflated statistics as BTV estimates tends to result in a higher number of
false negatives.
However, when a site consists of various formations or soil types, separate background data sets may
need to be established for each formation or soil type, therefore the project team may want to establish
separate BTVs for different formations. When it is not feasible (e.g., due to implementation complexities)
or desirable to establish separate background data sets for different geological formations present at a site
(e.g., large mining sites), the project team may decide to use the same BTV for all formations.. In this
case, a Q-Q plot of background data set collected from unimpacted areas may display discontinuities as
concentrations in different formations may vary naturally. In these scenarios, use a Q-Q plot and outlier
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test only to identify outliers (well separated from the rest of the data) which may be excluded from the
computation of BTV estimates.
Notes: The user specifies the allowable false positive error rate, a (=1-CC The false negative error rate
(declaring a location clean when in fact it is contaminated) is controlled by making sure that one is
dealing with a defensible/established background data set representing a background population and the
data set is free of outliers.
Let x\, X2, xn represent sampled concentrations of an established background data set collected from some
site-specific or general background reference area.
Upper Percentile. XQQS: Based upon an established background data set, a 95th percentile represents that
statistic such that 95% of the sampled data will be less than or equal to (<) x0.gs . It is expected that an
observation coming from the background population (or comparable to the background population) will
be < xo.gs with probability 0.95. A parametric percentile takes data variability into account.
Upper Prediction Limit (UPL): Based upon an established background data set, a 95% UPL (UPL95)
represents that statistic such that an independently collected observation (e.g., new/future) from the target
population (e.g., background, comparable to background) will be less than or equal to the UPL95 with CC
of 0.95. We are 95% sure that a single future value from the background population will be less than the
UPL95 with CC= 0.95. A parametric UPL takes data variability into account.
In practice, many onsite observations are compared with a BTV estimate. The use of a UPL95 to compare
many observations may result in a higher number of false positives; that is the use of a UPL95 to compare
many observations just by chance tends to incorrectly classify observations coming from the background
or comparable to background population as coming from the impacted site locations. For example, if
many (e.g., 30) independent onsite comparisons (e.g., Ra-226 activity from 30 onsite locations) are made
with the same UPL95, each onsite value may exceed that UPL95 with a probability of 0.05 just by
chance. The overall probability, aactuaiof at least one of those 30 comparisons being significant (exceeding
BTV) just by chance is given by:
aactuai = l-(l-a)k =1 - 0.9530 -1-0.21 = 0.79 (false positive rate).
This means that the probability (overall false positive rate) is 0.79 (and is not equal to 0.05) that at least
one of the 30 onsite locations will be considered contaminated even when they are comparable to
background. The use of a UPL95 is not recommended when multiple comparisons are to be made.
Upper Tolerance Limit (UTL): Based upon an established background data set, a UTL95-95 represents
that statistic such that 95% of observations (current and future) from the target population (background,
comparable to background) will be less than or equal to the UTL95-95 with CC of 0.95. A UTL95-95
represents a 95% UCL of the 95th percentile of the data distribution (population). A UTL95-95 is
designed to simultaneously provide coverage for 95% of all potential observations (current and future)
from the background population (or comparable to background) with a CC of 0.95. A UTL95-95 can be
used when many (unknown) current or future onsite observations need to be compared with a BTV. A
parametric UTL95-95 takes the data variability into account.
By definition a UTL95-95 computed based upon a background data set just by chance can classify 5% of
background observations as not coming from the background population with CC 0.95. This percentage
(false positive error rate) stays the same irrespective of the number of comparisons that will be made with
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a UTL95-95. However, when a large number of observations coming from the target population
(background, comparable to background) are compared with a UTL95-95, the number of exceedances
(not the percentage of exceedances) of UTL95-95 by background observations can be quite large. This
implies that a larger number (but not greater than 5%) of onsite locations comparable to background may
be falsely declared as requiring additional investigation which may not be cost-effective.
To avoid this situation, ProUCL provides a limit called USL which can be used to estimate the BTV
provided the background data set represents a single population free of outliers. The use of a USL is not
advised when the background data set may represent several geological formations/soil types.
Upper Simultaneous Limit (USL): Based upon an established background data set free of outiers and
representing a single statistical population (representing a single formation, representing the same soil
type, same aquifer), a USL95 represents that statistic such that all observations from the "established"
background data set are less than or equal to the USL95 with a CC of 0.95. Outliers should be removed
before computing a USL as outliers in a background data set tend to represent observations coming from a
population other than the background population represented by the majority of observations in the data set.
Since USL represents an upper limit on the largest value in the sample, that largest value should come from
the same background population. A parametric USL takes the data variability into account. It is expected that
all current or future observations coming from the background population (comparable to background
population, unimpacted site locations) will be less than or equal to the USL95 with CC, 0.95 (Singh and
Nocerino 2002). The use of a USL as a BTV estimate is suggested when a large number of onsite
observations (current or future) need to be compared with a BTV.
The false positive error rate does not change with the number of comparisons, as the USL95 is designed to
perform many comparisons simultaneously. Furthermore, the USL95 also has a built in outlier test (Wilks
1963), and if an observation (current or future) exceeds the USL95, then that value definitely represents
an outlier and does not come from the background population. The false negative error rate is controlled
by making sure that the background data set represents a single background population free of outliers.
Typically, the use of a USL95 tends to result in a smaller number of false positives than a UTL95-95,
especially when the size of the background data set is greater than 15.
3.7.2 Confidence Coefficient (CC) and Sample Size
This section briefly discusses sample sizes and the selection of CCs associated with the various upper
limits used to estimate BTVs.
• Higher statistical limits are associated with higher levels of CCs. For example, a 95% UPL is
higher than a 90% UPL.
• Higher values of a CC (e.g., 99%) tend to decrease the power of a test, resulting in a higher
number of false negatives - dismissing contamination when present.
Therefore, the CC should not be set higher than necessary.
• Smaller values of the CC (e.g., 0.80) tend to result in a higher number of false positives (e.g.,
declaring contamination when it is not present).
• In most practical applications, choice of a 95% CC provides a good compromise between
confidence and power.
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• Higher level of uncertainty in a background data set (e.g., due to a smaller background data set)
and higher values of critical values associated with smaller (n < 15-20) samples tend to dismiss
contamination as representing background conditions (results in higher number of false negatives;
identifying a location that may be dirty as background). This is especially true when one uses
UTLs and UPLs to estimate BTVs.
• Nonparametric upper limits based upon order statistics (e.g., the largest, the second largest, etc.)
may not provide the desired coverage as they do not take data variability into account.
Nonparametric methods are less powerful than the parametric methods; and they require larger
data sets to achieve power comparable to parametric methods.
3.2 Treatment of Outliers
The inclusion of outliers in a background data set tends to yield distorted and inflated estimates of BTVs.
Outlying observations which are significanly higher than the majority of the background data may not be
used in establishing background data sets and in the computation of BTV estimates. A couple of classical
outlier tests cited in environmental literature (Gilbert 1987; EPA 2006b, 2009; Navy 2002a, 2002b) are
available in the ProUCL software. The classical outlier procedures suffer from masking effects as they
get distorted by the same outlying observations that they are supposed to find! It is therefore
recommended to supplement outlier tests with graphical displays such as box plots, Q-Q plots. On a Q-Q
plot, elevated observations which are well-separated from the majority of data represent potential outliers.
It is noted that nonparametric upper percentiles, UPLs and UTLs, are often represented by higher order
statistics such as the largest value or the second largest value. When high outlying observations are
present in a background data set, the higher order statistics may represent observations coming from the
contaminated onsite/offsite areas. Decisions made based upon outlying observations or distorted
parametric upper limits can be incorrect and misleading. Therefore, special attention should be given to
outlying observations. The project team and the decision makers involved should decide about the proper
disposition of outliers, to include or not include them, in the computation of the decision making statistics
such as the UCL95 and the UTL95-95. Sometimes, performing statistical analyses twice on the same data
set, once using the data set with outliers and once using the data set without outliers, can help the project
team in determining the proper disposition of high outliers. Examples elaborating on these issues are
discussed in several chapters (Chapters 2, 4, 7) this document.
Notes: It should be pointed out that methods incorporated in ProUCL can be used on any data set with or
without NDs and with or without outliers. Do not misinterpret that ProUCL is restricted and can only be
used on data sets without outliers. It is not a requirement to exclude outliers before using any of the
statistical methods incorporated in ProUCL. The intent of the developers of the ProUCL software is to
inform the users on how the inclusion of occasional outliers coming from the low probability tails of the
data distribution can yield distorted values of UCL95, UPLs, UTLs, and various other statistics. The
decision limits and test statistics should be computed based upon the majority of data representing the
main dominant population and not by accommodating a few low probability outliers resulting in distorted
and inflated values of the decision statistics. Statistics computed based upon a data set with outliers tend
to represent those outliers rather than the population represented by the majority of the data set. The
inflated decision statistics tend to represent the locations with those elevated observations rather than
representing the main dominant population. The outlying observations may be separately investigated to
determine the reasons for their occurrences (e.g., errors or contaminated locations). It is suggested to
compute the statistics with and without the outliers, and compare the potential impact of outliers on the
decision making processes.
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Let Jti, X2, ..., xn represent concentrations of a contaminant/constituent of concern (COC) collected from
some site-specific or general background reference area. The data are arranged in ascending order and the
ordered sample (called ordered statistics) is denoted by X(i) < xp) < ... < X(n). The ordered statistics are used
as nonparametric estimates of upper percentiles, UPLs, UTLs and USLs. Also, let ji = In (x;); i = 1,2, ... ,
n, and y and sy represent the mean and standard deviation (sd) of the log-transformed data. Statistical
details of some parametric and nonparametric upper limits used to estimate BTVs are described in the
following sections.
3.3 Upper p*100% Percentiles as Estimates of BTVs
In most statistical textbooks (e.g., Hogg and Craig 1995), the/?* (e.g.,/? = 0.95) sample percentile of the
measured sample values is defined as that value, x , such that/?*100% of the sampled data set lies at or
below it. The carat sign over xp, indicates that it represents a statistic/estimate computed using the
sampled data. The same use of the carat sign is found throughout this guidance document. The
statistic x represents an estimate of the /?th population percentile. It is expected that about/?* 100% of the
population values will lie below the pth percentile. Specifically, x095 represents an estimate of the 95th
percentile of the background population.
3.3.1 Nonparametric p *100% Percentile
Nonparametric 95% percentiles are used when the background data (raw or transformed) do not follow a
discernible distribution at some specified (e.g., a = 0.05, 0.1) level of significance. Different software
packages (e.g., SAS, Minitab, and Microsoft Excel) use different formulae to compute nonparametric
percentiles, and therefore yield slightly different estimates of population percentiles, especially when the
sample size is small, such as less than 20-30. Specifically, some software packages estimate the /?th
percentile by using the p*nth order statistic, which may be a whole number between 1 and n or a fraction
lying between 1 and n, while other software packages compute the /?th percentile by the p*(n+\)ih order
statistic (e.g., used in ProUCL versions 4.00.02 and 4.00.04) or by the (pn+0.5) th order statistic. For
example, if n = 20, and p = 0.95, then 20*0.95 = 19, thus the 19th ordered statistic represents the 95th
percentile. If n = 17, and/? = 0.95, then 17*0.95= 16.15, thus the 16.15th ordered value represents the 95th
percentile. The 16.15th ordered value lies between the 16th and the 17th order statistics and can be
computed by using a simple linear interpolation given by:
X(16.15)=*(16)+0.15 (^(17)-X(16)). (3-1)
Earlier versions of ProUCL (e.g., ProUCL 4.00.02, 4.00.04) used thep*(n+\)th order statistic to estimate
the nonparametric/?* percentile. However, since most users are familiar with Excel and some consultants
have developed statistical software packages using Excel, and at the request of some users, it was decided
to use the same algorithm as incorporated in Excel to compute nonparametric percentiles. ProUCL 4.1
and higher versions compute nonparametric percentiles using the same algorithm as used in Excel 2007.
This algorithm is used on data sets with and without ND observations.
Notes: From a practical point of view, nonparametric percentiles computed using the various percentile
computation methods described in the literature are comparable unless the data set is small (e.g., n <20-
30) and/or comes from a mixed population consisting of some extreme high values. No single percentile
computation method should be considered superior to other percentile computation methods available in
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the statistical literature. In addition to nonparametric percentiles, ProUCL also computes several
parametric percentiles described as follows.
3.3.2 Normal p*l 00% Percentile
The sample mean, x . and sd, s, are computed first. For normally distributed data sets, the p* 100th sample
percentile is given by the following statement:
xp=x + Szp (3-2)
Here zp is the p* 100th percentile of a standard normal, N(0, 1), distribution, which means that the area
(under the standard normal curve) to the left of zp is p. If the distributions of the site and background data
are comparable, then it is expected that an observation coming from a population (e.g., site) comparable
to the background population would lie at or below the/?* 100% upper percentile, x , with probability p.
p-
3.3.3 Lognormalp *100% Percentile
To compute the pth percentile, xp , of a lognormally distributed data set, the sample mean, y , and sd, sy,
of log-transformed data, y are computed first. For lognormally distributed data sets, the p* 100th percentile
is given by the following statement:
(3-3)
zp is thep* 100th percentile of a standard normal, N(0,l), distribution.
3.3.4 Gamma p* 100% Percentile
Since the introduction of a gamma distribution, G (k, 9), is relatively new in environmental applications, a
brief description of the gamma distribution is given first; more details can be found in Section 2.3.3. The
maximum likelihood estimator (MLE) equations to estimate gamma parameters, k (shape parameter) and
9 (scale parameter), can be found in Singh, Singh, and laci (2002). A random variable (RV), X (arsenic
concentrations), follows a gamma distribution, G(k, 9), with parameters k > 0 and 9 > 0, if its probability
density function is given by the following equation:
f(x; k, 9) = —i— x^e-*16; x>0
"*— (3.4)
= 0; otherwise
The mean, variance, and skewness of a gamma distribution are: // = kQ, variance = o2 = kff, and
skewness = 2/v&- Note that as k increases, the skewness decreases, and, consequently, a gamma
distribution starts approaching a normal distribution for larger values of k (e.g., k > 10). In practice, k is
not known and a normal approximation may be used even when the MLE estimate of k is as small as 6.
Let k and 6 represent the MLEs of k and 9 respectively. The relationship between a gamma RV, X = G
(k, 9), and a chi-square RV, Y, is given by X = Y * 912, where Y follows a chi-square distribution with
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2k degrees of freedom (df). Thus, the percentiles of a chi-square distribution (as programmed in ProUCL)
can be used to determine the percentiles of a gamma distribution. In practice, k is replaced by its MLE.
Once an a* 100% percentile, y(a) 2k, of a chi-square distribution with 2k df is obtained, the a* 100%
percentile for a gamma distribution is computed using the following equation:
(3-5)
3.4 Upper Tolerance Limits
A UTL (l-a)-p (e.g., UTL95-95) based upon an established background data set represents that limit such
that/?* 100% of the observations (current and future) from the target population (background, comparable
to background) will be less than or equal to UTL with a CC, (1-a). It is expected that/?* 100% of the
observations belonging to the background population will be less than or equal to a UTL with a CC, (1-a).
A UTL (l-a)-p represents a (1-a) 700% UCL for the unknown pih percentile of the underlying
background population.
A UTL95-95 is designed to provide coverage for 95% of all observations potentially coming from the
background or comparable to background population(s) with a CC of 0.95. A UTL95-95 will be exceeded
by all (current and future) values coming from the background population less than 5% of the time with a
CC of 0.95, that is 5 exceedances per 100 comparisons (of background values) can result just by chance
for an overall CC of 0.95. Unlike a UPL95, a UTL95-95 can be used when many, or an unknown number
of current or future onsite observations need to be compared with a BTV. A parametric UTL95-95 takes
the data variability into account.
When a large number of comparisons are made with a UTL95-95, the number of exceedances (not the
percentage of exceedances) of the UTL95-95 by those observations can also be large just by chance. This
implies that just by chance, a larger number (but not larger than 5%) of onsite locations comparable to
background can be greater than a UTL95-95 potentially requiring unnecessary investigation which may
not be cost-effective. In order to avoid this situation, it is suggested to use a USL95 to estimate a BTV,
provided the background data set represents a single statistical population, free of outliers.
3.4.1 Normal Upper Tolerance Limits
First, compute the sample mean, x , and sd, s, using a defensible data set representing a single
background population. For normally distributed data sets, an upper (1 - a)* 100% UTL with coverage
coefficient, p, is given by the following statement.
UTL=x + K*s (3-6)
Here, K = K (n, a, p) is the tolerance factor and depends upon the sample size, n, CC = (1 - a), and the
coverage proportion = p. For selected values of n, p, and (1-a), values of the tolerance factor, K, have
been tabulated extensively in the various statistical books (e.g., Hahn and Meeker 1991). Those K values
are based upon the non-central t-distribution. Also, some large sample approximations (Natrella 1963) are
available to compute the K values for one-sided tolerance intervals (same for both UTLs and lower
tolerance limits). The approximate value of K is also a function of the sample size, n, coverage
coefficient, p, and the CC, (1 - a). For samples of small sizes, n< 30, ProUCL uses the tabulated (Hahn
and Meeker 1991) AT values. Tabulated K values are available only for some selected combinations ofp
(0.90, 0.95, 0.975) and (1-a) values (0.90, 0.95, 0.99). For sample sizes larger than 30, ProUCL computes
the K values using the large sample approximations, as given in Natrella (1963). The Natrella's
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approximation seems to work well for samples of sizes larger than 30. ProUCL computes these K values
for all valid values ofp and (1-a) and samples of sizes as large as 5000.
3.4.2 Lognormal Upper Tolerance Limits
The procedure to compute UTLs for lognormally distributed data sets is similar to that for normally
distributed data sets. First, the sample mean, y , and sd, sy, of the log-transformed data are computed. An
upper (1 - a)* 100% tolerance limit with tolerance or coverage coefficient, p is given by the following
statement:
UTL = exp(>> + K * s ) (3-7)
The K factor in (3-7) is the same as the one used to compute the normal UTL.
Notes: Even though there in no back-transformation bias present in the computation of a lognormal UTL,
a lognormal distribution based UTL is typically higher (sometimes unrealistically higher as shown in the
following example) than other parametric and nonparametric UTLs; especially when the sample size is
less than 20. Therefore, the use of lognormal UTLs to estimate BTVs should be avoided when skewness
is high (sd of logged data > 1 or 1.5) and sample size is small (e.g., n < 20-30).
3.4.3 Gamma Distribution Upper Tolerance Limits
Positively skewed environmental data can often be modeled by a gamma distribution. ProUCL software
has two goodness-of-fit tests: the Anderson-Darling (A-D) and Kolmogorov-Smirnov (K-S) tests for a
gamma distribution. These GOF tests are described in Chapter 2. UTLs based upon normal approximation
to the gamma distribution (Krishnamoorthy et al. 2008) have been incorporated in ProUCL 4.00.05 (EPA
2010d) and higher versions. Those approximations are based upon Wilson-Hilferty (WH)(Wilson and
Hilferty 1931) and Hawkins-Wixley (HW) (Hawkins and Wixley 1986) approximations.
Note: It should be pointed out that the performance of gamma UTLs and gamma UPLs based upon these
HW and WH approximations is not well-studied and documented. Interested researchers may want to
evaluate the performance of these gamma upper limits based upon HW and WH approximations.
A description of method to compute gamma UTLs is given as follows.
Let xi, x2, ..., xn represent a data set of size n from a gamma distribution, G(k, 6) with shape parameter, k
and scale parameter 6.
• According to the WH approximation, the transformation, Y = X1/3 follows an approximate normal
distribution. The mean, n and variance, a2 of the transformed normally distributed variable, Y are
given as follows:
ju = [6>1/3r(£ +1 / 3)] / r(£); and a2 = [02/3r(k + 2/ 3)] / r(£) - //2
• According to the HW approximation, the transformation, Y = X1/4 follows an approximate
normal distribution.
Let y and sy represent the mean and sd of the observations in the transformed scale (Y).
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Using the WH approximation, the gamma UTL (in original scale, X), is given by:
UTL = max o,y + K*sy (3-8)
Similarly, using the HW approximation, the gamma UTL in original scale is given by:
UTL= (y+K*sy)4 (3-9)
The tolerance factor, K is defined earlier in (3-6) while computing a UTL based upon normal distribution.
Note: For mildly skewed to moderately skewed gamma distributed data sets, HW and WH
approximations yield fairly comparable UTLs. However, for highly skewed data sets (k
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3.4.4.1 Determining the Order, r, of the Statistic, X(r), to Compute UTLp,(l-a)
Using the cumulative binomial probabilities, a number, r. 1 < r < n, is chosen such that the cumulative
i=r fn\
binomial probability: "V />'(! - p}("~'} becomes as close as possible to (1 - a). The binomial
%(*)
distribution (BD) based algorithm has been incorporated in ProUCL for data sets of sizes up to 2000. For
data sets of size, n >2000, ProUCL computes the rth (r. 1 < r < n) order statistic by using the normal
approximation (Conover, 1999) given by the equation (3-10).
p)+0.5 (3-10)
Depending upon the sample size, p, and (1 - a) the largest, the second largest, the third largest, and so
forth order statistic is used to estimate the UTL. As mentioned earlier for a given data set of size n, the rth
order statistic, X(r) may or may not achieve the specified CC, (1 - a). ProUCL uses the F-distribution based
probability statement to compute the CC achieved by the UTL determined by the rth order statistic.
3.4.4.2 Determining the Achieved Confidence Coefficient, CC achieve, Associated with X(r)
For a given data set of size, n, once the rth order statistic, X(r), has been determined, ProUCL can be used to
determine if a UTL computed using X(r) achieves the specified CC, (1 - a). ProUCL computes the
achieved CC by using the following approximate probability statement based upon the F-distribution with
vi and V2 degrees of freedom.
CCAcUm=(l-a.)= Probability (F(v^
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when the second largest value, X(n-i) is used to compute a UTL, and m=n-r+l when the rth order statistic,
X(r), is used to compute a UTL. For example, if the largest sample value, X(»), is used to compute a UTL95-
95, then a minimum sample size of 59 (see equation (3-12)) will be needed to achieve a confidence level
of 0.95 providing coverage to 95% of the observations coming from the target population. A UTL95-95
estimated by the largest value and computed based upon a data set of size less than 59 may not achieve
the desired confidence of 0.95 for the 95th percentile of the target population.
Note: The minimum sample size requirement of 59 cited in the literature is valid when the largest value,
X(n) (with m=l) in the data set is used to compute a compute a UTL95-95. For example, when the largest
order statistic, X(») (with m=l) is used to compute a nonparametric UTL95-95, the approximate minimum
sample size needed 0.25*5.99*1.95/0.05 ~ 58.4 (using equation (3-12)) which is rounded upward to 59;
and when the second largest value, X(n-i) (with m=2) is used to compute a UTL95-95, the approximate
minimum sample size needed = [(0.25*9.488*1.95)70.05] + 0.5 ~ 93. Similarly, to compute a UTL90-95
by the largest sample value, about 29 observations will be needed to provide coverage for 90% of the
observations from the target population with CC = 0.95. Other sample sizes for various values of p and
(I-a.) can be computed using equation, (3-12). In environmental applications, the number of available
observations from the target population is much smaller than 29, 59 or 93 and a UTL computed based
upon those data sets may not provide specified coverage with the desired CC. For specified values of CC,
(I-a.) and coverage, p, ProUCL 5.1 outputs the achieved CC by a computed UTL and the minimum
sample size needed to achieve the pre-specified CC.
3.4.4.4 Nonparametric UTL Based upon the Percentile Bootstrap Method
A couple of bootstrap methods to compute nonparametric UTLs are also available in ProUCL 5.1. Like
the percentile bootstrap UCL computation method, for data sets without a discernible distribution, one can
use percentile bootstrap resampling method to compute UTL^,^_a; =UTL/>, (1 - a). The TV bootstrapped
nonparametric pth percentiles, p, (i:=l,2,...,N), are arranged in ascending order: pl
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3.5 Upper Prediction Limits
Based upon a background data set, UPLs are computed for a single (UPLi) and k (UPLk) future
observations. Additionally, in groundwater monitoring applications, an upper prediction limit of the mean
of the k future observations, UPLk (mean) is also used. A brief description of parametric and
nonparametric upper prediction limits is provided in this section.
for a Single Future Observation: A UPLi computed based upon an established background data set
represents that statistic such that a single future observation from the target population (e.g., background,
comparable to background) will be less than or equal to the UPLi 95 with a CC of 0.95. A parametric UPL
takes the data variability into account. A UPLi is designed for a single future observation comparison;
however in practice users tend to use UPLi95 to perform many future comparisons which results in a high
number of false postives (observations declared contaminated when in fact they are clean).
When k>7 future comparisons are made with a UPLi, some of those future observations will exceed the
UPLi just by chance, each with probability 0.05. For proper comparison, a UPL needs to be computed
accounting for the number of comaprisons that will be performed. For example, if 30 independent onsite
comparisons (e.g., Pu-238 activity from 30 onsite locations) are made with the same background UPLi95,
each onsite value comparable to background may exceed that UPLi 95 with probability 0.05. The overall
probability of at least one of those 30 comparisons being significant (exceeding the BTV) just by chance
is given by:
aactuai = l-(l-a)k =1 - 0.9530 -1-0.21 = 0. 79 (false positive rate).
This means that the probability (overall false positive rate) is 0.79 (and not 0.05) that at least one of the 30
onsite observations will be considered contaminated even when they are comparable to background.
Similar arguments hold when multiple (=j, a positive integer) constituents are analyzed, and status (clean
or impacted) of an onsite location is determined based upon j comparisons (one for each analyte). The use
of a UPLi is not recommended when multiple comparisons are to be made.
3. 5. 1 Normal Upper Prediction Limit
The sample mean, x , and the sd, s, are computed first based upon a defensible background data set. For
normally distributed data sets, an upper (1 - a)* 100% prediction limit is given by the following well
known equation:
UPL = x + t((1_aun_1)} * s * VC + l/H) (3-13)
Here t(na) („_!« is a critical value from the Student's t-distribution with (n-\) df.
3.5.2 Lognormal Upper Prediction Limit
An upper (1 - a)*100% lognormal UPL is similarly given by the following equation:
UPL = exp(y +1((1_al(n_1}} *sy * J(\ + l/n)} (3-14)
Here t(na) („_!« is a critical value from the Student's t-distribution with (n-\) df.
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3.5.3 Gamma Upper Prediction Limit
Given a sample, xi, x2, ..., xn of size n from a gamma distribution G(k, 9), approximate (based upon WH
and HW approximations described earlier in Section 3.4.3, Gamma Distribution Upper Tolerance Limits),
(1 - a)*100% upper prediction limits for a future observation from the same gamma distributed
population are given by:
Wilson-Hilferty(WH)UPL= max\OAy + t^ w ,**s*Jl+y] (3-15)
^ \ U1 «M» !)) ^ Af /«/ J
(/ \4
y + f,,, *s *./!+ I/ (3-16)
^ ((l-a),(H-l)t 7 V / ft I ^ '
Here t^_a) (n_^ is a critical value from the Student's t-distribution with (n-\)df.
Note: As noted earlier, the performance of gamma UTLs and gamma UPLs based upon these WH and
HW approximations is not well-studied. Interested researchers may want to evaluate their performances
via simulation experiments. These approximations are also available in R script.
3.5.4 Nonparametric Upper Prediction Limit
A one-sided nonparametric UPL is simple to compute and is given by the following mth order statistic.
One can use linear interpolation if the resulting number, m, given below does not represent a whole
number (a positive integer).
UPL = X(m), where m = (n + 1) * (1 - a). (3-17)
For example, for a nonparametric data set of size n=25, a 90% UPL is desired. Then m = (26*0.90) =
23.4. Thus, a 90% nonparametric UPL can be obtained by using the 23rd and the 24th ordered statistics and
is given by the following equation:
UPL = X(23) + 0.4 * (X(24) -X(23) )
Similarly, if a nonparametric 95% UPL is desired, then m = 0.95 * (25 + 1) = 24.7, and a 95% UPL can
be similarly obtained by using linear interpolation between the 24th and 25th order statistics. However, if a
99% UPL needs to be computed, then m = 0.99 * 26 = 25.74, which exceeds 25, the sample size; for such
cases, the highest order statistic is used to compute the 99% UPL of the background data set. The largest
value(s) should be used with caution (as they may represent outliers) to estimate the BTVs.
Since nonparametric upper limits (e.g., UTLs, UPLs) are based upon higher order statistics, often the CC
achieved by these nonparametric upper limits is much lower than the specified CC of 0.95, especially
when the sample size is small. In order to address this issue, one may want to compute a UPL based upon
the Chebyshev inequality. In addition to various parametric and nonparametric upper limits, ProUCL
computes Chebyshev inequality based UPL.
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3.5.4.1 Upper Prediction Limit Based upon the Chebyshev Inequality
Like a UCL of the mean, the Chebyshev inequality can be used to compute a conservative but stable UPL
and is given by the following equation:
UPL = Jc+[^((1/00-1)*
This is a nonparametric method since the Chebyshev inequality does not require any distributional
assumptions. It should be noted that just like the Chebyshev UCL, a UPL based upon the Chebyshev
inequality tends to yield higher estimates of BTVs than the various other methods. This is especially true
when skewness is mild (sd of log-transformed data is low < 0.75), and the sample size is large (n > 30).
The user is advised to apply professional judgment before using this method to compute a UPL.
Specifically, for larger skewed data sets, instead of using a 95% UPL based upon the Chebyshev
inequality, the user may want to compute a Chebyshev UPL with a lower CC (e.g., 85%, 90%) to estimate
a BTV. ProUCL can compute a Chebyshev UPL (and all other UPLs) for any user specified CC in the
interval [0.5, 1].
3.5.5 Normal, Lognormal, and Gamma Distribution based Upper Prediction Limits for k
Future Comparisons
A UPLk95 computed based upon an established background data set represents that statistic such that k
future (next, independent and not belonging to the current data set) observations from the target
population (e.g., background, comparable to background) will be less than or equal to the UPLk95 with a
CC of 0.95. A UPLk95 for k (>1) future observations is designed to compare k future observations; we are
95% sure that "k" future values from the background population will be less than or equal to UPLk95
with CC of 0.95. In addition to UPLk, ProUCL also computes an upper prediction limit of the mean of k
future observations, UPLk (mean). A UPLk (mean) is commonly used in groundwater monitoring
applications. A UPLk controls the false positive error rate by using the Bonferroni inequality based critical
values to perform k future comparisons. These UPLs statisfy the relationship: UPLi
-------
UPL395 = \x
A lognormal distribution based UPLk (1 - a) for k future observations, xn+1,xn+2,...,xn+k is given by the
following equation:
A gamma distribution based UPLk for the next k > 7 (k future observations) are computed similarly using
the WH and HW approximations described in Section 3.4.3.
3. 5. 6 Proper Use of Upper Prediction Limits
It is noted that some users tend to use UPLs without taking their definition and intended use into
consideration; this is an incorrect application of a UPL. Some important points to note about the proper
use of UPLi and UPLk for k>7 are described as follows.
• When a UPLk is computed to compare k future observations collected from a site area or a group
of MW within an operating unit (OU), it is assumed that the project team will make a decision
about the status (clean or not clean) of the site (MWs in an OU) based upon those k future
observations.
• The use of an UPLk implies that a decision about the site-wide status will be made only after k
comparisons have been made with the UPLk. It does not matter if those k observations are
collected (and compared) simultaneously or successively. The k observations are compared with
the UPLk as they become available and a decision (about site status) is made based upon those k
observations.
• An incorrect use of a UPLi 95 is to compare many (e.g., 5, 10, 20, etc.) future observations. This
results in a higher than 0.05 false positive rate. Similarly, an inappropriate use of a UPLioo would
be to compare less than 100 (i.e., 10, 20, or 50 observations) future observations. Using a UPLioo
to compare 10 or 20 observations can potentially result in a high number of false negatives (a test
with reduced power), declaring contaminated areas comparable to background.
• The use of other statistical limits such as 95%-95% UTLs (UTL95-95) is preferred to estimate BTVs
and not-to-exceed values. The computation of a UTL does not depend upon the number of future
comparisons which will be made with the UTL.
3.6 Upper Simultaneous Limits
An (1 - a) * 100% upper simultaneous limit (USL) based upon an established background data set is meant
to provide coverage for all observations, xt, i = 1, 2, n simultaneously in the background data set. It is
implicitly assumed that the data set comes from a single background population and is free of outliers
(established background data set). A USL95 represents that statistic such that all observations from the
"established" background data set will be less than or equal to the USL95 with a CC of 0.95. It is expected
that observations coming from the background population will be less than or equal to the USL95 with a 95%
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CC. A USL95 can be used to perform any number (unknown) of comparisons of future observations. The
false positive error rate does not change with the number of comparisons as the purpose of the USL95 is to
perform any number of comparisons simultaneously.
Notes: If a background population is established based upon a small data set; as one collects more
observations from the background populations, some of the new background observations will exceed the
largest value in the existing data set. In order to address these uncertainties, the use of a USL is suggested,
provided the data set represents a single population without outliers.
3.6.1 Upper Simultaneous Limits for Normal, Lognormal and Gamma Distributions
The normal distribution based two-sided (1 - a) 100% simultaneous interval obtained using the first order
Bonferroni inequality (Singh and Nocerino 1995, 1997) is given as follows:
P(x-sdba
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Nonparametric USL: For nonparametric data sets, the largest value, X(n) is used to compute a
nonparametric USL. Just like a nonparametric UTL, a nonparametric USL may fail to provide the
specified coverage, especially when the sample size is small (e.g., <60). The confidence coefficient
actually achieved by a USL can be computed using the same process as used for a nonparametric UTL
described in Sections 3.4.4.2 and 3.4.4.3. Specifically, by substituting r = n in equation (3-11), the
confidence coefficient achieved by a USL can be computed, and by substituting m=l in equation (3-12),
one can compute the sample size needed to achieve the desired confidence.
Note: Nonparametric USLs, UTLs or UPLs should be used with caution; nonparametric upper limits are
based upon order statistics and therefore do not take the variability of the data set into account. Often
nonparametric BTVs estimated by order statistics do not achieve the specified CC unless the sample size
is fairly large.
Dependence of UTLs and USLs on the Sample Size: For smaller samples (n <10), a UTL tends to yield
impractically large values, especially when the data set is moderately skewed to highly skewed. For data
sets of larger sizes, the critical values associated with UTLs tend to stabilize whereas critical values
associated with a USL increase as the sample size increases. Specifically, a USL95 is less than a UTL95-
95 for samples of sizes, n <16, they are equal/comparable for samples of size 17, and a USL95 becomes
greater than a UTL95-95 as the sample size becomes greater than 17. Some examples illustrating the
computations of the various upper limits described in this chapter are discussed as follows.
Example 3-1. Consider the real data set used in Example 2-4 of Chapter 2 consisting of concentrations
for several constituents of potential concern, including aluminum, arsenic, chromium (Cr), and lead. The
computation of background statistics obtained using ProUCL for some of the metals are summarized as
follows.
Upper Limits Based upon a Normally Distributed Data Set: The aluminum data set follows a normal
distribution as shown in the following GOF Q-Q plot of Figure 3-1.
Normal Q-Q Plot for Aluminum
n = 24
Mean = 7789
Sd = 4264
Slope = 4293
Intercept = 7789
Correlation, R= 0.976
Shapiro-WilkTest
DilicalVal|0.05)-0.916
Dala Appeal Normal
Appro*. TestVdlue-D.939
p-Value-0.161
• Best Fil Line
Theoretical Quantiles (Standard Normal)
Figure 3-1. Normal Q-Q plot of Aluminum with GOF Statistics
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From the normal Q-Q plot shown in Figure 3-1, it is noted that the 3 largest values are higher (but not
extremely high) than the rest of the 21 observations. These observations may or may not come from the
same population as the rest of the 21 observations.
Table 3-1. BTV Estimated Based upon All 24 Observations
Aluminum
General Statistics
Total Number of Observations 24
Minimum 1710
Second Largest 154QO
Maximum 16200
Mean 7789
Coefficient of Variation 0.547
Mean of logged Data 8.79S
Number of Distinct Observations 24
First Quartile 405S
Median 7010
Third Quartile 10475
SD 4264
Skewness 0.542
SD of logged Data 0.61
Critical Values for Background Threshold Values (BTVs)
Tolerance Factor K (For UTL) 2,309
d2max for USL) 2.644
Shapiro Wilk Test Statistic
5% Shapiro Wilk Critical Value
Lilliefors Test Statistic
5% Lilliefors Critical Value
Normal GOF Test
0.939
0.916
0.109
0.181
Shapiro Wilk GOF Test
Data appear Normal at 5% Significance Level
Lilliefors GOF Test
Data appear Normal at 5% Significance Level
Data appear Normal at 5% Significance Level
Background Statistics Assuming Normal Distribution
95% UTL with 95% Coverage 17635
95%UPL|t) 15248
95% USL 19063
90% Percentile (z) 132-54
95% Percentile (z) 14803
93% Percentile (2) 17708
The classical outlier tests (Dixon and Rosner tests) did not identify these 3 data points as outliers. Robust
outlier tests, MCD (Rousseeuw and Leroy 1987), and PROP influence function (Singh and Nocerino,
1995) based tests identified the 3 high values as statistical outliers. The project team should decide
whether or not the 3 higher concentrations represent outliers. A brief discussion about robust outlier
methods is given in Chapter 7. The inclusion of the 3 higher values in the data set resulted in higher
upper limits. The various upper limits have been computed with and without the 3 high observations and
are summarized respectively, in Tables 3-1 and 3-2 as follows. The project team should make a
determination of which statistics (with outliers or without outliers) should be used to estimate BTVs.
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Table 3-2. BTV Estimated Based upon 21 Observations without 3 Higher Values
Aluminum
General Statistics
Total Number of Observations 21
Minimum 1710
Second Largest 11 SOD
Maximum 125DD
Mean 6669
Coefficient of Variation 0.482
Number of Distinct Observations 21
Number of Missing Observations 3
First Quartile 3900
Median 6350
Third Quartile 9310
SD 3215
Skewness 0.25
Mean of logged Data 8.676 SD of logged Data 0.549
Critical Values for Background Threshold Values (BTVs)
Tolerance Factor K (For UTL) 2.371
d2max for USL) 2.58
Normal GOF Test
Shapiro Wilk Test Statistic 0.955
5% Shapiro Wilk Critical Value 0.903
Lilliefors Test Statistic 0.12
5% Lilliefors Critical Value 0.193
Data appear Normal at
Shapiro Wilk GOF Test
Data appear Normal at 5V. Significance Level
Lilliefors GOF Test
Data appear Normal at 5*'= Significance Level
5% Significance Level
Background Statistics Assuming Normal Distribution
95% UTL with 95*4 Coverage 14291 90% Percerrtile (z) 10789
95%UPL
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Table 3-3. Lognormal Distribution Based UPLs, UTLs, and USLs
Chromium
General Statistics
Total Number of Observations 24
Minimum 3
Second Largest 20
Maximum 35.5
Mean 11.97
Coefficient of Variation 0.576
Mean of logged Data 2.334
Number of Distinct Observations
First Quartile
Median
Third Quartile
SD
Skewness
SD of logged Data
19
7.975
11
14.25
6.892
1.728
0.568
Gitical Values for Background Threshold Values (BTVs)
Tolerance Factor K (For UTL) 2.309
Lognormal
Shapiro Wilk Test Statistic 0.978
5% Shapiro Wilk Critical Value 0.91 6
Lilliefors Test Statistic 0.123
5% Lilliefons Critical Value 0.181
d2max for USL)
GOFTest
Shapiro Wilk Lognonnal GOF Test
Data appear Lognormal at 5% Significance Level
Lilliefors Lognonnal GOF Test
Data appear Lognormal at 5% Significance Level
2.644
Data appear Lognoimal at 5% Syiificance Level
Background Statistics assui
95*4 UTL with 95% Coverage 38.3
95%UPL(t) 27.37
95% U P L f or Next 5 Observations 43.96
95% LIPLfor Mean of 5 Observations 1 6.66
ning Lognormal Distribution
90% Percentile (z)
95% Percentile (z)
99% Percentile ft
95% USL
21.37
26.27
38.68
46.33
Example 3-3. Arsenic concentrations of the data set used in Example 2-4 follow a gamma distribution.
The background statistics, obtained using a gamma model, are shown in Table 3-4. Figure 3-3 is the
gamma Q-Q plot with GOF statistics.
Gamma Q-Q Plot for Arsenic
Aisenic
= 24
ear, = 2.1*83
tot-3-6616
els slai-0.5887
lope-1.0264
creep! - 0.01 S3
.rcijtion. R - i) 3770
dels dn-D a i ling Test
iiSi.jSisiif-0.5f1
CiilicalValue[0 05) - OL748
D ala appeal Gamma Dishibuted
• Best Fit Line
Theoretical Quantiles of Gamma Distribution
Figure 3-3. Gamma Q-Q plot of Arsenic with GOF Statistics
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Table 3-4. Gamma Distribution Based UPLs, UTLs, and USLs
Arsenic
General Statistics
Total Number of Observations 24
Minimum 8.66
Second Largest 3.7
Maximum 5.9
Mean 2.148
Coefficient of Variation 0.54
Mean of logged Data 0.639
Number of Distinct Observations 18
First Quartile 1.2
Median 2.05
Third Quartile 2.45
SD 1.159
Skewness 1.554
SD of logged Data 0.51
Critical Values for Background Threshold Values (BTVs)
Tolerance Factor K (For UTL) 2.309
d2maxforUSL} 2.644
Gamma GOF Test
A-D Test Statistic 0.341 Anderson-Darling Gamma GOF Test
5% A-D Critical Value 0.748 Detected data appear Gamma Distributed at 5% Significance Level
K-S Test Statistic 0114 Kolmogrov-Smimoff Gamma GOF Test
5% K-S Critical Value 0.179 Detected data appear Gamma Distributed at 5% Significance Level
Detected data appear Gamma Distributed at 5% Significance Level
khat(MLE)
Theta hat (MLE)
nu hat (MLE}
MLE Mean |bias corrected)
Gamma Statistics
4.153
0.517
199.3
2.142
k star (bias corrected MLE} 3.662
Theta star (bias corrected MLE) Q.587
nu star fbias corrected} 175.8
M LE Sd (bias corrected) 1.123
Background Statistics Assuming Gamma Distribution
95% Wilson Hiferty (WH) Approx. Gamma UPL 4.345
95% Hawkins Wixley (HW) Approx. Gamma UPL 4.397
95% WH Approx. Gamma UTL with 95% Coverage 5.382
95% HW .Approx. Gamma UTL with 95% Coverage 5.524
95%WHUSL 6.074
90% Percentile 3.654
95% Percentile 4.264
99% Percentile 5.574
95% HW USL 6.294
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Example 3-4. Lead concentrations of the data set used in Example 2-4 do not follow a discernible
distribution. The various nonparametric background statistics for lead are shown in Table 3-5.
Table 3-5. Nonparametric UPLs, UTLs, and USLs for Lead in Soils
Lead
General Statistics
Total Number of Observations 24
Minimum 4.9
Second Largest 38.5
Maximum 109
MCJI| 22.49
Coefficient of Variation 1 .193
Mean of logged Data 2.743
Number of Distinct Observations 1 8
First Quartile 10.43
Median 14
Third Quartile 19.25
SD| 26.83
Skewness 2.665
SD of logged Data 0.771
Critical Values for Background Threshold Values (BTVs)
Tolerance Factor K (For UTL) 2.309 d2max (For USL) 2.644
Nonparamelric Distribution Free Background Statistics
Data do not follow a Discernible Distribution (0.05)
Nonparametric Upper Limits for Background Threshold Values
Order of Statistic, r 24
r^praxirnate f 1 .263
95% Percentile Bootstrap UTL with 95% Coverage 109
95% UPL 106.4
mChebyshevUPL 104.6
95%ChebyshevUPL 141.8
95% USL 109
55% UTL with 55% Coverage 109
Confidence Coefficient (CC) achieved by UTL 0.708
95% BCA Bootstrap UTL with 95% Coverage 109
90% Percentile 44.81
95% Percentile 91.72
99% Percentile 106.6
Notes:
Note: As mentioned before, nonparametric upper limits are computed by higher order statistics, or by
some value in between (based upon linear interpolation) the higher order statistics. In practice,
nonparametric upper limits do not provide the desired coverage to the population parameter (upper
threshold) unless the sample size is large. From Table 3-5, it is noted that a UTL95-95 is estimated by the
maximum value in the data set of size 24. However, the CC actually achieved by UTL95-95 (and also by
USL95) is only 0.708. Therefore, one may want to use other upper limits such as 95% Chebyshev UPL =
141.8 to estimate a BTV.
Note: As mentioned earlier, for symmetric and mildly skewed nonparametric data sets (when sd of
logged data is <=0.5), one can use the normal distribution to compute percentiles, UPLs, UTLs and USLs.
Example 3-5: Why Use a Gamma Distribution to Model Positively Skewed Data Sets?
The data set considered in Example 2-2 of Chapter 2 is used to illustrate the deficiencies and problems
associated with the use of a lognormal distribution to compute upper limits. The data set follows a
lognormal as well as a gamma model; the various upper limits, based upon a lognormal and a gamma
model, are summarized as follows. The data set is highly skewed with sd of logged data = 1.68. The
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largest value in the data set is 169.8, the UTL95-95 and UPL95 based upon a lognormal model are 799.7
and 319 both of which are significantly higher than the maximum value of 169.8. UTL95-95s based upon
WH and HW approximations to gamma distributions are 245.3 and 285.6; UPLs based upon WH and HW
approximations are 163.5 and 178.2 which appear to represent more reasonable estimates of the BTV.
These statistics are summarized in Table 3-6 (lognormal) and Table 3-7 (gamma) below.
Table 3-6. Background Statistics Based upon a Lognormal Model
x
General Satisfies
Total Number of Observations 25
Minimum 0.349
Second Largest 164.3
Maximum 1S9.8
Mean 44.09
Coefficient of Variation 1.164
Mean of logged Data 2.S35
Number of Distinct Observations 25
First Quartile 5.093
Median 18.77
Third Quartile 72.62
SD 51.34
Skewness 1.294
SD of logged Data 1.68
Critical Values for Background Threshold Values (BTVs)
Tolerance Factor K (For UTL) 2.292
d2max for USL) 2.663
Lognonnal GOF Test
Shapiro Wilk Test Statistic 0.948 Shapiro Wilk Lognormal GOF Test
5% Shapiro Wilk Critical Value 0.918 Data appear Lognormal at 5% Significance Level
Lilliefors Test Statistic 0.135 Ljlliefors Lognormal GOF Test
5% Ulliefors Critical Value 0.177 Data appear Lognormal at 5% Significance Level
Data appear Lognonnal at 5"4 Significance Level
Background Statistics assuming Lognormal Distribution
95% UTL with 35% Coverage 799.7
35%UPLft) 319
90% Percentile ',z] 146.5
95% Percentile (2] 269.7
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Table 3-7. Background Statistics Based upon a Gamma Model
x
General Statistics
Total Number of Observations 25
Minimum 0.349
Second Largest 164.3
Maximum 163.8
Mean 44.09
Coefficient of Variation 1.164
Mean of logged Data 2.835
Number of Distinct Observations 25
First Quartile 5.093
Median 18.77
Third Quartile 72.62
SD 51.34
Skewness 1.294
SD of logged Data 1.68
Critical Values for Background Threshold Values (BTVs)
Tolerance Factor K (For UTL) 2.292
d2max for USL) 2.663
Gamma GOF Test
A-D Test Statistic 0.374 Anderson-Darling Gamma GOF Test
5% A-D Critical Value Q.794 Detected data appear Gamma Distributed at 5% Significance Level
K-S Test Statistic 0113 Kolmogrov-Smimoff Gamma GOF Test
5% K-S Critical Value 0.183 Detected data appear Gamma Distributed at 5% Significance Level
Detected data appear Gamma Distributed at 5% Significance Level
k hat (MLE)
Theta hat (MLE)
nu hat {MLE}
MLE Mean (bias corrected)
Gamma Statistics
0.643
68.58
32.15
44.03
k star (bias corrected M LE} 0.532
Theta star (bias corrected MLE) 74.42
nu star {bias corrected} 29.62
MLE Sd {bias corrected} 57.28
Background Statistics Assuming Gamma Distribution
95% Wilson Hilferty (WH) Approx. Gamma UPL 163,5
35% Hawkins Wbdey (HW) Approx. Gamma UPL 178.2
95% WH Approx. Gamma UTL with 95% Coverage 245,3
95% HW Approx. Gamma UTL with 95% Coverage 285.6
90% Percentile 115
95% Percentile 159.4
99% Percentile 266.8
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CHAPTER 4
Computing Upper Confidence Limit of the Population Mean
Based upon Left-Censored Data Sets Containing Nondetect
Observations
4.1 Introduction
Nondetect (ND) observations are inevitable in most environmental data sets. It should be noted that the
estimation of the mean and sd, and the computation of the upper limits (e.g., upper confidence limits
[UCLs], upper tolerance intervals [UTLs]) are two different tasks. For left-censored data sets with NDs,
in addition to the availability of good estimation methods, the availability of rigorous statistical methods
which account for data skewness is needed to compute the decision making statistics such as UCLs,
UTLs, and UPLs. For left-censored data sets consisting of multiple detection limits (DLs) or reporting
limits (RLs), ProUCL 4.0 (2007) and its higher versions offer methods to: 1) impute NDs using
regression on order statistics (ROS) methods; 2) perform GOF tests; 3) estimate the mean, standard
deviation (sd), and standard error of the mean; and 4) compute skewness adjusted upper limits (e.g.,
UCLs, UTLs, UPLs). Based upon KM (Kaplan and Meier 1958) estimates, and the distribution and
skewness of detected observations, several upper limit computation methods which adjust for data
skewness have also been incorporated in ProUCL 5.1.
For left-censored data sets with NDs, Singh and Nocerino (2002) compared the performances of the
various estimation methods (in terms of bias and MSB) to estimate the population mean, ^ , and sd, crl
including the MLE method (Cohen 1950, 1959), restricted MLE (RMLE) method (Perrson and Rootzen
1977); Expectation Maximization (EM) method (Gleit 1985), EPA Delta lognormal method (EPA 1991;
Hinton 1993), Winsorization method (Gilbert 1987), and regression on order statistics (ROS) method
(Helsel 1990). Singh, Maichle, and Lee (EPA 2006) performed additional simulation experiments to
study and evaluate the performances (in terms of bias and MSE) of KM and ROS methods for estimating
the population mean. They concluded that the KM method yields better estimates, in terms of bias, of
population mean in comparison with other estimation methods including the LROS (ROS on logged data)
method. Singh, Maichle, and Lee (EPA 2006) also studied the performances, in terms of coverage
probabilities, of some parametric and nonparametric UCL computation methods based upon ROS, KM,
and other estimation methods. They concluded that for skewed data sets, KM estimates based UCLs
computed using bootstrap methods (e.g., BCA bootstrap, bootstrap-t) and Chebyshev inequality perform
better than the Student's t statistic UCL and percentile bootstrap UCL computed using ROS and KM
estimates as described in Helsel (2005, 2012) and incorporated in NADA packages (2013).
As mentioned above, computing good estimates of the mean and sd based upon left-censored data sets
addresses only half of the problem. The main issue is computing decision statistics (UCL, UPL, UTL)
which account for NDs as well as uncertainty and data skewness inherently present in environmental data
sets. Until recently (ProUCL 4.0, 4.00.05, 4.1; Singh, Maichle, and Lee 2006), not much guidance was
available on how to compute the various upper limits (UCLs, UPLs, UTLs) based upon skewed left-
censored data sets with multiple DLs. For left-censored data sets, the existing literature (Helsel 2005,
2012) suggests computing upper limits using a Student's t-type statistic and percentile bootstrap methods
on KM and LROS estimates without adjusting for data skewness. Environmental data sets tend to follow
skewed distributions, and UCL95s and other upper limits computed using methods described in Helsel
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(2005, 2012) will under estimate the population parameters of interest including EPCs and background
threshold values.
In earlier versions of ProUCL (ProUCL versions 4 [2007, 2009, 2010]), all evaluated estimation methods
including the poor performing methods (MLE and RMLE, and Winsorization methods) and better
performing, in terms of bias in the mean estimate, estimation (KM method) and UCL computation
methods (BCA bootstrap, bootstrap-t) were incorporated in ProUCL version 4 (2007, 2009, 2010).
Currently, the KM estimation method is widely used in environmental applications to compute parametric
(when detected data follow a known distribution) and nonparametric upper limits needed to estimate
environmental parameters of interest such as the population mean and upper thresholds of a background
population. Note that the KM method is now included in a recent EPA RCRA groundwater monitoring
guidance document (2009).
Due to the poor performances and/or failure to correctly verify probability distributions for data sets with
multiple DLs, the parametric MLE and RMLE methods, the normal ROS and the Winsorization
estimation methods for computing upper limits are no longer available in ProUCL version 5.0/5.1. The
normal ROS method is available only under the Stats/Sample Sizes module of ProUCL 5.0/5.1 to impute
NDs based upon the normal distribution assumption for advanced users who may want to use the imputed
data in other graphical and exploratory methods such as scatter plots, box plots, cluster analysis and
principal component analysis (PCA). The estimation methods for computing upper limits retained in
ProUCL 5.0/5.1 include the two ROS (lognormal, and gamma) methods and the KM method. The KM
estimation method can be used on a wide-range of skewed data sets with multiple DLs and NDs
exceeding detected observations. Also, the substitution methods such as replacing NDs by half of their
respective DLs and the H-UCL method (EPA 2009 recommends its use in Chapter 15) have been retained
in ProUCL 5.0/5.1 for historical reasons, and academic and research purposes. Inclusion of the DL/2
method (substitution of 1A the DL for NDs) in ProUCL should not be inferred as a recommended method.
The developers of ProUCL are not endorsing the use of the DL/2 estimation method or H-UCL
computation method.
Note on the use of letter k (k): Not to get confused with the use of letter "k (k)" in this Chapter and in
Chapters 2, 3, 4, and 5. Following the standard statistical terminology, "k" is used to denote the shape
parameter of a gamma distribution, G(k, 9) as described in Chapter 2; "k" is used to represent future (next)
observations (Chapter 3 and 5), and "k" is used to represent the number of ND observations present in a
data set (Chapters 4 and 5).
Notes on Skewness of Left-Censored Data Sets: Skewness of a data set is measured as a function of sd, a
(or its estimate, cr) of log-transformed data. Like uncensored full data sets, a, or its estimate, a, of the
log-transformed detected data is used to get an idea about the skewness of a data set consisting of ND
observations. This information along with the distribution of detected observations is used to decide
which UCL should be used to estimate the EPC and other upper limits for data sets consisting of both
detects and NDs. For data sets with NDs, output sheets generated by ProUCL 5.0/5.1 display the sd,
-------
4.2 Pre-processing a Data Set and Handling of Outliers
Throughout this chapter (and in other chapters such as Chapters 2, 3, and 5), it has been implicitly
assumed that the data set under consideration represents a "single" statistical population as a UCL is
computed for the mean of a "single" statistical population. In addition to representing "wrong" values
(e.g., typos, lab errors), outliers may also represent observations coming from population(s) significantly
different from the main dominant population whose parameters (mean, upper percentiles) we are trying to
estimate based upon the available data set. The main objective of using a statistical procedure is to model
the majority of data representing the main dominant population and not to accommodate a few low
probability (coming from far and extreme tails) outlying observations potentially representing impacted
locations (site related or otherwise). Statistics such as a UCL95 of the mean computed using data sets
with occasional low probability outliers tend to represent locations exhibiting those elevated low
probability outlying observations rather than representing the main dominant population.
4.2.1 Assessing the Influence of Outliers and Disposition of Outliers
One can argue against "not using the outliers" while estimating the various environmental parameters
such as the EPCs and BTVs. An argument can be made that outlying observations are inevitable and can
be naturally occurring (not impacted by site activities) in some environmental media (and therefore in
data sets). For example, in groundwater applications, a few elevated values (coming from the far tails of
the data distribution with low probabilities) may be considered to be naturally occurring and as such may
not represent the impacted MW data values. However, the inclusion of a few outliers (impacted or
naturally occurring observations) tends to yield distorted and elevated values of the decision statistics of
interest (UCLs, UPLs, and UTLs); and those statistics tend not to represent the main dominant population
(MW concentrations). As mentioned earlier, instead of representing the main dominant population, the
inflated decision statistics (UCLs, UTLs) computed with outliers included, tend to represent those low
probability outliers. This is especially true when one is dealing with smaller data sets (n <20-30) and a
lognormal distribution is used to model those data sets.
To assess the influence of outliers on the various statistics (upper limits) of interest, it is suggested to
compute all relevant statistics using data sets with outliers and without outliers, and then compare the
results. This extra step often helps the project team/users to see the direct potential influence of outlier(s)
on the various statistics of interest (mean, UPLs, UTLs). This in turn will help the project team to make
informative decisions about the disposition of outliers. That is, the project team and experts familiar with
the site should decide which of the computed statistics (with outliers or without outliers) represent better
and more accurate estimate(s) of the population parameters (mean, EPC, BTV) under consideration.
4.2.2 Avoid Data Transformation
Data transformations are performed to achieve symmetry of the data set and be able to use parametric
(normal distribution based) methods on transformed data. In most environmental applications, the
cleanup decisions are made based on statistics and results computed in the original scale as the cleanup
goals need to be attained in the original scale. Therefore, statistics and results need to be back-
transformed in the original scale before making any cleanup decisions. Often, the back-transformed
statistics (UCL of the mean) in the original scale suffer from an unknown amount of transformation bias;
many times the transformation bias can be unacceptably large (for highly skewed data sets) leading to
incorrect decisions. The use of a log-transformation on a data set tends to accommodate outliers and hide
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contaminated locations instead of revealing them. Specifically, an observation that is a potential outlier
(representing a contaminated location) in the original raw scale may not appear to be an outlier in the log-
scale. This does not imply that the location with elevated concentrations in the original scale does not
represent an impacted location. This issue has been considered and illustrated throughout this guidance
document.
The use of a gamma model does not require any data transformation therefore whenever applicable the
use of a gamma distribution is suggested to model skewed data sets. In cases when a data set in the
original scale cannot be modeled by a normal or a gamma distribution, it is better to use nonparametric
methods rather than testing or estimating parameters in the transformed space. For data sets which do not
follow a discernible parametric distribution, nonparametric and computer intensive bootstrap methods can
be used to compute the upper limits needed to estimate environmental parameters. Several of those
methods are available in ProUCL 5.1(ProUCL 5.0) for data sets consisting of NDs with multiple DLs.
4.2.3 Do Not Use DL/2(t) UCL Method
In addition to environmental scientists, ProUCL is also used by students and researchers. Therefore, for
historical and comparison purposes, the substitution method of replacing NDs by half of the associated
DLs (DL/2) is retained in ProUCL 5.1.; that is the DL/2 GOF tests, UCL, UPL, and UTL computation
methods have been retained in ProUCL 5.0/5.1 for historical reasons, and comparison and academic
purposes. For data sets with NDs, output sheets generated by ProUCL display a message suggesting that
DL/2 is not a recommended method. It is suggested that the use of the DL/2 (t) UCL method (UCL
computed using Student's t-statistic) be avoided when estimating a EPC or BTVs, unless the data set
consists of only a small fraction of NDs (<5%) and the data are mildly skewed. The DL/2 UCL
computation method does not provide adequate coverage (Singh, Maichle, and Lee 2006) for the
population mean, even for censoring levels as low as 10% or 15%. This is contrary to statements (EPA
2006b) made that the DL/2 UCL method can be used for lower (< 20%) censoring levels. The coverage
provided by the DL/2 (t) UCL method deteriorates fast as the censoring intensity, percentage of NDs,
increases and/or data skewness increases.
4.2.4 Minimum Data Requirement
Whenever possible, it is suggested that a sufficient number of samples be collected to satisfy the
requirements for the data quality objectives (DQOs) for the site. Often, in practice, it is not feasible to
collect the number of samples as determined by DQOs-based sample size formulae. Therefore, some
rule-of-thumb minimum sample size requirements are described in this section. At the minimum, collect a
data set consisting of about 10 observations to compute reasonably reliable and accurate estimates of
EPCs (UCLs) and BTVs (UPLs, UTLs). The availability of at least 15 to 20 observations is desirable to
compute UCLs and other upper limits based upon re-sampling bootstrap methods. Some of these issues
have also been discussed in Chapter 1 of this Technical Guide. However, from a theoretical point of view,
ProUCL can compute various statistics (KM UCLs) based upon data sets consisting of at least 3 detected
observations. The accuracy of the decisions based upon statistics computed using such small data sets
remains questionable.
4.3 Goodness-of-Fit (GOF) Tests and Skewness for Left-Censored Data Sets
It is not easy to assess and verify the distribution of data sets with NDs, especially when multiple DLs are
present and those DLs exceed the detected values. One can perform GOF tests on detected data and
consider/expect that NDs (not the DLs) also follow the same distribution of detected data. For data sets
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with NDs, ProUCL has GOF tests for normal, lognormal, and gamma distributions which are also
supplemented with graphical Q-Q plots. GOF tests in ProUCL include: 1) exclude all NDs; 2) replace
NDs by their DL/2s; and 3) ROS methods. In the environmental literature (Helsel 2005, 2012), some
other graphs such as censored probability plots have also been described. However, the usefulness of
those graphs in the computation of decision making statistics is not clear. Some practitioners have
criticized that ProUCL does not offer censored probability plots, therefore, even though those graphs do
not provide additional useful information, ProUCL 5.1 now offers those graphs as well.
Formally, let x\, X2, ..., xn (including k NDs and (n-k) detected measurements) represent a random sample
of n observations obtained from a population under investigation (e.g., background area, or an area of
concern [AOC]). Out of the n observations, k: l
-------
Formally, let x\,x2, ...,xn represent « data values of a left-censored data set. Let p,KM and a^ represent
KM estimates of the mean and variance based upon such a data set with NDs. Let x[ denote the smallest xt. Then
Fx = \, x x j
] ' wlth Xo= ° t4'
Using the PLE (or KM) method, an estimate of the SE of the mean is given by the following equation.
Where k = number of ND observations, and
7=1
The KM variance is computed as follows:
(4'3)
U, \ in f = KM mean of the data, x
' (x)—KM
JU, 2\_f't/r = KM mean of the square of the data, x (second raw moment)
In addition to the KM mean, ProUCL computes both the SE of the mean given by (4-2) and the variance
given by (4-3). The SE is used to estimate EPCs (e.g., UCLs) whereas the variance is used to compute
BTV estimates (e.g., UTLs, USLs). The KM method in ProUCL can be used directly on left-censored
environmental data sets without requiring any flipping of data and back flipping of the KM estimates and
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other statistics (e.g., flipping LCL to compute a UCL) which may be burdensome for most users and
practitioners.
Note: Decision making statistics (e.g., UPLs and UTLs) used in background evaluations projects require
good estimates of the population standard deviation, sd. The decision statistics (e.g., UTLs) obtained
using the direct estimate of sd (Equation 4-3) and an indirect "back door" estimate of sd (Helsel 2012) can
differ significantly, especially for skewed data sets. An example illustrating this issue is described as
follows.
Example 4-1 (Oahu Data Set): Consider the moderately skewed well-cited Oahu data set (Helsel 2012).
A direct KM estimate of the sd obtained using equation (4-3) is o= 0.713; and an indirect KM estimate of
sd = sqrt (24)*SE = 4.899 * 0.165 = 0.807 (Helsel 2012, p 87). A UTL95-95 (direct) = 2.595 and a
UTL95-95 (based upon indirect estimate ofsd) = 2.812. The discrepancy between the two estimates ofsd
and upper limits (e.g., UTL95-95) computed using the two estimates increases with skewness.
Cautionary notes for NADA (2013) in R Users: It is well known that the KM method yields a good (in
terms of bias) estimate of the population mean (Singh, Maichle, and Lee 2006). However, the use of KM
estimates in the Student's t-statistic based UCL equation or percentile bootstrap method as included in
NADA packages do not guarantee that those UCLs will provide the desired (e.g., 0.95) coverage for the
population mean in all situations. Specifically, it is highly likely that for moderately skewed to highly
skewed data sets (determined using detected values) the Student's t-statistic or percentile bootstrap
method based UCLs computed using KM estimates will fail to provide the desired coverage to the
population mean, as these methods do not account for skewness. Several UCL (and other limits)
computation methods based upon KM estimates which adjust for data skewness are available in ProUCL
5.0 and ProUCL 5.1; those methods were not available in ProUCL 4.1.
4.5 Regression on Order Statistics (ROS) Methods
In this guidance document and in ProUCL software, LROS represents the ROS (also known as robust
ROS) method for a lognormal distribution and GROS represents the ROS method for a gamma
distribution. The ROS methods impute NDs based upon a hypothesized distribution such as a gamma or
a lognormal distribution. The "Stats/Sample Sizes" menu option of ProUCL 5.1 can be used to impute
and store imputed NDs along with the original detected values in additional columns generated by
ProUCL. ProUCL assigns self-explanatory titles for those generated columns. It is a good idea to store
the imputed values to determine the validity of the imputed NDs and assess the distribution of the
complete data set consisting of detects and imputed NDs. As a researcher, one may want to have access to
imputed NDs to be used by other methods such as regression analysis and PCA. Moreover, one cannot
easily perform multivariate methods on data sets with NDs; and the availability of imputed NDs makes it
possible for researchers to use multivariate methods on data sets with NDs. The developers believe that
statistical methods to evaluate data sets with NDs require further investigation and research. Providing the
imputed values along with the detected values may be helpful to practitioners conducting research in this
area. For data sets with NDs, ProUCL 5.0/ProUCL 5.1 also performs GOF tests on data sets obtained
using the LROS and GROS methods. The ROS methods yield a data set of size n with (n-k) original
detected observations and k imputed NDs. The full data set of size n thus obtained can be used to compute
the various summary statistics, and to estimate the EPCs and BTVs using methods described in Chapters
2 and 3 of this technical guidance document.
In a ROS method, the distribution (e.g., gamma, lognormal) of the (n-k) detected observations is assessed
first; and assuming that the £ND observations, x\, X2, ..., Xk follow the same distribution (e.g., gamma or a
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lognormal distribution when used on logged data) of the (n-k) detected observations, the NDs are imputed
using an OLS regression line obtained using the (n-k) pairs: (ordered detects, hypothesized quantiles).
Earlier versions of ProUCL software also included the normal ROS (NROS) method for computing the
various upper limits. The use of NROS on environmental data sets (with positive values) tends to yield
unfeasible and negative imputed ND values; and the use of negative imputed NDs yields biased and
incorrect results (e.g., UCL, UTLs). Therefore, the NROS method is no longer available in the
UCLs/EPCs and Upper Limits/BTVs modules of ProUCL version 5.0 and ProUCL 5.1. Instead, when
detected data follow a normal distribution, the use of KM estimates in normal equations is suggested for
computing the upper limits as described in Chapters 2 and 3.
4.5.1 Computation of the Plotting Positions (Percentiles) and Quantiles
Before computing the n hypothesized (lognormal, gamma) quantiles, q®; i:=k+l, k+2,...,n, and q^dif, i: =
1, 2, ..., k, the plotting positions (also known as percentiles) need to be computed for the n observations
with £NDs and (n-k) detected values. There are several methods available in the literature (Blom 1958;
Barnett, 1976; Singh and Nocerino, 1995, Johnson and Wichern, 2002) to compute the plotting positions
(percentiles). Note that plotting positions for the three ROS methods: LROS, GROS, and NROS are the
same. For a full data set of size n, the most commonly used plotting position for the ith observation
(ordered) is given by (/' - 3/s) / (« + %) or (i - 'Aj/n; i: =1,2,... ,n. These plotting positions are routinely used
to generate Q-Q plots based upon full uncensored data sets (Singh 1993; Singh and Nocerino 1995;
ProUCL 3.0 and higher versions). For the single DL case (with all observations below the DL reported as
NDs), ProUCL uses Blom's percentiles, (/' - %) / (n + %) for normal and lognormal distributions, and uses
empirical percentiles given by (i - %)/n for a gamma distribution. Specifically, for normal and lognormal
distributions, once the plotting positions have been obtained, the n normal quantiles, g® are computed
using the probability statement: P(Z < g®) = (/' - 3/8) /(« + %),/: = 1, 2, ...,«, where Z represents a
standard normal variate (SNV). The gamma quantiles are computed using the probability statement: P(X
-
-------
4.5.2 Computing OLS Regression Line to Impute NDs
An ordinary least squares (OLS) regression model is obtained by fitting a linear straight line to the (n-k)
ordered (in ascending order) detected values, x^ (perhaps after a suitable transformation), and the (n-k)
hypothesized (e.g., normal, gamma) quantiles, qpy, i:=k+l, k+2,...,n, associated with those (n-k) detected
ordered observations. The hypothesized quantiles are obtained for all of the n data values by using the
hypothesized distribution for the (n-k) detected observations. The quantiles associated with (n-k) detected
values are denoted by q^y, i:=k+l, k+2,...,n, and the k quantiles associated with ND observations are
denoted by q^y, i: = 1, 2, ..., k..
An OLS regression line is obtained first by using the (n - k) pairs, (q®, x®); /':= k + 1, k + 2, ...,«, where
Xft) are the (n-k) detected values arranged in ascending order. The OLS regression line fitted to the (n - k)
pairs (qft), Xft)); /':= k + \,k + 2, ...,n corresponding to the detected values is given by:
= a
i:= k + 1, k + 2,
n.
(4-4)
Table 4-1. Plotting Positions, Gamma and Lognormal (Normal) Quantiles (Q)
Pyrene
28
31
32
34
35
35
40
47
48
58
59
63
64
64
67
67
67
72
73
84
86
86
87
34
98
100
103
103
D_pyrene
a
1
1
1
a
a
1
1
1
a
1
1
1
1
1
1
1
1
1
1
a
1
1
1
1
1
1
1
Percentiles
0.01818162
0.063635671
0.090908101
§.118180531
§.048484321
0.096968641
Q. 163634582
0.181816202
0.199997822
0.109089721
0.238013937
0.257848432
0.277682927
0.297517422
0.317351916
0.337186411
0.357020906
0.376855401
0.396689895
0.41652439
0.218179443
0.455406297
Q.4744537Q8
0.493.50112
0.512548532
0.531595943
0.550643355
0.569690767
Gamma-Q(Hat)
4.339031664
14.41837445
20.46764835
26.60142257
11.07571144
21.82204786
37.09766155
41.41222876
45.80230241
24.54516765
55.25114882
60.33991965
65.54790077
70.88334133
76.3549565
81.97203137
87.74452837
93.68320235
99.79972782
1Q6.1G68416
50,27369978
119.0785779
125.7564975
132.6689087
139.8343146
147.2733809
155.009301
163.0682381
Normal-Q
-2.0928422
-1 .5249509
-1.3351838
-1.1841314
-1.6597307
-1.2990194
-0.9796292
•0.3084654
-0.841629
-1.2313836
•0.7127057
-0.6499928
-0.5897387
-0.5315541
-0.4751166
-04201.542
-0.3664333
•0.3137502
•0.2619243
-0.210793
•0.7783565
-0.1120136
•0.0640789
-0.016291
0.0314597
0.0792823
0.1272869
0.1755869
105
107
110
111
117
113
119
122
122
132
133
133
138
163
163
163
163
174
187
190
222
23S
273
289
306
333
453
29S2
1
1
1
1
Q
1
1
0
1
1
1
1
1
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
0.588738178
0.60778553
0.626833001
0.645880413
0.332463312
8.67836071
0.631793595
0.35261324
0.721551168
0.737875855
0.754200542
0.770525229
0.786849916
0.200733651
0.401587302
0.602380952
0.812301587
0.410714286
0.837662338
0.853896104
0.87012387
0.886363636
0.902537403
0.918831169
0.935064935
0.951238701
0.367532468
0.383766234
171.4798683
180.2780505
189.501663
199.1956546
80.62092045
216.3648821
224.8326032
86.44778239
243.5316694
254.6417323
266.4543419
279.06604.34
292.5350126
45.33627612
101.3388027
177.7401595
315.8763629
104.2387681
342.4104955
361.6446136
383.1239177
407.44B911
435.4987369
468.636129
509.1426864
561.2340357
634.6836503
759.9003157
0.2243003
0.2735521
0.323477
0.374222
-0.4331137
0.4631197
0.5009408
-0.3782748
0.5874557
0.63S8105
0.687768
0.7405777
0.7355388
-0.8387838
-0.2492407
0.2535148
0.8864096
-0.225708
0.9848957
1.0532907
1.1270053
1.2074141
1.2364944
1.3972525
1.5146142
1.6575784
1.8457049
2.1386067
When ROS is used on transformed data (e.g., log-transformed), then ordered values, x© ; /': = k + 1, k + 2,
..., n represent ordered detected data in that transformed scale (e.g., log-scale, Box-Cox (BC)-type
133
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transformation). Equation (4-4) is then used to impute or estimate the ND values. Specifically, for
quantile, q(ndi) corresponding to the ith ND, the imputed ND is given by X(ndi) = a + bq(ndi) ,• i:=l,2,...k.
When there is only a single DL and all values lying below the DL represent ND observations, then the
quantiles corresponding to those ND values typically are lower than the quantiles associated with the
detected observations. However, when there are multiple DLs, and when some of those DLs exceed
detected values, then quantiles, #(ndi) corresponding to some of those ND values might become greater
than the quantiles, q® associated with some of the detected values.
4.5.2.1 Influence of Outliers on Regression Estimates and Imputed NDs
Like all other statistics, it is well-known (Rousseeuw and Leroy 1987; Singh and Nocerino 1995; Singh
and Nocerino 2002) that presence of outliers (detects) also distorts the regression estimates of slope and
intercept which are used to impute NDs based upon a ROS method. It is noted that for skewed data sets
with outliers, the imputed values computed using the ROS method on raw data in the original scale
become negative (e.g., GROS method). Therefore, inclusion of outliers (e.g., impacted locations) can
yield distorted statistics and upper limits computed using the ROS method. This issue is also discussed
later in this chapter.
Note: It is noted that a linear regression line can be obtained even when only two detected observations
are available. Therefore, methods (e.g., ROS) discussed here and incorporated in ProUCL can be used on
data sets with 2 or more detected observations. However, to obtain a reliable OLS model (slope and
intercept) and imputed NDs for computation of defensible upper limits, enough (> 4-6 as a rule of thumb,
more are desirable) detected observations should be made available.
4.5.3 ROS Method for Lognormal Distribution
Let Org stand for the data in the original unit and Ln stand for the data in the natural logarithmic unit. The
LROS method may be used when the log-transformed detected data follow a lognormal distribution. For
the LROS method, the OLS model given by (4-4) is obtained using the log-transformed detected data and
the corresponding normal quantiles. Using the OLS linear model on log-transformed, detected
observations, the NDs in log-transformed scale are imputed corresponding to the k normal quantiles, #(ndi)
associated with the ND observations which are back-transformed in original, Org scale by exponentiation.
4.5.3.1 Fully Parametric Log ROS Method
Once the k NDs have been imputed, the sample mean and sd can be computed using the back-
transformation formula (El Shaarawi, 1989) given by equation (4-5) below. This method is called the
fully parametric method (Helsel, 2005). The mean, jj.Ln, and sd, oLn, are computed in log-scale using a
full data set obtained by combining the (n - k) detected log-transformed data values and the k imputed ND
(in log scale) values. Assuming lognormality, El-Shaarawi (1989) suggested estimating ju and erby back-
transformation using the following equations as one of the several ways of computing these estimates.
The estimates given by equation (4-5) are neither unbiased nor have minimum variance (Gilbert 1987).
Therefore, it is recommended to avoid the use of this version of ROS method on log-transformed data to
compute UCL95s and other statistics. This method is not available in the ProUCL software.
n 12), and a20rg = ^Org (e^(
-------
4.5.3.2 Robust ROS Method on Log-Transformed Data
The robust ROS method is performed on log-transformed data as described above. In the robust ROS
method, ND observations are first imputed in the log-scale, based upon a linear ROS model fitted to the
log-transformed detects and normal quantiles. The imputed NDs are transformed back in the original
scale by exponentiation. The process of using the ROS method based upon a lognormal distribution and
imputing NDs by exponentiation does not yield negative estimates for ND values; perhaps that is why it
got the name robust ROS (or LROS in ProUCL). This process yields a full data set of size n, and methods
described in Chapters 2 and 3 can be used to compute the decision statistics of interest including estimates
of EPCs and BTVs. If the detected observations follow a lognormal, the data set consisting of detects and
imputed NDs also follow a lognormal distribution. As expected, the process of imputing NDs using the
LROS method does not reduce the skewness of the data set and therefore, appropriate methods need to be
used to compute upper limits (Chapters 2 and 3) which provide specified (e.g., 0.95) coverage by
adjusting for skewness.
Note: The use of the robust ROS method has become quite popular. Helsel (2012) suggests the use of a
classical t-statistic or a percentile bootstrap method to compute a UCL of the mean based upon the full
data set obtained using the LROS method. These methods are also available in his NADA packages.
However, these methods do not adjust for skewness and for moderately skewed to highly skewed data
sets, and UCLs based upon these two methods fail to provide the specified coverage to the population
mean. For skewed data sets, methods described in Chapter 2 can be used on LROS data sets to compute
UCLs of the mean.
Example 4-3 (Oahu Data Set). Consider the Oahu arsenic data set of size 24 with 13 NDs. The detected
data set of size 11 follows a lognormal distribution as shown in Figure 4-1; this graph simply represents a
Q-Q plot of detects and does not account for NDs when computing quantiles. The censored probability
plot (new in ProUCL 5.1) is shown in Figure 4-2; its details can be found in the literature (Chapter 15 of
Unified Guidance, EPA 2009). A censored probability plot is also based upon detected observations and
it computes quantiles by accounting for NDs. The LROS data set consisting of 11 detects and 13 imputed
NDs also follows a lognormal distribution as shown in Figure 4-3. Summary statistics and LROS UCLs
are summarized in Table 4-2.
135
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Lognormal Q-Q Plot (Statistics using Detected Data) for Arsenic
Theoretical Quantiles (Standard Normal)
Arsenic
Tol.alNumberoIData = 24
Number or NDs=13
Ma«DL=2
N =11
Percent MDs = 542
Mean- -0.0255
Sd = 0.634
Slope = 0.631
Intercept 0.0255
Correlation, Ft = 0.333
Shapiro Wifc Test
E xact T esl H talisftr - 0.860
Critical Value(0.05,= 0.850
Data Appear Lognormal
Appro* TeslValue = a875
p Value^O 0882
Distribution Test Suspect
• Best Fit Line
Figure 4-1
Quantiles
Lognormal GOF Test on Detected Oahu Data Set - Does not Account for NDs to Compute
Lognormal Q-Q Plot for Arsenic
Statistics Displayed using Censored Quantiles and Detected Data
Theoretical Quantiles (Standard Normal)
Only Quantiles for Detects Displayed
Total Number of Data = 24
Number or Detects = 11
mum Censored Quamtile = 1.347
:enlNDs = 542
DL = 2
nx of D elects = -0.0255
tdvx of Detects = U634
lope = 0.559
ntercept =• tl 13T>
relation. R -- 0.97G
hapir o-Wik Ted
rilicalValue(0.05)-0.850
ata Appear Log normal
Appro* Test Value-0.07T.
p-Value-0.0882
Distribution Test Suspect
• 8est Fit Line
Figure 4-2. Lognormal Censored Probability Plot (Oahu Data) - Uses Only Detects but Accounts for NDs
to Compute Quantiles
Note: The two graphs displayed in Figures 4-1 and 4-2 provide similar information about data
distributions, as GOF tests simply use detected values (and not quantiles). Both graphs are okay without
any preference.
136
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Lognormal Q-Q Plot for Arsenic
Statistics using ROS Lognormal Imputed Estimates
NumbeiofNDs = 13
Mean =- -0.209
Set = 0.571
Slope -0.568
Intercept = 4.209
CoFielalion, R -= 0.363
Shapiro Wik Test
Ex
-------
The data set is moderately skewed with sd of logged detects equal to 0.694. All methods tend to yield
comparable results. One may want to use a 95% BCA bootstrap UCL or a bootstrap-t UCL to estimate the
EPC. However, the detected data follow a gamma distribution, therefore ProUCL recommends gamma
UCLs as shown in the following section.
4. 5. 3. 3 Gamma ROS Method
Many positively skewed data sets tend to follow a lognormal as well as a gamma distribution. Singh,
Singh, and laci (2002) noted that the gamma distribution is better suited to model positively skewed
environmental data sets. When a moderately skewed to highly skewed data set (uncensored data set or
detected values in a left-censored data set) follows a gamma, as well as, a lognormal distribution, the use
of a gamma distribution tends to result in more stable and realistic estimates of EPCs and BTVs
(Examples 2-2 and 3-2, Chapters 2 and 3). Furthermore, when using a gamma distribution to compute
decision statistics such as a UCL of the mean, one does not have to transform the data and back-transform
the resulting UCL into the original scale.
Let X(k+i)
-------
In the above equation, %^ represents a chi-square random variable with 2k degrees of freedom (df), and
Pi are the plotting positions (percentiles) obtained using the process described above. The process of
computing plotting positions, pt, i:=l,2,...,n, for left-censored data sets with multiple DLs has been
incorporated in ProUCL. The inverse chi-square algorithm function (AS91) from Best and Roberts (1975)
has been used to compute the inverse chi-square percentage points, zoi, as given by the above equations.
Using the OLS line (4-4) fitted to the (n - k) detected pairs, one can impute the k NDs resulting in a full
data set of size n = k + (n - k).
Notes about GROS for smaller values of k (e.g.. <): In the ProUCL 5.0 Technical Guide (and its earlier
versions) and ProUCL software, a suggestion was made that GROS may not be used when the shape
parameter, k is less than 0.1 or less than 0.5. However, during late 2014, some users pointed out that k
should be higher. Therefore, the latest version of ProUCL 5.1 now suggests that GROS may not be used
for values of k < 1.0. It should be pointed out that the GROS algorithm incorporated in ProUCL works
well for values of k > 2.
The GROS method incorporated in ProUCL does not appear to work well for smaller values of k or its
MLE estimate, k (e.g., <\). The algorithm used to compute gamma quantiles is not efficient enough and
does not perform well for smaller values of k. The developers thus far have not found time to look into
this issue. In January 2015, the developers of ProUCL requested the statistical community (via the
American Statistical Association's section on environmental statistics and/or personal communication) to
provide code/algorithms which may be used to improve the computation of gamma quantiles for smaller
values of k.
For now, GROS may not be used when the data set with detected observations (used to compute OLS
regression line) consists of outliers and/or is highly skewed (e.g., estimated values of k are small such as
<=1.0). When the estimated value (MLE) of the shape parameter, k, based upon detected data is small (<=
1.0), or when the data set consists of many tied NDs at multiple DLs with a high percentage of NDs
(>50%), the GROS tends to not perform well and often yields negative imputed NDs, due to outliers
distorting the OLS regression. Since environmental concentration data are non-negative, one needs to
replace the imputed negative values by a small positive value such as 0.1, 0.001. In ProUCL, negative
imputed values are replaced by 0.01. The use of such imputed values tends to yield inflated values of sd,
UCLs, andBTVestimates (e.g., UPLs, UTLs).
Preferred Method: Alternatively, when detected data follow a gamma distribution, one can use KM
estimates (described above) in gamma distribution based equations to compute UCLs (and other limits)
which account for data skewness, unlike KM estimates when used in normal UCL equations. This hybrid
gamma-KM method for computing upper limits is available in ProUCL 5.0/ProUCL 5.1. The details are
provided in Section 4.6. The hybrid KM-gamma method yields reasonable UCLs and accounts for NDs as
well as data skewness as demonstrated in Example 4-4.
/v*
Note: It is noted that when k >1, UCLs based upon the GROS method and gamma UCLs computed
using KM estimates tend to yield comparable UCLs from practical a point of view. This can also be seen
in Example 4-4 below.
139
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Example 4-4 (Oahu Data Set Continued): The detected data set of size 11 follows a gamma
distribution as shown in Figure 4-4. The GROS data consisting of 11 detects and 13 imputed NDs also
follows a gamma distribution as shown in Figure 4-5. Summary statistics and GROS UCLs are
summarized in Table 4-3 following Figure 4-5. Since the data set is only mildly skewed all methods
(GROS and Hybrid KM-Gamma) yield comparable results.
Gamma Q-Q Plot (Statistics using Detected Data) for Arsenic
Theoretical Quantiles of Gamma Distribution
ToialNumberofData-24
N =11
Percent NDs = 54%
Mean = 1.2X4
k star -1.701 8
theta star = 0.7255
Slope = 1.0345
Intscepl = -0 01 77
Correlation, R = Q9G38
Test Statistic = 0.254
Critical ValueED. 05] = 0.258
Data appear Gamma Distributed
Distribution Test Suspect
• Best Fit Line
Figure 4-4. Gamma GOF Test on Detected Concentrations of the Oahu Data Set
Gamma Q-Q Plot for Arsenic
Statistics using ROS Gamma Imputed Estimates
Theoretical Quantiles of Gamma Distribution
Imputed NDs Displayed in smaller font
Number o! NDs = 13
k star-1.8399
theta star = 0.5193
Slope = 1.0794
Correlation, R = 0.9752
Anderson-Darling Test
Tsst Statistic = 0.480
Critical Value(O.G5] = 0.755
Data appear Gamma Distributed
• Best Fit Line
Figure 4-5. Gamma GOF Test on GROS Data Obtained Using the Oahu Data Set
140
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Table 4-3. Summary Statistics and UCL95 Based upon Gamma ROS data
Minimum 0.119
Maximum 3.2
SD 0.758
k hat (MLE) 2.071
Theta hat (MLE) 0.461
nu hat {MLE) 99.41
MLE Mean (bias corrected) 0.956
MLE Sd (bias corrected) 3.704
.Approximate Chi Square Value (83.32, a) 67.S5
95% Gamma Approximate UCL (use when n>=50) 1.247
Mean 0.956
Median 0.7
CV 0.793
fc star {bias corrected M LE) 1.84
Theta star {bias corrected MLE) 0.519
nu star $>ias corrected) 88.32
.Adjusted Level of Significance $) 0.0392
Adjusted Chi Square Value (88.32, PS 66.38
35% Gamma Adjusted UCL Xise when n=:5C'; 1.271
Kaplan Meier (KM) Statistics Using Nomral Critical Values
Mean 0.949 Standard Error of Mean 0.165
0.713 95% KM (BCA) UCL 1.192
1.231 95% KM (Percentile Bootstrap) UCL 1.219
1.22 95% KM Bootstrap! UCL 1.374
1.443 95% KM Chebyshev UCL 1.6S7
1.977 99% KM Chebyshev UCL 2.588
SD
95% KM K UCL
95% KM (z) UCL
90% KM Chebyshev UCL
97.5% KM Chebyshev UCL
Gamma Kaplan-Meier (KM) Statistics
k hat (KM) 1.771 nuhat(KM) 85.02
.Approximate Chi Square Value (85.02, a) 64.77 Adjusted Chi Square Value (85.02, (5) S3.53
95% Gamma Approximate KM-UCL (use when ns=50) 1.246 35% Gamma Adjusted KM-UCL Xise when n=:5C) 1.27
Suggested UCL to Use
35% KM J; UCL 1.231
; Adjusted Gamma KM-UCL 1.27
95% GROS Adjusted Gamma UCL
1.271
ProUCL suggests using GROS UCL of 1.27.
4.6 A Hybrid KM Estimates and Distribution of Detected Observations Based
Approach to Compute Upper Limits for Skewed Data Sets - New in ProUCL
5.0/ ProUCL 5.1
The KM method yields good estimates of the population mean and sd. Since it is hard to verify and justify
the distribution of an entire left-censored data set consisting of detects and NDs with multiple DLs, it is
suggested that the KM method be used to compute estimates of the mean, sd, and standard error of the
mean. Depending upon the distribution and skewness of detected observations, one can use KM estimates
in parametric upper limit computation formulae to compute upper limits including UCLs, UPLs, UTLs,
and USLs. The use of this hybrid approach will yield more appropriate skewness adjusted upper limits
than those obtained using KM estimates in normal distribution based UCL and UTL equations.
Depending upon the distribution of detected data, ProUCLS.l (and its earlier version ProUCL 5.0)
computes upper limits using KM estimates in parametric (normal, lognormal, and gamma) equations to
compute the various upper limits. The use of this hybrid approach has also been suggested in Chapter 15
of EPA (2009) to compute upper limits using KM estimates in the lognormal distribution based equations
to compute the various upper limits.
ProUCL 5.1 and its earlier versions compute a 95% UCL of the mean based upon the KM method using:
1) the standard normal critical value, za and Student's t-critical value, ta,(n-i)', 2) bootstrap methods
including the percentile bootstrap method, the bias-corrected accelerated (BCA) bootstrap method, and
141
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bootstrap-t method, and 3) the Chebyshev inequality. Additionally, when detected observations of a left-
censored data set follow a gamma or a lognormal distribution, ProUCL 5.1 also computes KM UCLs and
other upper limits using a lognormal or a gamma distribution. The use of these methods yields skewness
adjusted upper limits. For a gamma distributed detected data, UCLs based upon the GROS and gamma
distribution on KM estimates are generally in good agreement unless the data set is highly skewed (with
estimated values of shape parameter, k50%)
with NDs tied at multiple DLs. The various UCL computation formulae based upon KM estimates and
incorporated in ProUCL 5.0/ProUCL 5.1 are described as follows.
4. 6. 1 Detected Data Set Follows a Normal Distribution
Based upon Student's t-statistic, a 95% UCL of the mean based upon the KM estimates is as follows:
KM UCL95 (t)=fi + t^^ ^ (4-8)
The above KM UCL (t) represents a good estimate of the EPC when detected data are normally
distributed or mildly skewed. However, KM UCLs, computed using a normal or t-critical value, do not
account for data skewness. The various bootstrap methods for left-censored data described in Section 4.7
can also be used on KM estimates to compute UCLs of the mean.
4. 6. 2 Detected Data Set Follows a Gamma Distribution
For highly skewed gamma distributed left-censored data with a large percentage of NDs and several NDs
tied at multiple RLs, the GROS method tends to yield impractical, negative imputed values for NDs. It is
also well known that the OLS estimates get distorted by outliers, therefore, GROS estimates and upper
limits also get distorted when outliers are present in a data set.
In order to avoid these situations, one can use the gamma distribution on KM estimates to compute the
various upper limits provided the detected data follow a gamma distribution. Using the properties of the
gamma distribution, an estimate of the shape parameter, k, is computed based upon a KM mean and a KM
variance. The mean and variance of a gamma distribution are given as follows:
Mean=£*0, and
Variance = k*(f
Substituting a KM mean, fiKM , and a KM variance, cr^ , in the above equations, an estimate of the
shape parameter, k, is computed by using the following equation:
Using /^ , cr^- , n, and kin equations (2-34) and (2-35), gamma distribution based approximate and
adjusted UCLs of the mean can be computed. Similarly, for gamma distributed left-censored data sets
with detected observations following a gamma distribution, KM mean and KM variance estimates can be
used to compute gamma distribution based upper limits described in Chapter 3. ProUCL 5.0/ProUCL 5.1
computes gamma distribution and KM estimates based UCLs and upper limits to estimate BTVs when
detected data follow a gamma distribution.
142
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Notes: It should be noted that the KM method does not require concentration data to be positive. In radio
chemistry, the DLs (or minimum detectable concentration [MDC]) for the various radionuclides are often
reported as negative values. Statistical models such as a gamma distribution cannot be used on data sets
consisting of negative values. However, the hybrid gamma-KM method described above can be used on
radionuclides data provided detected activities are all positive and follow a gamma distribution. One can
compute KM estimates using the entire data sets consisting of negative NDs and detected positive values.
Those KM estimates can be used to compute gamma UCLs described above provided fiKM >0.
4.6.3 Detected Data Set Follows a Lognormal Distribution
The EPA RCRA (2009) guidance document suggests computing KM estimates on logged data and
computing a lognormal H-UCL based upon the H-statistic. ProUCL computes lognormal and KM
estimates based UCLs and upper limits to estimate BTVs when detected data follow a lognormal
distribution. Like uncensored lognormally distributed data sets, for moderately skewed to highly skewed
left-censored data sets, the use of a lognormal distribution on KM estimates tends to yield unrealistically
high values of the various decision statistics; especially when the data sets are of sizes less than 30 to 50.
Example 4-5 (Oahu Data Set Continued): It was noted earlier that the detected Oahu data set follows a
gamma as well as a lognormal distribution. The hybrid normal, lognormal and gamma UCLs obtained
using the KM estimates are summarized in Table 4-4 as follows.
The hybrid Gamma UCL is 1.27, close to the UCL obtained using the GROS method of 1.271 (Example
4-4). The H-UCL as suggested in EPA (2009) is 1.155 which appears to be a little lower than the other
LROS BCA bootstrap UCL of 1.308 (Table 4-2).
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Table 4-4. UCL95 Based on Hybrid KM Method and Normal, Lognormal and Gamma Distribution
Kaplan-Meier (KM) Statistics using Normal Critical Values and either Nonparametric UCLs
Mean 0.949 Standard Error of Mean 0.165
§713 95% KM (BCA) UCL 1.228
1 .231 95% KM (Percentile Bootstrap) UCL 1 .21
1 .22 95% KM Bootstrap t UCL 1 .363
1 .443 95% KM Chebyshev UCL 1 .667
1 .977 99% KM Chebyshev UCL 2.5&B
SD
35% KM j; UCL
95% KM (z) UCL
30% KM Chebyshev UCL
97.5% KM Chebyshev UCL
Gamma GOF Tests on Detected Observations Only
A-D Test Statistic 0.787 Anderson-Darling GOF Test
5% A-D Critical Value 0.733 Detected Data Not Gamma Distributed at 5% Significance Level
K-S Test Statistic 0.254 KDlmogrov-Smirnoff GOF
5% K-S Critical Value 0.258 Detected data appear Gamma Distributed at 5% Significance Level
Detected data follow Appr Gamma Distribution at 5% Significance Level
Gamma Statistics on Detected Data Only
khat(MLE) 2.257
Theta hat (MLE) Q.548
nu hat (MLE) 49.65
MLE Mean (bias corrected) 1.236
k star (bias corrected MLE} 1.702
Theta star (bias corrected MLE} 0.727
nu star (bias corrected) 37.44
MLE Sd (bias corrected) 0.948
Gamma Kaplan-Meier (KM) Statistics
k hat {KM} 1.771 nu hat (KM) 85.02
Approximate Chi Square Value (85.02. a) 54.77 Adjusted Chi Square Value (85.02. P) 63.53
95% Gamma Approximate KM-UCL (use when n>=50) 1.24S 35% Gamma .Adjusted KM-UCL (use when n«5CJ 1.27
UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognonnally Distributed
KM Mean fogged) -0.236 35% H-UCL (KM -Log) 1 . 1 55
KM SD (logged) 0.547 95% Critical H Value (KM-Log) 2.023
KM Standard Error of Mean fogged) 0.137
Example 4-6. A real data set of size 55 with 18.8% NDs is considered next. The data set can be
downloaded from the ProUCL website. The minimum detected value is 5.2 and the largest detected value
is 79000, sd of detected logged data is 2.79 suggesting that the data set is highly skewed. The detected
data follow a gamma as well as a lognormal distribution as shown in Figures 4-6 and 4-7. It is noted that
GROS data set with imputed values follows a gamma distribution and LROS data set with imputed values
follows a lognormal distribution (results not included).
144
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Lognormal Q-Q Plot (Statistics using Detected Data) for A-DL
A-DL
Toial Number of Dat.
Number of NDs = 10
Max DL-124
18X
Percent ND s
Mean-7.031
Sd = 2.738
Slope = 2.789
Intercept-7 031
Correlation, R = 0.931
LJIiefors Test
Test Statistic = 0.104
CriticalVa!ue(& 051-0.132
Data Appear Lognormal
Theoretical Quantiles (Standard Normal)
Figure 4-6. Lognormal GOF Test on Detected TRS Data Set
Gamma Q-Q Plot (Statistics using Detected Data) for A-DL
Theoretical Quantiles of Gamma Distribution
AOL
Total Number of Data = 55
Number otNDs = 1Q
N = 45
Percent NDs = m
Mean-10556.1867
k star = 0 301S
theta star-34979.7238
Slope = 1 0745
Intercept--535.7060
Data appear Gam
• Best Fit Line
Figure 4-7. Gamma GOF Test on Detected TRS Data Set
145
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Table 4-5. Statistics and UCL95s Obtained Using Gamma and Lognormal Distributions
ArDL
Total Number of Observations
Number of Detects
Number of Distinct Detects
Minimum Detect
Maximum Detect 79000
Variance Detects 3.954E+S
Mean Detects 10556
Median Detects 1940
Skewness Detects 2.632
Mean of Logged Detects 7.831
General Satisfies
55
45
45
5.2
Number of Distinct Observations
Number of Non-Detects
Number of Distinct Non-Detects
Minimum Non-Detect
Maximum Non-Detect
Percent Non-Detects
SD Detects
CV Detects
KLirtosis Detects
SD of Logged Detects
53
10
8
3.8
124
18.18^;
13886
1.884
6.4%
2.7SS
Kaplan Meier {KM) Statistics using Normal Critical Values and other Nonparametnc UCLs
Mean 8638 Standard Error of Mean 2488
SD 18246 95% KM (BCA) UCL 13396
95% KM « UCL 12802 95% KM (Percentile Bootstrap} UCL 12792
35% KM (z) UCL 12731 35% KM Bootstrap t UCL 14509
30% KM Chebyshev UCL 16102 95% KM Chebyshev UCL 1S4S3
37.5% KM Chebyshev UCL 24176 93% KM Chebyshev UCL 33334
Gamma GOF Tests on Detected Observations Only
A-D Test Statistic 0.591 Anderson-Darling GOF Test
5% A-D Critical Value 0.86 Detected data appear Gamma Distributed at 5% Significance Level
K-S Test Statistic 0.115 Kolmogrov Smirnoff GOF
5% K-S Critical Value 0.143 Detected data appear Gamma Distributed at 5% Significance Level
Detected data appear Gamma Distributed at 5% Significance Level
Gamma Statistics on Detected Data Only
k hat (NILE) 0.307
Theta hat (MLE) 34333
nu hat (MLE) 27.67
MLE Mean tbias corrected) 10556
k star (bias corrected MLE} 0.302
Theta star (bias corrected MLE) 34980
nu star (bias corrected} 27.16
MLESd{bias corrected) 19216
Gamma Kaplan-Meier (KM) Statistics
k hat (KM) 0.224 nu hat (KM) 24.66
.Approximate Chi Square Value (24.66, a) 14.35 Adjusted Chi Square Value (24.66, P) 14.14
95% Gamma Approximate KM-UCL (use when ni=5D} 14844 95% Gamma Adjusted KM-UCL (use when n<50) 15066
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Table 4-5 (continued). Statistics and UCL95s Obtained Using Gamma and Lognormal Distributions
Gamma ROS Statistics using Imputed Non-Deteets
Minimum 0.1
Maximum 79000
SD 18415
khat(MLE) 0.138
Theta hat (MLE) 43697
nuhat(MLE) 21.74
MLE Mean (bias corrected) 8637
Approximate Chi Square Value (21 .S3. a] 12.26
35% Gamma .Approximate UCL fuse when n s=5C} 15426
Mean 8637
Median 588
CV 2.132
k star (bias corrected MLE} 0.139
Theta star (bias corrected MLE} 43402
nu star (bias corrected) 21 .S3
M LE Sd (bias corrected} 19361
Adjusted Level of Significance §} 0.0456
Adjusted Chi Square Value (21.83. p} 12.06
35% Gamma Adjusted UCL (use when n <50} 15675
lognoimal GOF Test on Detected Observations Only
Shapiro Wilk Test Statistic 0.333 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value 0.345 Detected Data Not Lognormal at 5% Significance Level
Ulliefors Test Statistic 0.104 Lilliefors GOF Test
5% Lilliefors Critical Value 0.132 Detected Data appear Lognormal at 5% Significance Level
Detected Data appear Approximate Lognormal at 5% Significance Level
Lognormal ROS Statistics Using Imputed Non-Detects
Mean in Original Scale 8638 Mean in Log Scale 5.383
SD in Original Scale 18414 SD in Log Scale 3.331
35% Percentile Bootstrap UCL 12853
95% Bootstrap t UCL 15032
35"% t UCL {assumes normality of RO S data) 12733
95% BCA Bootstrap UCL 13904
35% H-UCL (Log ROS) 1855231
UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognormally Distributed
KM Mean fogged) 6.03 95% H-UCL (KM -Log) 1173988
KM SD (logged) 3.286 95% Critical H Value (KM-Log) 5.7
KM Standard Error of Mean (logged) 0.449
Nonparametric Distribution Free UCL Statistics
Detected Data appear Gamma Distributed at 5% Significance Level
Suggested UCL to Use
35% KM (Chebyshev; UCL 19483
55% .Approximate Gamma KM-UCL 148^
95 vi G RO S Approximate Gamma UC L 15^26
From the above table, it is noted that the percentile bootstrap method on LROS method as described in
Helsel (2012) yields a lower value of the UCL95 = 12797, which is comparable to a KM (t)-UCL
=12802. The student's t statistic based upper limits (e.g., KM (t)-UCL) do not adjust for data skewness;
the two UCLs, bootstrap LROS UCL and KM(t)-UCL, appear to represent underestimates of the
population mean. As expected, H-UCL on the other hand, resulted in impractically large UCL values
(using both the LROS and KM methods). Based upon the data skewness, ProUCL suggested three UCLs
(e.g., Gamma UCL = 15426) out of several UCL methods available in the literature and incorporated in
ProUCL software.
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4.6.3.1 Issues Associated with the Use ofLognormal distribution to Compute a UCL of
Mean for Data Sets with Nondetects
Some drawbacks associated with the use of the lognormal distribution based UCLs on data sets with NDs
are discussed next.
Example 4-7. Consider the benzene data set (Benzene-H-UCL-RCRA.xls) of size 8 used in Chapter 21 of
the RCRA Unified Guidance document (EPA 2009). The data set consists of one ND value with DL of
0.5 ppb. In the RCRA guidance, the ND value was replaced by 0.5/2=0.25 to compute a lognormal H-
UCL. In this example, lognormal 95% UCLs (H-UCLs) are computed replacing the ND by the DL (0.5)
and also replacing the ND by DL/2=0.25. Normal and lognormal GOF tests using DL/2 for the ND value
are shown in Figures 4-8 and 4-9 as follows.
Benzene-ppb
Normal Q-Q Plot for Benzene-ppb Bnnan»-ppb
Statistics using DL/2 Substitution N^lacfNOs 1
1.1
0.5 05
0.25 • •
M«.zai
Sd-5.353
Slope * 4.032
Intercept -2. 931
: Correlation. R - 0 701
Shapro-Wilk Test
EHactTeslVakie = 0.521
Critical Val(0.05] = 081 8
Data Not Normal
Approx.Te$tVarue = 0.495
p-Value-1.8931E-5
NOTE: Proxy Methods
are not recommended
D Best Fit Line
V
.5 -1.0 -0.5 00 0.5 1.0 1.5
Theoretical Quantiles (Standard Normal)
DL/2 Substituted NDs Displayed in smaller font
Figure 4-8. Normal Q-Q Plot on Benzene Data with ND Replaced by DL/2
From the above Q-Q plot, it is easy to see that observation 16.1 ppb represents an outlier. The Dixon test
on logged data suggests that 2.779 (=ln(16.1)) is an outlier and observation 16.1 is an outlier in the
original scale. The outlier, 2.779 was accommodated by the lognormal distribution resulting in the
conclusion that the data set follows a lognormal distribution (Figure 4-9).
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Lognormal Q-Q Plot for Benzene-ppb
Statistics using DL/2 Substitution
Theoretical Quantiles (Standard Normal)
DL/2 Substituted NDs displayed in smaller font
Number ol NDs - 1
Mean - 0.204
Sd = 1.257
Sbpe-1.266
Intercept - 0.204
Correlate. R = 0.937
Shapiro-WtikTest
E^tTeslValue = 0.396
CrticalVal(0 051 = 0.818
Data Appear Lognormal
Appro*. TeslValue = 0.88
pValue = 0.192
NOTE: Proxy Methods
• Best Fit Line
Figure 4-9. Lognormal Q-Q Plot on Benzene Data with ND Replaced by DL/2
4.6.3.1.1
Impact of Using DL and DL/2 for Nondetects on UCL95 Computations
Lognormal distribution based H-UCLs computed by replacing ND by DL and by DL/2 are respectively
given in Tables 4-6 and 4-7 below.
Table 4-6. Lognormal 95% UCL (H-UCL) - Replacing ND by DL (=0.5)
Lognormal GOF Test
Shapiro Wilk Test Statistic 0.803 Shapiro Wilk Lognormal GOF Test
5% Shapiro Wilk Critical Value D.S18 Data Not Lognormal at 5% Significance Level
Ulliefors Test Statistic 0.273
5% Lilliefors Critical Value 0.313
Lilliefors Lognormal GOF Test
Data appear Lognormal at 5% Significance Level
Data appear Approximate Lognormal at 5% Significance Level
Lognormal Statistics
Minimum of Logged Data -0.693
Maximum of Logged Data 2.779
Assuming Lognormal Distribution
95*1 H-UCL 13.S2
95% Chebyshev (MVU E) UC L 6.496
99% Chebyshev {MVUE} UCL 11.86
Mean of logged Data 0.29
SD of logged Data 1.152
90% Chebyshev (MVU E) UCL 5.191
97.5% Chebyshev (MVU E) UCL 8.306
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Table 4-7. Lognormal 95% UCL (H-UCL) - Replacing ND by DL/2 (=0.25)
Lognoimal GOF Test
Shapiro Wilk Test Statistic 0.896 Shapiro Wilk Lognormal GOF Test
5% Shapiro Wilk Critical Value 0.818 Data appear Lognormal at 5% Significance Level
Ulliefors Test Statistic 0.255 Ljlliefors Lognormal GOF Test
5% Ljlliefors Critica! Value 0.313 Data appear Lognormal at 5% Significance Level
Data appear Lognormal at 5% Significance Level
Lognormal Statistics
Minimum of Logged Data -1.386 Mean of logged Data 0.204
Maximum of Logged Data 2.779 SDof logged Data 1.257
Assuming Lognomial Distribution
95% H-UCL 18.86 90% Chebyshev {MVUE} UCL 5.514
95% Chebyshev (MVUE) UCL 6.952 37.5% Chebyshev {MVUE} UCL 8.348
99% Chebyshev {MVUE} UCL 12.87
Note: 95% H-UCL (with ND replaced by DL/2) computed by ProUCL is in agreement with results
summarized in Chapter 21 of the RCRA Guidance (EPA 2009). However, it should be noted that the UCL
computed using the DL for ND is 13.62, and the UCL computed using DL/2 for ND is 18.86. Substitution
by DL/2 resulted in a data set with higher variability and a UCL higher than the one obtained using the
DL method. These two UCLs differ considerably confirming that the use of substitution methods should
be avoided.
From results summarized above, it is noted that replacing NDs reported as
-------
Table 4-9. Normal 95% UCL Computed by Replacing ND by DL = 0.5
Normal GOF Test
Shapiro Wilk Test Statistic 0.814 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value 0.803 Data appear Normal at 5% Significance Level
Lilliefors Test Statistic 0.269 Lilliefors GOF Test
5% Ulliefors Critical Value 0.335 Data appear Normal at 5% Significance Level
Data appear Normal at 5% Significance Level
Assuming Normal Distribution
95X Normal UCL 95X UCLs (Adjusted for Skewness)
3 E % Student s4 UC L 1.517 35% .Adjusted^ LT DC L (Chen-1995) 1.454
35% Modified^ UCL (Johnson-1978) 1.518
Table 4-10. Normal 95% UCL Computed by Replacing ND by DL/2 = 0.25
Normal GOF Test
Shapiro Wilk Test Statistic 0.875 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value 0.803 Data appear Normal at 5% Significance Level
Ulliefors Test Statistic 0.23S Ulliefors GOF Test
5% Ulliefors Critical Value 0.335 Data appear Normal at 5% Significance Level
Data appear Normal at 5% Significance Level
Assuming Normal Distribution
95% Normal UCL 95% UCLs (Adjusted for Skewness)
3E% Student s4 UCL 1.E1S 95% AdjustedCLT UCL (Chen-1335} 1.436
95% Modified^ UCL {Johnson-1978} 1.515
Note: The recommended UCL is the KM UCL= 1.523. It is noted that normal UCLs are not influenced by
changing a single ND from 0.5 (UCL95=1.517) to 0.25 (UCL95=1.516). Normal UCL95s without the
outlier appear to represent more realistic estimates of the EPC (population mean). The Lognormal UCL
based upon the data set with the outlier represents the outlying value(s) rather than representing the
population mean.
4.7 Bootstrap UCL Computation Methods for Left-Censored Data Sets
The use of bootstrap methods has become popular with the easy access to fast personal computers. As
described in Chapter 2, for full-uncensored data sets, repeated samples of size n are drawn with
replacement (that is each xl has the same probability = lln of being selected in each of the N bootstrap
replications) from the given data set of n observations. The process is repeated a large number of times, N
(e.g., 1000-2000), and each time an estimate, 6 of 9 (e.g., mean) is computed. These estimates are used
to compute an estimate of the SE of the estimate, 9 . Just as for the full uncensored data sets without any
NDs, for left-censored data sets, the bootstrap resamples are obtained with replacement. An indicator
variable, / (1 = detected value, and 0 = nondetected value), is tagged to each observation in a bootstrap
sample (Efron 1981).
Singh, Maichle, and Lee (EPA 2006) studied the performances, in terms of coverage probabilities, of four
bootstrap methods for computing UCL95s for data sets with ND observations. The four bootstrap
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methods included the standard bootstrap method, the bootstrap-t method, the percentile bootstrap method,
and the bias-corrected accelerated (BCA) bootstrap method (Efron and Tibshirani 1993; Manly 1997).
Some bootstrap methods, as incorporated in ProUCL, for computing upper limits on left-censored data
sets are briefly discussed in this section.
4.7.1 Bootstrapping Data Sets with Nondetect Observations
As before, let xndi, xnd2, ..., xndk, Xk+i, Xk+2, ..., x» be a random sample of size n from a population (e.g.,
AOC, or background area) with an unknown parameter 9 such as the mean, //, or the pth upper percentile
(used to compute bootstrap UTLs), xp, that needs to be estimated from the sampled data set with ND
observations. Let 6 be an estimate of 0, which is a function of k ND and (n - k) detected observations.
For example, the parameter, # , could be the population mean, //, and a reasonable choice for the
estimate, 9, might be the robust ROS, gamma ROS, or KM estimate of the population mean. If the
parameter, 0, represents the pth upper percentile, then the estimate, 9, may represent the pth sample
percentile, x , based upon a full data set obtained using one of the ROS methods described above. The
bootstrap method can then be used to compute a UCL of the percentile, also known as upper tolerance
limit. The computations of upper tolerance limits are discussed in Chapter 5.
An indicator variable, / (taking only two values: 1 and 0), is assigned to each observation (detected or
nondetected) when dealing with left-censored data sets (Efron 1981; Barber and Jennison 1999). The
indicator variables, 7,• :j:=l,2,...,n, represent the detection status of the sampled observations, Xj ;j: = 1,
2,..., n. A large number, N (1000, 2000) of two-dimensional bootstrap resamples, (xa, In ),j:=j: = 1, 2,...,
N, and /': = 1, 2,..., n, of size n are drawn with replacement. The indicator variable, /, takes on a value = 1
when a detected value is selected and / = 0 if a nondetected value is selected. The two-dimensional
bootstrap process keeps track of the detection status of each observation in a bootstrap re-sample. In this
setting, the DLs are fixed as entered in the data set, and the number of NDs vary from bootstrap sample to
bootstrap sample. There may be k\ NDs in the first bootstrap sample, fc NDs in the second sample, ..., and
kx NDs in the TV"1 bootstrap sample. Since the sampling is conducted with replacement, the number of
NDs, fa, i: = 1, 2, ..., N, in a bootstrap re-sample can take any value from 0 to n inclusive. This is typical
of a Type I left-censoring bootstrap process. On each of the N bootstrap resample, one can use any of the
ND estimation methods (e.g., KM, ROS) to compute the statistics of interest (e.g., mean, sd, upper
limits). It is possible that all (or most) observations in a bootstrap re-sample are the same. This is
specifically true, when one is dealing with small data sets. To avoid such situations (with all equal values)
it is suggested that there be at least 15 to 20 (preferably more) observations in the data set. As noted in
Chapter 2, it is not advisable to compute statistics based upon a bootstrap resample consisting of only a
few detected values such as < 4-5.
Let 9 be an estimate of 0 based upon the original left-censored data set of size n; if the parameter, 0
represents the population mean, then a reasonable choice for the estimate, 9, can be the sample ROS
mean, or sample KM mean. Similarly, calculate the sd using one of these methods for left-censored data
sets. The following two steps are common to all bootstrap methods incorporated in the ProUCL software.
Step 1. Let (jCji, xi2, ... , xin) represent the /'th bootstrap resample of size n with replacement from the
original left-censored data set (x\, x2, ..., x»). Note that an indicator variable (as mentioned above) is
tagged along with each data value, taking values 1 (if a detected value is chosen) and 0 (if a ND is chosen
in the resample). Compute an estimate of the mean (e.g., KM, and ROS) using the /'th bootstrap resample,
/: = 1,2, ...,7V.
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Step 2. Repeat Step 1 independently TV times (e.g., N = 2000), each time calculating new estimates (e.g.,
KM estimates) of the population mean. Denote these estimates (e.g., KM means, and ROS means) by
xl, X2, ..., XN . The bootstrap estimate of the population mean is given by the arithmetic mean, XB, of the
TVestimates xt C/VROS means or 7VKM means). The bootstrap estimate of the standard error is given by:
In general, a bootstrap estimate of 6 may be denoted by 9B (instead of XB). The estimate, 6B is the
arithmetic mean of the TV bootstrap estimates (e.g., KM mean, or ROS mean) given by 6?., i:=l,2,...N. If
the estimate, 9 , represents the KM estimate of, 6, then 6i (denoted by J. in the above paragraph) also
represents the KM mean based upon the ith bootstrap resample. The difference, 0B -9 , provides an
estimate of the bias of the estimate, 9 . After these two steps, a bootstrap procedure (percentile, BCA, or
bootstrap-t) is used similarly to the conventional bootstrap procedure on a full uncensored data set as
described in Chapter 2.
Notes: Just like for small uncensored data sets, for small left-censored data sets (<8-10) with only a few
distinct values (2 or 3), it is not advisable to use bootstrap methods. In these scenarios, ProUCL does not
compute bootstrap limits. However, due to the complexity of decision tables and lack of enough funding,
there could be some rare cases where ProUCL may recommend a bootstrap method based UCL which is
not computed by ProUCL (due to lack of enough data).
4.7.1.1 UCL of Mean Based upon Standard Bootstrap Method
Once the desired number of bootstrap samples and estimates has been obtained following the two steps
described above, a UCL of the mean based upon the standard bootstrap method can be computed as
follows. The standard bootstrap confidence interval is derived from the following pivotal quantity, t:
(4-10)
A (1 - a)*100% standard bootstrap UCL for #is given as follows:
aaB (4-11)
Here za is the upper a* critical value (quantile) of the standard normal distribution (SND). It is observed
that the standard bootstrap method does not adequately adjust for skewness, and the UCL given by the
above equation fails to provide the specified (1 - a)*100% coverage of the mean of skewed (e.g.,
lognormal and gamma) data distributions (populations).
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4.7.1.2 UCL of Mean Based upon Bootstrap-t Method
A (1 - a)* 100% UCL of the mean based upon the bootstrap-t method is given as follows.
(4-12)
It should be noted that the mean and sd used in equation (4-12) represent estimates (e.g., KM estimates,
ROS estimates) obtained using original left-censored data set. Similarly, the ^-cutoff value used in
equation (4-12) is computed using the pivotal ^-values based upon KM estimates or some other estimates
obtained using bootstrap re-samples. Typically, for skewed data sets (e.g., gamma, lognormal), the 95%
UCL based upon the bootstrap-t method performs better than the 95% UCLs based upon the simple
percentile and the BCA percentile methods. However, the bootstrap-t method sometimes results in
unstable and erratic UCL values, especially in the presence of outliers (Efron and Tibshirani 1993).
Therefore, the bootstrap-t method should be used with caution. In case this method results in erratic
unstable UCL values. The use of an appropriate Chebyshev inequality-based UCL is recommended.
Additional suggestions on this topic are offered in Chapter 2.
4. 7. 1. 3 Percentile Bootstrap Method
A detailed description of the percentile bootstrap method is given in Chapter 2. For left-censored data
sets, sample means are computed for each bootstrap sample using a selected method (e.g., KM, ROS),
which are arranged in ascending order. The 95% UCL of the mean is the 95th percentile and is given by:
95% Percentile - UCL = 95th%3c! ; /: = 1, 2, ..., N (4-13)
For example, when N = 1000, a simple 95% percentile-UCL is given by the 950th ordered mean value
given by ^(950) . It is observed that for skewed (lognormal and gamma) data sets, the BCA bootstrap
method performs (described below) slightly better (in terms of coverage probability) than the simple
percentile method.
4.7.1.4 Bias-Corrected Accelerated (BCA) Percentile Bootstrap Procedure
Singh, Maichle and Lee (2006) noted that for skewed data sets, the BCA method does represent a slight
improvement, in terms of coverage probability, over the simple percentile method. However, for
moderately skewed to highly skewed data sets with the sd of log -transformed data >1, this improvement
is not adequate and yields UCLs with a coverage probability lower than the specified coverage of 0.95.
The BCA UCL for a selected estimation method (e.g., KM, ROS) is given by the following equation:
(1- a) *100% UCLpRoc = BCA - UCL= x^ROC (4-14)
Here XpROC is the 012 100th percentile of the distribution of statistics given byxPROC; i: = 1, 2, ..., N, and
PROC is one of the many (e.g., KM, DL/2, ROS) mean estimation methods. Here 012 is given by the
following probability statement:
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a2 = O z0 + ^ ^- (4-15)
<1>(Z) is the standard normal cumulative distribution function and z(1 ~a) is the 100*(1 - a)th percentile of a
standard normal distribution. Also, z0 (bias correction) and a (acceleration factor) are given as follows:
z0=0
& \XPROC ,i < XPROC . , , , T ,AIS\
,/: =1, 2, ..., N (4-16)
\ >
-------
gamma or a lognormal distribution however, the detected data set without the outlier follows a lognormal
distribution.
Table 4-lla. Statistics Computed Using Outlier=2982
Pyrene
Total Number of Observations
Number of Detects
Number of Distinct Detects
Minimum Detect
Maximum Detect
Variance Detects 189219
Mean Detects 190.1
Median Detects 103
Skewness Detects 6.282
Mean of Logged Detects 4.711
General 9atistics
56
45
39
31
2332
Number of Distinct Observations
Number of Non-Defects
Number of Distinct Non-Detects
Minimum Non-Detect
Maximum Non-Detect
Percent Non-Detects
SD Detects
CV Detects
Kurtosis Detects
SD of Logged Detects
Kaplan Meier (KM) Satisfies using Normal Critical Values and other Nonparametric UCLs
Mean 164.1 Standard Error of Mean
SD 3S3.4 35% KM (BCA) UCL
95% KM |) UCL 252.2 35% KM (Percentile Bootstrap) UCL
250.7 35% KM Bootstrap t UCL
322 35% KM Chebyshev UCL
432.9 33% KM Chebyshev UCL
35% KM (z) UCL
90% KM Chebyshev UCL
37.5% KM Chebyshev UCL
Lognormal ROS Statistics Using Imputed Non-Detects
Mean in Original Scale 163.2
SD in Original Scale 333.1
95% t UC L (assumes normality of ROS data) 251.1
95% BCA Bootstrap UCL 322.1
95%H-UCL{LogROS) 170.4
Mean in Log Scale
SD in Log Scale
95% Percentile Bootstrap UCL
35% Bootstrap t UCL
44
11
8
28
174
13.64%
435
2.288
41
Q.8Q5
52.65
271.8
261
507,5
333.6
S87.3
4.537
S.843
262.S
5Q7.8
UCLs computed using the KM method and percentile bootstrap and t-statistic are 261 and 252.2. The
corresponding UCLs obtained using the LROS method are 262.6 and 251.2, which appear to
underestimate the population mean. The H-UCL based upon the LROS method is unrealistically lower
(170.4) than the other UCLs. Depending upon the data skewness (sd of detected logged data =0.81), one
can use the Chebyshev UCL95 (or Chebyshev UCL90) to estimate the EPC. Note that as expected, the
presence of one outlier resulted in a bootstrap-t UCL95 significantly higher than the various other UCLs.
Table 4-1 Ib has UCLs computed without the outlier. Exclusion of the outlier resulted in all comparable
UCL values. Any of those UCLs can be used to estimate the EPC.
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Table 4-1 Ib. Statistics Computed without Outlier=2982
Pyrene
General Statistics
Total Number of Observations 55 Number of Distinct Observations 43
Number of Detects 44 Number of Non-Detects 11
Number of Distinct Detects 38 Number of Distinct Non-Detects 8
Minimum Detect 31 Minimum Non-Detect 28
Maximum Detect 453 Maximum Non-Detect 174
Variance Detects 8226 Percent Non-Detects 20%
Mean Detects 126.6 SD Detects 90,7
Median Detects 103 CV Detects 0.716
Skewness Detects 1.735 Kurtosis Detects 3.483
Mean of Logged Detects 4.636 SD of Logged Detects 0.637
Kaplan-Meier (KM) Statistics using Normal Critical Values and other Nonparametric UCLs
Mean 112.3 Standard Error of Mean 11.84
SD 86.03 S5°= KM iBCA) UCL 134
35% KM ft) UCL 132.7 35% KM {Percentile Bootstrap) UCL 132.4
95% KM (z) UCL 132,3 95% KM Bootstrap t UCL 135.3
3fl% KM Chebyshev UCL 148.4 35% KM Chebyshev UCL 164,5
37.5% KM Chebyshev UCL 186.8 33% KM Chebyshev UCL 230.7
Lognormal GOF Test on Detected Observations Only
Shapiro Wilk Test Statistic Q.373 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value 0.344 Detected Data appear Lognormal at 5% Significance Level
Ulliefors Test Statistic O.Q9€5 Lilliefors GOF Test
5% Lilliefors Critical Value 0.134 Detected Data appear Lognormal at 5% Significance Level
Detected Data appear Lognormal at 5% Significance Level
Lognormal ROS Statistics Using Imputed Non Detects
Mean in Original Scale 112.4 Mean in Log Scale 4.43
SD in Original Scale 86.61 SD in Log Scale 0.677
95% t UCL {assumes normality of ROS data) 132 95% Percentile Bootstrap UCL 133
35% BCA Bootstrap UCL 135.7 35% Bootstrap t UCL 137
35%H-UCL{LogROS) 134.3
UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognormally Distributed
KM Mean flogged) 4.431 95% H-UCL (KM-Log) 135
KM SD (logged) 0.676 95% Critical H Value (KM-Log) 2.013
KM Standard Error of Mean flogged) 0.0356
The data set is not highly skewed with sd = 0.64 of logged detected data. Most methods (including H-
UCL) yield comparable results. Based upon data skewness, ProUCL recommends the use of a UCL95
based upon the KM BCA method (highlighted in blue in Table 4-1 Ib).
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4.9 Saving Imputed NDs Using Stats/Sample Sizes Module of ProUCL
Using this option, NDs are imputed based upon the selected distribution (normal, lognormal, or gamma)
of the detected observations. Using the menu option, "Imputed NDs using ROS Methods" ProUCL 5.1
can be used to impute and save imputed NDs along with the original data in additional columns
automatically generated by ProUCL. ProUCL assigns self-explanatory titles for those generated columns.
This option is available in ProUCL for researchers and advanced users who want to experiment with the
full data sets consisting of detected and imputed ND observations for other applications (e.g., ANOVA,
PCA).
4.10 Parametric Methods to Compute UCLs Based upon Left-Censored Data
Sets
Some researchers have suggested that parametric methods such as the expectation maximization (EM)
method and maximum likelihood method (MLE) cited earlier in this chapter would perform better than
the GROS method for data sets with NDs. As reported in ProUCL guidance and on ProUCL generated
output sheets, the developers do realize that the GROS method does not perform well when the shape
parameter, k, or its MLE estimate is small (<1). The GROS method appears to work fine when k is large
(> 2). However, for data sets with NDs and with many DLs, the developers are not sure if parametric
methods such as the MLE method and the EM method perform better than the GROS method and other
methods available in ProUCL. More research needs to be conducted to verify these statements. As noted
earlier, it is not easy (perhaps not possible in most cases) to correctly assess the distribution of a data set
containing NDs with multiple censoring points, a common occurrence in environmental data sets. If
distributional assumptions are incorrect, the decision statistics computed using this incorrect distribution
may also be incorrect. To the best of our knowledge, the EM method can be used on data sets with a
single DL. Earlier versions of ProUCL (e.g., ProUCL 4.0, 2007) had some parametric methods including
the MLE and RMLE methods; those methods were excluded from later versions of ProUCL due to their
poor performances.
The research in this area is limited; to the best of our knowledge, parametric methods (MLE and EM) for
data sets with multiple censoring points are not well-researched. The enhancement of these parametric
methods to accommodate left-censored data sets with multiple DLs will be a big achievement in
environmental statistical literature. The developers will be happy to include contributed better
performing methods in ProUCL.
4.11 Summary and Suggestions
Most of the parametric methods including the MLE, the RMLE, and the EM method assume that there is
only one DL. Like parametric estimates computed using uncensored data sets, MLE and EM estimates
obtained using a left-censored data set are influenced by outliers, especially when a lognormal model is
used. These issues are illustrated by an example as follows.
Example 4-9: Consider a left-censored data set of size 25 with multiple censoring points: <0.24, <0.24,
<1, <0.24, <15, <10, <0.24, <22, <0 .24, < 5.56, <6.61, 1.33, 168.6, 0.28, 0.47, 18.4, 0.48, 0.26, 3.29,
2.35, 2.46, 1.1, 51.97, 3.06, and 200.54. The data set appears to have 2 extreme outliers and 1
intermediate outlier as can be seen from Figure 4-10. From Figure 4-10 and the results of the Rosner
outlier test performed on the data set, it can be concluded that the 3 high detected values represent
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outliers. The Shapiro-Wilk test results performed on detected data shown in Figure 4-11 (censored
probability plot) suggest that the detected data set (with outliers) follows a lognormal distribution
accommodating the outliers.
Q-Q Plot for X *
Reported values used for nondetects TowNumbwofData-a
Number of N on-D elects = 1 1
180
150
120
X
30
60
30
0
-
Number cf Detects -14
DetecledMean-32.4?
Delected Sd = 66. 19
Slope (displayed data] - 34.52
! E8 6 Intercept (displayed data)" 20.64
Correlation. R = 0.653
Q Best Fit Line
51 97
15 1«4
-1 $ -1.2 -06 00 0.6 12 18
Theoretical Quantiles (Standard Normal)
NDs Displayed in smaller font
Figure 4-10. Exploratory Q-Q Plot to Identify Outliers Showing All Detects and Nondetects
Lognormal Q-Q Plot for X
Statistics Displayed using Censored Quantiles and Detected Data
Tola! Number otDat=r-25
Number d Detects = 14
Minimum Censored Quamlile = 1.964
Percent NDs = 44%
ManDL = 22
MearaofDeleclE = 1.259
SldvKol Detect! = 2.238
Slope = 2.64
Intercept = 0.57S
Correlation, R = 0.994
Shapiro-Wilk Test
ExactTest Sialistic = 0.333
Critical Vakje(0.05] = 0874
Dala Appear Lognwmal
Appro*. Test Value = 0.901
p-Vakw-0.116
• Best Fit Line
Theoretical Quantiles (Standard Normal)
Only Quantiles for Detects Displayed
Figure 4-11. Censored Q-Q Plot Showing GOF Test Results on Detected Log-transformed Data
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Statistics Computed with Outliers
From File: N D -D ala-f or M LE 1 xfc
General Statistics for Uncensaed Dataset
Variable NumObs tt Missing Minimum Maximum Mean SD SEM MAD/Q.675 Skewness Kurtosis CV
X 25 0 0.24 200,5 20.64 50.81 10.16 3.128 3.085 8.876 2.462
Percentiles for Uncensored Dataset
Variable NumObs ft Missing lOXile 2UZi\e 252ile(Q1]50£ile(Q2]75Xile(Q3] W%\\e 90%ile 95£ile 99%ile
X 25 0 0.24 0.256 0.28 2.35 10 15.68 39.88 145.3 182.9
Nonparametric estimates of the mean and sd using the KM method are summarized as follows.
From File: ND Data for MLE-1 .xb
General Statistics tor Censored Datasets (with NDs) using Kaplan Meet Method
Variable NumObs ft Missing Num Ds NumNDs % NDs Min ND Max ND KM Mean KM Var KM SD KM CV
X 25 0 14 11 44.00% 0.24 22 18.48 2528 50.28 2.72
MLE estimates of the mean and sd obtained using Minitab 16, UCL95, and a 95%-95% upper tolerance
limit based upon a lognormal distribution are summarized as follows. ML estimates in log scale are:
Parameter
Location
Scale
Estimate
-0.247900
2.71896
Standard
Error
0.641686
0.530176
Upper
Bound
0.807580
3.74710
Log Likelihood = -58. 151; MLE estimates in original raw scale are (back transformation):
Mean = 31. 45, SE of mean = 43.1279, and UCL95 = 300.041
The inclusion of outliers has resulted in inflated estimates, mean = 31.45, UCL95 = 300.41, and a
UTL95-95 = 346.54. The estimate of the mean based upon a data set with NDs should be smaller (e.g.,
KM mean = 18.48) than the mean estimate obtained using all NDs at their reported DLs, 20.64. For this
left-censored data set, the MLE of the mean based upon a lognormal distribution is 31.45 which appears
to be incorrect.
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Statistics Computed without Outliers
Detected data without the 2 extreme outliers also follow a lognormal distribution. MLE estimates,
UCL95, UTL95-95 computed without the outliers and lognormal distribution (using Minitab) are:
Estimates in log scale are provided as follows:
Standard 95% Upper
Parameter Estimate Error Bound
Location -0.561639 -0.561639 0.28616
Scale 2.02381 0.421546 2.85079
Log Likelihood = -38.56; MLE estimates in original raw scale are:
Mean = 4.42, SE of mean = 3.688, and UCL95 = 17.433, and UTL95-95 = 63.42
• Substantial differences are noted in the UCL95s ranging from 300.04 to 17.43, and in the UTL95-
95s ranging from 346.54 to 63.42.
It is not easy to verify the data distribution of a left-censored data set consisting of detects and NDs with
multiple DLs, therefore some poor performing estimation methods including the parametric MLE
methods and the Winsorization method are not retained in ProUCL 4.1 and higher versions. In ProUCL
5.1, emphasis is given on the use of nonparametric UCL computation methods and hybrid parametric
methods based upon KM estimates which account for data skewness in the computation of UCL95s. It is
recommended that one avoid the use of transformations to achieve symmetry while computing the upper
limits based upon left-censored data sets. It is not easy to correctly interpret statistics computed in the
transformed scale. Moreover, the results and statistics computed in the original scale do not suffer from
transformation bias.
When the sd of the log -transformed data, o, becomes >1.0, avoid the use of a lognormal model even when
the data appear to be lognormally distributed. Its use often results in unrealistic statistics of no practical
merit (Singh, Singh, and Engelhard 1997; Singh, Singh, and laci 2002). It is also recommended the user
identifies potential outliers representing observations coming from population(s) different from the main
dominant population and investigate them separately. Decisions about the disposition of outliers should
be made by all interested members of the project team.
It is recommended that the use of the DL/2 (t) UCL method be avoided, as the DL/2 UCL does not
provide the desired coverage (for any distribution and sample size) for the population mean, even for
censoring levels as low as 10% and 15%. This is contrary to the conjecture and assertion (EPA 2006a)
made that the DL/2 method can be used for lower (< 20%) censoring levels. The coverage provided by
the DL/2 (t) method deteriorates fast as the censoring intensity increases. The DL/2 (t) method is not
recommended by the authors or developers of this document and ProUCL software.
The use of the KM estimation method is a preferred method as it can handle multiple DLs. Therefore, the
use of KM estimates is suggested for computing decision statistics based upon methods which adjust for
data skewness. Depending upon the data set size, distribution of the detected data, and data skewness, the
various nonparametric and hybrid KM UCL95 methods including KM (BCA), bootstrap-t KM UCL,
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Chebyshev KM UCL, and Gamma-KM UCL based upon the KM estimates provide good coverages for
the population mean. Suggestions regarding the selection of a 95% UCL of the mean are provided to help
the user select the most appropriate 95% UCL. These suggestions are based upon the results of the
simulation studies summarized in Singh, Singh, and laci (2002) and Singh, Maichle, and Lee (2006). It is
advised that the project team collectively determine which UCL will be most appropriate for their site
project. For additional insight, the user may want to consult a statistician.
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CHAPTER 5
Computing Upper Limits to Estimate Background Threshold
Values Based upon Data Sets Consisting of Nondetect (ND)
Observations
5.1 Introduction
As described in Chapter 3, a BTV considered in this chapter represents an upper threshold parameter
(e.g., 95th) of the background population; which is used to perform point-by-point comparisons of onsite
observations. Estimation of BTVs and comparison studies require the computation of UPLs and UTLs
based upon left-censored data sets containing ND observations. Not much guidance is available in the
statistical literature on how to compute UPLs and UTLs based upon left-censored data sets of varying
sizes and skewness levels. Like UCLs, the use of Student's t-statistic and percentile bootstrap methods
based UPLs and UTLs are difficult to defend for moderately skewed to highly skewed data sets with
standard deviation (sd) of the log-transformed data exceeding 0.75-1.0. Since it is not easy to reliably
perform GOF tests on left-censored data sets; emphasis is given on the use of distribution-free
nonparametric methods including the KM, Chebyshev inequality, and other computer intensive bootstrap
methods to compute upper limits needed to estimate BTVs.
All BTV estimation methods for full uncensored data sets as described in Chapter 3 can be used on data
sets consisting of detects and imputed NDs obtained using ROS methods (e.g., GROS and LROS).
Moreover, all other comments about the use of substitution methods, disposition of outliers, and
minimum sample size requirements as described in Chapter 4 also apply to BTV estimation methods for
data sets with ND observations.
5.2 Treatment of Outliers in Background Data Sets with NDs
Just like full uncensored data sets, a few outlying observations present in a left-censored data set tend to
yield distorted estimates of the population parameters (means, upper percentiles, OLS estimates) of
interest. OLS regression estimates (slope and intercept) become distorted (Rousseeuw and Leroy 1987;
Singh and Nocerino 1995) by the presence of outliers. Specifically, in the presence of outliers, the ROS
method performed on raw data (e.g., GROS) tends to yield unfeasible imputed negative values for ND
observations. Singh and Nocerino (2002) suggested the use of robust regression methods to compute
regression estimates needed to impute NDs based upon ROS methods. Robust regression methods are
beyond the scope of ProUCL. It is therefore suggested that potential outliers be manually identified
where they may be present in a data set before proceeding with the computation of the various BTV
estimates as described in this chapter. As mentioned in earlier chapters, upper limits computed by
including a few low probability high outliers tend to represent locations with those elevated
concentrations rather than representing the main dominant population. It is suggested that relevant
statistics be computed using data sets with outliers and without outliers for comparison. This extra step
helps the project team to see the potential influence of outlier(s) on the various decision making statistics
(e.g., UCLs, UPLs, UTLs); and helps the project team in making informative decisions about the
disposition of outliers. That is, the project team and experts familiar with the site should decide which of
the computed statistics (with outliers or without outliers) represent more accurate estimate(s) of the
population parameters (e.g., mean, EPC, BTV) under consideration.
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A couple of classical outlier tests (Dixon and Rosner tests) are available in the ProUCL software. These
tests can be used on data sets with or without ND observations. Additionally, one can use graphical
displays such as Q-Q plots and box plots to visually identify high outliers in a left-censored data set. It
should be pointed out, that for environmental applications, it is the identification of high outliers (perhaps
representing contaminated locations and hot spots) that is important. The occurrence of ND (less than
values) observations and other low values is quite common in environmental data sets, especially when
the data are collected from a background or a reference area. For the purpose of the identification of high
outliers, one may replace ND values by their respective DLs or half of the DLs or may just ignore them
(especially when high reporting limits are associated with NDs) from the outlier tests. A similar approach
can be used to generate graphical displays, Q-Q plots and histograms. Except for the identification of high
outlying observations, the outlier test statistics, computed with NDs or without NDs, are not used in any
of the estimation and decision making processes. Therefore, for the purpose of testing for high outliers, it
does not matter how the ND observations are treated.
5.3 Estimating BTVs Based upon Left-Censored Data Sets
This section describes methods for computing upper limits (UPLs, UTLs, USLs, upper percentiles) that
may be used to estimate BTVs and other not-to-exceed levels from data sets with ND observations.
Several Student's t-type statistic and normal z-scores based methods have been described in the literature
(Helsel 2005; Millard and Neerchal 2002; USEPA 2007, 2010d, 2011) to compute UPLs and UTLs based
upon statistics (e.g., mean, sd) obtained using MLE, KM, or ROS methods. The methods used to
compute upper limits (e.g., UPL, UTL, and percentiles) based upon a Student's t-type statistic are also
described in this chapter; however, the use of such methods is not recommended for moderately skewed
to highly skewed data sets. These methods may yield reasonable upper limits (e.g., with proper coverage)
for normally distributed and mildly skewed to moderately skewed data sets with the sd of the detected
log-transformed data less than 1.0.
Singh, Maichle, and Lee (EPA 2006) demonstrated that the use of the t-statistic and the percentile
bootstrap method on moderately to highly skewed left-censored data sets yields UCL95s with coverage
probabilities much lower than the specified CC, 0.95. A similar pattern is expected in the behavior and
properties of the various other upper limits (e.g., UTLs, UPLs) used in the decision making processes of
the USEPA. It is anticipated that the performance (in terms of coverages) of the percentile bootstrap and
Student's t-type upper limits (e.g., UPLs, UTLs) computed using the KM and ROS estimates for
moderately skewed to highly skewed left-censored data sets (sd of detected logged data >1) would also be
less than acceptable. For skewed data sets, the use of the gamma distribution on KM estimates (when
applicable) or nonparametric methods, which account for data skewness, is suggested for computing BTV
estimates. A brief description of those methods is provided in the following sections.
5.3.1 Computing Upper Prediction Limits (UPLs) for Left-Censored Data Sets
This section describes some parametric and nonparametric methods for computing UPLs for left-censored
data sets.
5.3.1.1 UPLs Based upon Normal Distribution of Detected Observations and KM
Estimates
When detected observations in a data set containing NDs follow a normal distribution (which can be
verified by using the GOF module of ProUCL), one may use the normal distribution on KM estimates to
compute the various upper limits needed to estimate BTVs (also available in ProUCL 4.1). A (1 - a)* 100
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UPL for a future (or next) observation (observation not belonging to the current data set) can be computed
using the following KM estimates based equation:
+t
^2KM (l + l/n) (5-1)
((1-a ),(«-!))
Here t^^^^ is the critical value of the Student's t-distribution with (n-1) degrees of freedom
(df). If the distributions of the site data and the background data are comparable, then a new (next)
observation coming from the site population (e.g., site) should lie at or below the UPLi95 with probability
0.95. A similar equation can be developed for upper prediction limits for future k observations (described
in Chapter 3) and the mean of k future observations (incorporated in ProUCL 5.0/ProUCL 5.1).
5.3.1.2 UPL Based upon the Chebyshev Inequality
The Chebyshev inequality can be used to compute a reasonably conservative but stable UPL and is given
as follows:
UPL = Jc+[^((1/00-1)*(1+ 1/71)]^ (5-2)
The mean, x , and sd, sX: used in the above equation are computed using one of the estimation methods
(e.g., KM) for left-censored data sets. Just like the Chebyshev UCL, a UPL based upon the Chebyshev
inequality tends to yield higher estimate of BTVs than the other methods. This is especially true when
skewness is moderately mild (sd of log-transformed data is low < 0.75), and the sample size is large n >
30). It is advised to apply professional/expert judgment before using this method to compute a UPL.
Specifically, for larger skewed data sets, instead of using a 95% UPL based upon the Chebyshev
inequality, the user may want to compute a Chebyshev UPL with a lower CC (e.g., 85%, 90%) to estimate
a BTV. ProUCL can compute a Chebyshev UPL (and all other UPLs) for any user specified CC in the
interval [0.5, 1].
5.3.1.3 UPLs Based upon ROS Methods
As described earlier, ROS methods first impute k ND values using an OLS linear regression model
(Chapter 4). This results in a full data set of size n. For ROS methods (gamma, lognormal), ProUCL
generates additional columns consisting of (n - k) detected values and k imputed values of the k ND
observations present in a data set. Once, the ND observations have been imputed, the user may use any of
the available parametric and nonparametric BTVs estimation methods for full data sets (without NDs), as
described in Chapter 3. Those BTV estimation methods are not repeated here. The users may want to
review the behavior of the various ROS methods as described in Chapter 4.
5.3.1.4 UPLs when Detected Data are Gamma Distributed
When detected data follow a gamma distribution, methods described in Chapter 3 can be used on KM
estimates to compute gamma distribution based upper prediction limits for future k>l observations. These
limits are described below when k=l.
Wilson-Hilferty (WH) UPL = max OAy
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UPL = +1^^ * SyKM *
Here t^^^^ is a critical value from the Student's t-distribution with («-l) degrees of freedom (df),
and KM estimates are computed based upon the transformed y data as described in Chapter 3. All detects
and NDs are transformed to y-space to compute the KM estimates.
One of the advantages of using this method is that one does not have to impute NDs based upon the data
distribution using LROS or GROS method.
5.3.1.5 UPLs when Detected Data are Lognormally Distributed
When detected data follow a lognormal distribution, methods described in Chapter 3 can be used on KM
estimates to compute lognormal distribution based upper prediction limits for future k>l observations.
These limits are described below when k=l. An upper (1 - a)*100% lognormal UPL is given by the
following equation:
UPL = exp(y + f((1_«M»-D) *sy *
Here t^^^^ is a critical value from Student's t-distribution with (n-\) df, and y and sy represent the
KM mean and sd based upon the log-transformed data (detects and NDs), y. All detects and NDs are
transformed to y-space to compute the KM estimates.
5.3.2 Computing Upper p*100% Percentiles for Left-Censored Data Sets
This section briefly describes some parametric and nonparametric methods to compute upper percentiles
based upon left-censored data sets.
5.3.2.1 Upper Percentiles Based upon Standard Normal Z-Scores
In a left-censored data set, when detected data are normally distributed, one can use normal percentiles
and KM estimates (or some other estimates such as ROS estimates) of the mean and sdto compute the/?*
percentile given as given as follows:
XP=£KM+ZP^I£L (5-3)
Here zp is the p* 100th percentile of a standard normal, N (0, 1), distribution which means that the area
(under the standard normal curve) to the left of zp is p. If the distributions of the site data and the
background data are comparable, then an observation coming from a population (e.g., site) similar
(comparable) to that of the background population should lie at or below the />*100% percentile, with
probability/?.
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5.3.2.2 Upper Percentiles when Detected Data are Lognormally Distributed
When detected data follow a lognormal distribution, methods described in Chapter 3 can be used on the
KM estimates to compute lognormal distribution based upper percentiles. The lognormal distribution
based/?* percentile based upon KM estimates is given as follows:
In the above equation, y and sy represent the KM mean and sd based upon the log-transformed data
(detects and NDs), y. All detects and NDs are transformed to y-space to compute the KM estimates.
5.3.2.3 Upper Percentiles when Detected Data are Gamma Distributed
When detected data are gamma distributed, gamma percentiles can be computed similarly using the HW
and WH approximations to compute KM estimates. According to the WH approximation, the transformed
detected data Y = X1/3 follow an approximate normal distribution; and according to the HW
approximation, the transformed detected data Y = X1/4 follow an approximate normal distribution. Let y
and sy represent the KM mean and sd of the transformed data (detects and NDs), y. The percentiles based
upon the WH and HW transformations respectively are given as follows:
xp = max I (
Alternatively, following the process described in Section 4.6.2, one can use KM estimates to compute
KM estimates, k and 9 of the shape, k and scale, 6 parameters of the gamma distribution, and use the
chi-square distribution to compute gamma percentiles using the equation: X = Y * 9 12, where Y follows
a chi-square distribution with 2k degrees of freedom (df). This method does not require HW or WH
approximations to compute gamma percentiles. Once an a* 100% percentile, ya = y(a) 2k, of a chi-square
distribution with 2k dfis obtained, the a*100% percentile for a gamma distribution is computed using
the equation: xa = ya * 912. ProUCL 5.1 computes gamma percentiles using this equation based upon KM
estimates.
5.3.2.4 Upper Percentiles Based upon ROS Methods
As noted in Chapter 4, all ROS methods first impute k ND values using an OLS linear regression
(Chapter 4) assuming a specified distribution of detected observations. This process results in a full data
set of size n consisting of k imputed NDs and (n-k) detected original values. For ROS methods (normal,
gamma, lognormal), ProUCL generates additional columns consisting of the (n-k) detected values, and k
imputed ND values. Once, the ND observations have been imputed, an experienced user may use any of
the parametric or nonparametric percentile computation methods for full uncensored data sets as
described in Chapter 3. Those methods are not repeated in this chapter.
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5.3.3 Computing Upper Tolerance Limits (UTLs) for Left-Censored Data Sets
UTL computation methods for data sets consisting of NDs are described in this section.
5.3.3.1 UTLs Based on KM Estimates when Detected Data are Normally Distributed
Normal distribution based UTLs computed using KM estimates may be used when the detected data are
normally distributed (can be verified using GOF module of ProUCL) or moderately to mildly skewed,
with the sd of log-transformed detected data, a, less than 0.5-0.75. An upper (1 - a)* 100% tolerance limit
with tolerance or coverage coefficient, p, is given by the following statement:
(5-4)
Here K = K (n, a, p) is the tolerance factor used to compute upper tolerance limits and depends upon the
sample size, n, CC = (1- a), and the coverage proportion = p. The K critical values are based upon the
non-central t-distribution, and have been tabulated extensively in the statistical literature (Hahn and
Meeker 1991). For samples of sizes larger than 30, one can use Natrella's approximation (1963) to
compute the tolerance factor, K = K (n, a, p).
5.3.3.2 UTLs Based on KM Estimates when Detected Data are Lognormally Distributed
When detected data follow a lognormal distribution, methods described in Chapter 3 can be used on KM
estimates to compute lognormal distribution based upper tolerance limits. An upper (1 - a,)* 100%
tolerance limit with tolerance or coverage coefficient, p, is given by the following statement:
UTL =
The K factor in the above equation is the same as the one used to compute the normal UTL; y and sy
represent the KM mean and sd based upon the log-transformed data. All detects and NDs are transformed
to y-space to compute KM estimates.
5.3.3.3 UTLs Based on KM Estimates when Detected Data are Gamma Distributed
According to the WH approximation, the transformed detected data Y = X1/3 follow an approximate
normal distribution; and according to the HW approximation, the transformed detected data Y = X1/4
follow an approximate normal distribution when detected X data are gamma distributed. Let y and sy
represent the KM mean and sd based upon transformed data (detects and NDs), Y.
Using the WH approximation, the gamma UTL (in original scale, X), is given by:
UTL = max f 0, (j7 + K * sy )3
Similarly, using the HW approximation, the gamma UTL in original scale is given by:
UTL=(y+K*sy}*
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5.3.3.4 UTLs Based upon ROS Methods
As noted in Chapter 4, all ROS methods first impute k ND values using an OLS linear regression line
assuming a specified distribution of detected and nondetected observations. This process results in a full
data set of size n consisting of k imputed NDs and (n-k) detected original values. For ROS methods
(normal, gamma, lognormal), ProUCL generates additional columns consisting of the (n-k) detected
values, and k imputed ND values. Once, the ND observations have been imputed, an experienced user
may use any of the parametric or nonparametric UTL computation methods for full data sets as described
in Chapter 3 . Those methods are not repeated in this chapter.
Note: In the Stats/Sample Sizes module, using the General Statistics option for data sets with NDs, for
information and summary purposes, percentiles are computed using detects and nondetects, where
reported DLs are used for NDs. Those percentiles do not account for NDs. However, KM method based
upper limits such as the UTL95-95 account for NDs; therefore, sometimes, a UTL95-95 computed based
upon a ND method (e.g., KM method) may be lower than the 95% percentile computed using the General
Statistics option of Stats/Sample Sizes module.
5. 3. 4 Computing Upper Simultaneous Limits (USLs) for Left-Censored Data Sets
Parametric and nonparametric USL computation methods for are described as follows.
5.3.4.1 USLs Based upon Normal Distribution of Detected Observations and KM
Estimates
When detected observations follow a normal distribution (can be verified by using the GOF module of
ProUCL), one can use the normal distribution on KM estimates to compute a USL95 .
A one-sided (1 - a) 100% USL providing (1 - a) 100% coverage for all sample observations is given by:
USL=
Here (db2a)2 is the critical value of Max (Mahalanobis Distances) for 2*a level of significance.
5.3.4.2 USLs Based upon Lognormal Distribution of Detected Observations and KM
Estimates
When detected data follow a lognormal distribution, methods described in Chapter 3 can be used on the
KM estimates to compute lognormal distribution based USLs. Let y and sy represent the KM mean and
sd of the log-transformed data (detects and NDs), y; a (1 - a) 100% USL is given by as follows:
5. 3. 4. 3 USLs Based upon Gamma Distribution of Detected Observations and KM
Estimates
According to the WH approximation, the transformed detected data Y = X1/3 follow an approximate
normal distribution; and according to the HW approximation, the transformed detected data Y = X1/4
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follow an approximate normal distribution. Let y and sy represent the KM mean and sd of the
transformed data (detects and NDs), y. A gamma distribution based (using WH approximation), one-
sided (1 - a) 100% USL is given by:
A gamma distribution based (HW approximation) one-sided (1 - a) 100% USL is given as follows:
5.3.4.4
USLs Based upon ROS Methods
Once, the ND observations have been imputed, one can use parametric or nonparametric USL
computation methods for full data sets as described in Chapter 3.
Example 5-1 (Oahu Data Set). The detected data are only moderately skewed (sd of logged detects =
0.694) and follow a lognormal as well as a gamma distribution. The various upper limits computed using
ProUCL 5.1 are listed in Tables 5-1 through 5-3 as follows.
Table 5-1. Nonparametric and Normal Upper Limits Using KM Estimates
Arsenic
Total Number of Observations
Number of Distinct Observations
Number of Detects
Number of Distinct Detects
Minimum Detect
Maximum Detect
Variance Detected
Mean Detected
General Statistics
24
10
11
a
0.5
3.2
0.331
1.236
Mean of Detected Logged Data -0.0255
Number of Missing Observations 0
Number of Non-Detects 13
Number of Distinct Non-Detects 3
Minimum Non-Detect 0.9
Maximum Non-Detect 2
Percent Non-Detects 54.17%
SD Detected Q.3S5
SD of Detected Logged Data 0.694
Critical Values for Background Threshold Values (BTVs)
Tolerance Factor K (For UTL) 2.309
d2max for USL) 2.644
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Table 5-1 (continued). Nonparametric and Normal Upper Limits Using KM Estimates
Normal GOF Test on Detects Only
Shapiro Wilk Test Statistic §.777 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value Q.85 Data Not Normal at 5% Significance Level
IJIIiefors Test Statistic 0.273 Lilliefors GOF Test
5% Lilliefors Critical Value 0.2S7 Data Not Normal at 5*1 Significance Level
Data Not Normal at 5% Significance Level
Kaplan Meier (KM) Background Statistics Assuring Normal Distribution
Mean 8.949 SD 0.713
35% UTL95% Coverage 2.595 95%KMIJPL|) 2.196
95%KMChebyShevUPL 4.121 30% KM Percentile
-------
Table 5-2 (continued). Upper Limits Using GROS, KM Estimates and Gamma Distribution of
Detected Data
Minimum 0.119
Maximum 3.2
SD 0.758
k hat (HLE) 2.071
Theta hat {MLE} 0.461
nu hat {MLE} 33.41
MLE Mean |bias corrected) 0.956
35% Percentile of Chisquare (2k) 8.364
95% Percentile 2.32S
Mean
Median
CV
k star (bias corrected MLE}
Theta star (bias corrected MLE}
nu star {bias corrected)
MLE Sd {bias corrected)
30% Percentile
93% Percentile
The following statistics are computed using Gamma ROS Statistics on Imputed Data
Upper Limits using Wilson Hirferty (WH) and Hawkins Wbdey (HW) Methods
WH HW WH
95% Approx. Gamma UTL with 35% Coverage 3.149 3.239 35% Approx. Gamma UPL 2,384
95% Gamma USL 3.676 3.915
The following statistics are computed using gamma distribution and KM estimates
Upper Limits using Wilson Hilferty {WH) and Hawkins Wbdey (HW) Methods
k hat (KM) 1.771
WH HW
35% Approx. Gamma UTL with 95% Coverage 2.661 2.685
95% Gamma LISL 3.051 3.107
0.956
0.7
0.793
1.84
0,519
88.32
0.704
1.835
3,291
HW
2.436
nu hat {KM} 85.02
WH HW
35% Approx. Gamma UPL 2.087 2.077
Table 5-3. Upper Limits Using LROS method and KM Estimates and Lognormal Distribution of
Detected Data
Lognormal GOF Test on Detected Observations Only
Shapiro Wilk Test Statistic 0.86 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value 0.85 Detected Data appear Lognormal at 5% Significance Level
Lilliefons Test Statistic 0.223 Lilliefore GOF Test
5% Lilliefors Critical Value 0.267 Detected Data appear Lognormal at 5% Significance Level
Detected Data appear Lognormal at 5% Significance Level
Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Detects
Mean in Original Scale 0.972
SD in Original Scale 0.718
35% UTL35% Coverage 3.032
95% Bootstrap {%} UTL95% Coverage 3.2
9-0% Percentile (z) 1.686
99% Percentile {z} 3.0€2
Mean in Log Scale -0.209
SD in Log Scale 0.571
95% BCA UTL35% Coverage 3.2
35% UPL ft) 2.202
35% Percentile {z) 2.075
95% USL 3.671
Statistics using KM estimates on Logged Data and Assuming Lognormal Distribution
KM Mean of Logged Data -0.236 95% KM UTL fLognormal)35% Coverage 2.792
KM SD of Logged Data 0.547 95% KM UPL (Lognormal} 2.056
35% KM Percentile Lognormal {z} 1.942 95% KM USL {Lognormal} 3.354
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Example 5-2 A real data set of size 55 with 18.8% NDs is considered next. This data was used in Chapter
4 to illustrate the differences in UCLs computed using a lognormal and a gamma distribution. This data
set is considered here to illustrate the merits of the gamma distribution based upper limits. It can be seen
that the detected data follow a gamma as well as a lognormal distribution. The minimum detected value
is 5.2 and the largest detected value is 79000. The sd of the detected logged data is 2.79 suggesting that
the detected data set is highly skewed. Relevant statistics and upper limits including a UPL95, UTL95-
95, and UCL95 have been computed using both the gamma and lognormal distributions. The gamma
GOF Q-Q plot is shown as follows.
Gamma Q-Q Plot (Statistics using Detected Data) for A-DL
A-DL
Total Number oIOata = ffi
Number o!NDs = 10
MaxDL=124
No.45
Percent NDs = 18*
Mean- IfJbSG !«;.'
k star = 0.3018
Iheta stai - 34373.7238
Slope=1.0745
Intercept = -5357060
Correlalion,R = 0.9e44
Andeison-DailirgTest
Tesl Statistic = 0.591
CriticalValue(0 051 = 0.860
Data appeal Gamma DiitiibU
• Beit F* Line
Theoretical Quantiles of Gamma Distribution
Summary Statistics for Data Set of Example 5-2
ArDL
Total Number of Observations
Number of Distinct Observations
Number of Detects
Number of Distinct Detects
Minimum Detect
Maximum Detect
Variance Detected
Mean Detected
Mean of Detected Logged Data
General Statistics
55
53
45
45
5.2
7SCCC
3.954E+S
10556
7.031
Critical Values for Background Threshold Values
Tolerance Factor K (For UTL)
2.036
Number of Missing Observations 0
Number of Non-Detects 1 0
Number of Distinct Non-Detects S
Minimum Non-Detect 3.S
Maximum Non-Detect 124
Percent Non-Detects 18.1 8*4
SD Detected 19SS6
SD of Detected Logged Data 2.738
(BTVs)
d2maxfforUSL) 2.994
Mean of detects (=10556) reported above ignores all 18.18% NDs.
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KM Method Based Estimates of the Mean, SE of the Mean, and sd
Mean 8638
SD 18246
Standard Error of Mean 2488
KM mean (= 8638) reported above accounts for 18.18% NDs reported in the data set.
Notes: Direct estimate of KM sd = 18246
Indirect Estimate of KM sd (Helsel 2012) = 18451.5
The gamma GOF test results on detected data and various upper limits including UCLs obtained using the
GROS method and gamma distribution on KM estimates are provided in Table 5-4; and the lognormal
GOF test results on detected data and the various upper limits obtained using the LROS method and
lognormal distribution on KM estimates are provided in Table 5-5. Table 5-6 is a summary of the main
upper limits computed using the lognormal and gamma distribution of the detected data.
Table 5-4. Upper Limits Using GROS, KM Estimates and Gamma Distribution of Detected Data
Gamma GOF Tests on Detected Observations Only
A-D Test Statistic 0.531 Anderson-Darling GOF Test
5% A-D Critical Value 0.86 Detected data appear Gamma Distributed at 5% Significance Level
K-S Test Statistic 0115 Kblmogruv-Smimoff GOF
5% K-S Critical Value 0.143 Detected data appear Gamma Distributed at 5% Significance Level
Detected data appear Gamma Distributed at 5X Significance Level
Gamma Statistics on Detected Data Only
k hat (MLE) 0.307
Theta hat (MLE) 34333
nu hat (MLE) 27.67
MLE Mean {bias corrected) 18556
MLE Sd {bias corrected) 13216
k star jbias corrected MLE) 0.302
Theta star {bias corrected MLE) 34980
nu star (bias corrected) 27.16
35% Percentile of Chisquare (2k) 2.756
Upper Limits Computed Using Gamma ROS Method
Gamma ROS Statistics using Imputed Non-Detects
1.121
Mean 8642
Median 588
CV 2.13
k star {bias corrected MLE) 0.246
Theta star (bias corrected MLE) 35133
nu star teas corrected) 27.01
M LE Sd {bias corrected) 17440
90% Percentile 25972
35% Percentile 42055 33% Percentile 84376
The following statistics are computed using Gamma ROS Statistics on Imputed Data
Upper Limits using Wilson Hitferty (WH) and Hawkins Wijdey {HW) Methods
WH HW WH HW
95% Apprax. Garnrna UTL with 95% Coverage 47429 54346 95% Apprax. Gamma UPL 33332 35476
Minimum
Maximum 7500-0
SD 18412
k hat (MLE)
Theta hat (MLE) 35001
nu hat (MLE) 27.16
MLE Mean {bias corrected)
5""; Percentile of Chisquare (2k)
0.247
8642
2.33
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Upper Limits Computed Using Gamma Distribution and KM Estimates
The following statistics are computed using gamma distribution aid KM estimates
Upper Limits using Wilson HiHerty (WH) and Hawkins Wudey (HW) Methods
k hat (KM) 0.224 nu hat (KM) 24.66
WH HW WH HW
95'% Apprax. Gannma UTL with 95% Coverage 46378 54120 95'% Approx. Gamma UPL 32961 35195
95% UCL of the Mean Based upon GROS Method
Approximate Chi Square Value (27.01 .a) 1S. 16
95% Gamma Approximate UCL [use when ri>=5D) 14M5
.Adjusted Level of Significance f} 0.345S
Adjusted Chi Square Value (27.01, P) 15.93
95% Gamma Adjusted UCL (use when n<5<0) 14651
95% UCL of the Mean Using Gamma Distribution on KM Estimates
Gamma Kaplan-Meier (KM) Satisfies
k hat (KM) 0.224 nu hat (KM) 24.66
Approximate Chi Square Value (24.66, a] 14.35 Adjusted Chi Square Value (24.66. (3) 14.14
95% Gamma Approximate KM-UCL (use when n;=5G} 14844 95%Gamma Adjusted KM-UCL (use when n<50) 15066
Table 5-5. Upper Limits Using LROS and KM Estimates and Lognormal Distribution of Detected
Data
Lognormal GOF Test on Detected Observations Only
Shapiro Wilk Test Statistic 0.939 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value 0.945 Data Not Lognormal at 5% Significance Level
Lilliefors Test Statistic 0.104 Lilliefore GOF Test
5% Ulliefors Critical Value 0.132 Detected Data appear Lognormal at 5% Significance Level
Detected Data appear Approximate Lognormal at 5% Significance Level
Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Detects
Mean in Original Scale 863-8 Mean in Log Scale 5.983
SD in Original Scale 18414 SD in Log Scale 3.391
95% UTL95% Coverage 394791 95% BCA UTL95% Coverage 77530
95% Bootstrap (%} UTL95% Coverage 77530 95% UPL ft} 121584
90% Percentile (z) .3-0572 95% Percentile (z) 104784
99% Percentile (z) 1056400 95% USL 10156719
Statistics using KM estimates on Logged Data and Assuming Lognormal Distribution
KM Mean of Logged Data 6.03 95% KM UTL (Lognormal)95% Coverage 334181
KM SD of Logged Data 3.286 95% KM UPL (Lognormal) 106741
95% KM Percentile Lognomial (z} 92417
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95% UCL of the mean Using LROS and Lognormal Distribution on KM Estimates Methods
Lognormal ROS Statistics Using Imputed Non-Detects
Mean in Original Scale B63S Mean in Log Scale 5.SB3
SD in Original Scale 18414 SD in Log Scale 3.391
95% t UC L (assumes normality of RO S data) 12793
95% BCA Bootstrap UC L 13762
955; H-UCL [Log ROS) 1B55231
955; Percentile Bootstrap UCL 12676
955; Bootstrap! UCL 14659
UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognormally Distributed
KM Mean togged) 6.03 955; H-UCL (KM-Log) 11735BS
KM SD togged) 3.286 95%Critical H Value (KM-Log) 5.7
KM Standard Error of Mean Dogged) 0.449
Nonparametric upper percentiles are: 9340 (80%), 25320 (90%), 46040 (95%), and 77866 (99%). Other
upper limits, based upon the gamma and lognormal distribution, are described in Table 5-6. All
computations have been performed using the ProUCL software. In the following Table 5-6, method
proposed/described in the literature have been cited in the Reference Method of Calculation column.
Table 5-6. Summary of Upper Limits Computed using Gamma and Lognormal Distribution of
Detected Data: Sample Size = 55, No. of NDs=10, % NDs = 18.18%
Upper Limits
Min (detects)
Max (detects)
Mean (KM)
Mean (ROS)
95%Percentile(ROS)
UPL95 (ROS)
UTL95-95 (ROS)
UPL95 (KM)
UTL95-95 (KM)
UCL95 (ROS)
UCL (KM)
Gamma Distribution
Result
5.2
79,000
8,638
8,642
42,055
33,332
47,429
32,961
46,978
14,445
14,844
Reference/
Method of Calculation
~
~
~
~
~
WH- ProUCL
WH- ProUCL
WH-ProUCL
WH-ProUCL
ProUCL
ProUCL
Lognormal Distribution
Result
1.65
11.277
6.3
8,638
104,784
121,584
394,791
106,741
334,181
14,659
12,676
1,173,988
Reference/
Method of Calculation
Logged
Logged
Logged
~
~
Helsel(2012),EPA2009
Helsel(2012),EPA2009
EPA (2009)
EPA(2009)
bootstrap-t, ProUCL 5.0
percentile bootstrap, Helsel
(2012)
H-UCL, KM mean and sd
on logged data - EPA
(2009)
The statistics listed in Tables 5-4 and 5-5, and summarized in Table 5-6 demonstrate the need and merits
of using the gamma distribution for computing practical and meaningful estimates (upper limits) of the
decision parameters (e.g., mean, upper percentile) of interest.
Example 5.3. The benzene data set (Benzene-H-UCL-RCRA.xls) of size 8 used in Chapter 21 of the
RCRA Unified Guidance document (EPA 2009) was used in Section 4.6.3.1 to address some issues
associated with the use of lognormal distribution to compute a UCL of mean for data sets with
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nondetects. The benzene data set is used in this example to illustrate similar issues associated with the
computation of UTLs and UPLs based upon lognormal distribution using substitution methods.
Lognormal distribution based upper limits using ROS and KM methods are summarized in Table 5-7.
Table 5-7. Lognormal 95%-95% Upper Limits based upon LROS and KM Estimates
Lognormal GOF Test on Detected Observations Only
Shapiro Wilk Test Statistic 0.829 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value 0.803 Detected Data appear Lognormal at 5% Significance Level
Lilliefors Test Statistic 0.304 Ljlliefors GOF Test
5% Ulliefors Critical Value 0.335 Detected Data appear Lognormal at 5% Significance Level
Detected Data appear Lognonnal at 5% Sgnificanee Level
Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Detects
Mean in Original Scale 2.913 Mean in Log Scale 0.0326
SD in Original Scale 5.364 SD in Log Scale 1.443
95% UTL95% Coverage 109.2 95% BCA UTL95% Coverage 1S.1
95% Bootstrap {%} UTL95% Coverage 1S.1 95% UPL
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Table 5-9. Lognormal Distribution Based Upper Limits using DL/2 (=0.25) for ND
Mean of logged Data 0.204 SD of logged Data 1.257
Lognormal GOF Test
Shapiro Wilk Test Statistic 0.836 Shapiro Wilk Lognormal GOF Test
5% Shapiro Wilk Critical Value 0.818 Data appear Lognormal at 5% Significance Level
Ulliefors Test Statistic 0.255 Ulliefors Lognormal GOF Test
5% Ulliefors Critical Value 0.313 Data appear Lognormal at 5% Significance Level
Data appear Lognormal at 5% Significance Level
Background Statistics assuming Lognormal Distribution
95%UTLwith 35% Coverage 67.44 90% Pereentile (z) S.142
95%UPLft) 15,34 35% Pereentile {2} 9.S93
miJSL 15.73 99% Pereentile (z) 22.85
Note: Even though UPLs and UTLs computed using the lognormal distribution do not suffer from
transformation bias, a minor increase in the sd of logged data (from 1.152 to 1.257 above) causes a
significant increase in upper limits, especially in UTLs (from 52.5 to 67.44) computed using a small data
set (<15-20). This is particularly true when the data set consists of outliers.
Impact of Outlier, 16.1 ppb on the Computations of Upper Limits
Benzene data set without the outlier, 16.1 ppb, follows a normal distribution, and normal distribution
based upper limits without the outlier 16.1 are summarized as follows in Tables 5-10 (KM estimates), 5-
11 (ND by DL), and 5-12 (ND by DL/2).
Table 5-10 Normal Distribution Based Upper Limits Computed Using KM estimates
Normal GOF Test on Detects Only
Shapiro Wilk Test Statistic 0.847 Shapiro Wilk GOF Test
5*4 Shapiro Wilk Critical Value 0.788 Detected Data appear Normal at 5% Significance Level
Ulliefors Test Statistic Q.255 Ulliefors GOF Test
5°i Ulliefors Critical Value Q.362 Detected Data appear Normal at 5% Significance Level
Detected Data appear Normal at 5% Significance Level
Kaplan Meier (KM) Background Statistics Assuming Normal Distribution
Mean 1.086 SD 0.544
95% UTL95% Coverage 2.933 95%KMUPLft) 2.215
95% KM Chebyshev LI P L 3.519 30% KM Pereentile bj 1.732
95% KM Pereentile (z) 1.98 99% KM Pereentile (z) 2,35
95% KM USL 2.139
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Table 5-11 Normal Distribution Based Upper Limits Computed using DL for ND
Normal GOF Test
Shapiro Wilk Test Statistic 0.814 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value O.S03 Data appear Normal at 5% Significance Level
Ulliefors Test Statistic 0.263 Lilliefors GOF Test
5% Ulliefors Critical Value 0.335 Data appear Normal at 5% Significance Level
Data appear Normal at 5% Significance Level
Background Statistics Assuming Normal Distribution
95%UTLwith 95% Coverage 3.081 90% Percentile (z) 1.838
35%UPL|) 2.305 95% Percentile (z) 2.052
95%USL 2.224 99% Percentile (z) 2.452
Note: DL (=0.5) has been used for the ND value (does not accurately account for its ND status).
Therefore, upper limits are slightly higher than those computed using KM estimates.
Table 5-12 Normal Distribution Based Upper Limits Computed using DL/2 for ND
Normal GOF Test
Shapiro Wilk Test Statistic Q.875 Shapiro Wilk GOF Test
5% Shapiro Wilk Critical Value Q.SQ3 Data appear Normal at 5% Significance Level
Lilliefors Test Statistic Q.236 Lilliefors GOF Test
5% Ulliefors Critical Value 0.335 Data appear Normal at 5% Significance Level
Data appear Normal at 5X Significance Level
Background Statistics Assuming Normal Distribution
95%UTLwith 95% Coverage 3.2Q6 30% Percentile (z) 1.863
35%UPL{t) 2.36S 35% Percentile (z) 2.094
35% USL 2.2S 93% Percentile (z) 2.526
Note: DL/2 (=0.25) has been used for the ND value (does not accurately account for its ND status). The
use of DL/2 has increased the variance slightly which causes a slight increase in the various upper limits.
Therefore, upper limits are slightly higher than those computed using KM estimates and using DL for the
ND value. Based upon the benzene data set, normal UTL95-95 (= 2.93) computed using KM estimates
appears to represent a more realistic estimate of background threshold value.
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Example 5-4: The manganese (Mn) data set used in Chapter 15 of the Unified RCRA Guidance (2009)
has been used here to demonstrate how LROS method generates elevated BTVs. Summary statistics are
summarized as follows.
General Statistics for Censored Datasets (with NDs) using Kaplan Meief Method
Variable NumObs tt Missing Num Ds NumNDs % NDs Min ND Max ND KM Mean KM Vai KM SD KM CV
Mn 25 0 19 6 24.00% 2 5
General Statistics for Raw Dataset using Detected Data Onry
19.87
641
25.32
1274
Variable NumObs tt Missing Minimum Maximum Mean Median
Mn 19 0 3.3 106.3 25.46 12.6
Var
752.7
SD
27.44
MADALETSSkewness CV
9.34 1.942 1.079
Percentiles using all D elects (Ds) and N on-Detects (NDs)
Variable NumObs tt Missing IQXile 20%ile 25Xile(Q1] 50Xile(Q2) 75Xile(Q3) SOXrle 90Xile 95%ile 99Xile
Mn 25 0 2.52 5 5 10 21.6 25.06 50.52 72.48 99.32
The detected data follow a lognormal distribution, the maximum value in the data set is 106, and using the
LROS method (robust ROS method), one gets a 99% percentile = 183.4, and a UTL of 175. These
statistics are summarized in Table 5-13.
The detected data also follows a gamma distribution. Gamma-KM method based upper limits are
summarized as follows. The Gamma UTL95-95s (KM) are 92.5 (WH) and 99.32 (HW) and the 99%
percentiles are: 94.42 (WH) and 101.8 (HW). The Gamma UTL (KM) appears to represent a reasonable
estimate of BTV. These BTV estimates are summarized in Table 5-14.
Table 5-13 LROS and Lognormal KM Method Based Upper Limits
Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Detects
Mean in Original Scale 19.83 Mean in Log Scale 2.277
SDinOriginalS cale 25.87 S D in Log S cale 1.261
175.6 95% BCA UTL95% Coverage 106.3
106.3 95%UPL(t) 88.06
43.1 85% Percentile (z) 77.64
183.4 95%USL 280.4
95% UTL95% Coverage
95% Bootstrap (%) UTL95% Coverage
90% Percentile (z)
99% Percentile (z)
Statistics using KM estimates on Logged Data and Assuming Lognormal Drsfcrburjon
KM Mean of Logged Data 2.309 95% KM UTL (Lognormal)95% Coverage 151
KM S D of Logged D ata 1.182 95% KM U PL (Lognormal) 79.12
95% KM Percentile Lognormal (z) 70.31 95% KM USL (Lognormal) 234.1
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Table 5-14 Gamma KM Method Based Upper Limits
The following statistics are computed using gamma distribution and KM estuiates
U pper Limits using Wilson H ilferty (WH) and H awkins Wixley (HWJ Methods
k hat (KM) 0.616 mj hat (KM) 30.79
WH HW WH HW
95?; Approx. Gamma LITL with 95-t Coverage 92.4 99.32 95?; Approx. Gamma UPL 63.96 65.76
35% KM Gamma Percentile 59.5 60.7 95%Gamma(JSL 115.8 128.4
Notes: Even though one can argue that there is no transformation bias when computing lognormal
distribution based UTLs and UPLs, the use of a lognormal distribution on data with or without NDs often
yields inflated values which are not supported by the data set used to compute them. Therefore, its use
including LROS method should be avoided.
Before using a nonparametric BTV estimate, one should make sure that the detected data do not follow a
known distribution. When dealing with a data set with NDs, it is suggested to account for NDs and
determine the distribution of detected values instead of using a nonparametric UTL as used in Example
17-4 on page 17-21 of Chapter 17 of the EPA Unified Guidance, 2009. If detected data follow a
parametric distribution, one may want to compute a UTL using that distribution and KM estimates; this
approach will account for data variability instead defaulting to higher order statistics.
Summary and Recommendation
• It is recommended that occasional low probability outliers not be used in the computation of
decision making statistics. The decision making statistics (e.g., UCLs, UTLs, UPLs) should be
computed using observations representing the main dominant population. The use of a lognormal
distribution should be avoided in computing upper limits (UCLs, UTLs, UPLs) based upon data
sets with sd of detected logged data for moderately skewed to highly skewed data sets of sizes
smaller than 20-30. It is reasonable to state that, like uncensored data sets without NDs, the
minimum sample size requirement increases as the skewness increases.
• The project team should collectively make a decision about the disposition of outliers. It is often
helpful to compute decision statistics (upper limits) and hypothesis test statistics twice: once
including outliers, and once without outliers. By comparing the upper limits computed with and
without outliers, the project team can determine which limits are more representative of the site
conditions under investigation.
5.4 Computing Nonparametric Upper Limits Based upon Higher Order
Statistics
For full data sets without any discernible distribution, nonparametric UTLs and UPLs are computed using
higher order statistics. Therefore, when the data set consists of enough detected observations, and if some
of those detected data are larger than all of the NDs and the DLs, ProUCL computes USLs, UTLs, UPLs,
and upper percentiles by using nonparametric methods as described in Chapter 3. Since, nonparametric
UTLs, UPLs, USLs, and upper percentiles are represented by higher order statistics (or by some value in
between higher order statistic obtained using linear interpolation) every effort should be made to make
sure that those higher order statistics do not represent observations coming from population(s) other than
the main dominant (e.g., background) population under study.
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CHAPTER 6
Single and Two-sample Hypotheses Testing Approaches
Both single-sample and two-sample hypotheses testing approaches are used to make cleanup decisions at
polluted sites, and compare constituent concentrations of two (e.g., site versus background) or more (GW
in MWs) populations. This chapter provides guidance on when to use single-sample hypothesis test and
when to use two-sample hypotheses approaches. These issues were also discussed in Chapter 1 of this
Technical Guide. For interested users, this chapter presents a brief description of the mathematical
formulations of the various parametric and nonparametric hypotheses testing approaches as incorporated
in ProUCL. ProUCL software provides hypotheses testing approaches for data sets with and without ND
observations. For data sets containing multiple nondetects, a new two-sample hypothesis test, the Tarone-
Ware (T-W; 1978) test has been incorporated in the current ProUCL, versions 5.0 and 5.1. The developers
of ProUCL recommend supplementing statistical test results with graphical displays. It is assumed that
the users have collected an appropriate amount of good quality (representative) data, perhaps based upon
data quality objectives (DQOs). The Stats/Sample Sizes module can be used to compute DQOs based
sample sizes needed to perform the hypothesis tests described in this chapter.
6.1 When to Use Single Sample Hypotheses Approaches
When pre-established background threshold values and not-to-exceed values (e.g., USGS background
values, Shacklette and Boerngen 1984) exist, there is no need to extract, establish, or collect a background
or reference data set. Specifically, when not-to-exceed action levels or average cleanup standards are
known, one-sample hypotheses tests can be used to compare onsite data with known and pre-established
threshold values, provided enough onsite data needed to perform the hypothesis tests are available. When
the number of available site observations is less than 4-6, one might perform point-by-point site
observation comparisons with a BTV; and when enough onsite observations (> 8 to 10, more are
preferable) are available, it is suggested to use single-sample hypothesis testing approaches. Some recent
EPA guidance documents (EPA 2009) also recommend the availability of at least 8-10 observations to
perform statistical inference. Some minimum sample size requirements related to hypothesis tests are
also discussed in Chapter 1 of this Technical Guide.
Depending upon the parameter (e.g., the average value, juo, or a not-to-exceed action level, Ao),
representing a known threshold value, one can use single-sample hypothesis tests for the population mean
(t-test, sign test) or single-sample tests for proportions and percentiles. Several single-sample tests listed
below are available in ProUCL 5.1 and its earlier versions.
One-Sample t-Test: This test is used to compare the site mean,/^, with some specified cleanup standard, Cs
(jUo), where Cs represents a specified value of the true population mean, /u. The Student's t- test or UCL of
the mean is used (assuming normality of site data, or when the sample size is larger than 30, 50, or 100) to
verify the attainment of cleanup levels at a polluted sites (EPA 1989a, 1994). Note that the large sample
size requirement («= 30, 50, or 100) depends upon the data skewness. Specifically, as skewness increases
measured in terms of the sd, a, of the log-transformed data, the large sample size requirement also
increases to be able to apply the normal distribution and Student's t-statistic, due to the central limit
theorem (CLT).
One-Sample Sign Test or Wilcoxon Signed Rank (WSR) Test: These tests are nonparametric tests which
can also handle ND observations, provided all NDs and therefore their associated DLs are less than the
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specified threshold value, Cs. These tests are used to compare the site location (e.g., median, mean) with
some specified cleanup standard, Cs, representing the similar location measure.
One-Sample Proportion Test or Percentile Test: When a specified cleanup standard, Ao, such as a
preliminary remediation goal (PRG), or a compliance limit (CL) represents an upper threshold value of a
constituent concentration distribution rather than the mean threshold value, /n, a test for proportion or a
test for percentile (e.g., UTL95-95, UTL95-90) can be used to compare exceedances to the actionable
level. The proportion, p, of exceedances of A0 by site observations are compared to some pre-specified
allowable proportion, Po, of exceedances. One scenario where this test may be applied is following
remediation activities at an AOC. The proportion test can also handle NDs provided all NDs are below
the action level, A0.
It is beneficial to use DQO-based sampling plans to collect an appropriate amount of data. In any case, in
order to obtain reasonably reliable estimates and compute reliable test statistics, an adequate amount of
representative site data (at least 8 to 10 observations) should be made available to perform the single-
sample hypotheses tests listed above. As mentioned before, if only a small number of site observations are
available, instead of using hypotheses testing approaches, point-by-point site concentrations may be
compared with the specified action level, Ao. Individual point-by-point observations are not to be
compared with the average cleanup or threshold level, Cs. The estimated sample mean, such as a UCL95,
is compared with a threshold representing an average cleanup standard.
6.2 When to Use Two-Sample Hypotheses Testing Approaches
When BTVs, not-to-exceed values, and other cleanup standards are not available, then site data are
compared directly with the background data. In such cases, a two-sample hypothesis testing approach is
used to perform site versus background comparisons provided enough data are available from each of the
two populations. Note that this approach can be used to compare concentrations of any two populations
including two different site areas or two different MWs. The Stats/Sample Sizes module of ProUCL can
be used to compute DQO-based sample sizes for two-sample parametric and nonparametric hypothesis
testing approaches. While collecting site and background data, for better representation of populations
under investigation, one may also want to account for the size of the background area (and site area for
site samples) in sample size determinations. That is, a larger number (>10 to 15) of representative
background (or site) samples may need to be collected from larger background (or site) areas to capture
the greater inherent heterogeneity/variability typically present in larger areas.
The two-sample hypotheses approaches are used when the site parameters (e.g., mean, shape, distribution)
are compared with the background parameters (e.g., mean, shape, distribution). Specifically, two-sample
hypotheses testing approaches can be used to compare the average (also medians or upper tails)
constituent concentrations of two or more populations such as the background population and the
potentially contaminated site areas. Several parametric and nonparametric two-sample hypotheses testing
approaches, including Student's t-test, the Wilcoxon-Mann-Whitney (WMW) test, Gehan's test, and the
T-W test are included in ProUCL 5.1. Some details of those methods are described in this chapter for
interested users. It is recommended that statistical results and test statistics be supplemented with
graphical displays, such as the multiple Q-Q plots and side-by-side box plots as graphical displays do not
require any distributional assumptions and are not influenced by outlying observations and NDs.
Data Types: Analytical data sets collected from the two (or more) populations should be of the same type
obtained using similar analytical methods and sampling equipment. Additionally, site and background
data should be all discrete or all composite (obtained using the same design, pattern, and number of
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increments), and should be collected from the same medium (soil) at comparable depth levels (e.g., all
surface samples or all subsurface samples) and time (e.g., during the same quarter in groundwater
applications). Good sample collection methods and sampling strategies are described in Gerlach, R. W.,
and J. M. Nocerino (2003) and the ITRC Technical Regulatory guidance document (2012).
6.3 Statistical Terminology Used in Hypotheses Testing Approaches
The first step in developing a hypothesis test is to state the problem in statistical terminology by
developing a null hypothesis, Ho, and an alternative hypothesis, HA. These hypotheses tests result in two
alternative decisions: acceptance of the null hypothesis or the rejection of the null hypothesis based on the
computed hypothesis test statistic (e.g., t-statistic, WMW test statistic). The statistical terminologies
including error rates, hypotheses statements, Form 1, Form 2, and two-sided tests, are explained in terms
of two-sample hypotheses testing approaches. Similar terms apply to all parametric and nonparametric
single-sample and two-sample hypotheses testing approaches. Additional details may be found in EPA
guidance documents (2002b, 2006b), and MARSSIM (2000) or in statistical text books including Bain
and Engelhardt (1992), Hollander and Wolfe (1999), and Hogg and Craig (1995).
Two forms, Form 1 and Form 2, of the statistical hypothesis test are useful for environmental
applications. The null hypothesis in the first form (Form 1) states that the mean/median concentration of
the potentially impacted site area does not exceed the mean/median of the background concentration. The
null hypothesis in the second form (Form 2) of the test is that the concentrations of the impacted site area
exceed the background concentrations by a substantial difference, S, with S>0.
Formally, let Xi, X2, ..., Xn represent a random sample of size n collected from Population 1 (e.g.,
downgradient MWs or a site AOC) with mean (or median) fix, and Yi, Y2, ..., Ym represent a random
sample of size m from Population 2 (upgradient MWs or a background area) with mean (or median) /JY.
Let A = /Jx- JUY represent the difference between the two means (or medians).
6.3.1 Test Form 1
The null hypothesis (Ho): The mean/median of Population 1 (constituent concentration in samples
collected from potentially impacted areas (or monitoring wells)) is less than or equal to the mean/median
of Population 2 (concentration in samples collected from background (or upgradient wells) areas) with
Ho: A < 0.
The alternative hypothesis (HA). The mean/median of Population 1 (constituent concentration in samples
collected from potentially impacted areas) is greater than the mean of Population 2(background areas)
with HA: A > 0.
When performing this form of hypothesis test, the collected data should provide statistically significant
evidence that the null hypothesis is false leading to the conclusion that the site mean/median does exceed
background mean/median concentration. Otherwise, the null hypothesis cannot be rejected based on the
available data, and the mean/median concentration found in the potentially impacted site areas is
considered equivalent and comparable to that of the background areas.
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6.3.2 Test Form 2
The null hypothesis (Ho): The mean/median of Population 1 (constituent concentration in potentially
impacted areas) exceeds the mean/median or Population 2 (background concentrations) by more than S
units. Symbolically, the null hypothesis is written as Ho: A > S, where S>0.
The alternative hypothesis (HA): The mean/median of Population 1 (constituent concentration in
potentially impacted areas) does not exceed the mean/median of Population 2 (background constituent
concentration) by more than S (HA: A < S).
Here, S is the background investigation level. When S>0, Test Form 2 is called Test Form 2 with
substantial difference, S. Some details about this hypothesis form can be found in the background
guidance document for CERCLA sites (EPA 2002b).
6.3.3 Selecting a Test Form
The test forms described above are commonly used in background versus site comparison evaluations.
Therefore, these test forms are also known as Background Test Form 1 and Background Test Form 2
(EPA, 2002b). Background Test Form 1 uses a conservative investigation level of A = 0, but relaxes the
burden of proof by selecting the null hypothesis that the constituent concentrations in potentially impacted
areas are not statistically greater than the background concentrations. Background Test Form 2 requires a
stricter burden of proof, but relaxes the investigation level from 0 to S.
6.3.4 Errors Rates and Confidence Levels
Due to the uncertainties that result from sampling variation, decisions made using hypotheses tests will be
subject to errors, also known as decision errors. Decisions should be made about the width of the gray
region, A, and the degree of decision errors that is acceptable. There are two ways to err when analyzing
sampled data (Table 6-1) to derive conclusions about population parameters.
Type I Error: Based on the observed collected data, the test may reject the null hypothesis when in fact
the null hypothesis is true (a false positive or equivalently a false rejection). This is a Type I error. The
probability of making a Type I error is often denoted by a (alpha); and
Type II Error: On the other hand, based upon the collected data, the test may fail to reject the null
hypothesis when the null hypothesis is in fact false (a false negative or equivalently a false acceptance).
This is called Type II error. The probability of making a Type II error is denoted by /? (beta).
Table 6-1. Hypothesis Testing: Type I and Type II Errors
Decision Based on
Sample Data
Ho is not rejected
Ho is rejected
Actual Site Condition
Ho is True
Correct Decision: (1 - a)
Type I Error:
False Positive (a)
Ho is not true
Type II Error:
False Negative ((3)
Correct Decision: (1 - (3)
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The acceptable level of decision error associated with hypothesis testing is defined by two key
parameters: confidence level and power. These parameters are related to two error probabilities, a and/?.
Confidence level 100(1- oc)%: As the confidence level is lowered (or alternatively, as a is increased), the
likelihood of committing a Type I error increases.
Power 100(1 - fi)%: As the power is lowered (or alternatively, as /? is increased), the likelihood of
committing a Type II error increases.
Although a range of values in the interval (0, 1) can be selected for these two parameters, as the demand
for precision increases, the number of samples and the associated cost (sampling and analytical cost) will
generally also increase. The cost of sampling is often an important determining factor in selecting the
acceptable level of decision errors. However, unwarranted cost reduction at the sampling stage may incur
greater costs later in terms of increased threats to human health and the environment, or unnecessary
cleanup at a site area under investigation. The number of samples, and hence the cost of sampling, can be
reduced but at the expense of a higher possibility of making decision errors that may result in the need for
additional sampling, or increased risk to human health and the environment.
There is an inherent tradeoff between the probabilities of committing a Type I or a Type II error, a
simultaneous reduction in both types of errors can only occur by increasing the number of samples. If the
probability of committing a false positive error is reduced by increasing the level of confidence associated
with the test (in other words, by decreasing a), the probability of committing a false negative is increased
because the power of the test is reduced (increasing /?). The choice of a determines the probability of the
Type I error. The smaller the a-value, the less likely to incorrectly reject the null hypothesis (H0).
However, a smaller value for a also means lower power with decreased probability of detecting a
difference when one exists. The most commonly used a value is 0.05. With a = 0.05, the chance of
finding a significance difference that does not really exist is only 5%. In most situations, this probability
of error is considered acceptable.
Suggested values for the Two Types of Error Rates: Typically, the following values for error probabilities
are selected as the minimum recommended performance measures:
• For the Background Test Form 1, the confidence level should be at least 80% (a = 0.20) and the
power should be at least 90% (J3 = 0.10).
• For the Background Test Form 2, the confidence level should be at least 90% (a = 0.10) and the
power should be at least 80% (ft = 0.20).
Seriousness of the Two Types of Error Rates:
• When using the Background Test Form 1, a Type I error (false positive) is less serious than a Type II
error (false negative). This approach favors the protection of human health and the environment. To
ensure that there is a low probability of committing a Type II error, a Test Form 1 statistical test
should have adequate power at the right edge of the gray region.
• When the Background Test Form 2 is used, a Type II error is preferable to committing a Type I error.
This approach favors the protection of human health and the environment. The choice of the
hypotheses used in the Background Test Form 2 is designed to be protective of human health and the
environment by requiring that the data contain evidence of no substantial contamination.
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6.4 Parametric Hypotheses Tests
Parametric statistical tests assume that the data sets follow a known statistical distribution (mostly
normal); and that the data sets are statistically independent with no expected spatial and temporal trends
in the data sets. Many statistical tests (e.g., two-sample t-test) and models are only appropriate for data
that follow a particular distribution. Statistical tests that rely on knowledge of the form of the population
distribution of data are known as parametric tests. The most commonly used distribution for tests
involving environmental data is the normal distribution. It is noted that GOF tests which are used to
determine data set's distribution (e.g., S-W test for normality) often fail if there are not enough
observations, if the data contain multiple populations, or if there is a high proportion of NDs in the
collected data set. Tests for normality lack statistical power for small sample sizes. In this context, a
sample consisting of less than 20 observations may be considered a small sample. However, in practice,
many times it may not be possible, due to resource constraints, to collect data sets of sizes greater than 10.
This is especially true for background data sets, as the decision makers often do not want to collect many
background samples. Sometimes they want to make cleanup decisions based upon data sets of sizes even
smaller than 10. Statistics computed based upon small data sets of sizes < 5 cannot be considered reliable
to derive important decisions affecting human health and the environment.
6.5 Nonparametric Hypotheses Tests
Statistical tests that do not assume a specific statistical form for the data distribution(s) are called
distribution-free or nonparametric statistical tests. Nonparametric tests have good test performance for a
wide variety of distributions, and their performances are not unduly affected by NDs and the outlying
observations. In two-sample comparisons (e.g., t-test), if one or both of the data sets fail to meet the test
for normality, or if the data sets appear to come from different distributions with different shapes and
variability, then nonparametric tests may be used to perform site versus background comparisons.
Typically, nonparametric tests and statistics require larger size data sets to derive correct conclusions.
Several two-sample nonparametric hypotheses tests, the WMW test, Gehan test, and Tarone-Ware (T-W)
test, are available in ProUCL. Like the Gehan test, the T-W test is used for data sets containing NDs with
multiple RLs. The T-W test was new in ProUCL 5.0 and is also included in ProUCL 5.1.
The relative performances of different testing procedures can be assessed by comparing, />-values
associated with those tests. The p-value of a statistical test is defined as the smallest value of a (level of
significance, Type I error) for which the null hypothesis would be rejected based upon the given data sets
of sampled observations. The />-value of a test is sometimes called the critical level or the significance
level of the test. Whenever possible, critical values and/"-values have been computed using the exact or
approximate distribution of the test statistics (e.g., GOF tests, t-test, Sign test, WMW test, Gehan test, M-
K trend test).
Performance of statistical tests is also compared based on their robustness. Robustness means that the test
has good performance for a wide variety of data distributions, and that its performance is not significantly
affected by the occurrence of outliers. Not all nonparametric methods are robust and resistant to outliers.
Specifically, nonparametric upper limits used to estimate BTVs can get affected and misrepresented by
outliers. This issue has been discussed earlier in Chapter 3 of this Technical Guide.
• If a parametric test for comparing means is applied to data from a non-normal population and the
sample size is large, then a parametric test may work well, provided that the data sets are not heavily
skewed. For heavily skewed data sets, the sample size requirement associated with the CLT can
become quite large, sometimes larger than 100. A brief simulation study elaborating on the sample
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size requirements to apply the CLT on skewed data sets is given in Appendix B. For moderately
skewed (Chapter 4) data sets, the CLT ensures that parametric tests for the mean will work because
parametric tests for the mean are robust to deviations from normal distributions as long as the sample
size is large. Unless the population distribution is highly skewed, one may choose a parametric test
for comparing means when there are at least 25-30 data points in each group.
• If a nonparametric test for comparing means is applied on a data set from a normal population and the
sample size is large, then the nonparametric test will work well. In this case, the p-values tend to be a
little too large, but the discrepancy is small. In other words, nonparametric tests for comparing means
are only slightly less powerful than parametric tests with large samples.
• If a parametric test is applied on a data set from a non-normal population and the sample size is small
(< 20 data points), then the/"-value may be inaccurate because the CLT does not apply in this case.
• If a nonparametric test is applied to a data set from a non-normal population and the sample size is
small, then the /"-values tend to be too high. In other words, nonparametric tests may lack statistical
power with small samples.
Notes: It is suggested that the users supplement their test statistics and conclusions by using graphical
displays for visual comparisons of two or more data sets. ProUCL software has side-by-side box plots and
multiple Q-Q plots that can be used to graphically compare two or more data sets with and without ND
observations.
6.6 Single Sample Hypotheses Testing Approaches
This section describes the mathematical formulations of parametric and nonparametric single-sample
hypotheses testing approaches incorporated in ProUCL software. For the sake of interested users, some
directions to perform these hypotheses tests are described as follows. The directions are useful when the
user wants to manually perform these tests.
6.6.1 The One-Sample t-Testfor Mean
The one-sample t-test is a parametric test used for testing a difference between a population (site area,
AOC) mean and a fixed pre-established mean level (cleanup standard representing a mean concentration
level). The Stats/Sample Sizes module of ProUCL can be used to determine the minimum number of
observations needed to achieve the desired DQOs. The collected sample should be a random sample
representing the AOC under investigation.
6.6.1.1 Limitations and Robustness of One-Sample t-Test
The one-sample t-test is not robust in the presence of outliers and may not yield reliable results in the
presence of ND observations. Do not use this test when dealing with data sets containing NDs. Some
nonparametric tests described below may be used in cases where NDs are present in a data set. This test
may yield reliable results when performed on mildly or moderately skewed data sets. Note that levels of
skewness are discussed in Chapters 3 and 4. The use of a t-test should be avoided when data are highly
skewed (sd of log-transformed data exceeding 1, 1.5), even when the data set is of a large size such as
«=100.
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6.6.1.2 Directions for the One-Sample t-Test
Let Xj, x2, . . . , xn represent a random sample (analytical results) of size, n, collected from a population
(AOC). The use of the One-Sample t-Test requires that the data set follows a normal distribution; that is
when using a typical software package (e.g., Minitab), the user needs to test for the normality of the data
set. For the sake of users and to make sure that users do not skip this step, ProUCL verifies normality of
the data set automatically.
STEP 1: Specify an average cleanup goal or action level, fjto (G), and choose one of the following
combination of null and alternative hypotheses:
Form 1: Ho: site p <^ovs. HA: site ^ > ^o
Form 2: Ho: site ^i>^o vs. HA: site p < jUo
Two-Sided: Ho: site ju = jUo vs. HA: site ju ^ jUo.
Form 2 with substantial difference, S: Ho: site ^>^o+Svs. HA: site p < ^o + S, here S> 0.
STEP 2: Calculate the test statistic:
'.=
In the above equation, S is assumed to be equal to "0", except for Form 2 with substantial difference.
STEP 3: Use Student's t-table (ProUCL computes them) to find the critical value tn-i, i-a
Conclusion:
Form 1: If to > tn-i,a, then reject the null hypothesis that the site population mean is less than the cleanup
level, ^io
Form 2: If to < -tn-i,a, then reject the null hypothesis that the site population mean exceeds the cleanup
level, ^io
Two-Sided: If \to > tn-i, 0/2, then reject the null hypothesis that the site population mean is same as the
cleanup level, jUo
Form 2 with substantial difference, S: Ifto< -tn-i, i-a, then reject the null hypothesis that the site population
mean is more than the cleanup level, po + the substantial difference, S. Here, tn-i,a represents the critical
value from t-distribution with (n-\) degrees of freedom (df) such that the area to the right of tn-i,a under
the t-distribution probability density function is a.
6.6.1.3 P -values
In addition to computing critical values (some users still like to use critical values for a specified a),
ProUCL computes exact or approximate /^-values. A />-value is the smallest value for which the null
hypothesis is rejected in favor of the alternative hypotheses. Thus, based upon the given data set, the null
hypothesis is rejected for all values of a (the level of significance) greater than or equal to the />-value.
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The details of computing a p-value for a t-test can be found in any statistical text book such as Daniel
(1995). ProUCL computes /^-values for t-tests associated with each form of the null hypothesis.
Specifically, if the computed p-value is smaller than the specified value of, a, the conclusion is to reject
the null hypothesis based upon the collected data set.
6.6.1.4 Relation between One-Sample Tests and Confidence Limits of the Mean or
Median
There has been some confusion among the users whether to use a LCL or a UCL of the mean to determine
if the remediated site areas have met the cleanup standards. There is a direct relation between one sample
hypothesis tests and confidence limits of the mean or median. For example, depending upon the
hypothesis test form, a t-test is related to the upper or lower confidence limit of the mean, and a Sign test
is related to the confidence limits of the median. In confirmation sampling, either a one sample hypothesis
test (e.g., t-test, WSR test) or a confidence interval of the mean (e.g., LCL, UCL) can be used. Both
approaches result in the same conclusion.
These relationships have been illustrated for the t-test and the LCLs and upper UCLs for normally
distributed data sets. The use of a UCL95 to determine if a polluted site has attained the cleanup standard,
HO, after remediation is very common. If a UCL95 < HO, then it is concluded that the site meets the
standard. The conclusion based upon the UCL or LCL, or the interval (LCL, UCL) is derived from
hypothesis test statistics. For an example, while using a 95% lower confidence limit (LCL95), one is
testing hypothesis test Form 1, and when using UCL95, one is testing hypothesis Form 2.
For a normally distributed data set: x,,x2, . . . , xn ( e.g., collected after excavation), the UCL95 and
LCL95 are given as follows:
UCL95 = x+tn_^05
Objective: Does the site average, H, meet the cleanup level, /^o?
Form 1: Ho: site H HO
Form 2: Ho: site H>HO vs. HA: site H < Ho
Two-Sided: Ho: site H = HO vs. a HA: site H ^ Ho-
Based upon the t-test, conclusions are:
Form 1: If t> tn-i, o.os, then reject the null hypothesis in favor of the alternative hypothesis
Form 2: If to < -tn-i, o.os, then reject the null hypothesis in favor of the alternative hypothesis
Two-Sided: If \to > tn-i, 0.025, then reject the null hypothesis that the site population mean is same as the
cleanup level
Here tn-i, o.os represents a critical value from the right tail of the t-distribution with («-l) degrees of
freedom such that area to right of tn-i, o.os is 0.05.
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For Form 1, we have:
Reject Ho ift>tn.i,o.os , that is reject the null hypothesis when
Equivalently reject the null hypothesis and conclude that site has not met the cleanup standard when
x - tn_l 005*sd/ 4n > fJ0; or when LCL95>cleanup goal, po.
The site is concluded dirty when LCL95> po.
For Form 2, we have:
Reject Ho if t< -tn-i,o.o5 , that is reject the null hypothesis when
Equivalently reject the null hypothesis and conclude that site meets the cleanup standard when
x + tn_1005 * sd 1 4n < ju0 ; or when UCL95 < ju0.
The site is concluded clean when UCL95< jUo.
6.6.2 The One-Sample Test for Proportions
The one-sample test for proportions represents a test for evaluating the difference between the population
proportion, P, and a specified threshold proportion, P0. Based upon the sampled data set and sample
proportion,/), of exceedances of a pre-specified action level, Ao, by the n sample observations (e.g., onsite
observations); the objective is to determine if the population proportion (of exceedances of the threshold
value, Ao) exceeds the pre-specified proportion level, Po. This proportion test is equivalent to a sign test
(described next), when Po = 0.5. The Stats/Sample Sizes module of ProUCL can be used to determine the
minimum sample size needed to achieve pre-specified DQOs.
6.6.2.1 Limitations and Robustness
Normal approximation to the distribution of the test statistic is applicable when both (nPo) and n (1- Po)
are at least 5. For smaller data sets, ProUCL uses the exact binomial distribution (e.g., Conover, 1999) to
compute the critical values when the above statement is not true.
The Proportion test may also be used on data sets with ND observations, provided all ND values (DLs,
reporting limits) are smaller than the action level, Ao.
6.6.2.2 Directions for the One-Sample Test for Proportions
Let xh x2, . . . , xn represent a random sample (data set) of size, n, from a population (e.g., the site (e.g.,
exposure area) under investigation. Let Ao represent a compliance limit or an action level to be met by site
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data. It is expected (e.g., after remediation) that the proportion of site observations exceeding the action
level, Ao, is smaller than the specified proportion, P0.
Let B = number of site values in the data set exceeding the action level, Ao. A typical observed sample
value of B (based upon a data set) is denoted by b. It is noted that the random variable, B follows a
binomial distribution (BD) ~ B(«, P) with n equal to the number of trials and P being the unknown
population proportion (probability of success). Under the null hypothesis, the variable B follows a
binomial distribution (BD) ~ B(«, Po).
The sample proportion, p=b/n = (number of site values in the sample > Ao)/n
STEP 1: Specify a proportion threshold value, Po, and state the following null hypotheses:
Form 1: H0: P Po
Form 2: H0: P>Po vs. HA: P < Po
Two-Sided: H0: P = Po vs. HA: P+Po
STEP 2: Calculate the test statistic:
+ C-P
(6-2)
Where c =
Ip0(i-p0y
V /n
-0.5
n V'P > ° x(# of site values > A0)
and p = —
0.5.n
n
Here c is the continuity correction factor for use of the normal approximation.
Large Sample Normal Approximation
STEP 3: Typically, one should use BD (as described above) to perform this test. However, when both
(nPo) and n (1- Po) are at least 5, a normal (automatically computed by ProUCL) approximation may be
used to compute the critical values (z-values) and/"-values.
STEP 4: Conclusion described for the approximate test based upon the normal approximation:
Form 1: If ZQ > za, then reject the null hypothesis that the population proportion, P, of exceedances of
action level, Ao, is less than the specified proportion, Po.
Form 2: If ZQ < -za, then reject the null hypothesis that the population proportion, P, is more than the
specified proportion, Po.
Two-Sided: If ZQ > z«/2, then reject the null hypothesis that the population proportion, P, is the same as
the specified proportion, P0.
Here, z« represents the critical value of a standard normal variable, Z, such that area to the right of z«
under the standard normal curve is a.
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P-Values Based upon a Normal Approximation
As mentioned before, a/"-value is the smallest value for which the null hypothesis is rejected in favor of
the alternative hypotheses. Thus, based upon the given data, the null hypothesis is rejected for all values
of a (the level of significance) greater than or equal to the /"-value. The details of computing a/"-value for
the proportion test based upon large sample normal approximation can be found in any statistical text
book such as Daniel (1995). ProUCL computes large sample /"-values for the proportion test associated
with each form of null hypothesis.
6.6.2.3 Use of the Exact Binomial Distribution for Smaller Samples
ProUCL 5.0 also performs the proportion test based upon the exact binomial distribution when the sample
size is small and one may not be able to use the normal approximation as described above. ProUCL 5.0
checks for the availability of appropriate amount of data, and performs the tests using a normal
approximation or the exact binomial distribution accordingly.
STEP 1: When the sample size is small (e.g., < 30), and either (nPo), or n (1 - Po) is less than 5, one
should use the exact BD to perform this test. ProUCL 5.0 performs this test based upon the BD, when the
above conditions are not satisfied. In such cases, ProUCL 5.0 computes the critical values and/"-values
based upon the BD and its cumulative distribution function (CDF). The probability statements concerning
the computation of/"-values can be found in Conover (1999).
STEP 2: Conclusion Based upon the Binomial Distribution
Form 1: Large values of B cause the rejection of the null hypothesis. Therefore, reject the null hypothesis,
when B > b. Here b is obtained using the binomial cumulative probabilities based upon a BD (n, Po). The
critical value, b (associated with a) is given by the probability statement: P(B>b) = a, or equivalently,
P(B < b) = (1 - a). Since B is a discrete binomial random variable, the level, a may not be exactly
achieved by the critical value, b.
Form 2: For this form, small values of B will cause the rejection of the null hypothesis. Therefore, reject
the null hypothesis, when B < b. Here b is obtained using the binomial cumulative probabilities based
upon a BD(«, Po). The critical value, b is given by the probability statement: P(B) = a. As mentioned
before, since B is a discrete binomial random variable, the level, a may not be exactly achieved by the
critical value, b.
Two-Sided Alternative: The critical or the rejection region for the null hypothesis is made of two areas,
one in the right tail (of area ~ a2) and the other in the left tail (with area ~ ai), so that the combined area of
the two tails is approximately, a = a; + 0.2. That is for this hypothesis form, both small values and large
values of B will cause the rejection of the null hypothesis. Therefore, reject the null hypothesis, when B <
bi or B > \)2 Typically 0.1 and 0.2 are roughly equal, and in ProUCL, both are chosen to be equal to a /2; bi
and b2 are given by the probability statements: P (B < bi) ~ a/2, and P(B > bj) ~ a/2. B being a discrete
binomial random variable, the level, a may not be exactly achieved by the critical values, bi and b2.
P-Values Based upon Binomial Distribution as Incorporated In ProUCL: The probability statements for
computing a p-value for a proportion test based upon BD can be found in Conover (1999). Using the BD,
ProUCL computes /"-values for the proportion test associated with each form of null hypothesis. If the
computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis
based upon the collected data set used in the computations. There are some variations in the literature
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regarding the computation of p-values for a proportion test based upon the exact BD. Therefore, the p-
value computation procedure as incorporated in ProUCL 5.0 is described below.
Let b be the calculated value of the binomial random variable, B under the null hypothesis. ProUCL 5.0
computes the/>-values using the following probability (Prob) statements:
Form 1: p-value = Prob(B > b)
Form 2: p-value = Prob(B < b)
Two-sided Alternative:
For b>(n- b): .P-value = 2* Prob(B < b)
For b<(n- b): /'-value = 2*Prob(B > b)
6.6.3 The Sign Test
The Sign test is used to detect a difference between the population median and a fixed cleanup goal, C
(e.g., representing the desired median value). Like the WSRtest, the Sign test can also be used on paired
data to compare the location parameters of two dependent populations. This test makes no distributional
assumptions. The Sign test is used when the data are not symmetric and the sample size is small (EPA,
2006). The Stats/Sample Sizes module of ProUCL can be used to determine minimum number of
observations needed to achieve pre-specified DQOs associated with the Sign test.
6.6.3.1 Limitations and Robustness
Like the Proportion test, the Sign test can also be used on data sets with NDs, provided all values reported
as NDs are smaller than the cleanup level/action level, C. For data sets with NDs, the process to perform
a Sign test is the same as that for data sets without NDs, provided DLs associated with all NDs are less
than the cleanup level. Per EPA guidance document (2006), all NDs exceeding the action level are
discarded from the computation of Sign test statistic; also all observations, detects and NDs equal to the
action level are discarded from the computation of the Sign test statistic. Discarding of observations
(detects and NDs) will have an impact on the power of the test (reduced power). ProUCL has the Sign test
for data sets with NDs as described in USEPA (2006). However, the performance of the Sign test on data
sets with NDs requires some evaluation.
6.6.3.2 Sign Test in the Presence ofNondetects
A principal requirement when applying the sign test is that the cleanup level, C, should be greater than the
largest ND value; in addition all observations (detects and NDs) equal to the action level and all NDs
greater than or equal to the action level are discarded from the computation of the Sign test statistic.
6.6.3.3 Directions for the Sign Test
Let xb x2, . . . , xn represent a random sample of size n collected from a site area under investigation. As
before, S > 0 represents the substantial difference used in Form 2 hypothesis tests.
STEP 1: Letjux be the site population median.
State the following null and the alternative hypotheses:
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Form 1: H0: JUX C
Form 2: H0: JUX>C vs. HA: JUX < C
Two-Sided: H0: jux = C vs. HA: JUX^C
Form 2 with substantial difference, S: H0: jux >C+Svs. HA: jux < C + S
STEP 2: Calculate the n differences, dt = x. -C . If some of the 4=0, then reduce the sample size until
all the remaining \dt\>0. This means that all observations (detects and NDs) tied at C are ignored from the
computation. Compute the binomial random variable, B representing the number oft/. > 0, /': = l,2,...,n.
Note that under the null hypothesis, the binomial random variable, B follows a binomial distribution (BD)
~ BD (n, 5/2) where n represents the reduced sample size after discarding observations as described above.
Thus, one can use the exact BD to compute the critical values and /"-values associated with this test.
STEP 3: For n < 40, ProUCL computes the exact BD based test statistic, B; and
For n > 40, one may use the approximate normal test statistic given by,
B-n-S
(6.3)
The substantial difference, S =0, except for Form 2 hypotheses with substantial difference.
STEP 4: For n < 40, one can use the BD table as given in EPA (2006). These critical values are
automatically computed by ProUCL) to calculate the critical values. For n > 40, use the normal
approximation and the associated normal z critical values.
STEP 5: Conclusion when n < 40 (following EPA 2006):
Form 1: If B > BUFFER (n, 2a), then reject the null hypothesis that the population median is less than the
cleanup level, C.
Form 2: If B < BUFFER (n, 2a), then reject the null hypothesis that the population median is more than the
cleanup level.
Two-Sided: If B > BUFFER (n, a) or B < BUFFER (n, a) - 1, then reject the null hypothesis that the
population median is comparable to the cleanup level, C.
Form 2 with substantial difference, S: If B < BUFFER (n, 2a), then reject the null hypothesis that the
population median is more than the cleanup level, C + substantial difference, S.
ProUCL calculates the critical values andp-values based upon the BD (n, ^2) for both small samples and
large samples.
Conclusion: Large Sample Approximation when n>40
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Form 1: If zo > za, then reject the null hypothesis that population median is less than the cleanup level, C.
Form 2: If zo <- za, then reject the null hypothesis that the population median is greater than the cleanup
level, C.
Two-Sided: If zo > 7.0/2, then reject the null hypothesis that the population median is comparable to the
cleanup level, C.
Form 2 with substantial difference, S: If zo <- za, then reject the null hypothesis that the population
median is more than the cleanup level, C + substantial difference, S.
Here, za represents the critical value of a standard normal distribution (SND) such that area to the right of
za under the standard normal curve is a.
P-Values for One-Sample Sign Test
ProUCL calculates the critical values and p-values based upon: the BD(«, Yi) for small data sets; and
normal approximation for larger data sets as described above.
6.6.4 The Wilcoxon Signed Rank Test
The Wilcoxon Signed Rank (WSR) test is used for evaluating the difference between the location
parameter (mean or median) of a population and a fixed cleanup standard such as C, with Cs representing
a location value. It can also be used to compare the medians of paired populations (e.g., placebo versus
treatment). Hypotheses about parameters of paired populations require that data sets of equal sizes are
collected from the two populations.
6.6.4.1 Limitations and Robustness
For symmetric distributions, the WSR test appears to be more powerful than the Sign test. However,
WSR test tends to yield incorrect results in the presence of many tied values. On data sets with NDs, the
process to perform a WSR test is the same as that for data sets without NDs once all NDs are assigned
some surrogate value. However, like the Sign test, not much guidance is available in the literature for
performing WSR test on data sets consisting of ND observations. The WSR test for data sets with NDs as
described in USEPA (2006) and incorporated in ProUCL requires further investigation especially when
multiple DLs with NDs exceeding the detects are present in the data set.
For data sets with NDs with a single DL, DL, a surrogate value of DL/2 is used for all ND values (EPA,
2006). The presence of multiple DLs makes this test less powerful. It is suggested not to use this test
when multiple DLs are present with NDs exceeding the detected values. Per EPA (2006) guidance, when
multiple DLs are present, then all detects and NDs less than the largest DL may be censored which tends
to reduce the power of the test. In ProUCL 5.0, all NDs including the largest ND value are replaced by
half of their respective reporting limit values. All detected values are used as reported.
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6.6.4.2 Wilcoxon Signed Rank (WSR) Test in the Presence ofNondetects
Following the suggestions made in the EPA guidance document (2006), ProUCL uses the following
process to perform WSR test in the presence of NDs.
• For left-censored data sets with a single DL (it is preferred to have all detects greater than the
NDs), it is suggested (EPA, 2006) to replace all NDs by DL/2. This suggestion (EPA, 2006) has
been used in the WSR test as incorporated in ProUCL software. Specifically, if there are k ND
values with the same DL, then they are considered as "ties" and are assigned the average rank for
this group.
• The presence of multiple DLs makes this test less powerful. When multiple DLs are present, then
all NDs are replaced by half of their respective DLs. All detects are used as reported.
6.6.4.3 Directions for the Wilcoxon Signed Rank Test
Let xh x2, . . . , xn represent a random sample of size, n collected from a site area under investigation, and
C represent the cleanup level.
STEP 1: State/select one of the following null hypotheses:
Form 1: Ho: Site location < C vs. HA: Site location > C
Form 2: Ho: Site location > C vs. HA: Site location < C
Two-Sided: Ho: Site location = C vs. HA: Site location ^ C
Form 2 with substantial difference, S: Ho: Site location >C+Svs. Ha: Site location < C + S, here S > 0.
STEP 2: Calculate the deviations, dt = xt—C. If some dt =0, then reduce the sample size until all
\dt\ > 0. That is, ignore all observations withe/. =0.
STEP 3: Rank the absolute deviations, \dt\, from smallest to the largest. Assign an average rank to the
tied observations.
STEP 4: Let R be the signed rank of \dt\, where the sign of Ri is determined by the sign ofdt.
STEP 5: Test statistic calculations:
For n < 20, compute T+ = \, Rt , where T+ is the sum of the positive signed ranks.
{iJJjX)}
For n > 20, use a normal approximation and compute the test statistic given by
z0 = , \ [ (6-4)
Vv
Here var \T+ j is the variance of T+ and is given by
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var (T+ ) = y^ t,(tt2 -1); g = number of tied groups.
V / 24 48^;V; >* B P
STEP 6: Conclusion when n < 20:
Form 1: Larger values of the test statistic, T+, will cause the rejection of the Form 1 null hypothesis. That
is if T+ > — - wa = W(i-a), then reject the null hypothesis that the location parameter is less than the
cleanup level, C.
Form 2: Smaller values of the test statistic will cause the rejection of the Form 2 null hypothesis. If
r+ < wa, then reject the null hypothesis that the location parameter is greater than the cleanup level, C.
77(77 + 1)
Two-Sided Alternative: If T+ > — — waj2 or T+ < wa!2 , then reject the null hypothesis that the
location parameter is comparable to the action level, C.
Form 2 with substantial difference, S: lfT+ 20:
Form 1: Ifzo> za, then reject the null hypothesis that location parameter is less than the cleanup level, C.
Form 2: If zo < - za, then reject the null hypothesis that the location parameter is greater than the cleanup
level, C.
Two-Sided: If zo > Za/2, then reject the null hypothesis that the location parameter is comparable to the
cleanup level, C.
Form 2 with substantial difference, S: If zo <- za, then reject the null hypothesis that the location
parameter is more than the cleanup level, C + the substantial difference, S.
It should be noted that WSR can be used to compare medians (means when data are symmetric) of two
correlated (paired) data sets.
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Notes: The critical values, wa as tabulated in EPA (2006b) have been programmed in ProUCL. For
smaller data sets with n < 20 the/?-values are computed using the BD; and for larger data sets with n > 20
the normal approximation is used to compute the critical values and/"-values.
Example 6-1: Consider the aluminum and thallium concentrations of the real data set used in Example 2-
4 of Chapter 2. Please note that the aluminum data set follows a normal distribution and the thallium data
set does not follow a discernible distribution. One-sample t-test (Form 2), Proportion test (2-sided) and
WRS test (Form 1) results are shown below.
Single-sample t-Test, Ho.- Aluminum Mean Concentration > 10000
Date/Time of Computation
From File
Full Precision
Confidence Coefficient
Substantial Difference
.Action Level
3/9/2-013 8:46:40 AM
SuperFund^ds
OFF
95%
Q.OQQ
10QQQ.QOQ
Selected Null Hypothesis Mean ?= Action Level (Form 2)
Aftemative Hypothesis Mean < the Action Level
Aluminum
One Sample t-Test
HO: Sample Mean >= 10000 (Form 2)
Raw Statistics
Number of Valid Observations 24
Number of Distinct Observations 24
Minimum 1710
Maximum 16200
Mean 7789
Median 7012
SD 4264
SE of Mean 870.4
Test Value -2.54
Degrees of Freedom 23
Critical Value (0.05) -1.714
P-Value 0.00915
Conclusion with Alpha = 0.05
Reject HO. Conclude Mean < 10000
P-Value < Alpha (0.05)
Conclusion: Reject the null hypothesis and conclude that mean aluminum concentration <10000.
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Single-sample Proportion Test
Ho'. Proportion, P, of exceedances by thallium values exceeding the action level of 0.2 is equal to 0.1, vs.
HA'. Proportion of exceedances is not equal to 0.1.
Confidence Coefficient 95%
User Specified Proportion 0.100 (PQ of Exceedances of Action Level)
Action/compliance Limit 0.200
Select Null Hypothesis Sample Proportion, P of Exceedances of Action Level = User Specified Proportion (2 Sided Alternative)
Alternative Hypothesis Sample Proportion, P of Exceedances of Action Level o User Specified Proportion
Thallium
One Sample Proportion Test
Raw Statistics
Number of Valid Observations 24
Number of Distinct Observations 18
Minimum 0.0€6
Maximum 0.456
Mean 0.147
Median 0.07
SD 0.133
SE of Mean 0.0271
Number of Exceedances 6
Sample Proportion of Exceedances 0.25
HO: Sample Proportion = 0.1
.Approximate P-Value 0.0349
Conclusion with Alpha = 0.05
Reject HO. Conclude Sample Proportion <> 0.1
Conclusion: Proportion of thallium concentrations exceeding 0.2 is not equal to 0.1.
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Single-sample WRS Test
Ho'. Median of thallium concentrations <0.2
Confidence Coefficient
Substantial Difference
Action Level
95%
0.006
0.200
Selected Null Hypothesis Mean/Median <= Action Level (Form 1}
.Alternative Hypothesis Mean/Median > the Action Level
Thallium
One Sample Wilcoxon Signed Rank Test
Raw Statistics
Number of Valid Observations 24
Number of Distinct Observations 18
Minimum 0.066
Maximum 0.456
Mean 0.147
Median 0.07
SD 0.133
SE of Mean 0.0271
Number Above Action Level 6
Number Equal Action Level 0
Number Below Action Level 18
T-plus 93
T-minus 207
HO: Sample Mean/Median <= 0.2 (Form 1)
Large Sample z-Test Statistic -1.644
Critical Value (0.05) 1.645
P-Value 0.95
Conclusion with Alpha = 0.05
Do Not Reject HO. Conclude Mean/Median <= 0.2
P-Value > Alpha (0.05)
Conclusion: Do not reject the null hypothesis and conclude that median of thallium concentrations < 0.2.
Example 6-2: Consider the blood lead-levels data set discussed in the environmental literature (Helsel,
2013). The data set consists of several NDs. The box plot shown in Figure 6-1 suggests that median of
lead concentrations is less than the action level. The WSR tests the null hypothesis: Median lead
concentrations in blood > action level of 0.1
Box PloS for Blood Pb
Figure 6-1. Box Plot of Lead in Blood Data Comparing Pb Concentrations with the Action Level of 0.1
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Hood_Pb
One Sample Wilcoxon Signed Rank Test
Raw Statistics
Number of Valid Data 27
Number of Distinct Data 13
Number of Non-Detects 19
Number of Detects 8
Percent Non-Detects 70.37%
Minimum Non-detect 0.0137
Maximum Non-detect O.Q2
Minimum Detect 0.0235
Maximum Detect 0.2S3
Mean of Detects 0.107
HO: Sample Median >= 0.1 (Form 2)
Median of Detects O.Q776
SD of Detects 0.0911 , - , T . -,.... ,-~
Large Sample z-Test Statistic -3.66/
Median of Processed Data used in WSR 0.01 Critical Value {0.05} -1.645
Number Above .Action Level 4 P-Value 12291E-4
Number Equal .Action Level 0
Number Below .Action Level 23 Conclusion with Alpha = 0.05
T-plus 39 Reject HO. Conclude Mean/Median < 0.1
T-minus 339
P-Value < Alpha (0.05)
Conclusion: Both the graphical display and the WSR test suggest that median of lead concentrations in
blood is less than 0.1.
6.7 Two-sample Hypotheses Testing Approaches
The use of parametric and nonparametric two-sample hypotheses testing approaches is quite common in
environmental applications including site versus background comparison studies. Several of those
approaches for data sets with and without ND observations have been incorporated in the ProUCL
software. Additionally some graphical methods (box plots and Q-Q plots) for data sets with and without
NDs are also available in ProUCL to visually compare two or more populations.
Student's two-sample t-test is used to compare the means of the two independently distributed normal
populations such as the potentially impacted site area and a background reference area. Two cases arise:
1) the variances (dispersion) of the two populations are comparable, and 2) the variances of the two
populations are not comparable. Generally, a t-test is robust and not sensitive to minor deviations from the
assumptions of normality.
6.7.1 Student's Two-sample t-Test (Equal Variances)
6.7.1.1 Assumptions and their Verification
Xi, X2, ..., Xn represent site samples and Yi, ¥2, ... , Ym represent background samples that are collected at
random from the two independent populations. The validity of random samples and independence
assumptions may be confirmed by reviewing the procedures described in EPA (2006b). Let X and Y
represent the sample means of the two data sets. Using the GOF tests (available in ProUCL 5.0 under
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Statistical Tests Module), one needs to verify that the two data sets are normally distributed. If both m and
n are large (and the data are mildly to moderately skewed), one may make this assumption without further
verification (due to the CLT). If the data sets are highly skewed (skewness discussed in Chapters 3 and 4),
the use of nonparametric tests such as the WMW test supplemented with graphical displays is preferable.
6.7.1.2 Limitations and Robustness
The two-sample t-test with equal variances is fairly robust to violations of the assumption of normality.
However, if the investigator has tested and rejected normality or equality of variances, then nonparametric
procedures such as the WMW may be applied. This test is not robust to outliers because sample means
and standard deviations are sensitive to outliers. It is suggested that a t-test not be used on log-
transformed data sets as a t-test on log -transformed data tests the equality of medians and not the equality
of means. For skewed distributions there are significant differences between mean and median. The
Student's t- test assumes the equality of variances of the two populations under comparison; if the two
variances are not equal and the normality assumption of the means is valid, then the Satterthwaite's t-test
(described below) can be used.
In the presence of NDs, it is suggested to use a Gehan test or T-W (new in ProUCL 5.0) test. Sometimes,
users tend to use a t-test on data sets obtained by replacing all NDs by surrogate values, such as respective
DL/2 values, or DL values. The use of such methods can yield incorrect results and conclusions. The use
of substitution methods (e.g., DL/2) should be avoided.
6. 7. 1. 3 Guidance on Implementing the Student 's Two-sample t-Test
The number of site (Population 1), n and background (Population 2), m measurements required to conduct
the two-sample t-test should be calculated based upon appropriate DQO procedures (EPA [2006a,
2006b]). In case, it is not possible to use DQOs, or to collect as many samples as determined using DQOs,
one may want to follow the minimum sample size requirements as described in Chapter 1. The
Stats/Sample Sizes module of ProUCL can be used to determine DQOs based sample sizes. ProUCL also
has an F-test to verify the equality of two variances. ProUCL automatically performs this test to verify
the equality of two dispersions. The user should review the output for the equality of variances test
conclusions before using one of the two tests: Student's t-test or Satterthwaite's t-test. If some
measurements appear to be unusually large compared to the majority of the measurements in the data set,
then a test for outliers (Chapter 7) should be conducted. Once any identified outliers have been
investigated to determine if they are mistakes or errors and, if necessary, discarded, the site and
background data sets should be re-tested for normality using formal GOF tests and normal Q-Q plots.
The project team should decide the proper disposition of outliers. In practice, it is advantageous to carry
out the tests on data sets with and without the outliers. This extra step helps the users to assess and
determine the influence of outliers on the various test statistics and the resulting conclusions. This process
also helps the users in making appropriate decisions about the proper disposition (include or exclude from
the data analyses) of outliers.
6. 7. 1. 4 Directions for the Student 's Two-sample t-Test
X2, . . . , Xn represent a random sample collected from a site area (Population 1) and Yi, Y2, . . . ,
Ym represent a random data set collected from another independent population such as a background
population. The two data sets are assumed to be normally distributed or mildly skewed.
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STEP 1: State the following null and the alternative hypotheses:
Form 1: H0: /Jx -/JY < 0 vs. HA: /Jx -/JY > 0
Form 2: H0: fJx —fJY > 0 vs. HA: fJx — fJY < 0
Two-Sided: H0: fJx — fJY = 0 vs. HA: fJx —fJY ^ 0
Form 2 with substantial difference, S: H0: /Jx -/^ > S vs. HA: jUx -fa < S
STEP 2: Calculate the sample mean X and the sample variance Sx for the site (e.g., Population 1,
Sample 1) data and compute the sample mean Y and the sample variance Sy for the background data
(e.g., Population 2, Sample 2).
STEP 3: Determine if the variances of the two populations are equal. If the variances of the two
populations are not equal, use the Satterthwaite's test. Calculate the pooled sd, Sp and the t-test statistic, to.
(6-5)
(x-T)-s
to = - - , ' (6-6)
Here S = 0, except when used in Form 2 hypothesis with substantial difference, S > 0.
STEP 4: Compute the critical value tm+n-2,i-a such that 100(1 - a) % of the t-distribution with (m + n - 2)
dfis below tm+n.2,i-a.
STEP 5: Conclusion:
Form 1: Ifto> tm+n-2, i-a, then reject the null hypothesis that the site population mean is less than or equal
(comparable) to the background population mean.
Form 2: If to < -tm+n-2, i-a, then reject the null hypothesis that the site population mean is greater than or
equal to the background population mean.
Two-Sided: If to > tm+n-2,1-0/2, then reject the null hypothesis that the site population mean comparable to
the background population mean.
Form 2 with substantial difference, S: If to <- tm+n-2, i- a, then reject the null hypothesis that the site mean is
greater than or equal to the background population mean + the substantial difference, S.
6.7.2 The Satterthwaite Two-sample t-Test (Unequal Variances)
Satterthwaite's t-test is used to compare two population means when the variances of the two populations
are not equal. It requires the same assumptions as the two-sample t-test (described above) except for the
assumption of equal variances.
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6.7.2.1 Limitations and Robustness
In the presence of NDs, replacement by a surrogate value such as the DL or DL/2gives biased results. As
mentioned above, the use of these substitution methods should be avoided. Instead the use of
nonparametric tests such as the Gehan test or Tarone-Ware test is suggested when the data sets consist of
NDs. In cases where the assumptions of normality of means are violated, the use of nonparametric tests
such as the WMW test is preferred.
6.7.2.2 Directions for the Satterthwaite Two-sample t-Test
LetXi, X2, . . . , Xn represent random site (Population 1) samples and Yi, Y2, . . . , Ym represent random
background (Population 2) samples collected from two independent populations.
STEP 1: State the following null and the alternative hypotheses:
Form 1: H0: jUx - ]UY < 0 vs. HA: jUx - ]UY > 0
Form 2: H0: jux - JUY > 0 vs. HA: jUx - jUY < 0
Two-Sided: Ho: fj.x — fJ.Y = 0 vs. HA: £ix —//7 ^ 0
Form 2 with substantial difference, S: Ho: jUx — jL^^Svs. HA: jUx — Jur 0.
STEP 4: Use a t-table (ProUCL computes them) to find the critical value ti-a such that 100(1 - a)% of the
t-distribution with df degrees of freedom is below ti-a, where the Satterthwaite's Approximation for dfis
given by:
df =
SX | SY
n m
(6-8)
n (n-V) m (m-V)
STEP 5: Conclusion:
Form 1: If to > % i-a, then reject the null hypothesis that the site (Population 1) mean is less than or equal
(comparable) to the background (Population 2) mean.
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Form 2: \£to< -% i-a, then reject the null hypothesis that the site (Population 1) mean is greater than or
equal to the background (Population 2) mean.
Two-Sided: If to > % 1-0/2, then reject the null hypothesis that the site (Population 1) mean is comparable
to the background (Population 2) mean.
Form 2 with substantial difference, S: If to < -% ;- a, then reject the null hypothesis that the site
(Population 1) mean is greater than or equal to the background (Population 2) mean + the substantial
difference, S.
P-Values for Two-sample t-Test
A /rvalue is the smallest value for which the null hypothesis is rejected in favor of the alternative
hypotheses. Thus, based upon the given data, the null hypothesis is rejected for all values of a (the level
of significance) greater than or equal to the /"-value. ProUCL computes (based upon an appropriate t-
distribution) /^-values for two-sample t-tests associated with each form of the null hypothesis. If the
computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis
based upon the collected data set used in the various computations.
6.8 Tests for Equality of Dispersions
This section describes a test that verifies the assumption of the equality of two variances. This assumption
is needed to perform a simple two-sample Student's t-test described above.
6.8.1 The F- Test for the Equality of Tw o- Variances
An F-test is used to verify whether the variances of two populations are equal. Usually the F-test is
employed as a preliminary test, before conducting the two-sample t-test for the equality of two means.
The assumptions underlying the F-test are that the two-samples represent independent random samples
from two normal populations. The F-test for equality of variances is sensitive to departures from
normality. There are other statistical tests such as the Levene's test (1960) which also tests the equality of
the variances of two normally distributed populations. However, the inclusion of the Levene test will not
add any new capability to the software. Therefore, taking the budget constraints into consideration, the
Levene's test has not been incorporated in the ProUCL software.
Moreover, it should be noted that, although it makes sense to first determine if the two variances are equal
or unequal, this is not a requirement to perform a t-test. The t-distribution based confidence interval or
test for Hi - fi2 based on the pooled sample variance does not perform better than the approximate
confidence intervals based upon Satterthwaite's test. Hence testing for the equality of variances is not
required to perform a two-sample t-test. The use of Welch-Satterthwaite's or Cochran's method is
recommended in all situations (see, for example, F. Hayes [2005]).
6.8.1.1 Directions for the F-Test
Let Xl5 X2, . . . , Xn represent the n data points from site (Population 1) and Y1; Y2, . . . , Ym represent the
m data points from background (Population 2). To manually perform an F-test, one can proceed as
follows:
STEP 1: Calculate the sample variances S% (for the X's) and Sy (for the Y's)
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22 22
STEP 2: Calculate the variance ratios Fx= SX/SY and FY= SY fsx • Let F equal the larger of these two
values. If F = Fx, then let k = n - 1 and q = m - 1. If F = Fy, then let k = m - 1 and q = n- 1.
STEP 3: Using a table of the F- distribution (ProUCL 5.0 computes them), find a cutoff, U =f1_a/2(k, q)
associated with the F distribution with k and q degrees of freedom for some significance level, a. If the
calculated F value > U, conclude that the variances of the two populations are not equal.
P-Values for Two-sample Dispersion Test for Equality of Variances
ProUCL computes /"-values for the two-sample F-test based upon an appropriate F-distribution. If the
computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis
based upon the collected data sets.
Example 6-3: Consider a real manganese data set collected from an upgradient well (Well 1) and two
downgradient MWs (Wells 2 and 3). The side-by-side box plots comparing concentrations of the three
wells are shown in Figure 6-2. The two-sample t-test comparing the manganese concentrations of the two
downgradient MWs are summarized in Table 6-2.
Box Plot for Mn
Figure 6-2. Box Plots Comparing Concentrations of Three Wells: One Upgradient and Two
Downgradient
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Table 6-2. T-Test Comparing Mn in MW8 vs. MW9
HQ: Mean Mn concentrations of MW 8 and MW9 are comparable
Selected Null Hypothesis Sample 1 Mean = Sample 2 Mean (Two Sided Aftemative)
Alternative Hypothesis Sample 1 Mean <> Sample 2 Mean
Sample 1 Data: Mn-89{8)
Sample 2 Data: Mn-89(9)
Raw Statistics
Number of Valid Observations
Number of Distinct Observations
Minimum
Maximum
Mean
Median
SD
SE of Mean
Sample 1 vs Sample 2 Two-Sample t-Test
Sample 1
16
16
1270
4600
1998
1750
B3S.S
209.7
Sample 2
16
15
1050
3080
1968
2055
500.2
125
HO: Mean of Sample 1 = Mean of Sample 2
t-Test
Value
0.123
0.123
Method DF
Pooled (Equal Variance) 30
Welch-Satterthwaite (Unequal Variam 24.5
Pooled SD: 690.548
Conclusion with Alpha = D.G5C
Student t [Pooled): Do Not Reject HO. Conclude Sample 1 = Sample 2
Welch-Satterthwalte: Do Not Reject H2. Conclude Sample 1 = Sample 2
Lower C.Val Upper C.V'al
t (0.025) t (0.975} P-Value
-2.042 2.042 0.903
-2.0€4 2.064 0.903
Test of Equality of Variances
Variance of Sample 1 703523
Variance of Sample 2 250190
Numerator DF Denominator DF
15 15
Conclusion with .Alpha = 0.05
Two variances appear to be equal
F-Test Value
2.812
P-Value
0,054
Conclusion: The variances of the two populations are comparable, both the t-test and Satterthwaite test
lead to the conclusion that there are no significant differences in the mean manganese concentrations of
the two downgradient monitoring wells.
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6.9 Nonparametric Tests
When the data do not follow a discernible distribution, the use of parametric statistical tests may lead to
inaccurate conclusions. Additionally, if the data sets contain outliers or ND values, an additional level of
uncertainty is faced when conducting parametric tests. Since most environmental data sets tend to consist
of observations from two or more populations including some outliers and ND values, it is unlikely that
the current wide-spread use of parametric tests is justified, given that these tests may be adversely
affected by outliers and by the assumptions made for handling ND values. Several nonparametric tests
have been incorporated in ProUCL that can be used on data sets consisting of ND observations with
single and multiple DLs.
6.9.1 The Wilcoxon-Mann-Whitney (WMW) Test
The Mann-Whitney (M-W) (or WMW) test (Bain and Engelhardt, 1992) is a nonparametric test used for
determining whether a difference exists between the site and the background population distributions.
This test is also known as the WRS test. The WMW test statistic tests whether or not measurements
(location, central) from one population consistently tend to be larger (or smaller) than those from the
other population based upon the assumption that the dispersion/shapes of the two distributions are
roughly the same (comparable).
6.9.1.1 Advantages and Disadvantages
The main advantage of the WMW test is that the two data sets are not required to be from a known type
of distribution. The WMW test does not assume that the data are normally distributed, although a normal
distribution approximation is used to determine the critical value of the WMW test statistic for large
sample sizes. The WMW test may be used on data sets with NDs provided the DL or the reporting limit
(RL) is the same for all NDs. If NDs with multiple DLs are present, then the largest DL is used for all ND
observations. Specifically, the WMW test handles ND values by treating them as ties. Due to these
constraints, other tests such as the Gehan test and theTarone-Ware test are better suited to perform two-
sample tests on data sets consisting of NDs. The WMW test is more resistant to outliers than two-sample
t-tests discussed earlier. It should be noted that the WMW test does not place enough weight on the larger
site and background measurements. This means, a WMW may lead to the conclusion that two populations
are comparable even when the observations in the right tail of one distribution (e.g., site) are significantly
larger than the right tail observations of the other population (e.g., background). Like all other tests, it is
suggested that the WMW test results be supplemented with graphical displays.
6.9.1.2 WMW Test in the Presence ofNondetects
If there are t ND values with a single DL, then they are considered as "ties" and are assigned the average
rank for this group. If more than one DL is present in the data set, then WMW test censors all of the
observations below the largest DL, and are treated as NDs at the largest DL. This of course results in loss
of power associated with WMW test.
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6. 9. 1. 3 WMW Test Assumptions and Their Verification
The underlying assumptions of the WMW test are:
• The soil sample measurements obtained from the site and background areas are statistically and
spatially independent (not correlated). This assumption requires: 1) that an appropriate probability-
based sampling design strategy be used to determine (identify) the sampling locations of the soil
samples for collection, and 2) those soil sampling locations are spaced far enough apart that a spatial
correlation among concentrations at different locations is not likely to be present.
The probability distribution of the measurements from a site area (Population 1) is similar to (e.g.,
including variability, shape) the probability distribution of measurements collected from a
background or reference area (Population 2). The assumption of equal variances of the two regions:
site vs. background should also be evaluated using descriptive statistics and graphical displays such as
side-by-side box plots. The WMW test may result in an incorrect conclusion if the assumption of
equality of variability is not met.
6.9.1.4 Directions for the WMW Test when the Number of Site and Background
Measurements is small (n < 20 or m <20)
i, X-2, . . . ,Xn represent systematic and random site samples (Group 1, Sample 1) and Yi, ¥2, . . . , Ym
represent systematic and random background samples (Group 2, Sample 2) collected from two
independent populations. It should be noted that instead of 20, some texts suggest to use 10 as a small
sample size for the two populations.
STEP 1: Let fj.x represent site (Population 1) median and JUY represent the background (Population 2)
median. State the following null and the alternative hypotheses:
Form I: Ho. jux- JUY <0 vs.HA: JUX-JUY >0
Form 2: H0: JUX-JUY > 0 vs. HA: JUX-JUY < 0
Two-Sided: H0: jux- JUY = 0 vs. HA: jux- JUY ^0
Form 2 with substantial difference, S: H0: JUX-JUY >S vs. HA: JUX-JUY
-------
• If a few less-than values (NDs) occur (say, < 10%), and if all such values are less than the
smallest detected measurement in the pooled data set, then treat all NDs as tied values at the
reported DL or at an arbitrary (when no DL is reported) value less than the smallest detected
measurement. Assign the average of the ranks that would otherwise be assigned to these tied less-
than values (the same procedure as for tied detected measurements). Today with the availability
of advanced technologies and instruments, instead of reporting NDs as less-than values, NDs are
typically reported at DL levels below which the instrument cannot accurately measure the
concentrations present in a sample. The use of DLs is particularly helpful when NDs are reported
with multiple DLs (RLs).
• If between 10% and 40% of the pooled data set are reported as NDs, and all are less than the
smallest detected measurement, then one may use the approximate WMW test procedure
described below provided enough (e.g., n > 10 and m > 10) data are available. However, the use
of the WMW test is not recommended in the presence of multiple DLs or RLs with NDs larger
than the detected values.
STEP 3: Calculate the sum of the ranks of the n site measurements. Denote this sum by Ws and then
calculate the Mann-Whitney (M-W), ^/-statistic as follows:
U = Ws-n(n + l)/2 (6-9)
The test proposed by Wilcoxon based upon the rank sum, Ws is called the WRS test. The test based upon
the f/-statistic given by (6-9) was proposed by Mann and Whitney and is called the WMW test. These two
tests are equivalent tests and yield the same results and conclusions. ProUCL outputs both statistics;
however the conclusions are derived based upon the U-statistic and its critical and p-values. Mean and
variance of the U-statistic are given as follows:
Notes: Note the difference between the definitions of U and Ws. Obviously the critical values for Ws and
U are different. However, critical values for one test can be derived from the critical values of the other
test by using the relationship given by the above equation (6-9). These two tests (WRS test and WMW
test) are equivalent tests, and the conclusions derived by using these test statistics are equivalent. For data
sets of small sizes (with m or n <20), ProUCL computes exact as well as normal distribution based
approximate critical values. For large samples with n and m both greater than 20, ProUCL computes
normal distribution based approximate critical values and/>-values.
STEP 4: For specific values of n, m, and a, find an appropriate WMW critical value, wa, from the table
as given in EPA (2006) and also in Daniel (1995). These critical values have been incorporated in the
ProUCL software.
STEP 5: Conclusion:
Form 1 : lfU>nm-wa, then reject the null hypothesis that the site population median is less than or equal
to the background population median.
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Form 2: If U< wa, then reject the null hypothesis that the site population median is greater than or equal
to the background population median.
Two-Sided: If U > nm - Wa/2 or U< Wa/2, then reject the null hypothesis that the site population median
(location) is comparable to that of the background population median (location).
Form 2 with substantial difference, S: If U< wa, then reject the null hypothesis that the site population
median is greater than or equal to the background population median + the substantial difference, S. S
takes a positive value only for this form of the hypothesis with substantial difference, in all other forms of
the null hypothesis, S = 0.
P-Values for Two-sample WMW Test for Small Samples
For small samples, ProUCL computes only approximate (as computed for large samples) />-values for the
WMW test. Details of computing approximate /"-values are given in the next section for larger data sets.
If the computed p-value is smaller than the specified value of, a, the conclusion is to reject the null
hypothesis based upon the collected data set.
6.9.1.5 Directions for the WMW Test when the Number of Site and Background
Measurements is Large (n > 20 andm > 20)
It should be noted that some texts suggest that both n and m needs to be >10 to be able to use the large
sample approximation. ProUCL uses large sample approximations when «>20 and m>20.
STEP 1: As before, let fj.x represent the site and JUY represent the background population medians
(means). State the following null and the alternative hypotheses:
Form 1: H0: JUX-JUY < 0 vs. HI: jux-juY > 0
Form 2: H0: jux- JUY > 0 vs. HI: jux- JUY < 0
Two-Sided: H0: JUX-JUY = 0 vs. Hi: jux-juY ^0
Form 2 with substantial difference, S: Ho: jux — jUY > S vs. jux — JUY < S
Note that when the Form 2 hypothesis is used with substantial difference, S, the value S is added to all
observations in the background data set before ranking the combined data set of size (n+rri). For data sets
with NDs, the Form 2 hypothesis test with substantial difference, S is not incorporated in ProUCL 5.0.
STEP 2: List and rank the pooled set of n + m site and background measurements from smallest to
largest, keeping track of which measurements came from the site and which came from the background
area. Assign the rank of 1 to the smallest value among the pooled data, the rank of 2 to the second
smallest value among the pooled data, and so forth. All observations tied at a give value, x0, are assigned
the average rank of the observations tied at x0. The same process is used for all tied values.
• The WMW test is not recommended when many NDs observations with multiple DLs and /or
NDs exceeding the detected values are present in the data sets. Other tests such as the T-W and
Gehan tests also available in ProUCL 5.0 are better suited for data sets consisting of many NDs
with multiple DLs and/or NDs exceeding detected values.
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• It should however be noted these nonparametric tests (WMW test, Gehan test, and T-W test)
assume that the shape (variability) of the two data distributions (e.g., background and site) are
comparable. If this assumption is not met, these tests may lead to incorrect test statistics and
conclusions.
STEP 3: Calculate the sum of the ranks of the site (Population 1) measurements. Denote this sum by Ws.
ProUCL 5.1 computes the WMW test statistics by adjusting for tied observations using equation (6-11);
that is the large sample variance of the WMW test statistic is computed using equation (6-11) which
adjusts forties.
STEP 4: When no ties are present, calculate the approximate WMW test statistic, Zo as follows:
(
nm(n +
12
The above test statistic, Z0 is equivalent to the following approximate Z0 statistic based upon the Mann-
Whitney f/-statistic:
U-nm/2
nm
(n + m + i)
.12
When ties are present in the combined data set of size (n+rri), the adjusted large sample approximate test
value, Z0 is computed by using the following equation:
=
(nm 10 andn > 10
Let Xi, X2, . . . , Xn represent data points from the site population and Yi, ¥2, . . . , Ym represent
background data from the background population. Like the WMW test, this test also assumes that the
variabilities of the two distributions (e.g., background vs. Site, MW1 vs. MW2) are comparable. Since we
are dealing with data sets consisting of many NDs, the use of graphical methods such as the side-by-side
box plots and multiple Q-Q plots is also desirable to compare the spread/variability of the two data
distributions. For data sets of sizes larger than 10 (recommended), a test based upon normal
approximations is described in the following.
STEP 1: Let jux represent the site and JUY represent the background population medians. State the
following null and the alternative hypotheses:
Form 1: H0: jux-juY < 0 vs. HA: JUX-JUY > 0
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Form 2: H0: jux- JUT > 0 vs. HA: JUX-JUT < 0
Two-Sided: H0: jux- JUY = 0 vs. HA: jux- JUY ^0
For data sets with NDs, the Form 2 hypothesis test with substantial difference, S is not incorporated in
ProUCL 5.0/5.1. The user may want to adjust their background data sets accordingly to perform this
hypothesis test form.
STEP 2: List the combined m background and n site measurements, including the ND values, from
smallest to largest, where the total number of combined samples is N = m + n. The DLs associated with
the ND (or less-than values) observations are used when listing the TV data values from smallest to largest.
STEP 3: Determine the N ranks, Ri, R2, ... , Rn, for the N ordered data values using the method described
in the example given below.
STEP 4: Compute the N scores, a(Ri), a(R2), ... , a(Rn), using the formula a(Ri) = 2R, - N - 7, where / is
successively set equal to 7, 2, ... , N.
STEP 5: Compute the Gehan statistic, G, as follows:
G =
(6-12)
mn
Where s ' or
u=°
/z; = 7 if the ith datum is from the site population
hi = 0 if the ith datum is from the background population
N = n + m
a(Rt) = 2 R,•- N-l, as indicated above.
STEP 6: Use the normal z-table to get the critical values.
STEP 7: Conclusion based upon the approximate normal distribution of the G-statistic:
Form 1: If G > z;_a, then reject the null hypothesis that the site population median is less than or equal to
the background population median.
Form 2: If G <- z;_«, then reject the null hypothesis that the site population median is greater than or equal
to the background population median.
Two-Sided: If |G| >z;_o/2, then reject the null hypothesis that the site population median is same as the
background population median.
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P-Values for Two-sample Gehan Test
For the Gehan's test, p-values are computed using a normal approximation for the Gehan's G-statistic.
The /"-values can be computed using the simple procedure as used for computing large sample /"-values
for the two-sample nonparametric WMW test. ProUCL computes p-values for the Gehan test for each
form of the null hypothesis. If the computed p-value is smaller than the specified value of, a, the
conclusion is to reject the null hypothesis based upon the collected data set used in the various
computations.
6.9.3 Tarone-Ware (T-W) Test
Like the Gehan test, the T-W test (1978) is a nonparametric test which can be used to test for the
differences between the distributions of two populations (e.g., two sites, site versus background, two
monitoring wells) when the data sets have multiple censoring points and DLs. The T-W test as described
below has been incorporated in ProUCL 5.0 and 5.1. It is noted that the Gehan and T-W tests yield
comparable test results.
6.9.3.1 Limitations and Robustness
The T-W test can be used when the background and/or site data sets contain multiple NDs with different
DLs and NDs exceeding detected values.
• If the censoring mechanisms are different for the site and background data sets, then the test
results may be an indication of this difference in censoring mechanisms (e.g., high DLs due to
dilution effects) rather than an indication that the null hypothesis is rejected.
• Like the Gehan test, the T-W test can be used when many ND observations or multiple DLs may
be present in the two data sets; conclusions derived using this test may not be reliable when
dealing with samples of small sizes (<10). Like the Gehan test, the T-W test described below is
based upon the normal approximation of the T-W statistic and should be used when enough (e.g.,
m>10 and n>10) site and background (or monitoring well) data are available.
6.9.3.2 Directions for the Tarone-Ware Test when m > 10 andn > 10
Let Xi, X2, . . . , Xn represent n data points from the site population and Yi, Y2, . . . , Ym represent sample
data from the background population. Like the Gehan test, this test also assumes that the variabilities of
the two data distributions (e.g., background vs. site, monitoring wells) are comparable. One may use
exploratory graphical methods to informally verify this assumption. Graphical displays are not affected
by NDs and outlying observations.
STEP 1: Let jux represent the site and //7 represent the background population medians. The following
null and alternative hypotheses can be tested:
Form 1: H0: jux- JUY < 0 vs. HA: jux- JUY > 0
Form 2: H0: jux-juY > 0 vs. HA: JUX-JUY < 0
Two-Sided: H0: jux- JUY = 0 vs. HA: jux- JUY ^0
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STEP 2: Let TV denote the number of distinct detected values in the combined background and site data
set of size (n+rri) including the ND values. Arrange the N distinct detected measurements in the
combined data set in ascending order from smallest to largest. Note that N will be less than n+m. Let
Zj < z2 < z3 <...< ZN represent N distinct ordered detected values in the data set of size, (n+m).
STEP 3: Determine the N ranks, Ri, R2, ..., RN, for the N ordered distinct detected data values:
Zj < z2 < z3 <... < ZN in the combined data set of size (n+rri).
STEP 4: Count the number, «,, 1=1,2, ..., TV of detects and NDs (reported as DLs or reporting limits) less
than or equal to z; in the combined data set of size (n+rri). For each distinct detected value, z, compute ct
= number of detects exactly equal to z*; 1=1,2,... .N
STEP 5: Repeat Step 4 on the site data set. That is count the number, mt ,1=1,2,... .N of detects and NDs
(reported as DLs or reporting limits) less than or equal to z* in site data set of size, («). Also, for each
distinct detected value, z,, compute dt = number of detects in the site data set exactly equal to zt;
i=l,2, ....N. Finally, compute, li,1=1,2, ....N, the number of detects and NDs (reported as DLs or reporting
limits) less than or equal to z* in background data set of size (rri).
STEP 6: Compute the expected value and variance of detected values in the site data set of size, n, using
the following equations:
ESite(Detectioii) = ci *mt /nt (6-13)
VSlte (Detection) = c, * (n, - c, )m,l, /(n? (n, -1)) (6-14)
STEP 7: Compute the normal approximation of the TW test statistic using the following equation:
T-W= l=l , (6-15)
STEP 8: Conclusion based upon the approximate normal distribution of the T-W statistic:
Form 1 : If T-W> z;_«, then reject the null hypothesis that the site population median is less than or equal to
the background population median.
Form 2: If T-W<- zi.a, then reject the null hypothesis that the site population median is greater than or
equal to the background population median.
Two-Sided: If \T-W\ >z 1.0/2, then reject the null hypothesis that the site population median is same as the
background population median.
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P-Values for Two-sample T-W Test
Critical values and p-values for the T-W test are computed following the same procedure as used for the
Gehan test. ProUCL computes normal distribution based approximate critical values and /"-values for the
T-W test for each form of the null hypothesis. If the computed p-value is smaller than the specified value
of, a, the conclusion is to reject the null hypothesis based upon the data set used in the computations.
Example 6-5. The copper (Cu) and zinc (Zn) concentrations data with NDs (from Millard and Deverel
1988) collected from groundwater of the two zones, Alluvial Fan and Basin Trough, is used to perform
the Gehan and T-W tests using ProUCL 5.0. Box plots comparing Cu in the two zones are shown in
Figure 6-3 and box plots comparing Zn concentrations in the two zones are shown in Figure 6-4.
Box Plot for Cu
Figure 6-3. Box plots Comparing Cu in Two Zones: Alluvial Fan versus Basin Trough
Box Plot for Zn
Figure 6-4. Box Plots Comparing Zn in Two Zones: Alluvial Fan versus Basin Trough
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Table 6-4. Gehan Test Comparing the Location Parameters of Copper (Cu) in Two Zones
H0: Cu concentrations in two zones. Alluvial Fan and Basin Trough, are comparable
Selected Null Hypothesis Sample 1 Mean/Median = Sample 2 Mean/Median (Two Sided Alternative}
Alternative Hypothesis Sample 1 Mean/Median <> Sample 2 Mean/Median
Sample 1 Data: Cu (alluvial fan)
Sample 2 Data: Cu (basin trough)
Raw Statistics
Number of Valid Data
Number of Missing Observations
Number of Non-Detects
Number of Detect Data
Minimum Non-Detect
Maximum Non-Detect
Percent Non-detects
Minimum Detect
Maximum Detect
Mean of Detects
Median of Detects
SD of Detects
Sample 1
65
3
17
48
1
20
26.15%
1
20
4.14S
2
4.005
Sample 2
43
1
14
35
1
15
28.57%
1
23
5.229
3
5.214
Sample I vs Sample 2 Gehan Test
HO: Mean of Sample 1 = Mean erf background
Gehan 2 Test Value -1.372
Lower Critical z {0.025} -1.96
Upper Critical z (0.975) 1.96
P-Value 0.17
Conclusion with jftJpha = 0.05
Do Not Reject HO, Cooduefe Sample 1 = Sample 2
P-Value >= alpha (0.05)
Conclusion: Based upon the box plots shown in Figure 6-3 and the Gehan test summarized in Table 6-4,
the null hypothesis is not rejected, and it is concluded that the mean/median Cu concentrations in
groundwater from the two zones are comparable.
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Table 6-5. Tarone-Ware Comparing Location Parameters of Zinc Concentrations
H0: Zn concentrations in groundwaters of Alluvial Fan = groundwaters of Basin Trough
Selected Null Hypothesis Sample 1 Mean/Median = Sample 2 Mean/Median (Two Sided Alternative)
Alternative Hypothesis Sample 1 Mean/Median <> Sample 2 Mean/Median
Sample 1 Data: Zn{alluvial fan)
Sample 2 Data: Zn (basin trough)
Raw Statistics
Sample 1 Sample 2
Number of Valid Data 67 50
Number of Missing Observations 1 0
Number of Non-Detects 16 4
Number of Detects 51 46
Minimum Non-Detect 3 3
Maximum Non-Detect 10' 10
Percent Non-detects 23.88% 8.00°;
Minimum Detect 5 3
Maximum Detect 620 90
Mean of Detects 27.88 23.13
Median of Detects 11 20
SD of Detects 85.02 19.03
Sample 1 vs Sample 2 Tanme-Ware Ted:
HO: Mean/Median of Sample 1 = Mean/Median of Sample 2
TW Statistic -2.113
Lower TW Critical Value{0.025) -1.96
Upper TW Critical Value {0.975} 1.9€
P-Value
Conclusion with Alpha = 0,05
Reject HO, Conclude Sample 1 <> Sample 2
P-Value < alpha (0.05)
Conclusion: Based upon the box plots shown in Figure 6-4 and the T-W test results summarized in Table
6-5, the null hypothesis is rejected, and it is concluded that the Zn concentrations in groundwaters of two
zones are not comparable (p-value = 0.0346).
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CHAPTER 7
Outlier Tests for Data Sets with and without Nondetect Values
Due to resource constraints, it is not possible (nor needed) to sample an entire population (e.g., reference
area) of interest under investigation; only parts of the population are sampled to collect a random data set
representing the population of interest. Statistical methods are then used on sampled data sets to draw
conclusions about the populations under investigation. In practice, a sampled data set can consist of some
wrong/incorrect values, which often result from transcription errors, data-coding errors, or instrument
breakdown errors. Such wrong values could be outlying (well-separated, coming from 'low' probability
far tails), with respect to the rest of the data set; these outliers need to be fixed and corrected (or removed)
before performing a statistical method. However, a sampled data set can also consist of some correct
measurements that are extremely large or small relative to the majority of the data, and therefore those
low probability extreme values are suspected of misrepresenting the main dominant background
population from which they were collected. Typically, correct extreme values represent observations
coming from population(s) other than the main dominant population; and such observations are called
outliers with respect to the main dominant population.
In practice, the boundaries of an environmental population (background) of interest may not be well-
defined and the selected population actually may consist of areas (concentrations) not belonging to the
main dominant population of interest (e.g., reference area). Therefore, a sampled data set may consist of
outlying observations coming from population(s) not belonging to the main dominant background
population of interest. Statistical tests based on parametric methods generally are more sensitive to the
existence of outliers than are those based on nonparametric distribution-free methods. It is well-known
(e.g., Rousseeuw and Leroy 1987; Barnett and Lewis 1994; Singh and Nocerino 1995) that the presence
of outliers in a data set distorts the computations of all classical statistics (e.g., sample mean, sd, upper
limits, hypotheses test statistics, GOF statistics, OLS regression estimates, covariance matrices, and also
outlier test statistics themselves) of interest. Outliers also lead to both Types I and Type II errors by
distorting the test statistics used for hypotheses testing. Statistics computed using a data set with outliers
lack statistical power to address the objective/issue of interest (e.g., use of a BTV to identify
contaminated locations). The use of such distorted statistics (e.g., two-sample tests, UCL95, UTL95-95)
may lead to incorrect cleanup decisions which may not be cost-effective or protective of human health
and the environment.
A distorted estimate (e.g., UCL95) computed by accommodating a few low probability outliers (coming
from far tails) tends to represent the population area represented by those outliers and not the main
dominant population of interest.
It is also well-known that classical outlier tests such as the Rosner Test suffer from masking effects
(Huber 1981; Rousseeuw and Leroy 1987; Barnett and Lewis 1994; Singh and Nocerino 1995, and
Marona, Martin, and Yohai 2006); this is especially true when outliers are present in clusters of data
points and /or the data set represents multiple populations. Masking means that the presence of some
outliers hides the presence of other intermediate outliers. The use of robust and resistant outlier
identification methods is recommended in the presence of multiple outliers. Several modern robust outlier
identification methods exist in the statistical literature cited above. However, robust outlier identification
procedures are beyond the scope of the ProUCL software and this technical guidance document. In order
to compute robust and resistant estimates of the population parameters of interest (e.g., EPCs, BTVs),
EPA NERL-Las Vegas, NV developed a multivariate statistical software package, Scout 2008, Version
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1.0 (EPA 2009d) consisting of several univariate and multivariate robust outlier identification and
estimation methods. Scout software can be downloaded from the following EPA website:
http://archivc.cpa.gov/csd/archivc-scoutAvcb/html/
7.1 Outliers in Environmental Data Sets
In addition to representing contaminated locations, outliers in an environmental data set occur due to non-
random, random and seasonal fluctuations in the environment. Outliers tests identify statistical outliers
present in a data set. The variabilities of data sets originating from environmental applications are much
higher than the variabilties of data sets collected from other applications such as the biological and
manufacturing processes, therefore, in environmental applications, not all outliers identified by a
statistcial test may represent real physical outliers. Typically, extreme statistical outliers in a data set
represent non-random situations potentially representing impacted locations; extreme outliers should not
be included in statistical evaluations. Mild and intermediate statistical outliers may be present due to
random natural fluctuations and variability in the environment; those outlying observations may be
retained in statistical evaluations such as estimating BTVs. Based upon site CSM and expert knowledge,
the project team should make these determinations.
The use of graphical displays is very helpful in distingushing between extreme statistical outliers (real
physical outliers) and intermediate statistical outliers. It is suggested that outlier tests be supplemented
with exploratory graphical displays such as Q-Q plots and box plots (Johnson and Wichern 2002;
Hoaglin, Moseteller and Tukey 1983). ProUCL has several of these graphical methods which can be used
to identify multiple outliers potentially present in a data set. Graphical displays provide additional insight
into a data set that cannot be revealed by tests statistics (e.g., Rosner test, Dixon test, S-W test). Graphical
displays help identify observations that are much larger or smaller than the bulk (majority) of the data.
The statistical tests alone cannot determine whether a statistical outlier should be investigated further.
Based upon historical and current site and regional information, graphical displays, and outlier test
results, the project team and the decision makers should decide about the proper disposition of outliers to
include or not to include them in the computation of the various decision making statistics such as UCL95
and UTL95-95. Performing statistical analyses twice on the same data set, once using the full data set
with outliers and once using the data set without high/extreme outliers coming from the far tails, helps the
project team in determining the proper disposition of those outliers. Several examples illustrating these
issues have been discussed in this technical guidance document (e.g., Chapters 2 through 5).
Some Notes
Note 1: In practice, extreme outliers represent: 1) low probability observations possibly coming from the
extreme far tails of the distribution of the main population under consideration, with low to negligible
probability, or 2) observations coming from population(s) different from the main dominant population of
interest. On a normal exploratory Q-Q plot, observations well-separated (sticking out, significantly higher
than the majority of the data) from the majority of observations represent extreme physical outliers; and
the presence of a few high outlying observations distorts the normality of a data set. That is, many data
sets follow a normal distribution after the removal of identified outliers.
Note 2 (about Normality): Rosner and Dixon outlier tests require normality of a data set without the
suspected outliers. Literature about these outlier tests is somewhat confusing and users tend to believe that
the original data (with outliers) should follow a normal distribution. A data set with outliers very seldom
follow a normal distribution as the presence of outliers tends to destroy the normality of a data set.
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Note 3: Methods incorporated in ProUCL can be used on any data set with or without NDs, and with or
without the outliers. In the past, some practitioners have mis-stated that ProUCL software is restricted and
can be used only on data sets without outliers. Just like any other software, it is not a requirement to
exclude outliers before using any of the statistical methods incorporated in ProUCL. However, it is the
intent of the developers of the ProUCL software to inform the users on how the inclusion of a. few low
probability outliers can yield distorted UCL95; UPLs, UTLs, as well as other statistics. The outlying
observations should be investigated separately to determine the reasons for their occurrences (e.g., errors
or contaminated locations). It is suggested that statistics are computed with and without the outliers
followed by evaluation of the potential impact of outliers on the decision making processes.
7.2 Outliers and Normality
The presence of outliers in a data set destroys the normality of the data set (Wilks 1963; Barnett and
Lewis 1994; Singh and Nocerino 1995). It is highly likely that a data set which contains outliers will not
follow a normal distribution unless the outliers are present in clusters. The classical outlier tests, Dixon
and Rosner tests, assume that the data set without the suspected outliers follow a normal distribution; that
is for both Rosner and Dixon tests, the data set representing the main body of the data obtained after
removing the outliers, and not the original data set with outliers needs to follow a normal distribution.
There appears to be some confusion among some practitioners (Helsel and Gilroy 2012) who mistakenly
assume that one can perform Dixon and Rosner tests only when the data set, including outliers, follows a
normal distribution, which is only rarely true.
As noted earlier, a lognormal model tends to accommodate outliers (Singh, Singh, and Engelhardt 1997),
and a data set with outliers can follow a lognormal distribution. This does not imply that the outlier
potentially representing the impacted location does not exist! Those impacted locations may need further
investigations. Outlier tests should be performed on raw data, as the cleanup decision needs to be made
based upon concentration values in the raw scale and not in the log-scale or some other transformed scale
(e.g., cube root). Outliers are not known in advance. ProUCL has normal Q-Q plots which can be used to
get an idea about the number of outliers or mixture populations potentially present in a data set. This can
help a user to determine the suspected number of outliers needed to perform the Rosner test. Since the
Dixon and Rosner tests may not identify all potential outliers present in a data set, the data set obtained,
even without the identified outliers, may not follow a normal distribution. Over the last 25 years, several
modern iterative robust outlier identification methods have been developed (Rousseeuw and Leroy 1987;
Singh and Nocerino 1995) which are beyond the scope of ProUCL. Some of those methods are available
in the Scout 2008 version 1.0 software (EPA 2009d).
7.3 Outlier Tests for Data Sets without Nondetect Observations
A couple of classical outlier tests discussed in the environmental literature (EPA 2006b, and Gilbert 1987)
and included in ProUCL software are described as follows. It is noted that these classical tests suffer from
masking effects and may fail to identify potential outliers present in a data set. This is especially true
when multiple outliers or multiple populations (e.g., various AOCs of a site) may be present in a data set.
Such scenarios can be revealed by using exploratory graphical displays including Q-Q and box plots.
7.3.1 Dixon's Test
Dixon's Extreme Value test (1953) can be used to test for statistical outliers when the sample size is less
than or equal to 25. Initially, this test was derived for manual computations. This test is described here for
historical reasons. It is noted that Dixon's test considers both extreme values that are much smaller than
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the rest of the data (Case 1) and extreme values that are much larger than the rest of the data (Case 2).
This test assumes that the data without the suspected outlier are normally distributed; therefore, one may
want to perform a test for normality on the data without the suspected outlier. However, since the Dixon
test may not identify all potential outliers present in a data set, the data set obtained after excluding the
identified outliers may still not follow a normal distribution. This does not imply that the identified
extreme value does not represent an outlier.
7. 3. 1. 1 Directions for the Dixon 's Test
Steps described below are provided for interested users, as ProUCL performs all of the operations
described as follows:
STEP 1: Let X(i), X(2), . . . , X(n) represent the data ordered from smallest to largest. Check that the data
without the suspect outlier are normally distributed. If normality fails, then apply a different outlier
identification method such as a robust outlier identification procedure. Avoid the use of a data
transformation, such as a log-transformation, to achieve normality so that the data meet the criteria to
use the Dixon test. All cleanup and remediation decisions are made based upon the data set in raw scale.
Therefore, outliers, perhaps representing isolated contaminated locations, should be identified in the
original scale. As mentioned before, the use of a log -transformation tends to hide and accommodate
outliers (instead of identifying them).
STEP 2: X(i) is a potential outlier (Case 1): Compute the test statistic, C, where
X(n) ~
for8V> forl4
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7.3.2 Rosner's Test
An outlier test developed by Rosner (1975, 1983) can be used to identify up to 10 outliers in data sets of
sizes > 25. The details of the test can be found in Gilbert (1987). Like the Dixon test, the critical values
associated with the Rosner test are computed using the normal distribution of the data set without the k
(<10) suspected outliers. The assumption here is that the data set without the suspected outliers follows a
normal distribution, as a data set with outliers tends not to follow a normal distribution. A graphical
display, such as a Q-Q plot, can be used to identify suspected outliers needed to perform the Rosner test.
Like the Dixon test, the Rosner test also suffers from masking.
7.3.2.1 Directions for the Rosner's Test
To apply Rosner's test, first determine an upper limit, r0, on the number of outliers (r0 < 10), then order
the r0 extreme values from most extreme to least extreme. Rosner's test statistic is computed using the
sample mean and sample sd.
STEP 1: Let Xi, X2, . . . , Xn represent the ordered data points. By inspection, identify the maximum
number of possible outliers, ro. Check that the data are normally distributed (without outliers).
A data set with outliers seldom passes the normality test.
STEP 2: Compute the sample mean, r , and the sample sd, s, for all the data. Label these values J(0) and
5(0), respectively. Determine the value that is farthest from J(0) and label this observation
j/0). Delete _y(0) from the data and compute the sample mean, labeled xm, and the sample sd,
labeled sm. Then determine the observation farthest from x(~Y) and label this observationj/1-1.
Delete _y(1) and compute x(2) and s(2). Continue this process until r0 extreme values have been
eliminated. After carrying out the above process, we have:
r*(0),s(0), v(0)l; \x(l},s(l}, v(1)l: ..., \x(r^l),s(r^l), v^^l where
, -x(0)2 ,and /° is the farthest value x(f).
n-i j=\ \n-y=i
The above formulae for ;tw and s^ assume that the data have been re-numbered after each
outlying observation is deleted.
y(r-l) _ j(r-l) I
STEP 3: To test if there are "r" outliers in the data, compute: Rr = — and compare Rr to
i3
the critical value Ar in the tables from any statistical literature. If Rr> hr, conclude that there
are r outliers.
First, test if there are r0 outliers (compare Rr _\ to Ar_j). If not, then test if there are r0 - 1
outliers (compare Rr _2 to Xr _2). If not, then test if there are ro - 2 outliers, and continue, until
either it is determined that there are a certain number of outliers or that there are no outliers.
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7.4 Outlier Tests for Data Sets with Nondetect Observations
In environmental studies, identification of detected high outliers, coming from the right tail of the data
distribution and potentially representing impacted locations, is important as locations represented by those
extreme high values may require further investigation. Therefore, for the purpose of the identification of
high outliers, one may replace the NDs by their respective DLs, DL/2, or may just ignore them (especially
when elevated DLs are associated with NDs and/or when the number of detected values is large) from any
of the outlier test (e.g., Rosner test) computations, including the graphical displays such as Q-Q plots.
Both of these procedures, ignoring NDs or replacing them by DL/2, for identification of outliers are
available in ProUCL for data sets containing NDs. Like uncensored full data sets, outlier tests on data sets
with NDs should be supplemented with graphical displays. ProUCL can be used to generate Q-Q plots
and box plots for data sets with ND observations.
Notes: Outlier identification procedures represent exploratory tools and are used for pre-processing of a
data set to identify outliers or multiple populations that may be present in a data set. Except for the
identification of high outlying observations, the outlier identification statistics, computed with NDs or
without NDs, are not used in any of the estimation and decision making process. Therefore, for the
purpose of the identification of high outliers, it should not matter how the ND observations are treated. To
compute test statistics (e.g., Gehan test) and decision statistics (e.g., UCL95, UTL95-95), one should
follow the procedures as described in Chapters 4 through 6.
Example 7-1. Consider a lead data set of size 10 collected from a Superfund site. The site data set
appears to have some outliers. Since the data set is of small size, only the Dixon test can be used to
identify outliers. The normal Q-Q plot of the lead data is shown in Figure 7-1 below. Figure 7-1
immediately suggests that the data set has some outliers. The Dixon test cannot directly identify all
outliers present in a data set, only robust methods can identify multiple outliers. Multiple outliers may be
identified one at a time iteratively by using the Dixon test on data sets after removing outliers identified in
previous iterations. However, due to masking, the iterative process based upon the Dixon test may or may
not be able to identify multiple outliers.
Q-Q Plot for OS_Lead
Intercept = 268.2
Correlation, R = 0.67
Theoretical Quantiles (Standard Normal)
Figure 7-1. Normal Q-Q Plot Identifying Outliers
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Table 7-1. Dixon Outlier Test Results for Site Lead Data Set
Dixon's Outlier Test for OS Lead
Number of Observations = 10
10% critical value: 0.409
5% critical value: 0.477
1% critical value: 0.597
t. Observation Value 1940 is a Potential Outlier (Upper Tail)?
Test Statistic: 0.836
For 10% significance level. 194Q is an outlier.
For 5% significance level, 194C is an outlier.
For 1% significance level. 194(1 is an outlier.
2. Observation Value 19.7 is a Potential Outlier (Lower Tail)?
Test Statistic: 0.013
For 10% significance level, 19.7 is not an outlier.
For 5% significance level, 19.7 is not an outlier.
For 1 % significance level, 19.7 is not an outlier.
Example 7-2. Consider She's (1997) pyrene data set of size n=56 with 11 NDs. The Rosner test results
on data without the 11 NDs are summarized in Table 7-2, and the normal Q-Q plot without NDs is shown
in Figure 7-2 below.
3000
Q-Q Plot for Pyrene
Nondetects not displayed
2500
2000
a
c
1000
500
'
0
»
....?.- ' ' '
•
459
3GB ^
Pyrene
Total Number of Data-SE
Number of Non-Deleete*11
Number of Detects -45
Detected Mean-190.1
Defected Sd = 435
Slope (displayed data] -224.4
Intercept (displayed data^ 130.1
Conelalion, R = 0 506
D Best Fit Line
-1.6 -1.2 -0.6 0.0 0.6 1.2 1.8
Theoretical Quantiles (Standard Normal)
NDs Displayed in smaller font
Figure 7-2. Normal Q-Q Plot of Pyrene Data Set Excluding NDs
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Table 7-2. Rosner Test Results on Pyrene Data Set Excluding NDs
Rosner's Outlier Test for 10 Outliers in Pyrene
Total N 56
Number NDs 11
Number Detects 45
Mean of Detects 190.1
SD of Detects 435
Number of data 45
Number of suspected outliers 10
s not included in the following:
tt
1
2
3
4
5
6
7
8
9
10
Mean
190.1
12S.6
118.9
111S
109.1
104.6
100.3
9668
93.3
9061
Potential
sd outlier
430.1 2982
90.7 459
75.7 333
68.74 306
62.43 289
56.1 273
4965 238
4478 222
40.17 190
37.21 187
Obs.
Number
45
44
43
42
41
40
39
3B
37
36
Test
value
6.491
3.665
2.828
2.796
2.SS1
3.001
2.773
2.798
2.408
2.59
Critical Critical
value (5%) value (1%)
3.09 3.44
3.08
343
3.07 3.41
306
3.05
3.038
3.026
3.014
3.002
2.99
3.4
3.39
3.378
3366
3354
3342
3.33 |
For 5:i significance level, there are 2 Potential Outliers
2982. 459
For 1 *= Signrficance Level, there are 2 Potential Outliers
2982. 459
Example 7-3. Consider the aluminum data set of size 28 collected from a Superfund site. The normal Q-
Q plot is shown in Figure 7-3 below. Figure 7-3 suggests that there are 4 outliers (at least the
observation=30,000) present in the data set. The Rosner test results are shown in Table 7-3. Due to
masking, the Rosner test could not even identify the outlying observation of 30,000.
Q-Q Plot for Aluminum
Slope-7193
Intercept-102E
Corte^fon, R =
Theoretical Quantiles (Standard Normal)
Figure 7-3. Normal Q-Q Plot of Aluminum Concentrations
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Table 7-3. Rosner Test Results on Pyrene Data Set Excluding NDs
Rosner's Outlier Test for Aluminum
Mean 10284
Standard Deviation 7449
Number of data 28
Number of suspected outliers 10
8
1
2
3
4
5
6
7
8
9
10
Mean
10284
9553
S359
8358
7789
7423
70€1
5Sfi9
S377
S102
sd
7315
6490
5822
5050
42S4
3356
3637
3215
3000
2S12
Potential
outlier
30000
25000
240(10
22000
162QO
15400
15300
12500
11600
10700
Obs.
Number
26
25
27
28
10
13
6
14
21
12
Test
Critical
Critical
value value (5%) value (1 %}
2.695
2,38
2.584
2.702
1.973
2.016
2.2S5
1.814
1.741
1.635
2.88
2.86
2.84
2.82
2.8
2.776
2.752
2.728
2.704
2.S8
3.2
3.18
3.16
3.14
3.11
3.082
3.054
3.026
2.998
2.97
For 5% Significance Level, there is no Potential Outlier
For 1 ;= Significance Level, there is no Potential Outlier
As mentioned earlier, there are robust outlier identification methods which can be used to identify
multiple outliers/multiple populations present in a data set. Several of those methods are incorporated in
Scout 2008 (EPA 2009d). A couple of formal (with test statistics) robust graphs based upon the PROP
influence function and MCD method (Singh and Nocerino 1995) are shown in Figures 7-4 and 7-5. The
details of these methods are beyond the scope of ProUCL. The two graphs suggest that there are several
outliers present including the elevated value of 30,000. All observations exceeding the horizontal lines
displayed at critical values of the Largest Mahalanobis Distance (MD) (Wilks 1963; Barnett and Lewis
1994) represent outliers.
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Index Plot of MDs using PROP Estimate
Initial Eitlmatai: OKG
Influence Aipns G UK
MDDiitnbution: Bell
Figure 7-4. Robust Index Plot of MDs Based Upon the PROP Influence Function
Index Plot of MDs using MCD Estimate
Initul S.teof V • r
•h'v«lueof ||n*pt1)f2)-tE
lmti*l Sublet* = 500
Bed Retained Subieti = 11
Index of Observations
Figure 7-5. Robust Index Plot of MDs Based upon the MCD Method
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CHAPTER 8
Determining Minimum Sample Sizes for User Specified Decision
Parameters and Power Assessment
This chapter describes mathematical formulae used to determine data quality objectives (DQOs)-based
minimum sample sizes required by estimation, and hypothesis testing approaches used to address
statistical issues for environmental projects (EPA 2006a, 2006b). The sample size determination
formulae for estimation of the unknown population parameters (e.g., mean, percentiles) depend upon the
pre-specified values of the decision parameters: CC, (1-a), and the allowable error margin, A, between the
estimate and the unknown true population parameter. For example, if the environmental problem requires
the calculation of the minimum number of samples required to estimate the true unknown population
mean, A would represent the maximum allowable difference between the estimate of the sample mean
and the unknown population mean. Similarly, for hypotheses testing approaches, sample size
determination formulae depend upon the pre-specified values of the decision parameters chosen while
defining and describing the DQOs associated with an environmental project. The decision parameters
associated with hypotheses testing approaches include the Type I false positive error rate, a; and the Type
II false negative error rate, /?=! -power; and the allowable width, A, of the gray region. For values of the
parameter of interest (e.g., mean, proportion) lying in the gray region, the consequences of committing the
two types of errors described in Chapter 6 are not significant from both the human health and the cost
effectiveness points of view.
Even though the same symbol, A, has been used to denote the allowable error margin in an estimate (e.g..
of mean) and the width of the gray region associated with the various hypothesis testing approaches, there
are differences in the meanings of the error margin and width of the gray region. A brief description of
these terminology is provided in this chapter. The user is advised to consult the already existing EPA
guidance documents (EPA 2006a, 2006b; MARSSIM 2000) for the detailed description of the terms with
interpretation used in this chapter. Both parametric (assuming normality) and nonparametric (distribution
free) DQOs-based sample size determination formulae as described in EPA guidance documents
(MARSSIM 2000; EPA 2002c, 2006a, 2006b, and 2009) are available in the ProUCL software. These
formulae yield minimum sample sizes needed to perform statistical methods meeting pre-specified DQOs.
The Stats/ Sample Sizes module of ProUCL has the minimum sample size determination methods for
most of the parametric and nonparametric one-sided and two-sided hypotheses testing approaches
available in ProUCL.
ProUCL includes the DQOs-based parametric minimum sample size formula to estimate the population
mean, assuming that the sample mean follows a normal distribution or assuming that the criteria is met
due to the CLT]. ProUCL outputs a non-negative integer as the minimum sample size. This minimum
sample size is calculated by rounding the value, obtained by using a sample size formula, upward. For all
sample size determination formulae incorporated in ProUCL, it is implicitly assumed that samples (e.g.,
soil, groundwater, sediment samples) are randomly collected from the same statistical population (e.g.,
AOC or MW), and therefore the sampled data (e.g., analytical results) represent independently and
identically distributed (i.i.d) observations from a single statistical population. During the development of
the Stats/Sample Sizes module of ProUCL, emphasis was given to assure that the module is user friendly
with a straight forward unambiguous mechanism (e.g., graphics user interface [GUIs]) to input desired
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decision parameters (e.g., a, /? error rates, width, A of the gray region) needed to compute the minimum
sample size for a selected statistical application.
Most of the sample size formulae available in the literature and incorporated in ProUCL) require an
estimate (e.g., preliminary from other sites and pilot studies or based upon actual collected data) of the
population variability. In practice, the population variance,
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size computed in retrospect, the user may want to collect additional samples to assure that the test
achieves the desired power.
• It should be pointed out that there could be differences in the sample sizes computed in the two
different stages due to the differences in the values of the estimated variability. Specifically, the
preliminary estimate of the variance computed using information from similar sites could be
significantly different from the variance computed using the available data already collected from
the study area under investigation which will yield different values of the sample size.
Sample size determination methods in ProUCL can be used for both stages. The only difference will be in
the input value of the standard deviation/variance. It is the users' responsibility to input a correct value
for the standard deviation during the two stages.
8.1 Sample Size Determination to Estimate the Population Mean
In exposure and risk assessment studies, a UCL95 of the population mean is used to estimate the EPC
term. Listed below are several variations of methods available in the literature to compute the minimum
sample size, n, needed to estimate the population mean with specified confidence coefficient (CC), (1 -
a), and allowable/tolerable error margin (allowable absolute difference between the estimate and the
parameter), A in an estimate of the mean.
8. 1. 1 Sample Size Formula to Estimate Mean without Considering Type II (ft) Error Rate
The sample size can be computed using the following normal distribution based equation (when
population variance is known),
« =
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normal distribution (which can be assumed due to the CLT). ProUCL does not compute minimum sample
sizes required to estimate the population median. While estimating the mean, the symbol A represents the
allowable error margin (+/-) in the mean estimate. For example for A = 10, the sample size is computed to
assure that the error in the estimate will be within ±10 units of the true unknown population mean with
specified CC of (1 -a).
For estimation of the mean, the most commonly used formula to compute the sample size, n, is given by
(8-2) above; however, under normal theory, the use of t-distribution based formula (8-3) is more
appropriate to compute n. It is noted that the difference between the sample sizes obtained using (8-2) or
(8-3) is not significant. They usually differ by only 2 to 3 samples (Blackwood 1991; Singh, Singh, and
Engelhardt 1999). It is a common practice to address this difference by using the following adjusted
formula (Kupper and Hafner 1989; Bain and Engelhardt 1991) to compute the minimum sample size
needed to estimate the mean for specified CC, (1 - a), and margin of error, A.
« = AWA2 + zi2W2 (8-4)
To be able to use a normal (instead oft-critical value) distribution based critical value, as used in (8-4), a
similar adjustment factor is used in other sample size formulae described in the following sections (e.g.,
two-sample t-test, WRS test). This adjustment is also used in various sample size formulae described in
EPA guidance documents (MARSSIM 2000; EPA 2002c, 2006a, 2006b). ProUCL uses equation (8-4) to
compute sample sizes needed to estimate the population mean for specified values of CC, (1- a), and error
margin, A. An example illustrating the sample size determination to estimate the mean is given as
follows.
Example 8-1. Sample Size for estimation of the mean (CC = 0.95, s = 25, error margin, A = 10)
I i Sample Size for Estimation of Mean
Based on Specified Values of Decision Paiameters/DQQs (DataQuaRy Objectives]
Date^Time of Computation 2/26^2010 12:12:37 PM
User Selected Options
Confidence Coefficient 95%
Allowable Error Margin 10
Estimate of Standard Deviation 25
Approximate Minimum Sample Size
95% Confidence Coefficient: 26
8.1.2 Sample Size Formula to Estimate Mean with Consideration to Both Type I (a) and Type
II (ft) Error Rates
This scenario corresponds to the single-sample hypothesis testing approach. For specified decision error
rates, a and ft, and width, A, of the gray region, ProUCL can be used to compute the minimum sample
size based upon the assumption of normality. ProUCL also has nonparametric minimum sample size
determination formulae to perform Sign and WSR tests. The nonparametric Sign test and WSR test are
used to perform single sample hypothesis tests for the population location parameter (mean or median).
A brief description of the standard terminology used in the sample size calculations associated with
hypothesis testing approaches is described first as follows.
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a = False Rejection Rate (Type I Decision Error), i.e., the probability of rejecting
the null hypothesis when in fact the null hypothesis is true
/? = False Acceptance Rate (Type II Decision Error), i.e., the probability of not
rejecting the null hypothesis when in fact the null hypothesis is false
z;_« = a value from a standard normal distribution for which the proportion of the
distribution to the left of this value is 1 - a
zi-p = a value from a standard normal distribution for which the proportion of the
distribution to the left of this value is 1 - /?
A = width of the gray region (specified by the user); in a gray region, decisions are "too close to
call", a gray region is that area where the consequences of making a decision error (Type I or
Type II) are relatively minor.
The user is advised to note the difference between the gray region (associated with hypothesis testing
approaches) and error margin (associated with estimation approaches).
Example illustrating the above terminology: Let the null and alternative hypotheses be: H0: p < Cs, and
HA: /J > Cs. The width, A, of the gray region for this one sided alternative hypothesis is A = jUi - Cs, where
Cs is the cleanup standard specified in the null hypothesis, and jUi (>G) represents an alternative value
belonging to the parameter value set determined by the alternative hypothesis. Note that the gray region
lies to the right (e.g., see Figure 8-1) of the cleanup standard, Cs, and for all values of ju in the interval,
(Cs, jUi], with length of the interval = width of gray region= A = jUi - Cs. The consequences of making an
incorrect decision (e.g., accepting the null hypothesis when in fact it is false) will be minor.
8.2 Sample Sizes for Single-Sample Tests
8.2.1 Sample Size for Single-Sample t-test (Assuming Normality)
This section describes formulae to determine the minimum number of samples, n, needed to conduct a
single-sample t-test, for 1-sided as well as two-sided alternatives, with pre-specified decision error rates
and width of the gray region. This hypothesis test is used when the objective is to determine whether the
mean concentration of an AOC exceeds an action level (AL); or to verify the attainment of a cleanup
standard, Cs (EPA 1989a). In the following, s represents an estimate (e.g., an initial guess, historical
estimate, or based upon expert knowledge) of the population sd, a.
Three cases/forms of hypothesis testing as incorporated in ProUCL are described as follows:
237
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8.2.1.1 Case I (Right-Sided Alternative Hypothesis, Form 1)
Ho: site mean, n AL or a Cs
Gray Region: Range of the mean concentrations where the consequences of deciding that the site mean is
less than the AL when in fact it is greater (that is a dirty site is declared clean) are not significant. The
upper bound of the gray region, A, is defined as the alternative mean concentration level, //; (> Cs), where
the human health and environmental consequences of concluding that the site is clean (when in fact it is
not clean) are relatively significant. The false acceptance error rate, /?, is associated with this upper bound
(jj.i) of the gray region: A=,U;_ Cs. These are illustrated in Figure 8-1 below (EPA 2006a). A similar
explanation of the gray region applies to other single-sample Form 1 right-sided alternative hypotheses
tests (e.g., Sign test, WSRtest) considered later in this chapter.
•3
— M
*. .=
2 ^
= .i
w
1
0.9 -
0.8
0.7 -
0.6 -
0.5-
0.4
0.3 -
0.1-
0.1 -
I
Baseline
I i i
0
40
Alternative
i I
Tolerable False
Rejection Decision
EnqrBates
Tolerable False
Acceptance Decision
Error Rates
Gray Region
Relatively Large
Decision Error Rates
are Considered
Tolerable
140 160 180 200
Action Level
True Value of the Parameter (Mean Concentration, ppm)
Diagram Where the Alternative Condition Exceeds the Action Level
Figure 8-1. Gray Region for Right-Sided (Form 1) Alternative Hypothesis Tests (EPA 2006a)
8.2.1.2 Case II (Left-Sided Alternative Hypothesis, Form 2)
Ho: site mean, p >AL or Cs vs. HA: site mean, p
-------
(when in fact it is not dirty) would be costly requiring unnecessary cleaning of a site. The false
acceptance rate, ft, is associated with that lower bound (pii) of the gray region, A= Cs - //;. These are
illustrated in Figure 8-2.
A similar explanation of the gray region applies to other single-sample left-sided (left-tailed) alternative
hypotheses tests including the Sign test and WSR test.
l-
0.9-
0.8-
0.7-
0.6-
0.5-
0.4-
0.3-
0.2-
0.1-
0
• Alteniative
^_^:
Baseline
Gray Region
Relatively Large
Decision Error Rites
are Considered
Tolerable
Tolerable False
Rejection Decision
Error Rates
Tolerable False
Acceptance Decision
En or Rates
20 40 60 80 100 120 140 160 180 200
Action Level
True Value of the Parameter (Menu Concentration, ppm)
Diagram Where the Alternative Condition Falls Below the Action Level
Figure 8-2. Gray Region for Left-Sided (Form 2) Alternative Hypothesis Tests (EPA 2006a)
The minimum sample size, n, needed to perform the single-sample one-sided t-test (both Forms 1 and 2
described above) is given by
(8-5)
8.2.1.3 Case III (Two-Sided Alternative Hypothesis)
Ho: site mean, p = Cs; vs. HA: site mean, p ^ Cs
The minimum sample size for specified performance (decision) parameters is given by:
239
-------
(8-6)
_a_
A = width of the gray region, A= abs (Cs- jUi), abs represents the absolute value operation.
In this case, the gray region represents a two-sided region symmetrically placed around the mean
concentration level equal to Cs, or AL; consequences of committing the two types of errors in this gray
region would be minor (not significant). A similar explanation of the gray region applies to other single-
sample two-sided (two-tailed) alternative hypotheses tests such as the Sign test and WSRtest.
In equations (8-5) and (8-6), the computation of the estimated variance, s2 depends upon the project stage.
Specifically,
s2 = a preliminary estimate of the population variance (e.g., estimated from similar sites, pilot
studies, expert opinions) which is used during the planning stage; or
s2 = actual sample variance of the collected data to be used when assessing the power of the test
in retrospect based upon collected data.
Note: ProUCL outputs the estimated variance based upon the collected data on single sample t-test output
sheet; ProUCL 5.1 sample size GUI draws users' attention to input an appropriate estimate of variance,
the user should input an appropriate value depending upon the project stage/data availability.
The following example: "Sample Sizes for Single-sample t-Test" discussed in Guidance on Systematic
Planning Using the Data Quality Objective Process (EPA 2006a, page 49) is used here to illustrate the
sample size determination for a single-sample t-test. For specified values of the decision parameters, the
minimum number of samples is given by n > 8.04. For a one-sided alternative hypothesis, ProUCL
computes the minimum sample size to be 9 (rounding up), and a sample size of 1 1 is computed for a two-
sided alternative hypothesis.
Example 8-2. Sample Size for Single-sample t-Test Sample Sizes (a = 0.05, ft = 0.2, s = 10.41, A = 10)
! Sample Sizes for Single Sample t Test
Based on Specified Values of Decision Paiameters/DQOs (Data Quafty Objectives)
D ate/T ime of Computation 2/26/2010 12:41:58 PM
User Selected Options
False Rejection Rate [Alpha] 0.05
False Acceptance R ate [B eta] 0.2
Width of Grav Region [Delta] 10
E stimate of S tandard D eviation 10.41
Approximate Minimum Sample Size
Single Sided Alternative Hypothesis: 9
Two Sided Alternative Hypothesis: 11
5.2.2 Single Sample Proportion Test
This section describes formulae used to determine the minimum number of samples, n, needed to
compare an upper percentile or proportion, P, with a specified proportion, P0 (e.g., proportion of
240
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exceedances, proportion of defective items/drums, proportion of observations above the specified AL),
for user selected decision parameters. The details are given in EPA guidance document (2006a). Sample
size formulae for three forms of the hypotheses testing approach are described as follows.
5.2.2.1 Case I (Right-Sided Alternative Hypothesis, Form 1)
Ho: population proportion < specified value (Po) vs. HA: population proportion > specified value (Po)
Gray Region: Range of true proportions where the consequences of deciding that the site proportion, P, is
less than the specified proportion, P0, when in fact it is greater (that is a dirty site is declared clean) are not
significant. The upper bound of the gray region, A, is defined as the alternative proportion, Pi (> Po),
where the human health and environmental consequences of concluding that the site is clean (when in fact
it is not clean) are relatively significant. The false acceptance error rate, /?, is associated with this upper
bound (Pi) of the gray region (A=P;_/V-
8.2.2.2 Case II (Left-Sided Alternative Hypothesis, Form 2)
Ho: population proportion > specified value (Po) vs. HA: population proportion < specified value (Po)
Gray Region: Range of true proportions where the consequences of deciding that the site proportion, P, is
greater than or equal to the specified proportion, Po, when in fact it is smaller (a clean site is declared
dirty) are not considered significant. The lower bound of the gray region is defined as the alternative
proportion, Pi (< P0), where the consequences of concluding that the site is dirty (when in fact it is not
dirty) would be costly requiring unnecessary cleaning of a clean site. The false acceptance rate, (3, is
associated with that lower bound (Pi) of the gray region (A= Po- Pi).
The minimum sample size, n, for the single-sample proportion test (for both cases I and II) is given by
n =
P -P
•M •'O
(8-7)
8.2.2.3 Case III (Two-Sided Alternative Hypothesis)
Ho: population proportion = specified value (Po)vs. HA: population proportion^ specified value (Po)
The following procedure is used to determine the minimum sample size needed to conduct a two-sided
proportion test.
a =
P ~P
•M •'O
when Pi = Po + A; and
P -P
•M •'O
for right-sided alternative;
for left-sided alternative;
241
-------
when Pi = Po - A
Po = specified proportion
Pi = outer bound of the gray region.
A = width of the gray region = \Po - Pi =abs (Po - Pi)
The sample size, n, for two-sided proportion test (Case III) is given by
n = max(a,b) (8-8)
An example illustrating the single-sample proportion test is considered next. This example: "Sample
Sizes for Single-sample Proportion Test" is also discussed in EPA 2006a (page 59). For this example, for
the specified decision parameters, the number of samples is given by n > 365. However, ProUCL
computes the sample size to be 419 for the right-sided alternative hypothesis, 368 for the left-sided
alternative hypothesis, and 528 for the two-sided alternative hypothesis.
Example 8-3. Output for Single-Sample proportion test sample size (a = 0.05, /? = 0.2, Po = 0.2, A = 0.05)
j I Sample Sizes for Single Sample Proportion Test
Based on Specified Values of Decision Parameters/DQOs (Data QuaEty Objectives)
D ate/T ime of Computation 2/26/2010 12:50:52 PM
User Selected Options
False Rejection Rate [Alpha] 0.05
False Acceptance R ate [B eta] 0.2
Width of G ray R egion [D elta] 0.05
Proportion/Action Level [PO] 0.2
Approximate Minimum Sample Size
Right Sided Alternative Hypothesis: 419
Left Sided Alternative Hypothesis: 368
Two Sided Alternative Hypothesis: max(471,528)
Notes: The correct use of the Sample Size module, to determine the minimum sample size needed to
perform a proportion test, requires that the users have some familiarity with the single-sample hypothesis
test for proportions. Specifically the user should input feasible values for the specified proportion, Po, and
width, A, of the gray region. The following example shows the output screen when unfeasible values are
selected for these parameters.
Example 8-4. Output - Single-sample Proportion Test Sample Sizes (a = 0.05, ft = 0.2, P0 = 0.7, A = 0.8)
i Sample Sizes for Single Sample Proportion Test
Based on Specified Values of Decision Pararneters/DQOs (Data Quafty Objectives)
Date/Time of Computation 2/26/2010 12:55:51 PM
User Selected Options
False Rejection Rate [Alpha] 0.05
False Acceptance Rate [Beta] 0.2
Width of Gray Region [Delta] 0.8
Proportion/Action Level [PO] 0.7
Approximate Minimum Sample Size
Right Sided Alternative Hypothesis: Not Feasible - Please check your Decision Pararneters/DOOs
Left Sided Alternative Hypothesis: Not Feasible - Please check your Decision Parameters/DQQs
Two Sided Alternative Hypothesis: Not Feasible - Please check your Decision Pararneters/DQOs
242
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8.2.3 Nonparametric Single-sample Sign Test (does not require normality)
The purpose of the single-sample nonparametric Sign test is to test a hypothesis involving the true
location parameter (mean or median) of a population against an AL or Cs without assuming normality of
the underlying population. The details of sample size determinations for nonparametric tests can be found
inConover(1999).
8. 2. 3. 1 Case I (Right-Sided Alternative Hypothesis)
Ho: population location parameter < specified value, Cs vs. HA: population location parameter >
specified value, Cs
A description of the gray region associated with the right-sided Sign test is given in Section 8.2.1.1.
8. 2. 3. 2 Case II (Left-Sided Alternative Hypothesis)
Ho: population location parameter > specified value, Cs vs. HA: population location parameter
< specified value, Cs
A description of the gray region associated with this left-sided Sign test is given in Section 8.2. 1 .2.
The minimum sample size, n, for the single-sample one-sided (both left-sided and right-sided) Sign test is
given by the following equation:
- - i /o r,\
n = — - - — — - , where (8-9)
4(SignP-0.5)
SignP = <&\ — (8-10)
{sdj
A = width of the gray region
sd = an estimate of the population (e.g., reference area, AOC, survey unit) standard deviation
Some guidance on the selection of an estimate of the population sd, a, is given in Section 8.1.1 above.
-------
n =
(
4(SignP-0.5)
In the following example, ProUCL computes the sample size to be 35 for a single-sided alternative
hypothesis and 43 for a two-sided alternative hypothesis for default values of the decision parameters.
Note: Like the parametric t-test, the computation of the standard deviation (sd) depends upon the project
stage. Specifically,
sd2 (used to compute P in equation (8-10)) = a preliminary estimate of the population variance
(e.g., estimated from similar sites, pilot studies, expert opinion) which is used during the
planning stage; and
sd2 (used to compute P) = sample variance computed using the actual collected data to be used
when assessing the power of the test in retrospect based upon the collected data.
ProUCL outputs the sample variance based upon the collected data on the Sign test output sheet; and
ProUCL 5.1 sample size GUI draws user's attention to input an appropriate estimate, sd2, the user should
input an appropriate value depending upon the project stage/data availability.
Example 8-5. Output for Single-sample Sign Test Sample Sizes (a = 0.05, ft = 0.1, sd = 3, A = 2)
I Sample Sizes for Single Sample Sign Test
Based on SpecifiedValues of Decision Parameters/DQOs [Data Quaiy Objectives]
Date/Time of Computation 2/26/201012:15:27 PM
User Selected Options
False Rejection Rate [Alpha] 0.05
False Acceptance Rate [Beta] 0.1
Width of G ray R egion [D elta] 2
Estimate of Standard Deviation 3
Approximate Minimum Sample Size
Single Sided Alternative Hypothesis: 35
Two Sided Alternative Hypothesis: 43
8.2.4 Nonparametric Single Sample Wilcoxon Sign Rank (WSR) Test
The purpose of the single WSR test is similar to that of the Sign test described above. This test is used to
compare the true location parameter (mean or median) of a population against an AL or Cs without
assuming normality of the underlying population. The details of this test can be found in Conover (1999)
andEPA(2006a).
8.2.4.1 Case I (Right-Sided Alternative Hypothesis)
Ho: population location parameter < specified value, Cs vs. HA: population location parameter >
specified value, Cs
A description of the gray region associated with this right-sided test is given in Section 8.1.2.1.
244
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8.2.4.2 Case II (Left-Sided Alternative Hypothesis)
Ho: population location parameter > specified value, Cs vs. HA: population location parameter <
specified value, Cs
A description of the gray region associated with this left-sided (left-tailed) test is given in Section 8.1.2.2.
The minimum sample size, n, needed to perform the single-sample one-sided (both left-sided and right-
sided) WSRtest is given as follows.
n = \.\6
£*
2
J
(8-11)
Where:
sd2 = a preliminary estimate of the population variance which is used during the planning stage;
and
sd2 = actual sample variance computed using the collected data to be used when assessing the
power of the test in retrospect based upon collected data
Note: ProUCL 5.0 sample size GUI draws user's attention to input an appropriate estimate, sd2; the user
should input an appropriate value depending upon the project stage/data availability.
8.2.4.3 Case III (Two-Sided Alternative Hypothesis)
Ho: population location parameter = specified value, Cs;vs. HA: population location parameter =£
specified value, Cs
A description of the gray region associated with the two-sided WSRtest is given in Section 8.1.2.3.
The sample size, n, needed to perform the single-sample two-sided WSRtest is given by:
Zl ~"+Zl " 7' " (8-12)
A" 2
V J
Where:
sd2 = a preliminary estimate of the population variance (e.g., estimated from similar sites) which
is used during the planning stage; and
sd2 = sample variance computed using actual collected data to be used to assess the power of the
test in retrospect.
Note: ProUCL 5.0 sample size GUI draws user's attention to input an appropriate estimate, sd2, the user
should input an appropriate value depending upon the project stage/data availability.
245
-------
The following example: "Sample Sizes for Single-sample Wilcoxon Signed Rank Test" is discussed in
the EPA 2006a (page 65). ProUCL computes the sample size to be 10 for a one-sided alternative
hypothesis, and 14 for a two-sided alternative hypothesis.
Example 8-6. Output for Single-sample WSR Test Sample Sizes (a = 0.1, ft = 0.2, sd = 130, A = 100)
i j Sample Sizes (or Single Sample WilcoHon Signed Rank Ted
B ased on S pecified Values of D ecision Paiameters/D Q Q s (D ata Q uaAy Objectives)
D ate/T ime of Computation 2/26/2010 1:13:58 PM
User Selected Options
False Rejection Rate [Alpha] 0.1
False Acceptance Rate [Beta] 0.2
Width of Q ray R egion [D elta] 100
Estimate of Standard Deviation 130
Approximate Minimum Sample Size
Single Sided Alternative Hypothesis: 10
Two Sided Alternative Hypothesis: 14
8.3 Sample Sizes for Two-Sample Tests for Independent Sample
This section describes minimum sample size determination formulae needed to compute sample sizes
(same number of samples (n=m) from two populations) to compare the location parameters of two
populations (e.g., reference area vs. survey unit, two AOC, two MW) for specified values of the decision
parameters. ProUCL computes sample sizes for one-sided as well as two-sided alternative hypotheses.
The sample size formulae described in this section assume that samples are collected following the simple
random or systematic random sampling (e.g., EPA 2006a) approaches. It is also assumed that samples are
collected randomly from two independently distributed populations (e.g., two different uncorrelated
AOCs); and samples (analytical results) collected from each of population represent independently and
identically distributed observations from their respective populations.
8.3.1 Parametric Two-sample t-test (Assuming Normality)
The details of the two-sample t-test can be found in Chapter 6 of this ProUCL Technical Guide.
8.3.1.1 Case I (Right-Sided Alternative Hypothesis)
Ho: site mean, jUi < background mean, jU2VS. HA: site mean, jUi > background mean, jU2
Gray Region: Range of true concentrations where the consequences of deciding the site mean is less than
or equal to the background mean (when in fact it is greater), that is, a dirty site is declared clean, are
relatively minor. The upper bound of the gray region is defined as the alternative site mean concentration
level, jUi (> jU2), where the human health, and environmental consequences of concluding that the site is
clean (or comparable to background) are relatively significant. The false acceptance rate, /?, is associated
with the upper bound of the gray region, A.
246
-------
8. 3. 1 2 Case // (Left-Sided Alternative Hypothesis)
Ho: site mean, jUi > background mean, jU2VS. HA: site mean, jUi < background mean, jU2
Gray Region: Range of true mean values where consequences of deciding the site mean is greater than or
equal to the background mean (when in fact it is smaller); that is, a clean site is declared a dirty site, are
considered relatively minor. The lower bound, //; (< ^2) of the gray region, is defined as the
concentration where consequences of concluding that the site is dirty would be too costly, potentially
requiring unnecessary cleanup. The false acceptance rate is associated with the lower bound of the gray
region.
The minimum sample sizes (equal sample sizes for both populations) for the two-sample one-sided t-test
(both cases I and II described above) are given by:
m = n = 2(zl_a+zl_p)2 ^ +^ (8-13)
The decision parameters used in equations (8-13) and (8-14) have been defined earlier in Section 8.1.1.2.
A = width (e.g., difference between two means) of the gray region
Sp = a preliminary estimate of the common population standard deviation, a, of the two
populations (discussed in Chapter 6). Some guidance on the selection of an estimate of the
population sd, a, is given above in Section 8.1.2.
Sp = pooled standard deviation computed using the actual collected data to be used when
assessing the power of the test in retrospect.
8.3.1.3 Case III (Two-Sided Alternative Hypothesis)
Ho: site mean, pi = background mean, /Li2vs. HA: site mean, //; ^ background mean, /j.2
The minimum sample sizes for specified decision parameters are given by:
(8-14)
The following example: "Sample Sizes for Two-sample t Test" is discussed in the EPA 2006a guidance
document (page 68). According to this example, for the specified decision parameters, the minimum
number of samples from each population comes out to be m = n > 4.94. ProUCL computes minimum
sample sizes for the two populations to be 5 (rounding up) for the single sided alternative hypotheses and
7 for the two-sided alternative hypothesis.
Note: Sp represents the pooled estimate of the populations under comparison. During the planning stage,
the user inputs a preliminary estimate of variance while computing the minimum sample sizes; and while
assessing the power associated with the t-test, the user inputs the pooled standard deviation, Sp, computed
using the actual collected data.
247
-------
Sp = a preliminary estimate of the common population standard deviation (e.g., estimated from
similar sites, pilot studies, expert opinion) which is used during the planning stage; and
Sp = pooled standard deviation computed using the collected data to be used when assessing the
power of the test in retrospect.
ProUCL outputs the pooled standard deviation, Sp, based upon the collected data on the two sample t-test
output sheet; ProUCL 5.1 sample size GUI draws user's attention to input an appropriate estimate of the
standard deviation, the user should input an appropriate value depending upon the project stage/data
availability.
Example 8-7. Output for two-sample t-test sample sizes (a = 0.05, ft = 0.2, sp = 1.467, A = 2.5)
j Sample Sizes for Two Sample (Test
Based on Specified Values of Decision Parameters/DQOs(Data QuaHy Objectives)
D ate/T ime of Computation 2/26/2010 1:17:57 PM
User Selected Options
False R ejection R ate [Alpha] 0.05
False Acceptance R ate [B eta] 0.2
Width of G ray R egion [D elta] 2.5
Estimate of Pooled SD 1.467
Approximate Minimum Sample Size
Single Sided Alternative Hypothesis: 5
Two Sided Alternative Hypothesis: 7
8.3.2 Wilcoxon-Mann-Whitney (WMW) Test (Nonparametric Test)
The details of the two-sample nonparametric WMW can be found in Chapter 6; this test is also known as
the two-sample WRS test.
8.3.2.1 Case I (Right-Sided Alternative Hypothesis)
Ho: site median < background median vs. HA: site median > background median
The gray region for the WMW Right-Sided alternative hypothesis is similar to that of the two-sample li-
test described in Section 8.1.3.1.
8.3.2.2 Case II (Left-Sided Alternative Hypothesis)
Ho: site median > background median vs. HA: site median < background median
The gray region for the WMW left-sided alternative hypothesis is similar to that of two-sample t-test
described in Section 8.1.3.2.
The sample sizes n and m, for one-sided two-sample WMW tests are given by
248
-------
(8-15)
Here:
sd2 =a preliminary estimate of the common variance,
-------
Example 8-8. Output for Two-sample WMW Test Sample Sizes (a = 0.05, ft = 0.1, s = 3, A = 2)
! Sample Sizes for Two SampleWilcoxon Mann Whitney Test
Based on Specified Values of Decision Paramelers/DQQs (Data Quafcy Objectives)
Date/Time of Computation 2/26/201012:18:47 PM
User Selected Options
False R ejection R ate [Alpha] 0.05
False Acceptance R ate [B eta] 0.1
Width of G ray R egion [D elta] 2
Estimate of Standard Deviation 3
Approximate Minimum Sample Size
Single Sided Alternative Hypothesis: 46
Two Sided Alternative Hypothesis: 56
8. 3. 3 Sample Size for WMW Test Suggested by Noether(198 7)
For the two-sample WRS test (WMW test), the MARSSIM guidance document (EPA 2000) uses the
following combined sample size formula suggested by Noether (1987). The combined sample size,
N=(m+n) equation for the one-sided alternative hypothesis defined in Case I (Section 8.3.2.1) and Case II
(Section 83.2.2) above is given as follows:
--
N = m + n = - - — , where
3(P-0.5)
A = Width of the gray region
sd = an estimate of the common standard deviation of the two populations.
P =
-------
Example: An example illustrating these sample size calculations is discussed as follows. In the following
example, ProUCL computes the sample size to be 46 for the single sided alternative hypothesis and 56 for
the two sided alternative hypothesis when the user selects the default values of the decision parameters.
Using Noether's formula (as used in MARSSIM document), the combined sample size, N= m + n
(assuming m = n) is 87 for the single sided alternative hypothesis, and 107 for the two sided alternative
hypothesis.
Output for two sample WMW Test sample sizes (a = 0.05, /? = 0.1, s = 3, A = 2)
Sample Sizes for Two S ample WilcoKon-M ann-WHney Test
B ased on S pecified Values of D ecision Parameters/D Q 0 s (D ata Q uafty Objectives)
D ate/T ime of Computation 7/23/2010 11:58:40 AM
User Selected Options
False R ejection R ate [Alpha] 0.05
False Acceptance R ate [B eta] 0.1
Width of Gray Region [Delta] 2
Estimate of Standard Deviation 3
Approximate Minimum Sample Size
Single Sided Alternative Hypothesis: 46
Two Sided Alternative Hypothesis: 56
MARSSIM WRS Test (Noether, 1987)
Approximate Minimum Sample Size
Single Sided Alternative Hypothesis: 87
Two Sided Alternative Hypothesis: 107
8.4 Acceptance Sampling for Discrete Objects
ProUCL can be used to determine the minimum number of discrete items that should be sampled, from a
lot consisting of n discrete items, to accept or reject the lot (drums containing hazardous waste) based
upon the number of defective items (e.g., mean contamination above an action level, not satisfying a
characteristic of interest) found in the sampled items. This acceptance sampling approach is specifically
useful when the sampling is destructive, that is an item needs to be destroyed (e.g., drums need to be
sectioned) to determine if the item is defective or not. The number of items that need to be sampled is
determined for the allowable number of defective items, d= 0, 1, 2, ...,«. The sample size determination
is not straight forward as it involves the use of the beta and hypergeometric distributions. Several
researchers (Scheffe and Tukey 1944; Laga and Likes 1975; Hahn and Meeker 1991) have developed
statistical methods and algorithms to compute the minimum number of discrete objects that should be
sampled to meet specified (desirable) decision parameters. These methods are based upon nonparametric
tolerance limits. That is, computing a sample size so that the associated UTL will not exceed the
acceptance threshold of the characteristic of interest. The details of the terminology and algorithms used
for acceptance sampling of lots (e.g., a batch of drums containing hazardous waste) can be found in the
RCRA guidance document (EPA 2002c).
In acceptance sampling, sample sizes based upon the specified values of decision parameters can be
computed using the exact beta distribution (Laga and Likes 1975) or the approximate chi-square
distribution (Scheffe and Tukey 1944). Exact as well as approximate algorithms have been incorporated
251
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in ProUCL 4.1 and higher versions of ProUCL. It is noted that the approximate and exact results are often
in complete agreement for most values of the decision parameters. A brief description now follows.
8.4.1 Acceptance Sampling Based upon Chi-square Distribution
The sample size, n, for acceptance sampling using the approximate chi-square distribution is given by:
m-\ ( (l + p) ] 2f ,
n = ~j~ + \ 4/1 \ *«(2/M) (8'17)
v V / J
Where:
m = number of non-conforming defective items (always >\,m = \ implies '0' exceedance rule)
p=\- proportion
proportion = pre-specified proportion of non-conforming items
a = 1 - confidence coefficient, and
Xa 2m = th£ cumulative percentage point of a chi-square distribution with 2m df, the area to the
left of xl,im is «.
8.4.2 Acceptance Sampling Based upon Binomial/Beta Distribution
Let x be a random variable with arbitrary continuous probability density function f(x). Let xi p\ = \-a (8-18)
The statement given by (8-18) implies that the interval (xr, xn+l_s) contains at least a proportion, p, of
the distribution with the probability, (1 - a). The interval, (xr,xn+l_s), whose endpoints are the rth
smallest and sth largest observations in a sample size of n, is a nonparametric 100p% tolerance interval
with a confidence coefficient of (1 - a), and xr and xn+i-s are the lower and upper tolerance limits
respectively.
xn+l-s
The variable z = f(x)dx has the following beta probability density function:
x,
g(z) = — -z™(\-z)m-\ 0
-------
The probability P (z >p) can be expressed in terms of binomial distribution as follows:
n}ptl-p"-t (8-20)
For given values of m, p and a, the minimum sample size, n, for acceptance sampling is obtained by
solving the inequality:
P(z>p)>l-a (8-21)
Where:
#2 = number of non-conforming items (always greater than 1)
p = 1 -proportion
proportion = pre-specified proportion of non-conforming items; and
a = 1 - confidence coefficient.
An example output generated by ProUCL is given as follows.
Example 8-9. Output Screen for Sample Sizes for Acceptance Sampling (default options)
! Acceptance Sampling for Pre-specified Proportion of Non-confomwig Items
Based on Specified Values of Decision Paiameters/DQOs
D ate/T ime of Computation 2/26/2010 12:20:34 PM
User Selected Options
Confidence Coefficient 0.95
Pre-specified proportion of non-conforming items in the lot 0.05
Number of allowable non-conforming items in the lot 0
Approximate Minimum Sample Size
Exact Binomial/Beta Distribution 59
Approximate Chisquare Distribution (Tukey-Scheffe) 59
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CHAPTER 9
Oneway Analysis of Variance Module
Both parametric and nonparametric Oneway Analysis of Variance (ANOVA) methods are available in
ProUCL 5.0 under the Statistical Tests module. A brief description of Oneway ANOVA is described in
this chapter.
9.1 Oneway Analysis of Variance (ANOVA)
In addition to the two-sample hypothesis tests, ProUCL software has Oneway ANOVA to compare the
location (mean, median) parameters of more than two populations (groups, treatments, monitoring wells).
Both classical and nonparametric ANOVA are available in ProUCL. Classical Oneway ANOVA assumes
the normality of all data sets collected from the various populations under comparison; classical ANOVA
also assumes the homoscedasticity of the populations that are being compared. Homoscedasticity means
that the variances (spread) of the populations under comparisons are comparable. Classical Oneway
ANOVA represents a generalization of the two-sample t-test (Chapter 6). ProUCL has GOF tests to
evaluate the normality of the data sets but a formal F-test to compare the variances of more than two
populations has not been incorporated in ProUCL. The users may want to use graphical displays such as
side-by-side box plots to compare the spreads present in data sets collected from the populations that are
being compared. A nonparametric Oneway ANOVA test: Kruskal-Wallis (K-W) test is also available in
ProUCL. The K-W test represents a generalization of the two-sample WMW test described in Chapter 6.
The K-W test does not require the normality of the data sets collected from the various
populations/groups. However, for each group, the distribution of the characteristic of interest should be
continuous and those distributions should have comparable shapes and variabilities.
9.1.1 General Oneway ANOVA Terminology
Statistical terminology used in Oneway ANOVA is described as follows:
g: number of groups, populations, treatments under comparison
/': an index used for the ith group, / = 1, 2, ... , g
nf. number of observations in the ith group
j: an index used for they* observation in a group; for the ith,j = 1,2, ...,«,
xtf. the j* observation of the response variable in the /'* group
n: total number of observations= nl+n2+...
x. . = sum of all observations in the ith group
j=i
J. = mean of the observations collected from the ith group
x = mean of all, nt (the observations)
jut = true (unknown) mean of the /'th group
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In Oneway ANOVA, the null hypothesis, Ho, is stated as: the g groups under comparison have equal
means (medians) and that any differences in the sample means/medians are due to chance. The alternative
hypothesis, HA is stated as: the means/medians of the g groups are not equal.
The decision to reject or accept the null hypothesis is based upon a test statistic computed using the
available data collected from the g groups.
9.2 Classical Oneway ANOVA Model
The ANOVA model is represented by a regression model in which the predictor variables are the
treatment or group variables. The Oneway ANOVA model is given as follows:
xtj=»t+etj (9-1)
Where ^ is the population mean (or median) of the ith group, and errors, e^, are assumed to be
independently and normally distributed with mean = 0 and with a constant variance, a2. All observations
in a given group have the same expectation (mean) and all observations have the same variance regardless
of the group. The details of Oneway ANOVA can be found in most statistical books including the text by
Kunteretal. (2004).
The null and the alternative hypotheses for Oneway ANOVA are given as follows:
HA : At least one of the means (or medians) is not equal to others
Based upon the available data collected from the g groups, the following statistics are computed. ProUCL
summarizes these results in an ANOVA Table.
• Sum of Squares Between Groups is given by:
g ,
99 —'Vwfv—vl (Q 7^
00Between Groups £^ ' \ ' I \y~^>
i=l
• Sum of Squares Within Groups is given by:
99 =VV(r -rV CQ T>
^Within Groups Zj Zj V W ') \ '
• Total Sum of Squares is given by:
(9-4)
Between Groups Degrees of Freedom (df): g-1
Within Groups df. n-g
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etween Groups
• Total df. n-1
• Mean Squares Between Groups is given by:
ss,
yl fC^ L
^ Between Groups ~ ^ \^~^7
• Mean Squares Within Groups:
TI v-r> Within Groups ,„ ,,
Within Groups \ J
n-g
• Scale estimate is given by:
S = ^MSmthm Groups (9-7)
• R2 is given by:
ee
7?2 = 1 mthin Groups ,„ „,
OO ^ '
• Decision statistic, F, is given by:
T^ t^, <• <• Between Groups /r. r.\
F Statistic = — (9-9)
Within Groups
Under the null hypothesis, the F-statistic given in equation (9-9) follows the F(g.i), (n-g) distribution with
(g-1) and (n-g) degrees of freedom, provided the data sets collected from the g groups follow normal
distributions. ProUCL software computes/>-values using the F distribution, F(g.i), (n-g).
Conclusion: The null hypothesis is rejected for all levels of significance, a >/rvalue.
9.3 Nonparametric Oneway ANOVA (Kruskal-Wallis Test)
Nonparametric Oneway ANOVA or the K-W test (Kruskal and Wallis 1952, Hollander and Wolfe 1999)
represents a generalization of the two-sample WMW, test which is used to compare the equality of
medians of two groups. Like the WMW test, analysis for the K-W test is also conducted on ranked data,
therefore, the distributions of the g groups under comparisons do not have to follow a known statistical
distribution (e.g., normal). However, distributions of the g groups should be continuous with comparable
shapes and variabilities. Also the g groups should represent independently distributed populations.
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The null and alternative hypotheses are defined in terms of medians, /w* of the g groups:
(9-10)
Hn :m, =
01
HA : At least one of the g medians is not equal to others
While performing the K-W test, all n observations in the g groups are arranged in ascending order with
the smallest observation receiving the smallest rank and the largest observation getting the highest rank.
All tied observations receive the average rank of those tied observations.
K-W Test on Data Sets with NDs: It should be noted that the K-W test may be used on data sets with NDs
provided all NDs are below the largest detected value. All NDs are considered as tied observations
irrespective of reporting limits (RLs) and receive the same rank. However, the performance of the K-W
test on data sets with NDs is not well studied; therefore, it is suggested that the conclusion derived using
the K-W test statistics be supplemented with graphical displays such as side-by-side box plots. Side-by-
side box plots can also be used as an exploratory tool to compare the variabilities of the g populations
based upon the g data sets collected from those populations.
The K-W ANOVA table displays the following information and statistics:
• Mean Rank of the ith Group, R^ : Average of the ranks (in the combined data set of size, «) of the
Hi observations in the ith group.
• Overall Mean Rank, R : Average of the ranks of all n observations.
• Z-value of each group are computed using the following equation (Standardized Z):
Z,= ., *- ., (9-H)
12
n = total number of observations = nl+n2+ ... + n
Hi = observation in the ith group
g = number of groups
Zj given by (9-11) represents standardized normal deviates. The Z, can be used to determine the
significance of the difference between the average rank of the ith group and the overall average rank, R, of
the combined data set of sized n.
• Kruskal-Wallis H-Statistic (without ties) is given by:
- (9-12)
n(n + l)
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K-W H-Statistic adjusted for ties is given by:
H
H
adj-ties
i
&
S',3-',
n -n
(9-13)
Where tt = number of tied values in ith group
For large values of n, the H-statistic given above follows an approximate chi-square distribution with (g-
1) degrees of freedom, /'-values associated with the H-statistic given by (9-12) and (9-13) are computed
by using a chi-square distribution with (g-1) degrees of freedom. The /"-values based upon a chi-square
approximation test are fairly accurate when the number of observations, n, is large such as > 30.
Conclusion: The null hypothesis is rejected in favor of the alternative hypothesis for all levels of
significance, a >/>-value.
Example 9-1. Consider Fisher's famous Iris data set (Fisher 1936) with 3 iris species. The classical
Oneway ANOVA results comparing petal widths of 3 iris species are summarized as follows.
Classical Oneway ANOVA
Date/Time erf Computation 3/2/2013 1:25:29 PM
From File FULLIRISjds
Full Precision OFF
pt-width
Group Obs
1 50
2 50
3 50
Grand Statistics (All data) 153
Mean SD Variance
0.246 0.105 0.0111
1.326 0.198 0.0391
2.026 0.275 0.07.54
1.133 0.762 0.581
Classical One-Way Analysis of Variance Table
Source SS DOF
Between Groups 80.41 2
Within Groups 6.157 147
Total 36.57 149
MS V.R.(FStat) P-Value
40.21 360 0
0.0419
Pooled Standard Deviation 0.205
R-Sq 0.929
Note: A p-value <= 0.05 (or some other selected level) suggests that there are significant differences in
mean/median characteristics of the various groups at 0.05 or other selected level of significance
A p-value > 0.05 (or other selected level) suggests that mean/median characteristics of the various groups are comparable.
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Example 9-2 (Iris Data). The K-W Oneway ANOVA results comparing petal widths of 3 iris species are
summarized as follows.
Nonparametric Oneway ANGVATKnlskai-Wallis Test)
Date/Time of Computation 3/2/2013 123:12 P M
From File FLJLLIRISjds
Full Precision OFF
pt-width
Group
1
2
3
Overall
Obs
50
50
50
150
Median
0.2
1.3
2
1,3
Ave Rank
25,5
76.43
124.5
75,5
Z
-3.9€7
0.195
9.771
K-W(H-Stat) DOF P-Value {Apprax. Chisquare)
129.9 2 0
131.2 2 0 {Adjusted for Ties}
Note: A p value <= 0.05 (or some other selected level) suggests that there are significant differences in
mean/median characteristics of the various groups at 0.05 or other selected level of significance
A p value > 0.05 (or other selected level) suggests that mean/median characteristics of the various groups are comparable
259
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CHAPTER 10
Ordinary Least Squares Regression and Trend Analysis
Trend tests and ordinary least squares (OLS) regression methods are used to determine trends (e.g.,
decreasing, increasing) in time series data sets. Typically, OLS regression is used to determine linear
relationships between a dependent response variable and one or more predictor (independent) variables
(Draper and Smith 1998); however statistical inference on the slope of the OLS line can also be used to
determine trends in the time series data used to estimate an OLS line. A couple of nonparametric
statistical tests, the Mann-Kendall (M-K) test and the Theil-Sen (T-S) test to perform trend analysis have
also been incorporated in ProUCL 5.0/ProUCL 5.1. Methods to perform trend analysis and OLS
Regression with graphical displays are available under the Statistical Tests module of ProUCL 5.1. In
environmental monitoring studies, OLS regression and trend tests can be used on time series data sets to
determine potential trends in constituents' concentrations over a defined period of time. Specifically, the
OLS regression with time or a simple index variable as the predictor variable can be used to determine a
potential increasing or decreasing trend in mean concentrations of an analyte over a period of time. A
significant positive (negative) slope of the regression line obtained using the time series data set with
predictor variable as a time variable suggests an upward (downward) trend. A brief description of the
classical OLS regression as function of the time variable, T(t), is described as follows. It should however
be noted that the OLS regression and associated graphical displays can be used to determine a linear
relation for any pair of dependent variable, Y, and independent variable, X. The independent variable
does not have to be a time variable.
10.1 Ordinary Least Squares Regression
The linear regression model for a response variable, Y and a predictor (independent) variable, t is given as
follows:
Y = bn + b,t + e:
(10-1)
E[Y] = b0+ bj = mean response at t
In (10-1), variable e is a random variable representing random measurement error in the response
variable, Y (concentrations). The error variable, e, is assumed to follow a normal distribution, N (0, o2),
with mean 0 and unknown variance, a2. Let (tit yj; i: =1, 2,....n represent the paired data set of size n,
where yt is the measured response when the predictor variable, t =t\. It is noted that multiple observations
may be collected at one or more values of the prediction variable, t. Using the regression model (10-1) on
this data set, we have:
l ! (10-2)
E[yt ] = b0+ blti = mean response when t = tt
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For each fixed value, tt of the predictor variable, t, the random error,et is normally distributed with
NfO,^2). Random errors, et, are independently distributed. Without the random error, e, all points will lie
exactly on the population regression line estimated by the OLS line. The OLS estimates of the intercept,
b0 and slope, bi are obtained by minimizing the residual sum of squares. The details of deriving the OLS
estimates, b0 and \ of the intercept and slope can be found in Draper and Smith (1998).
The OLS regression method can be used to determine increasing or decreasing trends in the response
variable Y (e.g., constituent concentrations in a MW) over a time period (e.g., quarters during a 5 year
time period). A positive statistically significant slope estimate suggests an upward trend and a statistically
significant negative slope estimate suggests a downward or decreasing trend in the mean constituent
concentrations. The significance of the slope estimate is determined based upon the normal assumption
of the distribution of error terms, ef, and therefore, of responses, yit i:=l,2,...,n
ProUCL computes OLS estimates of parameters bo and bf, performs inference about the slope and
intercept estimates, and outputs the regression ANOVA table including the coefficient of determination,
R2, and estimate of the error variance, a2. Note that R2 represents the square of the Pearson correlation
coefficient between the dependent response variable, y, and the independent predictor variable, t.
ProUCL also computes confidence intervals and prediction intervals around the OLS regression line; and
can be used to generate scatter plots of n pairs, (t, y), displaying the OLS regression line, confidence
interval for mean responses, and prediction interval band for individual observations (e.g., future
observations).
General OLS terminology and sum of squares computed using the collected data are described as follows:
(10-3)
The OLS estimates of slope and intercept are given as follows:
=Sty/Stt; and
(10-4)
The estimated OLS regression line is given by: y = b0 + bj and error estimates also called residuals are
given by ef =yi—yi', i = 1,2,...., n . It should be noted that for each /', j>. represents the mean response at
value, tt of the predictor variable, t, for i:=l,2,... ,n.
The residual sum of squares is given by:
SSE = £(yt-yt? (10-5)
z=l
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Estimate of the error variance, a2, and variances of the OLS estimates, b0 and 1\ are given as follows:
0; and 3) or Ho: bv =0 vs. the
alternative, Hi: bv < 0 . Under the null hypothesis, the test statistic is obtained by dividing the regression
estimate by its SE:
t = bJSE(b^) (10-7)
Under normality of the responses, yt (and the random errors, e,), the test statistic given in (10-7) follows a
Student's t-distribution with (n-2) degrees of freedom (df). A similar process is used to perform inference
about the intercept, b0 of the regression line. The test statistic associated with the OLS estimate of the
intercept, b0 also follows a Student's t-distribution with (n-2) degrees of freedom.
P -values: ProUCL computes and outputs t-distribution based />-values associated with the two-sided
alternative hypothesis, Hi: 4^ ^0. The /"-values are displayed on the output sheet as well as on the
regression graph generated by ProUCL.
Note: ProUCL displays residuals including standardized residuals on the OLS output sheet. Those
residuals can be imported (copying and pasting) in an excel data file to assess the normality of those OLS
residuals. The parametric trend evaluations based upon the OLS slope (significance, confidence interval)
are valid provided the OLS residuals are normally distributed. Therefore, it is suggested that the user
assesses the normality of OLS residuals before drawing trend conclusions using a parametric test based
upon the OLS slope estimate. When the assumptions are not met, one can use graphical displays and
nonparametric trend tests, M-K and T-S tests, to determine potential trends in time series data set.
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10.1.1 Regression ANOVA Table
The following statistics are displayed on the regression ANOVA table.
Sum of Squares Regression (SSR): SSR represents that part of the variation in the response variable, Y,
which is explained by the regression model, and is given by:
Sum of Squares Error (SSE): SSE represents that part of the variation in the response variable, Y, which
is attributed to random measurement errors, and is given by:
Sum of Squares Total (SST): SST is the total variation present in the response variable, Y and is equal to
the sum of SSR and SSE.
SST = yj-y2=SSR + SSE (10-9)
z=l
Regression Degrees of Freedom (df): 1 (1 predictor variable)
Error df. n-2; and Total df. n-1
Mean Sum of Squares (MS) Regression (MSR): is given by SSR divided by the regression df which is
equal to 1 in the present scenario with only one predictor variable.
MSR = SSR
Mean Sum of Squares Error (MSB): is given by SSE divided by the error degrees of freedom
MSE=SSE_
n-2
MSE represents an unbiased estimate of the error variance,
-------
MSE
(HMO,
P -value: The overall p-value associated with the regression model is computed using the Fi,(n-2)
distribution of the test- statistic given by equation (10-10).
R2: represents the variation explained in the response variable, Y, by the regression model, and is given
b:
(10-11)
SST
Adjusted R square (Adjusted R2): The adjusted R2 is considered a better measure of the variation
explained in the response variable, Y, and is given by:
R2 =1 (n~l}SSE
ad]usted (n-2JSST
10.1.2 Confidence Interval and Prediction Interval around the Regression Line
ProUCL also computes confidence and prediction intervals around the regression line and displays these
intervals along with the regression line on the scatter plot of the paired data used in the OLS regression.
ProUCL generates, when selected, a summary table displaying these intervals and residuals.
Confidence Interval (LCL. UCL): represents a band within which the estimated mean responses, j>. , are
expected to fall with specified confidence coefficient, (1-a). Upper and lower confidence limits (LCL and
UCL) are computed for each mean response estimate, j>. , observed at value, tt, of the predictor variable, t.
These confidence limits are given by:
Where the estimated standard deviation, sd(yf) , of the mean response, j). , is given by:
Sd[y,]= MSE(- + ^-^-)-i = l,2,...,n
\ n sa
A confidence band can be generated by computing the confidence limits given by (10-12) for each value,
tt of the predictor variable, t; i: =1,2, ...n.
Prediction Limits (LPL. UPL): represents a band within which a predicted response (and not the mean
response), j>0, for a specified new value, to ,of the predictor variable, t, is expected to fall. Since the
variances of the individual predicted responses are higher than the variances of the mean responses, a
prediction band around the OLS line is wider than the confidence band. The LPL and UPL comprising the
prediction band are given by:
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± t
, with =
(10-13)
Where the estimated standard deviation, sd(y0~), of a new response, y0 ,(or the individual response for
existing observations) is given by:
n
S
Like the confidence band, a prediction band around the OLS line can be generated by computing the
prediction limits given by (10-13) for each value, t\, of the predictor variable, t, and also other values oft
(within the experiment range) for which the response, y, was not observed.
Notes: Unlike M-K and T-S trend tests, multiple observations may be collected at one or more values of
the predictor variable. Specifically, OLS can be performed on data sets with multiple measurements
collected at one or more values of the predictor variable (e.g., sampling time variable, t).
Example 10-1. Consider the time series data set for sulfate as described in RCRA Guidance (EPA 2009).
The OLS graph with relevant test statistics is shown in Figure 10-1 below. The positive slope estimate,
33.12, is significant with ap-value of 0 suggesting that there is an upward trend in sulfate concentrations.
Classical Regression
Slope
Intercept
Scale Estimate
P-value (Reg)
P-vatue (Slope)
22
33.1230
-2,503 27S8
07372
0.8586
47.4355
0.0000
SDofS
Staroiardized S
Approxi mate p-val ue
Confidence Coefficient
Red = Prediction Interval
137.0000
35.3977
5.2546
Figure 10-1. OLS Regression of Sulfate as a Function of Time
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Number Reported ^-values) 22
Dependendant Variable Sulfate
Independent Variable Date
Regression Estimates and Inference Table
Parameter Estimates Std. Error T-values p-values
intercept -2503 410.7 -€.095 5.8853E-6
Date 33.12 4.422 7.49 3.1763E-7
OLS ANOVA TaWe
Source of Variation SS DOF MS F-Value P-Value
Regression 12S230 1 126230 56.1
Bror 45003 20 2250
Total 171233 21
R Square 0.737
Adjusted R Square 0.724
Sqrt(MSE) = Scale 47.44
10.2 Trend Analysis
Time Series Data Set: When the predictor variable, t, represents a time variable (or an index variable), the
data set ftt, yj; i:=l,2,....n is called a time series data set, provided values of the variable, t, satisfy:
ti
-------
Handling Nondetects: The trend module in ProUCL 5.1 does not recognize a nondetect column consisting
of zeros and ones. For data sets consisting of nondetects with varying DLs, one can replace all NDs with
half of the lowest DL (DL/2) or by replacing all NDs by a single value lower than the lowest DL. When
multiple DLs are present in a data set, the use of substitution methods should be avoided. Replacing NDs
by their respective DLs or by their DL/2 values is like performing trend test on DLs or on DL/2s,
especially when the percentage of NDs present in the data set is high.
10.2.1 Mann-Kendall Test
The M-K trend test is a nonparametric test which is used on a time series data set, (tit yt); i:=l,2,... .n as
described earlier. As a nonparametric procedure, the M-K test does not require the underlying data to
follow a specific distribution. The M-K test can be used to determine increasing or decreasing trends in
measurement values of the response variable, y, observed during a certain time period. If an increasing
trend in measurements exists, then the measurement taken first from any randomly selected pair of
measurements should, on average, have a lower response (concentration) than the measurement collected
at a later point.
The M-K statistic, S, is computed by examining all possible distinct pairs of measurements in the time
series data set and scoring each pair as follows. It should be noted that for a measurement data set of size,
n, there are n(n-l)/2 distinct pairs, (yj,yt) withj>/, which are being compared.
• If an earlier measurement, yt, is less in magnitude than a later measurement, j;, then that pair is
assigned a score of 1;
• If an earlier measurement value is greater in magnitude than a later value, the pair is assigned a
score of-1; and
• Pairs with identical (y* = yj) measurements values are assigned a score of 0.
The M-K test statistic, S, equals the sum of scores assigned to all pairs. The following conclusions are
derived based upon the values of the M-K statistic, S.
• A positive value of S implies that a majority of the differences between earlier and later
measurements are positive suggesting the presence of a potential upward and increasing trend
overtime.
• A negative value for S implies that a majority of the differences between earlier and later
measurements are negative suggesting the presence of a potential downward/decreasing trend.
• A value of S close to zero indicates a roughly equal number of positive and negative scores
assigned to all possible distinct pairs, (yj,yt) withy>/, suggesting that the data do not exhibit any
evidence of an increasing or decreasing trend.
When no trend is present in time series measurements, positive differences in randomly selected pairs of
measurements should balance negative differences. In other words, the expected value of the test statistic
S, E[S], should be close to '0' when the measurement data set does not exhibit any evidence of a trend.
To account for randomness and inherent variability in measurements, the statistical significance of the M-
K test statistic is determined. The larger the absolute value of S, the stronger the evidence for a real
increasing or decreasing trend. The M-K test in ProUCL can be used to test the following hypotheses:
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Null Hypothesis, Ho: Data set does not exhibit sufficient evidence of any trends (stationary
measurements) vs.
• HA: Data set exhibits an upward trend (not necessarily linear); or
• HA: Data set exhibits a downward trend(not necessarily linear); or
• HA: Data set exhibits a trend (two-sided alternative - (not necessarily linear)).
Under the null hypothesis of no trend, it is expected that the mean value of S =0; that is E[S] =0.
Notes: The M-K test in ProUCL can be used for testing a two-sided alternative, HA, stated above. For a
two-sided alternative hypothesis, the />-values (exact as well as approximate) reported by ProUCL need to
be doubled.
10.2.1.1 Large Sample Approximation for M-K Test
When the sample size n is large, the exact critical values for the statistic S are not readily available.
However, as a sum of identically-distributed random quantities, the distribution of S tends to
approximately follow a normal distribution by the CLT. The exact /"-values for the M-K test are available
for sample sizes up to 22 and have been incorporated in ProUCL. For samples of sizes larger than 22, a
normal approximation to S is used. In this case, a standardized ^-statistic, denoted by Z is computed by
using the expected mean value and sd of the test statistic, S.
The sd ofS, sd(S) is computed using the following equation:
(10-14)
Where n is the sample size, g represents the number of groups of ties (if any) in the data set, and /}• is the
number of tied observations in the f1 group of ties. If no ties or NDs are present, the equation reduces to
the simpler form:
(10-15)
The standardized S statistic denoted by Z for an increasing (or decreasing) trend is given as follows:
Z = ^^- ifS>0;
sd(S)
Z = 0 if S = 0;and (10-16)
ifs<0
sd(S)
Like the S statistic, the sign of Z determines the direction of a potential trend in the data set. A positive
value of Z suggests an upward (increasing) trend and a negative value of Z suggests a downward or
decreasing trend. The statistical significance of a trend is determined by comparing Z with the critical
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value, z«, of the standard normal distribution; where za represents that value such that the area to the right
of za under the standard normal curve is a.
10.2.1.2 Step-by-Step Procedure to perform the Mann-Kendall Test
The M-K test does not require the availability of an event or a time variable. However, if graphical trend
displays (e.g., T-S line) are desired, the user should provide the values for a time variable. When a time or
an event variable is not provided, ProUCL generates an index variable and displays the time-series graph
using the index variable.
Step 1. Order the measurement data: yi, y?, ...., yn by sampling event or time of collection. If the
numerical values of data collection times (event variable) are not known, the user should enter data values
according to the order they were collected. Next, compute all possible differences between pairs of
measurements, (yj-yi) for/ > /'. For each pair, compute the sign of the difference, defined by:
sgr
0
Step 2. Compute the M-K test statistic, S, given by the following equation:
(10-18)
In the above equation the summation starts with a comparison of the very first sampling event against
each of the subsequent measurements. Then the second event is compared with each of the samples taken
after it (i.e., the third, fourth, and so on). Following this pattern is probably the most convenient way to
ensure that all distinct pairs have been considered in computing S. For a sample of size n, there will be
«(«-l)/2 distinct pairs, (i,j) withy>/.
Step 3. For n<23, the tabulated critical levels, acp (tabulated />-values) given in Hollander and Wolfe
(1999), have been incorporated in ProUCL. If S > 0 and a > acp, conclude there is statistically significant
evidence of an increasing trend at the a significance level. If S < 0 and a> acp, conclude there is
statistically significant evidence of a decreasing trend. If a < acp, conclude that data do not exhibit
sufficient evidence of any significant trend at the a level of significance .
Specifically, the M-K test in ProUCL tests for one-sided alternative hypothesis as follows:
Ho: no trend vs. HA: upward trend
or
Ho: no trend vs. HA: downward trend
ProUCL computes tabulated /"-values (for sample sizes <23) based upon the sign of the M-K statistic, S,
as follows:
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IfS>0, the tabulated p-value (acp) is computed for Ho: no trend, vs. HA: upward trend
IfS<0, the tabulated p-value (acp) is computed for Ho: no trend vs. HA. downward trend
If the p-value is larger than the specified a (e.g., 0.05), the null hypothesis of no trend is not rejected.
Step 4. For n > 22, large sample normal approximation is used for S, and a standardized S is computed.
Under the null hypothesis of no trend, E(S) =0, and the sd is computed using equations (10-14) or (10-15).
When ties are present, sd(S) is computed by adjusting for ties as given in (10-14). Standardized S,
denoted by Z is computed using equation (10-16).
Step 5. For a given significance level (a), the critical value za is determined from the standard normal
distribution.
If Z >0, a critical value and p-value are computed for Ho: no trend, vs. HA: upward trend.
If Z<0, a critical value and p-value are computed for Ho: no trend vs. HA: downward trend
If the p-value is larger than the specified a (e.g., 0.05), the null hypothesis of no trend is not rejected.
Specifically, compare Z against this critical value, za. If Z>0 and Z > za, conclude there is a statistically
significant evidence of an increasing trend at an a-level of significance. If Z<0 and Z < -za, conclude
there is statistically significant evidence of a decreasing trend. If neither exists, conclude that the data do
not exhibit sufficient evidence of any significant trend. For large samples, ProUCL computes the p-value
associated with Z.
Notes: As mentioned, the M-K test in ProUCL can be used for testing a two-sided alternative, HA stated
above. For a two-sided alternative hypothesis, /"-values (both exact and approximate) reported by ProUCL
need to be doubled.
Example 10-2. Consider a nitrate concentration data set collected over a period of time. The objective is
to determine if there is a downward trend in nitrate concentrations. No sampling time event values were
provided. The M-K test has been used to establish a potential trend in nitrate concentrations. However, if
the user also wants to see a trend graph, ProUCL generates an index variable and displays the trend graph
along with OLS line and the T-S nonparametric line (based upon the index variable) as shown in Figure
10-2 below. Figure 10-2 displays all the statistics of interest. The M-K trend statistics are summarized as
follows.
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Mann-Kendall Trend Test Analysis
User Selected Options
Date/Time of Computation
From Rle
Full Precision
Confidence Coefficient
Level of Significance
3/2/2013 3:45:38 PM
Trend-data forNitrate_a jds
OFF
0.95
0.05
Nitrate
General Satisfies
Number Values
Number Values Missing
Number Values Reported (n)
Minimum
Maximum
Mean
Geometric Mean
Median
Standard Deviation
2M
2
202
9.312
19.96
14.29
14.2
13.96
1.688
Mann-Kendall Test
Test Value (S) -4684
Critical Value (0.05) -1.645
Standard Deviation of S 960.5
Standardized Value of S -4.876
Approximate p-value 5.4240E-7
Statistically significant evidence of a decreasing
trend at the specified level of significance.
Mann-Kendall Trend Test
Mann-Kendall Trend Analysis
Confidence Coefficient
Level of Significance
StaraJard Deviation of S
Standardized Value of S
Test Value (SI
Appx. Critical Value (0.05)
202.0000
03500
0.0500
-1.6449
OOCOO
OLS Regres sion Ljne (Blue)
OLS Regression Slope -0.0102
OLS Regression. Intercept 15.3308
Theil-Sen Trend line (I
Thai-Sen Slope
Theil-Sen Intercept
Statistically significant evidence
-0.0093
14.9034
Generated Index
Figure 10-2. Trend Graph with M-K Test Results and OLS Line and Nonparametric Theil-Sen Line
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10.2.2 Theil - Sen Line Test
The details of T-S test can be found in Hollander and Wolfe (1999). The T-S test represents a
nonparametric version of the parametric OLS regression analysis and requires the values of the time
variable at which the response measurements were collected. The T-S procedure does not require
normally distributed trend residuals and responses as required by the OLS regression procedure. It is also
not critical that the residuals be homoscedastic (having equal variance over time). For large samples,
even a relatively mild to modest slope of the T-S trend line can be statistically significantly different from
zero. It is best to first identify whether or not a significant trend (slope) exists, and then determine how
steeply the concentration levels are increasing (or decreasing) over time for a significant trend.
New in ProUCL 5.1: This latest ProUCL 5.1 version computes yhat values and residuals based upon the
Theil-Sen nonparametric regression line. ProUCL outputs the slope and intercept of the T-S trend line,
which can be used to compute residuals associated with the T-S regression line.
Unlike the M-K test, actual concentration values are used in the computation of the slope estimate
associated with the T-S trend test. The test is based upon the idea that if a simple slope estimate is
computed for every pair (n(n-l)/2 pairs in all) of distinct measurements in the sample (known as the set of
pairwise slopes), the average of this set of n(n-l)/2 slopes would approximate the true unknown slope.
Since the T-S test is a nonparametric test, instead of taking an arithmetic average of the pairwise slopes,
the median slope value is used as an estimate of the unknown population slope. By taking the median
pairwise slope instead of the mean, extreme pairwise slopes - perhaps due to one or more outliers or other
errors - are ignored and have little or negligible impact on the final slope estimator.
The T-S trend line is also nonparametric because the median pairwise slope is combined with the median
concentration value and the median of the time values to construct the final trend line. Therefore, the T-S
line estimates the change in median concentration over time and not the mean as in linear OLS regression;
the parametric OLS regression line described in Section 10.1 estimates the change in the mean
concentration overtime (when the dependent variable represents the time variable).
Averaging of Multiple Measurements at Sampling Events: In practice, when multiple observations are
collected/reported at one or more sampling events (times), one or more pairwise slopes may become
infinite, resulting in a failure to compute the T-S test statistic. In such cases, the user may want to pre-
process the data before using the T-S test. Specifically, to assure that only one measurement is available
at each sampling event, the user pre-processes the time series data by computing average, median, mode,
minimum, or maximum of the multiple observations collected at those sampling events. The T-S test in
ProUCL 5.1 provides the option of averaging multiple measurements collected at the various sampling
events. This option also computes M-K test and OLS regression statistics using the averages of multiple
measurements collected at the various sampling event.
Note: The OLS regression and M-K test can be performed on data sets with multiple measurements taken
at the various sampling time events. However, often it is desirable to use the averages (or median) of
measurements taken at the various sampling events to determine potential trends present in a time-series
data set.
272
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10.2.2. 1 Step-by-Step Procedure to Compute Theil-Sen Slope
Step 1. Order the data set by sampling event or time of collection of those measurements. Let yi, j2, ...,yn
represent ordered measurement values. Consider all possible distinct pairs of measurements, (yit y}) for/ >
/'. For each pair, compute the simple pairwise slope estimate given by:
j , .
m = — — for j
For a time-series data set of size n, there are N=n(n-\)/2 such pairwise slope estimates, /%. If a given
observation is a ND, one may use half of the DL or the RL as its estimated concentration. Alternatively,
depending upon the distribution of detected values (also called the censored data set), the users may want
to use imputed estimates of ND values obtained using the GROS or LROS method.
Step 2. Order the N pairwise slope estimates, /% from the smallest to the largest and re-label them as
m(\), m(2),..., m(N). Determine the T-S estimate of slope, Q, as the median value of this set of TV ordered
slopes. Computation of the median slope depends on whether N is even or odd. The median slope is
computed using the following algorithm:
Q =
(10-19)
.. ,r
if N = even
Step 3. Arrange the n measurements in ascending order from smallest to the largest value: y(l), y(2), ... ,
y(n). Determine the median measurement using the following algorithm:
y= ( \ / (10-20)
K(n/2) +^((n+2)/2) )/ ._f
\ { > ^ >V if n = even
Similarly, compute the median time, t of the n ordered sampling times: t\, h, to tn by using the same
median computation algorithm as used in (10-19) and (10-20).
Step 4. Compute the T-S trend line using the following equation:
10.2.2.2 Large Sample Inference for Theil - Sen Test Based upon Normal Approximation
As described in Step 2 above, order the N pairwise slope estimates, /% in ascending order from smallest
to the largest: #?(!), m(2),..., m(N). Compute S given in (10-18) and its sd given below:
273
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(10-21)
ProUCL can be used to test the following hypotheses:
Ho: Data set does not exhibit sufficient evidence of any trends (stationary measurements) vs.
I. HA: Data set exhibits a trend (two-sided alternative)
II. HA: Data set exhibits an upward trend; or
III. HA: Data set exhibits a downward trend.
Case I. Testing for the null hypothesis, Ho.- Time series data set does not exhibit any trend, vs. the two-
sided alternative hypothesis, HA: Data Set exhibits a trend.
• Compute the critical value, Ca using the following equation:
Ca=Za/Sd(S)
Compute Mi andA/2 as:
and
M, =
Obtain the Mf largest and Mf largest slopes, (^(MI) ) and yn(M^ j , from the set consisting of
all n(n-l)/2 slopes. Then the probability of the T-S slope, Q, lying between these two slopes is
given by the statement:
On ProUCL output, (m(M^ j is labeled as LCL and (m(M^J is labeled as UCL.
• Conclusion: If 0 belongs to the interval, (m,M),m,M)), conclude that T-S test slope is
insignificant; that is, conclude that there is no significant trend present in the time series data set.
Cases II and III: Test for an upward (downward) trend with Null hypothesis, Ho.- Time series data set does
not exhibit any trend, vs. the alternative hypothesis, HA: data set exhibits an upward (downward) trend.
• For specified level of significance, a, compute the following:
Ca=Za*sd(S)
'N-Cn
and M9 =
274
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• Obtain the Mf largest and Mf largest slopes, (jn(M^\ and (m(M^\from the set consisting of
all n(n-l)/2 slopes.
• Conclusion:
If WMf) ) > 0 , then the data set exhibits a significant upward trend.
If y^(M2) } < 0 5 then the data set exhibits a significant downward trend.
Example 10-3. Time series data (time event, concentration) were collected from several groundwater
MWs on a Superfund site. The objective is to determine potential trends present in concentration data
collected quarterly from those wells over a period of time. Some missing sampling events (quarters) are
also present. ProUCL handles the missing values, computes trend test statistics and generates a time series
graph along with the OLS and T-S lines.
Theil-Sen Trend Line and OLS Regression Line
0 LS Regression Line (Hue)
OLS Regression Slope 34.0343
OLS Regression Intercept -2.585 1296
TheM-Sen Trend line (Red)
Theil-S£fl Slope 31.4286
Thai-Sen Intercept -2.349.1429
M2
276260
633740
20.0000
LCL of Slope
UCLd Slope 4215S7
Figure 10-3. Time Series Plot and OLS and Theil-Sen Results with Missing Values
The Excel output sheet, generated by ProUCL and showing all relevant results, is shown as follows:
Approximate inference for Theil-Sen Trend Test
Mann-Kendall Statistic IS) 72
Standard Deviation of S 1 S.24
Standardized Value at S
Vd -Amiss
General Statistics
Number of Events
Number Values Observations
Number Values Missing
Number Values Reported in)
Minimum
Maximum
Mean
Geometric Mean
Median
Standard Deviation
14
16
2
14
450
700
536. S
533.6
525
62.99
3.SS3
Approximate p^alue 4.9562E-5
Number of Slopes
Theil-Sen Slope
51
31.43
Theil-Sen Intercept -2349
MT 3D.5
One-sided 95% lower limit of Slope 21.36
95% LCL of Slope (0.025) 20
35% UCL of Slope (0.975) 42.16
Statistically significant evidence of an increasing
trend at the specified level of significance.
275
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Notes: As with other statistical tests (e.g., Shapiro-Wilk and Lilliefors GOF tests for normality), it is very
likely, that based upon a given data set, the three trend tests described here will lead to different trend
conclusions. It is important that the user verifies the underlying assumptions required by these tests (e.g.,
normality of OLS residuals). A parametric OLS slope test is preferred when the underlying assumptions
are met. Conclusions derived using nonparametric tests supplemented with graphical displays are
preferred when OLS residuals are not normally distributed. These tests can also yield different results
when the data set consists of missing values and/or there are gaps in the time series data set. It should be
pointed out that an OLS line (therefore slope) can become significant even by the inclusion of an extreme
value (e.g., collected after skipping of several intermediate sampling events) extending the domain of the
sampling events time interval. For example, a perfect OLS line can be generated using two points at two
extreme ends resulting in a significant slope; whereas nonparametric trend tests are not as influenced by
such irregularities in the data collection and sampling events. In such circumstances, the user should draw
a conclusion based upon the site CSM, expert and historical site knowledge and expert opinions.
10.3 Multiple Time Series Plots
The Time Series Plot option of the Trend Analysis module can generate time series plots for multiple
groups/wells comparing concentration levels of those groups over a period of time. Time series plots are
also useful for comparing concentrations of a MW during multiple periods (every 2 years, 5 years, ...)
collected quarterly, semi-annually. This option can also handle missing sampling events. However, the
number of observations in each group should be the same, sharing the same time event variable (if
provided). An example time series plot comparing concentrations of three MWs during the same period of
time is shown as follows.
Time-Series Trend Analysis
6 iverits.'Time Periods
Missing Values
OL Regression Line (Blue)
h l-Sen Trend LJne (Red)
Time-Series Trend Analysis
Th l
LS Regress!,
LS Regress
LS Regression Slope 5.533.1131
LS Regression Intercept 23,392.5536
heil-Sen Slope 3,623.7500
heil-Sen Intercept 25,178,1250
LS Regression Slope
.S Regression Intercept
heil-Sen Slope
heil-Ser Intercept
-2.3839
548716
-6.1500
61.9000
Figure 10-4. Time Series Plot Comparing Concentrations of Multiple Wells over a Period of Time
This option is specifically useful when the user wants to compare the concentrations of multiple groups
(wells) and the exact sampling event dates are not available (data only option). The user may just want to
graphically compare the time-series data collected from multiple groups/wells during several quarters
(every year, every 5 years, ...). Each group (e.g., well) defined by a group variable must have the same
number of observations and should share the same sampling event values (when available). That is the
276
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number of sampling events and values (e.g., quarter ID, year ID, etc.) for each group (well) must be the
same for this option to work. However, the exact sampling dates (not needed to use this option) in the
various quarters (years) do not have to be the same as long as the values of the sampling quarters
(1,3,5,6,7,9, etc.) used in generating the time-series plots for the various groups (wells) match. Using the
geological and hydrological information, this kind of comparison may help the project team in identifying
non-compliance wells (e.g., with upward trends in constituent concentrations) and associated reasons.
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CHAPTER 11
Background Incremental Sample Simulator (BISS)
Simulating BISS Data from a Large Discrete Background Data
The Background Incremental Sample Simulator (BISS) module was incorporated in ProUCLS.O at the
request of the Office of Superfund Remediation and Technology Innovation (OSRTI). However, this
module is currently under further investigation and research, and therefore it is not available for general
public use. This module may be released in a future version of the ProUCL software, along with strict
conditions and guidance for how it is applied. The main text for this chapter is not included in this
document for the release to general public. Only a brief placeholder write-up is provided here. It is
assumed that the user is familiar with the incremental sampling methodology (ISM) ITRC (2012)
document and terminologies associated with the ISM approach. Those terminologies (e.g., sample
support, decision unit [DU], replicated etc.) are not described in this chapters.
The following scenario describes the site or project conditions under which the BISS module could be
useful: Suppose there is a long history of soil sample collection at a site. In addition to having a large
amount of site data, a robust background data set (at least 30 samples from verified background
locations), has also been collected. Comparison of background data to on-site data has been, and will
continue to be, an important part of this project's decision-making strategy. All historical data is from
discrete samples, including the background data. There is now a desire to switch to incremental sampling
for the site. However, guidance for incremental sampling makes it clear that it is inappropriate to compare
discrete sample results to incremental sample results. That includes comparing a site's incremental results
directly to discrete background results.
One option is to recollect all background data in the form of incremental samples from background
decision units (DUs) that are designed to match site DUs in geology, area, depth, target soil particle size,
number of increments, increment sample support. If project decision-making uses a BTV strategy to
compare site DU results one at a time against background, then an appropriate number (the default is no
less than 10) of background DU incremental samples would need to be collected to determine the BTV
for the population of background DUs. However, if the existing discrete background data show
background concentrations to be low (in comparison to site concentrations) and fairly consistent relative
standard deviation, RSD <1, there is a second option described as follows.
When a robust discrete background data set that meets the above conditions already exists, the following
is an alternative to automatically recollecting ALL background data as incremental samples.
Step 1. Identify 3 background DUs and collect at least 1 incremental sample from each for a minimum of
3 background incremental samples.
Step 2. Enter the discrete background data set (n > 30) and the >3 background incremental samples into
the BISS module (the BISS module will not run unless both data sets are entered).
• The BISS module will generate a specified (default is 7) simulated incremental samples from the
discrete data set.
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• The module will then run a t-test to compare the simulated background incremental data set (e.g.,
with n = 7) to the actual background incremental data set (n > 3).
o If the t-test finds no difference between the 2 data sets, the BISS module will combine
the 2 data sets and determine the statistical distribution, mean, standard deviation,
potential UCLs and potential BTVs for the combined data set. Only this information will
be supplied to the general user. The individual values of the simulated incremental
samples will not be provided.
o If the t-test finds a difference between the actual and simulated data sets, the BISS
module will not combine the data sets nor provide a BTV.
o In both cases, the BISS module will report summary statistics for the actual and
simulated data sets.
Step 3. If the BISS module reported out statistical analyses from the combined data set, select the BTV to
use with site DU incremental sample results. Document the procedure used to generate the BTV in project
reports. If the BISS module reported that the simulated and actual data sets were different, the historical
discrete data set cannot be used to simulate incremental results. Additional background DU incremental
samples will need to be collected to obtain a background DU incremental data set with the number of
results appropriate for the intended use of the background data set.
The objective of the BISS module is to take advantage of the information provided by the existing
background discrete samples. The availability of a large discrete data set collected from the background
areas with geological formations and conditions comparable to the site DU(s) of interest is a requirement
for successful application of this module. There are fundamental differences between incremental and
discrete samples. For example, the sample support (defined in ITRC [2012]) of discrete and incremental
samples are very different. Sample support has a profound effect on sample results so samples with
different sample supports should not be compared directly, or compared with great caution.
Since incremental sampling is a relatively new approach, the performance of the BISS module requires
further investigation. If you would like to try this strategy for your project, or if you have questions,
contact Deana Crumbling, crumbling.deana@epa.gov.
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280
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APPENDIX A
Simulated Critical Values for Gamma GOF Tests, the Anderson-
Darling Test and the Kolmogorov-Smirnov Test &
Summary Tables of Suggestions and Recommendations for
UCL95S
Updated Critical Values of Gamma GOF Test Statistics (New in ProUCL 5.0)
For values of the gamma distribution shape parameter, k < 0.2, critical values of the two gamma empirical
distribution tests (EDF) GOF tests: Anderson-Darling (A-D) and Kolmogorov Smirnov (K-S) tests
incorporated in ProUCL 4.1 and earlier versions have been updated in ProUCL 5.0. Critical values
incorporated in earlier versions of ProUCL were simulated using the gamma deviate generation algorithm
(Whittaker 1974) available at the time and with the source code provided in the book Numerical Recipes
in C, the Art of Scientific Computing (Press et al. 1990). It is noted that the gamma deviate generation
algorithm available at the time was not very efficient, especially for smaller values of the shape
parameter, k < 0.1. For small values of the shape parameter, k, significant discrepancies were found in the
critical values of the two gamma GOF test statistics obtained using the two gamma deviate generation
algorithms: Whitaker (1974) and Marsaglia and Tsang (2000).
Even though, discrepancies were identified in critical values of the two GOF tests for value of k < 0.1, for
comparison purposes, critical values of the two tests have also been re-generated for k=0.2. For values of
k < 0.2, critical values for the two gammas EDF GOF tests have been re-generated and tables of critical
values of the two gamma GOF tests have been updated in this Appendix A. Specifically, for values of the
shape parameter, k (e.g., k < 0.2), critical values of the two gamma GOF tests have been generated using
the more efficient gamma deviate generation algorithm as described in Marsaglia and Tsang (2000) and
Best (1983). Detailed description about the implementation of Marsaglia and Tsang's algorithm to
generate gamma deviates can be found in Kroese, Taimre, and Botev (2011). It is noted that for values of
k > 0.1, the simulated critical values obtained using Marsaglia and Tsang's algorithm (2000) are in
general agreement with the critical values of the two GOF test statistics incorporated in ProUCL 4.1 for
the various values of the sample size considered. Therefore, those critical values for values of k > 0.2
have not been updated in tables as summarized in this Appendix A. The developers double checked the
critical values of the two GOF tests by using MatLab to generate gamma deviates. Critical values
obtained using MatLab code are in general agreement with the newly simulated critical values
incorporated in critical value tables summarized in this appendix.
Simulation Experiments
The simulation experiments performed are briefly described here. The experiments were carried out for
various values of the sample size, n = 5(25)1, 30(50)5, 60(100)10, 200(500)100, and 1000. Here the
notation n=5(25)l means that n takes values starting at 5 all the way up to 25 at increments of 1 each;
n=30(50)5 means that n takes values starting at 30 all the way up to 50 at increments of 5 each, and so on.
Random deviates of sample size n were generated from a gamma, (k, 0), population. The considered
values of the shape parameter, k, are: 0.025, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, and 50.0. These
values of k cover a wide range of values of skewness, 2/Vk. The distributions of the Kolmogorov-Smirnov
(K-S) test statistic, D, and the Anderson-Darling (A-D) test statistic, A2, do not depend upon the scale
281
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parameter, 6, therefore, the scale parameter, 6, has been set equal to 1 in all of the simulation experiments.
A typical simulation experiment can be described in the following four steps.
Step 1. Generate a random sample of the specified size, n, from a gamma, G (k, 1), distribution. For
values of k>0.2, the algorithm as outlined in Whittaker (1974) was used to generate the gamma
deviates; and for values of k < 0.2, Marsaglia and Tsang's algorithm (2000) has been used to
generate gamma deviates.
Step 2. For each generated sample, compute the MLEs of k and 6 (Choi and Wette 1969), and the K-S
and the A-D test statistics (Anderson and Darling, 1954; D'Agostino and Stephens 1986;
Schneider and Clickner 1976) using the incomplete gamma function (details can be found in
Chapter 2 of this document).
Step 3. Repeat Steps 1 and 2, a large number (iterations) of times. For values ofk>0.2, 20,000 iterations
were used to compute critical values. However, since generation of gamma deviates are quite
unstable for smaller values of k (<0.1), 500,000 iterations have been used to obtain the newly
generated critical values of the two test statistics based upon Marsaglia and Tsang's algorithm.
Step 4. Arrange the resulting test statistics in ascending order. Compute the 90%, 95%, and 99%
percentiles of the K-S test statistic and the A-D test statistic.
The resulting raw 10%, 5%, and 1% critical values for the two tests are summarized in Tables 1 through 6
as follows. The critical values as summarized in Tables 1-6 are in agreement (up to 3 significant digits)
with all available exact or asymptotic critical values (note that critical values of the two GOF tests are not
available for values of k<\). It is also noted that the critical values for the K-S test statistic are more stable
than those for the A-D test statistic. It is hoped that the availability of the critical values for the GOF tests
for the gamma distribution will result in the frequent use of more practical and appropriate gamma
distributions in environmental and other applications.
Note on computation of the gamma distribution based decision statistics and critical values: While
computing the various decision statistics (e.g., UCL and BTVs), ProUCL uses biased corrected estimates,
kstar, K , and theta star, 0* (described in Section 2.3.3) of the shape, k, and scale, 0, parameters of the
gamma distribution. It is noted that the critical values for the two gamma GOF tests reported in the
literature (D'Agostino and Stephens 1986; Schneider and Clickner 1976; Schneider 1978) were computed
using the MLE estimates, k and 0, of the two gamma parameters, k and$. Therefore, the critical values
of A-D and K-S tests incorporated in ProUCL have also been computed using the MLE estimates: khat,
k, and theta hat, 0, of the two gamma parameters, k and 0.
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Table A-l. Critical Values for A-D Test Statistic for Significance Level = 0.10
n\k 0.025 0.05 0.1 0.2 0.5 1 2 5 10
20
50
5
6
7
8
9
10
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
60
70
80
90
100
200
300
400
500
1000
0.919726
0.923855
0.924777
0.928382
0.928959
0.930055
0.934218
0.934888
0.935586
0.936246
0.937456
0.937518
0.937751
0.938503
0.938587
0.939277
0.940150
0.941743
0.943737
0.945107
0.947909
0.947922
0.948128
0.948223
0.949613
0.951013
0.951781
0.952429
0.953464
0.955133
0.956040
0.957279
0.802558
0.819622
0.829767
0.834365
0.840361
0.847992
0.864609
0.866151
0.866978
0.869658
0.870368
0.871858
0.874119
0.874483
0.875008
0.875990
0.876204
0.882689
0.885557
0.885878
0.887142
0.887286
0.890153
0.891061
0.891764
0.892197
0.892833
0.893123
0.893406
0.898383
0.898554
0.898937
0.715363
0.735533
0.746369
0.758146
0.765446
0.771909
0.792009
0.795984
0.796929
0.799900
0.800417
0.801716
0.803861
0.804803
0.805412
0.806629
0.807918
0.811964
0.814862
0.817072
0.817778
0.818568
0.820774
0.822280
0.823067
0.823429
0.824216
0.826133
0.826715
0.827845
0.827995
0.828584
0.655580
0.670716
0.684718
0.694671
0.701756
0.707396
0.727067
0.727392
0.729339
0.731904
0.732093
0.733548
0.735995
0.736736
0.737239
0.738236
0.738591
0.741572
0.743736
0.747438
0.748890
0.749399
0.749930
0.750605
0.751452
0.752461
0.752765
0.753696
0.754433
0.755130
0.755946
0.757750
0.612
0.625
0.635
0.641
0.648
0.652
0.663
0.665
0.666
0.668
0.67
0.669
0.671
0.67
0.671
0.672
0.673
0.674
0.676
0.677
0.677
0.677
0.679
0.679
0.68
0.68
0.681
0.682
0.682
0.683
0.683
0.684
0.599
0.61
0.618
0.624
0.629
0.632
0.642
0.642
0.644
0.643
0.645
0.645
0.646
0.646
0.645
0.647
0.648
0.65
0.65
0.651
0.651
0.652
0.652
0.653
0.654
0.654
0.654
0.654
0.655
0.655
0.655
0.655
0.594
0.603
0.609
0.616
0.62
0.623
0.63
0.632
0.632
0.634
0.633
0.633
0.634
0.636
0.635
0.635
0.636
0.637
0.638
0.637
0.639
0.64
0.64
0.641
0.641
0.642
0.642
0.642
0.641
0.641
0.643
0.643
0.591
0.599
0.607
0.612
0.614
0.618
0.624
0.626
0.626
0.626
0.626
0.627
0.628
0.628
0.629
0.628
0.629
0.629
0.631
0.631
0.632
0.632
0.632
0.633
0.633
0.634
0.633
0.634
0.634
0.635
0.635
0.635
0.589
0.599
0.606
0.61
0.613
0.616
0.622
0.624
0.623
0.623
0.625
0.626
0.626
0.627
0.627
0.627
0.627
0.628
0.629
0.629
0.63
0.63
0.631
0.63
0.631
0.631
0.631
0.631
0.633
0.633
0.632
0.632
0.589
0.598
0.604
0.609
0.613
0.615
0.621
0.622
0.623
0.624
0.624
0.624
0.626
0.625
0.625
0.626
0.626
0.627
0.628
0.628
0.628
0.629
0.629
0.63
0.63
0.629
0.63
0.631
0.631
0.631
0.631
0.631
0.588
0.598
0.605
0.608
0.612
0.614
0.621
0.621
0.622
0.623
0.624
0.624
0.624
0.625
0.625
0.625
0.625
0.626
0.627
0.628
0.629
0.629
0.629
0.63
0.629
0.63
0.63
0.63
0.63
0.631
0.631
0.63
283
-------
Table A-2. Critical Values for K-S Test Statistic for Significance Level = 0.10
n\k 0.025 0.050 0.10 0.2 0.50 1.0 2.0 5.0 10.0 20.0
50.0
5
6
7
8
9
10
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
60
70
80
90
100
200
300
400
500
1000
0.382954
0.359913
0.336053
0.315927
0.300867
0.286755
0.238755
0.232063
0.225072
0.218863
0.213757
0.209044
0.204615
0.199688
0.195776
0.192131
0.188048
0.172990
0.160170
0.150448
0.142187
0.135132
0.123535
0.114659
0.107576
0.101373
0.096533
0.068958
0.056122
0.048635
0.043530
0.030869
0.377607
0.352996
0.329477
0.312018
0.296565
0.283476
0.237248
0.228963
0.222829
0.216723
0.211493
0.205869
0.201904
0.197629
0.193173
0.189663
0.185450
0.169910
0.158322
0.148475
0.140171
0.133619
0.122107
0.113414
0.106191
0.100267
0.095061
0.067898
0.055572
0.048048
0.042949
0.030621
0.370075
0.343783
0.321855
0.305500
0.290030
0.276246
0.231259
0.224049
0.218089
0.212018
0.206688
0.202242
0.197476
0.193503
0.188985
0.185566
0.181905
0.166986
0.155010
0.145216
0.137475
0.130496
0.119488
0.110949
0.104090
0.097963
0.093359
0.066258
0.054295
0.047103
0.042053
0.029802
0.358618
0.332729
0.312905
0.295750
0.280550
0.268807
0.223045
0.216626
0.211438
0.205572
0.201002
0.196004
0.191444
0.187686
0.182952
0.179881
0.176186
0.161481
0.150173
0.140819
0.133398
0.126836
0.116212
0.107529
0.100923
0.095191
0.090566
0.064542
0.052716
0.045745
0.040913
0.028999
0.346
0.319
0.301
0.284
0.27
0.257
0.214
0.208
0.202
0.197
0.192
0.187
0.183
0.179
0.175
0.172
0.169
0.155
0.144
0.135
0.127
0.121
0.111
0.103
0.096
0.091
0.086
0.062
0.05
0.044
0.039
0.028
0.339
0.313
0.294
0.278
0.264
0.251
0.209
0.203
0.197
0.192
0.187
0.183
0.179
0.175
0.171
0.168
0.165
0.151
0.14
0.132
0.124
0.118
0.108
0.1
0.094
0.089
0.084
0.06
0.049
0.043
0.038
0.027
0.336
0.31
0.29
0.274
0.26
0.248
0.206
0.2
0.194
0.189
0.184
0.18
0.176
0.172
0.169
0.165
0.162
0.149
0.138
0.13
0.122
0.116
0.107
0.099
0.093
0.088
0.083
0.059
0.048
0.042
0.038
0.027
0.334
0.307
0.288
0.272
0.258
0.246
0.204
0.198
0.193
0.188
0.183
0.179
0.175
0.171
0.167
0.164
0.161
0.147
0.137
0.128
0.121
0.115
0.106
0.098
0.092
0.087
0.082
0.059
0.048
0.042
0.037
0.026
0.333
0.307
0.288
0.271
0.257
0.245
0.204
0.198
0.192
0.187
0.182
0.178
0.174
0.17
0.167
0.163
0.16
0.147
0.136
0.128
0.121
0.115
0.105
0.098
0.092
0.086
0.082
0.058
0.048
0.042
0.037
0.026
0.333
0.307
0.287
0.271
0.257
0.245
0.203
0.197
0.192
0.187
0.182
0.178
0.174
0.17
0.166
0.163
0.16
0.147
0.136
0.128
0.121
0.115
0.105
0.097
0.091
0.086
0.082
0.058
0.048
0.041
0.037
0.026
0.333
0.307
0.287
0.271
0.257
0.245
0.203
0.197
0.192
0.187
0.182
0.178
0.174
0.17
0.166
0.163
0.16
0.147
0.136
0.128
0.121
0.115
0.105
0.097
0.091
0.086
0.082
0.058
0.048
0.041
0.037
0.026
284
-------
Table A-3. Critical Values for A-D Test Statistic for Significance Level = 0.05
n\k 0.025 0.05 0.1 0.2 0.5 1 2 5 10 20
50
5
6
7
8
9
10
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
60
70
80
90
100
200
300
400
500
1000
1.151052
1.163733
1.164504
1.164753
1.165715
1.165767
1.166499
1.166685
1.168544
1.168987
1.169801
1.169916
1.170231
1.170651
1.170815
1.171897
1.173062
1.174361
1.174900
1.177053
1.178564
1.178640
1.179045
1.179960
1.180934
1.183445
1.183507
1.184370
1.186474
1.186711
1.186903
1.188089
0.993916
1.015175
1.027713
1.033965
1.039023
1.051305
1.072701
1.072764
1.074729
1.076805
1.078026
1.080724
1.082101
1.083139
1.084161
1.085896
1.086184
1.095072
1.095964
1.097870
1.099630
1.100960
1.103255
1.105666
1.106509
1.106661
1.107269
1.108491
1.112771
1.113282
1.114064
1.114697
0.867326
0.892648
0.910212
0.926242
0.936047
0.945231
0.971851
0.976822
0.979261
0.982322
0.983408
0.985352
0.988749
0.989794
0.990147
0.991640
0.991848
1.000576
1.000838
1.004925
1.006416
1.007896
1.009514
1.013808
1.014011
1.015090
1.015433
1.018998
1.019934
1.020022
1.020267
1.020335
0.775584
0.801734
0.822761
0.835780
0.847305
0.855135
0.883252
0.883572
0.885946
0.889231
0.891016
0.892498
0.895978
0.896739
0.897642
0.898680
0.899874
0.903940
0.907253
0.909633
0.911353
0.912084
0.914286
0.914724
0.914808
0.915898
0.917512
0.920264
0.920502
0.920551
0.921806
0.923848
0.711
0.736
0.752
0.762
0.771
0.777
0.793
0.796
0.798
0.8
0.803
0.803
0.805
0.804
0.805
0.806
0.807
0.809
0.812
0.813
0.813
0.814
0.816
0.817
0.819
0.818
0.818
0.821
0.822
0.823
0.822
0.824
0.691
0.715
0.728
0.736
0.743
0.748
0.763
0.763
0.766
0.767
0.769
0.768
0.77
0.771
0.769
0.772
0.773
0.775
0.776
0.779
0.777
0.78
0.779
0.78
0.782
0.783
0.783
0.784
0.784
0.785
0.785
0.785
0.684
0.704
0.715
0.724
0.73
0.736
0.747
0.75
0.749
0.753
0.752
0.752
0.754
0.756
0.755
0.755
0.756
0.758
0.76
0.759
0.761
0.763
0.763
0.763
0.763
0.765
0.765
0.766
0.766
0.766
0.767
0.768
0.681
0.698
0.71
0.719
0.723
0.729
0.739
0.741
0.742
0.743
0.742
0.745
0.745
0.746
0.747
0.746
0.747
0.746
0.75
0.751
0.753
0.754
0.753
0.754
0.754
0.755
0.754
0.756
0.757
0.757
0.756
0.757
0.679
0.698
0.708
0.715
0.722
0.725
0.737
0.739
0.739
0.739
0.741
0.742
0.743
0.744
0.744
0.744
0.745
0.745
0.748
0.748
0.748
0.75
0.751
0.751
0.75
0.752
0.752
0.751
0.755
0.754
0.753
0.753
0.679
0.697
0.707
0.716
0.721
0.725
0.735
0.737
0.738
0.739
0.74
0.741
0.743
0.74
0.742
0.742
0.743
0.744
0.747
0.747
0.748
0.748
0.749
0.749
0.751
0.75
0.75
0.751
0.751
0.751
0.752
0.752
0.678
0.697
0.708
0.715
0.721
0.724
0.734
0.735
0.737
0.738
0.74
0.739
0.741
0.743
0.741
0.742
0.742
0.744
0.745
0.746
0.747
0.748
0.748
0.749
0.748
0.751
0.75
0.75
0.752
0.752
0.752
0.75
285
-------
Table A-4. Critical Values for K-S Test Statistic for Significance Level = 0.05
n\k 0.025 0.05 0.1 0.2 0.5 1 2 5 10 20 50
5
6
7
8
9
10
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
60
70
80
90
100
200
300
400
500
1000
0.425015
0.393430
0.367179
0.348874
0.331231
0.315236
0.262979
0.255659
0.247795
0.240719
0.235887
0.229517
0.224925
0.219973
0.215140
0.211022
0.207233
0.187026
0.176132
0.165449
0.156286
0.148646
0.135915
0.126014
0.118350
0.111619
0.106157
0.070489
0.061746
0.053335
0.047696
0.034028
0.416319
0.384459
0.361553
0.342809
0.325179
0.311210
0.260524
0.251621
0.244721
0.237832
0.232558
0.227125
0.221654
0.217725
0.212869
0.208355
0.204154
0.187026
0.174396
0.163501
0.154614
0.146991
0.134711
0.124810
0.116873
0.110232
0.104696
0.074659
0.061067
0.052747
0.047419
0.033719
0.405292
0.374897
0.353471
0.335397
0.317725
0.303682
0.253994
0.246493
0.240192
0.233566
0.227223
0.222103
0.217434
0.212415
0.207622
0.203870
0.200009
0.183312
0.170208
0.159727
0.151477
0.143731
0.131391
0.122186
0.114417
0.107708
0.102748
0.072990
0.059533
0.051917
0.046238
0.032830
0.388127
0.364208
0.342709
0.323081
0.308264
0.294373
0.245069
0.238415
0.231881
0.226194
0.220341
0.214992
0.209979
0.205945
0.201004
0.197443
0.193701
0.177521
0.165130
0.154749
0.146553
0.139040
0.127762
0.118044
0.111066
0.104276
0.099320
0.070805
0.057851
0.050257
0.044893
0.031659
0.372
0.349
0.327
0.309
0.294
0.281
0.234
0.227
0.221
0.215
0.21
0.205
0.2
0.196
0.192
0.188
0.184
0.169
0.157
0.148
0.139
0.132
0.121
0.113
0.105
0.1
0.095
0.067
0.055
0.048
0.043
0.03
0.364
0.341
0.32
0.301
0.287
0.274
0.228
0.221
0.215
0.209
0.204
0.199
0.195
0.191
0.187
0.183
0.18
0.165
0.153
0.144
0.136
0.129
0.118
0.11
0.103
0.097
0.092
0.065
0.054
0.047
0.042
0.03
0.36
0.336
0.315
0.297
0.282
0.27
0.224
0.218
0.212
0.206
0.201
0.196
0.192
0.188
0.184
0.18
0.177
0.162
0.151
0.141
0.133
0.127
0.116
0.108
0.101
0.095
0.091
0.064
0.053
0.046
0.041
0.029
0.358
0.333
0.313
0.295
0.28
0.267
0.222
0.216
0.21
0.204
0.199
0.194
0.19
0.186
0.182
0.178
0.175
0.16
0.149
0.14
0.132
0.126
0.115
0.107
0.1
0.094
0.09
0.064
0.052
0.045
0.041
0.029
0.358
0.332
0.312
0.294
0.279
0.267
0.222
0.215
0.209
0.203
0.199
0.194
0.189
0.185
0.182
0.178
0.175
0.16
0.149
0.139
0.132
0.125
0.115
0.106
0.1
0.094
0.089
0.064
0.052
0.045
0.04
0.029
0.357
0.332
0.311
0.294
0.279
0.266
0.221
0.215
0.209
0.203
0.198
0.193
0.189
0.185
0.181
0.178
0.174
0.16
0.148
0.139
0.132
0.125
0.114
0.106
0.099
0.094
0.089
0.064
0.052
0.045
0.04
0.029
0.357
0.332
0.311
0.293
0.279
0.266
0.221
0.214
0.208
0.203
0.198
0.193
0.189
0.185
0.181
0.177
0.174
0.16
0.148
0.139
0.131
0.125
0.114
0.106
0.099
0.094
0.089
0.063
0.052
0.045
0.04
0.029
286
-------
Table A-5. Critical Values for A-D Test Statistic for Significance Level = 0.01
n\k 0.025 0.05 0.1 0.2 0.5 1 2 5 10 20
50
5
6
7
8
9
10
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
60
70
80
90
100
200
300
400
500
1000
1.749166
1.751877
1.752404
1.752700
1.758051
1.759366
1.762174
1.763292
1.763403
1.763822
1.764890
1.765012
1.765021
1.765611
1.765703
1.766530
1.766655
1.771265
1.772614
1.772920
1.774318
1.775401
1.777021
1.780583
1.782174
1.786462
1.788600
1.789565
1.791785
1.796178
1.799037
1.810595
1.518258
1.543508
1.556906
1.561426
1.567347
1.575002
1.593432
1.596448
1.599618
1.599735
1.603396
1.604198
1.604737
1.605233
1.609641
1.609644
1.609908
1.617605
1.620179
1.622877
1.624156
1.630356
1.630972
1.634413
1.636678
1.637946
1.639307
1.640278
1.640656
1.641470
1.642244
1.642639
1.258545
1.305996
1.332339
1.358108
1.372050
1.384541
1.418705
1.422813
1.425118
1.435826
1.441772
1.443435
1.446116
1.448791
1.449964
1.451442
1.451659
1.462230
1.465890
1.468763
1.469148
1.471192
1.474981
1.477148
1.481082
1.483922
1.484231
1.486139
1.489654
1.491079
1.491158
1.492652
1.068746
1.123216
1.162744
1.187751
1.210845
1.218849
1.263841
1.273189
1.273734
1.274053
1.278280
1.279990
1.281092
1.284002
1.288792
1.289696
1.290311
1.295794
1.296988
1.304213
1.308833
1.311004
1.312242
1.313856
1.315184
1.316508
1.318003
1.318714
1.322935
1.323876
1.328415
1.328852
0.945
0.99
1.019
1.044
1.058
1.071
1.1
1.112
1.11
1.116
1.115
1.118
1.126
1.119
1.125
1.126
1.127
1.133
1.136
1.138
1.141
1.142
1.144
1.145
1.15
1.149
1.149
1.156
1.154
1.158
1.155
1.157
0.905
0.946
0.979
0.99
1.007
1.018
1.048
1.047
1.053
1.054
1.059
1.056
1.057
1.062
1.059
1.065
1.064
1.072
1.072
1.076
1.074
1.079
1.079
1.079
1.085
1.086
1.085
1.089
1.09
1.093
1.089
1.092
0.89
0.928
0.951
0.97
0.984
0.994
1.018
1.019
1.023
1.027
1.026
1.031
1.031
1.036
1.034
1.035
1.038
1.044
1.045
1.046
1.048
1.053
1.054
1.055
1.055
1.056
1.054
1.059
1.058
1.057
1.057
1.06
0.883
0.918
0.944
0.961
0.967
0.981
1.002
1.007
1.008
1.015
1.013
1.016
1.017
1.023
1.017
1.02
1.021
1.023
1.027
1.03
1.036
1.034
1.032
1.038
1.036
1.038
1.042
1.041
1.043
1.043
1.047
1.043
0.882
0.916
0.938
0.955
0.968
0.977
0.999
1.004
1.004
1.006
1.01
1.012
1.013
1.014
1.02
1.015
1.017
1.023
1.025
1.027
1.03
1.029
1.032
1.031
1.033
1.034
1.035
1.031
1.038
1.039
1.04
1.035
0.879
0.911
0.935
0.956
0.969
0.975
0.997
1
1.003
1.005
1.006
1.005
1.013
1.011
1.012
1.012
1.014
1.019
1.021
1.023
1.026
1.028
1.029
1.031
1.032
1.031
1.033
1.032
1.033
1.035
1.034
1.036
0.879
0.912
0.938
0.953
0.967
0.973
0.999
0.999
1
1.003
1.008
1.009
1.008
1.013
1.013
1.013
1.013
1.018
1.018
1.022
1.024
1.025
1.03
1.028
1.029
1.033
1.032
1.033
1.031
1.034
1.034
1.031
287
-------
Table A-6. Critical Values for K-S Test Statistic for Significance Level = 0.01
n\k 0.025 0.050 0.10 0.2 0.50 1.0 2.0 5.0 10.0 20.0 50.0
5
6
7
8
9
10
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
60
70
80
90
100
200
300
400
500
1000
0.495311
0.464286
0.437809
0.412467
0.390183
0.373002
0.310445
0.302682
0.294519
0.285220
0.277810
0.271994
0.266096
0.260430
0.254210
0.249574
0.246298
0.220685
0.208407
0.196230
0.185995
0.176191
0.161519
0.149283
0.139831
0.132254
0.126224
0.085150
0.073232
0.063283
0.056181
0.040020
0.482274
0.454103
0.426463
0.404538
0.383671
0.368362
0.307559
0.298348
0.289320
0.280990
0.275460
0.268927
0.262728
0.256537
0.252405
0.246722
0.242298
0.222267
0.206958
0.193613
0.183011
0.173662
0.158802
0.148241
0.138103
0.130746
0.123308
0.088338
0.072401
0.062708
0.056147
0.039807
0.467859
0.441814
0.411589
0.392838
0.375103
0.358647
0.300791
0.290148
0.283394
0.276126
0.269173
0.261936
0.256686
0.251727
0.245607
0.240947
0.236164
0.217254
0.202296
0.188617
0.179728
0.170513
0.155658
0.144542
0.135441
0.127231
0.121414
0.086339
0.071096
0.061239
0.054822
0.038938
0.449435
0.423777
0.398890
0.379962
0.361937
0.348328
0.289751
0.280643
0.274722
0.265561
0.260992
0.253878
0.247915
0.242711
0.236271
0.233143
0.228867
0.209442
0.194716
0.182935
0.173141
0.163792
0.150458
0.139590
0.131479
0.123253
0.117441
0.083391
0.068521
0.059235
0.053042
0.036987
0.431
0.402
0.38
0.36
0.343
0.328
0.274
0.266
0.259
0.252
0.246
0.24
0.235
0.23
0.225
0.221
0.216
0.199
0.185
0.173
0.164
0.156
0.143
0.132
0.124
0.117
0.111
0.079
0.065
0.056
0.05
0.036
0.421
0.391
0.369
0.349
0.333
0.318
0.266
0.258
0.251
0.245
0.238
0.233
0.228
0.223
0.218
0.214
0.21
0.193
0.179
0.168
0.158
0.151
0.138
0.128
0.12
0.114
0.108
0.077
0.063
0.054
0.049
0.035
0.414
0.385
0.362
0.344
0.327
0.312
0.261
0.253
0.246
0.24
0.234
0.228
0.223
0.219
0.215
0.21
0.206
0.189
0.176
0.165
0.156
0.148
0.136
0.126
0.118
0.111
0.106
0.075
0.062
0.053
0.048
0.034
0.41
0.382
0.36
0.34
0.323
0.309
0.258
0.251
0.244
0.237
0.232
0.226
0.221
0.216
0.212
0.208
0.204
0.187
0.174
0.163
0.154
0.146
0.134
0.124
0.117
0.11
0.105
0.074
0.061
0.053
0.047
0.034
0.41
0.381
0.358
0.339
0.323
0.308
0.257
0.25
0.243
0.236
0.231
0.225
0.22
0.216
0.211
0.207
0.203
0.186
0.173
0.162
0.154
0.146
0.134
0.124
0.116
0.11
0.104
0.074
0.061
0.053
0.047
0.033
0.408
0.38
0.357
0.339
0.322
0.308
0.257
0.249
0.242
0.236
0.23
0.225
0.22
0.215
0.211
0.207
0.203
0.186
0.173
0.162
0.153
0.146
0.133
0.124
0.116
0.109
0.104
0.074
0.061
0.053
0.047
0.033
0.408
0.38
0.357
0.338
0.322
0.307
0.256
0.249
0.242
0.236
0.23
0.225
0.219
0.215
0.21
0.206
0.203
0.185
0.172
0.162
0.153
0.145
0.133
0.124
0.116
0.11
0.104
0.074
0.06
0.053
0.047
0.033
288
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DECISION SUMMARY TABLES
Table A-7. Skewness as a Function of a (or its MLE, sy = 6), sd of \og(X)
Standard Deviation of Skewness
Logged Data
a < 0.5 Symmetric to mild skewness
0.5 < a < 1.0 Mild skewness to moderate skewness
1.0 < a < 1.5 Moderate skewness to high skewness
1.5 < a < 2.0 High skewness
9 0 < < ^ 0 Very high skewness (moderate probability of
~ ' outliers and/or multiple populations)
> - „ Extremely high skewness (high probability of
~ ' outliers and/or multiple populations)
Table A-8. Summary Table for the Computation of a 95% UCL of the Unknown Mean, fii,
of a Gamma Distribution
k (Skewness 0 , „. 0
Sample Size, n Suggestion
Bias Adjusted)
Approximate gamma 95% UCL (Gamma KM or
k* > 1.0 n>=50 GROS)
,-* ,n Adjusted gamma 95% UCL (Gamma KM or GROS)
K ^1.0 H ~~J(J
£* Jt. 95% UCL based upon bootstrap-t
~ ' or Hall's bootstrap method*
Adjusted gamma 95% UCL (Gamma KM) if
k* <1.0 n>\5,n<50 available, otherwise use approximate gamma 95%
UCL(Gamma KM)
k* <1.0 n > 50 Approximate gamma 95% UCL (Gamma KM)
*In case the bootstrap-t or Hall's bootstrap methods yield erratic, inflated, and unstable UCL values, the
UCL of the mean should be computed using an adjusted gamma UCL.
289
-------
Table A-9. Summary Table for the Computation of a 95% UCL of the Unknown Mean,
of a Lognormal Population
/v
a
a <0.5
0.5 < a <1.0
1.0 < a <1.5
1.5 < a <2.0
2.0 < £ <2.5
2.5 < a <3.0
3.0< <7<3.5**
A . f\ — **
a >3.5
Sample Size, n
For all n
For all n
n<25
n>25
n<20
20<«<50
«>50
«<20
20<«<50
50<«<70
«>70
«<30
30<«<70
70<«< 100
n> 100
«<15
15<«<50
50<«< 100
100<«<150
n> 150
For all n
Suggestions
Student's t, modified-t, or H-UCL
H-UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
97.5% or 99% Chebyshev (Mean, Sd) UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
99% Chebyshev (Mean, Sd) UCL
97.5% Chebyshev (Mean, Sd) UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
99% Chebyshev (Mean, Sd)
97.5% Chebyshev (Mean, Sd) UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
Bootstrap-t or Hall's bootstrap method*
99% Chebyshev(Mea«, Sd)
97.5% Chebyshev (Mean, Sd) UCL
95% Chebyshev (Mean, Sd) UCL
H-UCL
Use nonparametric methods*
*In the case that Hall's bootstrap or bootstrap-t methods yield an erratic unrealistically large UCL value,
UCL of the mean may be computed based upon the Chebyshev inequality: Chebyshev (Mean, Sd) UCL
** For highly skewed data sets with a exceeding 3.0, 3.5, it is suggested that the user pre-process the
data. It is very likely that the data include outliers and/or come from multiple populations. The population
partitioning methods may be used to identify mixture populations present in the data set.
290
-------
Table A-10. Summary Table for the Computation of a 95% UCL of the Unknown Mean, fii,
Based upon a Skewed Data Set (with all Positive Values) without a Discernible
Distribution, Where a is the sd of Log-transformed Data
/*,
a
a <0.5
0.5 < a <1.0
1.0 < a < 1.5
1.5 < a <2.0
2.0 < a <2.5
2.5 < a <3.0
3.0<£<3.5**
3.5«
Sample Size, n
For all n
For all n
For all n
n<20
20
-------
292
-------
APPENDIX B
Large Sample Size Requirements to use the Central Limit
Theorem on Skewed Data Sets to Compute an Upper Confidence
Limit of the Population Mean
As mentioned earlier, the main objective of the ProUCL software funded by the USEPA is to compute
accurate and defensible decision statistics to help the decision makers in making reliable decisions which
are cost-effective, and protective of human health and the environment. ProUCL software is based upon
the philosophy that rigorous statistical methods can be used to compute the correct estimates of the
population parameters (e.g., site mean, background percentiles) and decision making statistics including
the upper confidence limit (UCL) of the population mean, the upper tolerance limit (UTL), and the upper
prediction limit (UPL) to help decision makers and project teams in making decisions. The use and
applicability of a statistical method (e.g., Student's t-UCL, CLT-UCL, adjusted gamma-UCL, Chebyshev
UCL, bootstrap-t UCL) depend upon data size, data skewness, and data distribution. ProUCL computes
decision statistics using several parametric and nonparametric methods covering a wide-range of data
variability, skewness, and sample size. A couple of UCL computation methods described in the statistical
text books (e.g., Hogg and Craig, 1995) based upon the Student's t-statistic and the Central Limit
Theorem (CLT) alone cannot address all scenarios and situations commonly occurring in the various
environmental studies.
Moreover, the properties of the CLT and Student's t-statistic are unknown when NDs with varying DLs
are present in a data set - a common occurrence in data sets originating from environmental applications.
The use of a parametric lognormal distribution on a lognormally distributed data set tends to yield
unstable impractically large UCLs values, especially when the standard deviation (sd) of the log-
transformed data is greater than 1.0 and the data set is of small size such as less than 30-50 (Hardin and
Gilbert 1993; Singh, Singh, and Engelhardt, 1997). Many environmental data sets can be modeled by a
gamma as well as a lognormal distribution. Generally, the use of a gamma distribution on gamma
distributed data sets yields UCL values of practical merit (Singh, Singh, and laci 2002). Therefore, the use
of gamma distribution-based decision statistics such as UCLs, upper prediction limits (UPLs), and UTLs
should not be dismissed just because it is easier to use a lognormal model. The advantages of computing
the gamma distribution-based decision statistics have been discussed in Chapters 2 through 5 of this
technical guidance document.
Since many environmental decisions are made based upon a 95% UCL (UCL95) of the population mean,
it is important to compute UCLs and other decision making statistics of practical merit. In an effort to
compute correct and appropriate UCLs of the population mean and other decision making statistics, in
addition to computing the Student's t statistic and the CLT based decision statistics (e.g., UCLs, UPLs),
significant effort has been made to incorporate rigorous statistical methods based UCLs in ProUCL
software covering a wide-range of data skewness and sample sizes (Singh, Singh, and Engelhardt 1997;
Singh, Singh, and laci 2002). It is anticipated that the availability of the statistical limits in the ProUCL
covering a wide range of environmental data sets will help decision makers in making more informative
and defensible decisions at Superfund and RCRA sites.
It is noted that even for skewed data sets, practitioners tend to use the CLT or Student's t-statistic based
UCLs of the mean for samples of sizes 25-30 (large sample rule-of-thumb to use CLT). However, this
rule-of-thumb does not apply to moderately skewed to highly skewed data sets, specifically when a (sd of
293
-------
the log-transformed data) starts exceeding 1. It should be noted that the large sample requirement depends
upon the skewness of the data distribution under consideration. The large sample requirement for the
sample mean to follow an approximate normal distribution increases with skewness. It is noted that for
skewed data sets, even samples of size greater 100 may not be large enough for the sample mean to
follow an approximate normal distribution (Figures B-l through B-7 below) and the UCLs based upon the
CLT and Student's t statistics fail to provide the desired 95% coverage of the population mean for samples
of sizes as large as 100 as can be seen in Figures B-l through B-7.
Noting that the Student's t-UCL and the CLT-UCL fail to provide the specified coverage of the
population mean of skewed distributions, several researchers, including Chen (1995), Johnson (1978),
Kleijnen, Kloppenburg, and Meeuwsen (1986), and Sutton (1993), proposed adjustments for data
skewness in the Student's t statistic and the CLT. They suggested the use of a modified-t-statistic and
skewness adjusted CLT for positively skewed distributions (for details see Chapter 2 of this Technical
Guide). From statistical theory, the CLT yields UCL results slightly smaller than the Student's t-UCL and
the adjusted CLT, and the Student's t-statistic yield UCLs smaller than the modified t-UCLs (details in
Chapter 2 of this document). Therefore, only the modified t-UCL has been incorporated in the simulation
results described in the following. Specifically, if a UCL95 based upon the modified t-statistic fails to
provide the specified coverage to the population mean, then the other three UCL methods, Student's t-
UCL, CLT-UCL, and the adjusted CLT-UCL, will also fail to provide the specified coverage of the
population mean. The simulation graphs summarized in this appendix suggest that the skewness adjusted
UCLs such as the Johnson's modified-t UCL (and therefore Student's t-UCL and CLT-UCL) do not
provide the specified coverage to the population mean even for mildly to moderately skewed (a in [0.5,
1.0]) data sets. The coverage of the population mean provided by these UCLs becomes worse (much
smaller than the specified coverage) for highly skewed data sets.
The graphical displays, shown in Figures B-l through B-7, cover mildly, moderately, and highly skewed
data sets. Specifically, Figures B-l through B-7 compare the UCL95 of the mean based upon parametric
and nonparametric bootstrap methods and also UCLs computed using the modified-t UCL for mildly
skewed (G(5,50), LN(5,0.05)); moderately skewed (G(2,50), LN(5,1)); and highly skewed (G(0.5, 50),
G(l,50), and LN(5,1.5)) data distributions. From the simulation results presented in Figures B-l through
B-7, it is noted that for skewed distributions, as expected the UCLs based on the modified t-statistic (and
therefore UCLs based upon the CLT and the Student's t-statistic) fail to provide the desired 95% coverage
of the population mean of gamma distributions: G(0.5,50), G(l,50), G(2,50); and of lognormal
distributions: LN(5,0.5), LN(5,1), LN(5,1.5) for samples of sizes as large as 100; and the large sample
size requirement increases as the skewness increases.
The use of the CLT-UCL and Student's t-UCL underestimate the population mean/ EPC for most skewed
data sets.
294
-------
u
D
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m
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ns
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BB -
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ra B2 -
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0 7S -
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— * — Modified-t
—A— 96% Chebysfiev
— • — BoDtstrap-t
— *— Bootstrap BCA
— o — Approximate Gamma
—B— Adjusted Gamma
i
i
I
10 20 30 40 50 60 7D 00 90 100
Figure B-l. Graphs of Coverage Probabilities by 95% UCLs of the mean of G (£=0.50, 9=50)
295
-------
O
D
10
en
4)
CB
£ 90
c
Q.
v
m
a
o
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82 -
78
n
0
10
20
30
70
• Max Test
• Modified-t
• 95% Chebyshev
• Bootstrap-!
• Bootstrap BCA
-Approximate Gamma
-Adjusted Gamma
90
40 50 60
Figure B-2. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(£=1.00, 9=50)
100
Coverage Percentage by 95% UCLs
D CD OD CQ CO
J CO CU -fr QD
I
f>—
/h^^
/ -*-g H—-~— -____
n
/ ^s— __-»- —•— :zzzzrrrrrr^jrrz===^^
/ __ »— ^_____ ^^^^_ j>_ ^ _______^--- — —
"/
— '^— Max Test
— *— Modified-t
& 95% Chebyshev
— •— Bootstrap-t
— *r- Bootstrap BCA
— e— Approximate Gamma
— B— Adjusted Gamma
~+
0 10 20 30 40 50 80 70 80 90 100
Sample Size
Figure B-3. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(£=2.00, 9=50)
296
-------
1DO -
w
_j
O
05
rt
QJ
IS
o
o
92 -
BB -
Max Test
Modified-t
95% Chebyshev
Baotstrap-t
Bootstrap BCA
Approximate Garnma
Adjusted Gamma
0
10
20
30
70
90
40 5Q BO
Figure B4. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(£=5.00, 9=50)
100
100 -
o
3
Ol
ft
"c
o
80 -
CD
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