4>EPA
United States
Environmental Protection
Agency
ProUCL Version 4.00.02
User Guide
RESEARCH AND DEVELOPMENT
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EPA/600/R-07/038
April 2007
www.epa.gov
ProUCL Version 4.00.02
User Guide
Prepared for:
Brian Schumacher
U.S. Environmental Protection Agency
Office of Research and Development
National Exposure Research Laboratory
Environmental Sciences Division
Technology Support Center
Characterization and Monitoring Branch
944 E. Harmon Ave.
Las Vegas, NV89119
Prepared by:
Anita Singh, Ph.D.1
Robert Maichle1
Ashok K. Singh, Ph.D.2
Sanghee E. Lee1
Narain Armbya1
1 Lockheed Martin Environmental Services
1050 E. Flamingo Road, Suite N240
Las Vegas, NV 89119
Department of Hotel Management
University of Nevada, Las Vegas
Las Vegas, NV 89154
Editor and Technical Direction by:
John Nocerino
U.S. Environmental Protection Agency
Office of Research and Development
National Exposure Research Laboratory
Environmental Sciences Division
Characterization and Monitoring Branch
944 E. Harmon Ave.
Las Vegas, NV89119
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.
U.S. Environmental Protection Agency
Office of Research and Development
Washington, DC 20460 129cmb07
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Notice
The United States Environmental Protection Agency (EPA) through its Office of Research and
Development (ORD) funded and managed the research described here. It has been peer reviewed by the
EPA and approved for publication. Mention of trade names and commercial products does not constitute
endorsement or recommendation by the EPA for use.
ProUCL software was developed by Lockheed Martin under a contract with the EPA and is made
available through the EPA Technical Support Center in Las Vegas, Nevada. Use of any portion of
ProUCL that does not comply with the ProUCL User 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 and is certified free of
viruses.
With respect to ProUCL distributed software and documentation, neither the 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.
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Changes from ProUCL 4.0 (version 4.00.00) to ProUCL 4.00.002
Although extensive changes were made in the code for ProUCL 4.0 (version 4.00.00) to produce ProUCL
4.00.02, those changes are transparent to the users. Most of those changes were made so that
ProUCL 4.00.02 is compatible with our developing statistical software, Scout (e.g., both programs share
the same statistical libraries). ProUCL will also reside as a separate module in Scout as a research tool.
There is a very minor correction of a displayed value in one of the hypothesis tests, the two sample t-test.
The p-value associated with the t-test was computed in two different ways: one way is correct and the
other way, although it produced subtle differences, is incorrect. The incorrect method has been removed
from ProUCL 4.00.02.
Several extra warning messages have been added to ProUCL 4.00.02, mainly in regard to attempting tests
when a data set is very small (n < 5), when the number of detected values is small (e.g., only zero, one, or
two), or when all of the values are non-detected values. For an example, some screens depicting those
warning messages are included in the newly added Section 2.11 (page 40) of this ProUCL 4.00.02 User
Guide.
The only software files that were changed from ProUCL version 4.0 (4.00.00) to version 4.0.02 were
updates in the ProUCL.exe file, and updates to the StatsLib.dll file to produce a more advanced
ScoutLib.dll file. Very minor changes were made to this ProUCL 4.00.02 User Guide, including: changes
to avoid inappropriate user inputs (warnings), changes to the title page, the inclusion of an
acknowledgement page, and the inclusion of a contact information page.
No changes were made to the ProUCL 4.0 Technical Guide; that is, the ProUCL 4.0 Technical Guide is
still applicable to the ProUCL 4.00.02 software and User Guide.
Contact Information for ProUCL 4.00.002
The ProUCL software is 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 ProUCL should be addressed to:
Felicia Barnett, (HSTL)
US EPA, Region 4
61 Forsyth Street, S.W.
Atlanta, GA 30303-8960
barnett.felicia@epa.gov
(404) 562-8659
Fax: (404) 562-8439
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Executive Summary
Statistical inference, including both estimation and hypotheses testing approaches, is routinely used to:
1. Estimate environmental parameters of interest, such as exposure point concentration
(EPC) terms, not-to-exceed values, and background level threshold values (BTVs) for
contaminants of potential concern (COPC),
2. Identify areas of concern (AOC) at a contaminated site,
3. Compare contaminant concentrations found at two or more AOCs of a contaminated site,
4. Compare contaminant concentrations found at an AOC with background or reference
area contaminant concentrations, and
5. Compare site concentrations with a cleanup standard to verify the attainment of cleanup
standards.
Several exposure and risk management and cleanup decisions in support of United States Environmental
Protection Agency (EPA) projects are often made based upon the mean concentrations of the COPCs. A
95% upper confidence limit (UCL95) of the unknown population (e.g., an AOC) arithmetic mean (AM),
[j.\, can be used to:
• Estimate the EPC term of the AOC under investigation,
• Determine the attainment of cleanup standards,
• Compare site mean concentrations with reference area mean concentrations, and
• Estimate background level mean contaminant concentrations. The background mean
contaminant concentration level may be used to compare the mean of an area of concern.
It should be noted that it is not appropriate to compare individual point-by-point site
observations with the background mean concentration level.
It is important to compute a reliable and stable UCL95 of the population mean using the available data.
The UCL95 should approximately provide the 95% coverage for the unknown population mean, ^. Based
upon the available background data, it is equally important to compute reliable and stable upper
percentiles, upper prediction limits (UPLs), or upper tolerance limits (UTLs). These upper limits based
upon background (or reference) data are used as estimates of BTVs, compliance limits (CL), or not-to-
exceed values. These upper limits are often used in site (point-by-point) versus background comparison
evaluations.
Environmental scientists often encounter trace level concentrations of COPCs when evaluating sample
analytical results. Those low level analytical results cannot be measured accurately and, therefore, are
typically reported as less than one or more detection limit (DL) values (also called nondetects). However,
practitioners need to obtain reliable estimates of the population mean, //;, and the population standard
deviation, GI, and upper limits including the UCL of the population mass or mean, the UPL, and the UTL
based upon data sets with nondetect (ND) observations. Additionally, they may have to use hypotheses
testing approaches to verify the attainment of cleanup standards, and compare site and background
concentrations of COPCs as mentioned above.
Background evaluation studies, BTVs, and not-to-exceed values should be estimated based upon
defensible background data sets. The estimated BTVs or not-to-exceed values are then used to identify the
COPCs, to identify the site AOCs or hot spots, and to compare the contaminant concentrations at a site
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with background concentrations. The use of appropriate statistical methods and limits for site versus
background comparisons is based upon the following factors:
o Objective of the study,
o Environmental medium (e.g., soil, groundwater, sediment, air) of concern,
o Quantity and quality of the available data,
o Estimation of a not-to-exceed value or of a mean contaminant concentration,
o Pre-established or unknown cleanup standards and BTVs, and
6. Sampling distributions (parametric or nonparametric) of the concentration data sets
collected from the site and background areas under investigation.
In background versus site comparison evaluations, the environmental population parameters of interest
may include:
• Preliminary remediation goals (PRGs),
• Soil screening levels (SSLs),
• RBC standards,
• BTVs, not-to-exceed values, and
• Compliance limit, maximum concentration limit (MCL), or alternative concentration
limit (ACL), frequently used in groundwater applications.
When the environmental parameters listed above are not known or pre-established, appropriate upper
statistical limits are used to estimate those parameters. The UPL, UTL, and upper percentiles are used to
estimate the BTVs and not-to-exceed values. Depending upon the site data availability, point-by-point site
observations are compared with the estimated (or pre-established) BTVs and not-to-exceed values. If
enough site and background data are available, two-sample hypotheses testing approaches are used to
compare site concentrations with background concentrations levels. These statistical methods can also be
used to compare contaminant concentrations of two site AOCs, surface and subsurface contaminant
concentrations, or upgradient versus monitoring well contaminant concentrations.
The ProUCL Version 4.0 (ProUCL 4.0) is an upgrade of ProUCL Version 3.0 (EPA, 2004). ProUCL 4.0
contains statistical methods to address various environmental issues for both full data sets without
nondetects and for data sets with NDs (also known as left-censored data sets).
ProUCL 4.0 contains:
o Rigorous parametric and nonparametric (including bootstrap methods) statistical methods
(instead of simple ad hoc or substitution methods) that can be used on full data sets
without nondetects and on data sets with below detection limit (BDL) or ND
observations.
2. State-of-the-art parametric and nonparametric UCL, UPL, and UTL computation
methods. These methods can be used on full-uncensored data sets without nondetects and
also on data sets with BDL observations. Some of the methods (e.g., Kaplan-Meier
method, ROS methods) are applicable on left-censored data sets having multiple
detection limits. The UCL and other upper limit computation methods cover a wide range
of skewed data sets with and without the BDLs.
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3. Single sample (e.g., Student's t-test, sign test, Proportion test, Wilcoxon Singed Rank
test) and two-sample (Student's t-test, Wilcoxon-Mann-Whitney test, Gehan test, quantile
test) parametric and nonparametric hypotheses testing approaches for data sets with and
without ND observations. These hypothesis testing approaches can be used to: verify the
attainment of cleanup standards, perform site versus background comparisons, and
compare two or more AOCs, monitoring wells (MWs).
4. The single sample hypotheses testing approaches are used to compare site mean, site
median, site proportion, or a site percentile (e.g., 95th) to a compliance limit (action level,
regularity limit). The hypotheses testing approaches can handle both full-uncensored data
sets without nondetects, and left-censored data sets with nondetects. Simple two-sample
hypotheses testing methods to compare two populations are available in ProUCL 4.0,
such as two-sample t-tests, Wilcoxon-Mann-Whitney (WMW) Rank Sum test, quantile
test, Gehan's test, and dispersion test. Variations of hypothesis testing methods (e.g.,
Levene's method to compare dispersions, generalized WRS test) are easily available in
most commercial and freely available software packages (e.g., MINITAB, R).
5. ProUCL 4.0 also includes graphical methods (e.g., box plots, multiple Q-Q plots,
histogram) to compare two or more populations. ProUCL 4.0 can also be used to display
a box plot of one population (e.g., site data) with compliance limits or upper limits (e.g.,
UPL) of other population (background area) superimposed on the same graph. This kind
of graph provides a useful visual comparison of site data with a compliance limit or
BTVs. Graphical displays of a data set (e.g., Q-Q plot) should be used to gain insight
knowledge contained in a data set that may not otherwise be clear by looking at simple
test statistics such as t-test, Dixon test statistic, or Shapiro-Wilk (S-W) test statistic.
6. ProUCL 4.0 can process multiple contaminants (variables) simultaneously and has the
capability of processing data by groups. A valid group column should be included in the
data file.
7. ProUCL 4.0 provides GOF test for data sets with nondetects. The user can create
additional columns to store extrapolated (estimated) values for nondetects based upon
normal ROS, gamma ROS, and lognormal ROS (robust ROS) methods.
ProUCL 4.0 retains all of the capabilities of ProUCL 3.0, including goodness-of-fit (GOF) tests for a
normal, lognormal, and a gamma distribution and computation of UCLs based upon full data sets without
nondetects. Graphical displays and GOF tests for data sets with BDL observations have also been
included in ProUCL 4.0. It is re-emphasized that the computation of appropriate UCLs, UPLs, and other
limits is based upon the assumption that the data set under study represents a single a single population.
This means that the data set used to compute the limits should represent a single statistical population. For
example, a background data set should represent a defensible background data set free of outlying
observations. ProUCL 4.0 includes simple and commonly used classical outlier identification procedures,
such as the Dixon test and the Rosner test. These procedures are included as an aid to identify outliers.
These simple classical outlier tests often suffer from masking effects in the presence of multiple outliers.
Description and use of robust and resistant outlier procedures is beyond the scope of ProUCL 4.0.
It is suggested that the classical outlier procedures should always be accompanied by graphical displays
including box plots and Q-Q plots. The use of a Q-Q plot is useful to identify multiple or mixture samples
that might be present in a data set. However, the decision regarding the proper disposition of outliers (e.g.,
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to include or not to include outliers in statistical analyses; or to collect additional verification samples)
should be made by members of the project team and experts familiar with site and background conditions.
Guidance on the disposition of outliers and their accommodation in a data set by using a transformation
(e.g., lognormal distribution) is discussed in Chapter 1 of this User Guide.
ProUCL 4.0 has improved graphical methods, which may be used to compare the concentrations of two or
more populations such as:
o Site versus background populations,
o Surface versus subsurface concentrations,
o Concentrations of two or more AOCs, and
o Identification of mixture samples and/or potential outliers
These graphical methods include multiple quantile-quantile (Q-Q) plots, side-by-side box plots, and
histograms. Whenever possible, it is desirable to supplement statistical results with useful visual displays
of data sets. There is no substitute for graphical displays of a data set. For example, in addition to
providing information about the data distribution, a normal Q-Q plot can also help 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, and jumps and breaks of significant magnitude
may suggest the presence of observations from multiple populations in the data set. It is suggested that
analytical outlier tests (e.g., Rosner test) and goodness-of-fit (G.O.F.) tests (e.g., SW test) should always
be supplemented with the graphical displays such as Q-Q plot and box plot.
ProUCL 4.0 serves as a companion software package for Calculating Upper Confidence Limits for
Exposure Point Concentrations at Hazardous Waste Sites (EPA, 2002a) and Guidance for Comparing
Background and Chemical Concentrations in Soil for CERCLA Sites (EPA, 2002b). ProUCL 4.0 is also
useful to verify the attainment of cleanup standards (EPA, 1989). ProUCL 4.0 can also be used
to perform two-sample hypotheses tests and to compute various upper limits often needed in
groundwater monitoring applications (EPA, 1992 and EPA, 2004).
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Acronyms and Abbreviations
%NDs
ACL
A-D, AD
AM
AOC
BC
BCA
BDL
BTV
BW
CERCLA
CL
CLT
CMLE
COPC
CV
DL
DL/2 (t)
DL/2 Estimates
DQO
EA
EOF
EM
EPA
EPC
Percentage of Nondetect observations
alternative concentration limit
Anderson-Darling test
arithmetic mean
area(s) of concern
Box-Cox-type transformation
bias-corrected accelerated bootstrap method
below detection limit
background threshold value
Black and White (for printing)
Comprehensive Environmental Response, Compensation, and
Liability Act
compliance limit
central limit theorem
Cohen's maximum likelihood estimate
contaminant(s) of potential concern
Coefficient of Variation
detection limit
UCL based upon DL/2 method using Student's t-distribution
cutoff value
estimates based upon data set with nondetects replaced by half
of the respective detection limits
data quality objective
exposure area
empirical distribution function
expectation maximization
Environmental Protection Agency
exposure point concentration
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FP-ROS (Land)
UCL based upon fully parametric ROS method using Land's H-
statistic
Gamma ROS (Approx.) UCL based upon Gamma ROS method using the gamma
Gamma ROS (BCA)
GOF, G.O.F.
approximate-UCL method
UCL based upon Gamma ROS method using the bias-corrected
accelerated bootstrap method
goodness-of-fit
H-UCL
UCL based upon Land's H-statistic
ID
IQR
identification code
interquartile range
K
KM (%)
KM (Chebyshev)
KM(t)
KM(z)
K-M, KM
K-S, KS
Next K, Other K, Future K
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 cutoff value
UCL based upon Kaplan-Meier estimates using standard normal
distribution cutoff value
Kaplan-Meier
Kolmogorov-Smirnov
LN
Log-ROS Estimates
lognormal distribution
estimates based upon data set with extrapolated nondetect values
obtained using robust ROS method
MAD
Maximum
MCL
Mean
Median
Minimum
MLE
MLE (t)
Median Absolute Deviation
Maximum value
maximum concentration limit
classical average value
Median value
Minimum value
maximum likelihood estimate
UCL based upon maximum likelihood estimates using Student's
t-distribution cutoff value
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MLE (Tiku)
Multi Q-Q
MVUE
ND
NERL
NumNDs
NumObs
ORD
PRO
Q-Q
RBC
RCRA
ROS
RU
SD, Sd, sd
SSL
S-W, SW
UCL
UCL95, 95% UCL
UPL
UPL95, 95% UPL
USEPA
UTL
Variance
WMW
WRS
WSR
UCL based upon maximum likelihood estimates using the
Tiku's method
multiple quantile-quantile plot
minimum variance unbiased estimate
nondetect or nondetects
National Exposure Research Laboratory
Number of Nondetects
Number of Observations
Office of Research and Development
preliminary remediation goals
quantile-quantile
risk-based cleanup
Resource Conservation and Recovery Act
regression on order statistics
remediation unit
substantial difference
standard deviation
soil screening levels
Shapiro-Wilk
upper confidence limit
95% upper confidence limit
upper prediction limit
95% upper prediction limit
United States Environmental Protection Agency
upper tolerance limit
classical variance
Wilcoxon-Mann-Whitney
Wilcoxon Rank Sum
Wilcoxon Signed Rank
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Acknowledgements
We wish to express our gratitude and thanks to all of the many people that reviewed, tested, and
gave some very helpful suggestions for the development of ProUCL 4.0. We wish to especially
acknowledge: Nadine Adkins, Ray Bienert, Cheri L. Butler, Lucia Casabo, John P. Christopher,
Marion Edison, Evan Englund, Tim Ehli, Kuen Huang-Farmer, Dolores Gardner, Philip E.
Goodrum, Scott A. Grubisich, Dennis R. Helsel, Jennifer Hubbard, Richard O. Gilbert, Sarah
Levinson, Jayne Michaud, Jeff Myers, Yassine Nachti,Gareth Pearson, Marian Olsen, Kristina
Rayner, Nancy Rios-Jafolla, Shahrokh Rouhani, Peter Rousseeuw, Dan Sahagun, Terry W.
Schulz, Mike Schum, Tom Siard, Christopher Sibert, Bob Stewart, Rick Sugatt, J. Keith Tolson,
Robert Tucker, John Warren, and Edward P. Wosika.
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Table of Contents
Notice iii
Changes from ProUCL 4.0 (4.00.00) to ProUCL 4.00.02 iv
Contact Information for ProUCL 4.00.02 iv
Executive Summary v
Acronyms and Abbreviations ix
Acknowledgements xii
Introduction 1
The Need for ProUCL Software 1
ProUCL 4.0 Capabilities 2
ProUCL Applications 4
ProUCL Methods 5
Background versus Site Comparison Evaluations 7
Graphical Capabilities 11
Technical Guide 12
Minimum Hardware Requirements 12
Software Requirements 12
Installation Instructions 13
Getting Started 13
Chapter 1 Guidance on the Use of Statistical Methods and Associated Minimum
Sample Size Requirements 15
1.1 Background Data Sets 16
1.2 Site Data Sets 17
1.3 Discrete Samples or Composite Samples? 18
1.4 Upper Limits and Their Use 18
1.5 Point-by-Point Comparison of Site Observations with BTVs, Compliance Limits,
and Other Threshold Values 21
1.6 Hypothesis Testing Approaches and Their Use 23
1.6.1 Single Sample Hypotheses - BTVs and Not-to-Exceed Values are Known
(Pre-established) 23
1.6.2 Two-sample Hypotheses - When BTVs and Not-to-Exceed
Values are Unknown 24
1.7 Minimum Sample Size Requirements 25
1.7.1 Minimum Sample Size for Estimation and Point-by-Point Site
Observation Comparisons 26
1.7.2 Minimum Sample Size Requirements for Hypothesis Testing Approaches 26
1.8 Sample Sizes for Bootstrap Methods 27
1.9 Statistical Analyses by a Group ID 27
1.10 Use of Maximum Detected Values as Estimates of Upper Limits 27
1.10.1 Use of Maximum Detected Values to Estimate BTVs and
Not-to-Exceed Values 28
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1.10.2 Use of Maximum Detected Values to Estimate EPC Terms 28
1.10.3 Samples with Nondetect Observations 29
1.10.4 Avoid the Use of DL/2 Method to Compute UCL95 29
1.10.5 Samples with Low Frequency of Detection 30
1.11 Other Applications of Methods in ProUCL 4.0 30
1.11.1 Identification of COPCs 31
1.11.2 Identification ofNon-Compliance Monitoring Wells 31
1.11.3 Verification of the Attainment of Cleanup Standards, Cs 31
1.11.4 Using BTVs (Upper Limits) to identify Hot Spots 32
Chapter 2 Entering and Manipulating Data 33
2.1 Creating a New Data Set 33
2.2 Opening an Existing Data Set 33
2.3 Input File Format 34
2.4 Number Precision 34
2.5 Entering and Changing a Header Name 35
2.6 Saving Files 36
2.7 Editing 36
2.8 Handling Nondetect Observations 37
2.9 Caution 38
2.10 Summary Statistics for Data Sets with Nondetect Observations 39
2.11 Warning Messages and Recommendations for Datasets with Insufficient
Amount of Data 40
2.12 Handling Missing Values 42
2.13 User Graphic Display Modification 44
2.13.1 Graphics Tool Bar 44
2.13.2 Drop-Down Menu Graphics Tools 44
Chapter 3 Select Variables Screen 46
3.1 Select Variables Screen 46
3.1.1 Graphs by Groups 48
Chapter 4 Summary Statistics 50
4.1 Summary Statistics with Full Data Sets 50
4.2 Summary Statistics withNDs 52
Chapters Estimating Nondetects Using ROS Methods 56
Chapter 6 Graphical Methods (Graph) 58
6.1 Box Plot 59
6.2 Histogram 61
6.3 Multi-QQ 63
6.3.1 Multi-QQ (Full) 63
6.3.2 Multi-QQ (withNDs) 64
Chapter 7 Simple Classical Outlier Tests 68
7.1 Outlier Test for Full Data Set 70
7.2 Outlier Test for Data Set with NDs 70
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Chapter 8 Goodness-of-Fit (G.O.F.) Tests 74
8.1 ROS Estimated (Est.) NDs - Saving Extrapolated NDs 77
8.2 Goodness-of-Fit Tests with Full Data Sets 78
8.2.1 GOF Tests for Normal and Lognormal Distribution 78
8.2.2 GOF Tests for Gamma Distribution 81
8.3 Goodness-of-Fit Tests Excluding NDs 83
8.3.1 Normal and Lognormal Options 84
8.3.2 Gamma Distribution Option 87
8.4 Goodness-of-Fit Tests with Log-ROS Estimates 89
8.4.1 Normal or Lognormal Distribution (Log-ROS Estimates) 90
8.4.2 Gamma Distribution (Log-ROS Estimates) 93
8.5 Goodness-of-Fit Tests with DL/2 Estimates 95
8.5.1 Normal or Lognormal Distribution (DL/2 Estimates) 96
8.5.2 Gamma Distribution (DL/2 Estimates) 99
8.6 Goodness-of-Fit Tests Statistics 101
Chapter 9 Single Sample and Two-Sample Hypotheses Testing Approaches 104
9.1 Single Sample Hypotheses Tests 104
9.1.1 Single Sample Hypothesis Testing for Full Data without Nondetects 105
9.1.1.1 Single Sample t-Test 105
9.1.1.2 Single Sample Proportion Test 107
9.1.1.3 Single Sample Sign Test 109
9.1.1.4 Single Sample Wilcoxon Signed Rank (WSR) Test Ill
9.1.2 Single Sample Hypothesis Testing for Data Sets with Nondetects 114
9.1.2.1 Single Proportion Test on Data Sets with NDs 114
9.1.2.2 Single Sample Sign Test with NDs 116
9.1.2.3 Single Sample Wilcoxon Signed Rank Test with NDs 118
9.2 Two-Sample Hypotheses Testing Approaches 121
9.2.1 Two-Sample Hypothesis Tests for Full Data 122
9.2.1.1 Two-Sample t-Test without NDs 124
9.2.1.2 Two-Sample Wilcoxon-Mann-Whitney (WMW) Test without NDs 126
9.2.1.3 Two-Sample Quantile Test for Full Data without NDs 129
9.2.2 Two-Sample Hypothesis Testing for Data Sets without Nondetects 132
9.2.2.1 Two-Sample Wilcoxon-Mann-Whitney Test with Nondetects 132
9.2.2.2 Two-Sample Gehan Test for Data Sets with Nondetects 136
9.2.2.3 Two-Sample Quantile Test for Data Sets with Nondetects 138
Chapter 10 Background Statistics 142
10.1 Background Statistics for Full Data Sets without Nondetects 143
10.1.1 Normal or Lognormal Distribution 143
10.1.2 Gamma Distribution 147
10.1.3 Nonparametric Methods 149
10.1.4 All Statistics Option 151
10.2 Background Statistics with NDs 153
10.2.1 Normal or Lognormal Distribution 153
10.2.2 Gamma Distribution 158
10.2.3 Nonparametric Methods (with NDs) 160
10.2.4 All Statistics Option 162
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Chapter 11 Computing Upper Confidence Limits (UCLs) of Mean 166
11.1 UCLs for Full Data Sets 167
11.1.1 Normal Distribution (Full Data Sets without NDs) 167
11.1.2 Gamma, Lognormal, Nonparametric, All Statistics Option
(Full Data without NDs) 170
11.2 UCL for Data Sets with NDs 175
Chapter 12 Windows 186
Chapter 13 Help 188
Chapter 14 Handling the Output Screens and Graphs 190
Copying Graphs 190
Printing Graphs 191
Printing Non-graphical Outputs 192
Saving Output Screens as Excel Files 193
Chapter 15 Recommendations to Compute a 95% UCL (Estimate of EPC Term)
of the Population Mean, /Ji, Using Symmetric and Positively Skewed
Full Data Set without any Nondetects 196
15.1 Normally or Approximately Normally Distributed Data Sets 196
15.2 Gamma Distributed Skewed Data Sets 197
15.3 Lognormally Distributed Skewed Data Sets 198
15.4 Data Sets without a Discernable Skewed Distribution - Nonparametric Methods for
Skewed Data Sets 200
15.5 Should the Maximum Observed Concentration be Used as an Estimate of
the EPC Term? 203
Chapter 16 Recommendations to Compute a 95% UCL of the Population Mean,
Pi, Using Data Sets with Nondetects with Multiple Detection Limits 206
16.1 General Recommendations and Suggestions 206
16.2 Recommended UCL95 Methods for Normal (Approximate Normal) Distribution 208
16.3 Recommended UCL95 Methods for Gamma Distribution 209
16.4 Recommended UCL95 Methods for Lognormal Distribution 209
16.5 Recommended Nonparametric UCL Methods 210
Glossary 214
References 216
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Introduction
The Need for ProUCL Software
Statistical inferences about the sampled populations and their parameters are made based upon defensible
and representative data sets of appropriate sizes collected from the populations under investigation.
Statistical inference, including both estimation and hypotheses testing approaches, is routinely used to:
1. Estimate environmental parameters of interest such as exposure point concentration
(EPC) terms, not-to-exceed values, and background level threshold values (BTVs) for
contaminants of potential concern (COPC),
2. Identify areas of concern (AOC) at a contaminated site,
3. Compare contaminant concentrations found at two or more AOCs of a contaminated site,
4. Compare contaminant concentrations found at an AOC with background or reference
area contaminant concentrations,
5. Compare site concentrations with a cleanup standard to verify the attainment of cleanup
standards.
Statistical inference about the sampled populations and their parameters are made based upon defensible
and representative data sets of appropriate sizes collected from the populations under investigation.
Environmental data sets originated from the Superfund and RCRA sites often consist of observations
below one or more detection limits (DLs). In order to address the statistical issues arising in: exposure and
risk assessment applications; background versus site comparison and evaluation studies; and various other
environmental applications, several graphical, parametric, and nonparametric statistical methods for data
sets with nondetects and without nondetects have been incorporated in ProUCL 4.0.
Exposure and risk management and cleanup decisions in support of United States Environmental
Protection Agency (EPA) projects are often made based upon the mean concentrations of the COPCs. A
95% upper confidence limit (UCL95) of the unknown population (e.g., an AOC) arithmetic mean (AM),
Hi, can be used to:
• Estimate the EPC term of the AOC under investigation,
• Determine the attainment of cleanup standards,
• Compare site mean concentrations with reference area mean concentrations, and
• Estimate background level mean contaminant concentrations. The background mean contaminant
concentration level may be used to compare the mean of an AOC. It should be noted that it is not
appropriate to compare individual point-by-point site observations with the background mean
concentration level.
It is important to compute a reliable and stable UCL95 of the population mean using the available data.
The UCL95 should approximately provide the 95% coverage for the unknown population mean, H\- Based
upon the available background data, it is equally important to compute reliable and stable upper
percentiles, upper prediction limits (UPLs), or upper tolerance limits (UTLs). These upper limits based
upon background (or reference) data are used as estimates of BTVs, compliance limits (CL), or not-to-
exceed values. These upper limits are often used in site (point-by-point) versus background comparison
evaluations.
Environmental scientists often encounter trace level concentrations of COPCs when evaluating sample
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analytical results. Those low level analytical results cannot be measured accurately, and therefore are
typically reported as less than one or more detection limit (DL) values (also called nondetects). However,
practitioners often need to obtain reliable estimates of the population mean, //;, the population standard
deviation, ah and upper limits, including the upper confidence limit (UCL) of the population mass or
mean, the UPL, and the UTL based upon data sets with nondetect (ND) observations. Hypotheses testing
approaches are often used to verify the attainment of cleanup standards, and compare site and background
concentrations of COPCs.
Background evaluation studies, BTVs, and not-to-exceed values should be estimated based upon
defensible background data sets. The estimated BTVs or not-to-exceed values are then used to identify the
COPCs, to identify the site AOCs or hot spots, and to compare the contaminant concentrations at a site
with background concentrations. The use of appropriate statistical methods and limits for site versus
background comparisons is based upon the following factors:
1. Objective of the study,
2. Environmental medium (e.g., soil, groundwater, sediment, air) of concern,
3. Quantity and quality of the available data,
4. Estimation of a not-to-exceed value or of a mean contaminant concentration,
5. Pre-established or unknown cleanup standards and BTVs, and
6. Sampling distributions (parametric or nonparametric) of the concentration data sets collected
from the site and background areas under investigation.
In background versus site comparison evaluations, the environmental population parameters of interest
may include:
• Preliminary remediation goals (PRGs),
• Soil screening levels (SSLs),
• Risk-based cleanup (RBC) standards,
• BTVs, not-to-exceed values, and
• Compliance limit, maximum concentration limit (MCL), or alternative concentration limit (ACL),
frequently used in groundwater applications.
When the environmental parameters listed above are not known or have not been pre-established,
appropriate upper statistical limits are used to estimate the parameters. The UPL, UTL, and upper
percentiles are used to estimate the BTVs and not-to-exceed values. Depending upon the site data
availability, point-by-point site observations are compared with the estimated (or pre-established) BTVs
and not-to-exceed values. If enough site and background data are available, two-sample hypotheses
testing approaches are used to compare site concentrations with background concentrations levels. These
statistical methods can also be used to compare contaminant concentrations of two site AOCs, surface and
subsurface contaminant concentrations, or upgradient versus monitoring well contaminant concentrations.
ProUCL 4.0 Capabilities
ProUCL Version 4.0 (ProUCL 4.0) is an upgrade of ProUCL Version 3.0 (EPA, 2004). ProUCL 4.0
contains statistical methods to address various environmental issues for both full data sets without
nondetects and for data sets with NDs (also known as left-censored data sets).
ProUCL 4.0 contains:
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o Rigorous parametric and nonparametric (including bootstrap methods) statistical methods
(instead of simple ad hoc or substitution methods) that can be used on full data sets
without nondetects and on data sets with below detection limit (BDL) or nondetect (ND)
observations.
o State-of-the-art parametric and nonparametric UCL, UPL, and UTL computation
methods. These methods can be used on full-uncensored data sets without nondetects and
also on data sets with BDL observations. Some of the methods (e.g., Kaplan-Meier
method, ROS methods) are applicable on left-censored data sets having multiple
detection limits. The UCL and other upper limit computation methods cover a wide range
of skewed data sets with and without the BDLs.
o Single sample (e.g., Student's t-test, sign test, proportion test, Wilcoxon Singed Rank
test) and two-sample (Student's t-test, Wilcoxon-Mann-Whitney test, Gehan test, quantile
test) parametric and nonparametric hypotheses testing approaches for data sets with and
without ND observations. These hypothesis testing approaches can be used to: verify the
attainment of cleanup standards, perform site versus background comparisons, and
compare two or more AOCs, monitoring wells (MWs).
o The single sample hypotheses testing approaches are used to compare site mean, site
median, site proportion, or a site percentile (e.g., 95th) to a compliance limit (action level,
regularity limit). The hypotheses testing approaches can handle both full-uncensored data
sets without nondetects, and left-censored data sets with nondetects. Simple two-sample
hypotheses testing methods to compare two populations are available in ProUCL 4.0,
such as two-sample t-tests, Wilcoxon-Mann-Whitney (WMW) Rank Sum test, quantile
test, Gehan's test, and dispersion test. Variations of hypothesis testing methods (e.g.,
Levene's method to compare dispersions, generalized WRS test) are easily available in
most commercial and freely available software packages (e.g., MINITAB, R).
o ProUCL 4.0 includes graphical methods (e.g., box plots, multiple Q-Q plots, histogram)
to compare two or more populations. Additionally, ProUCL 4.0 can also be used to
display a box plot of one population (e.g., site data) with compliance limits or upper
limits (e.g., UPL) of other population (background area) superimposed on the same
graph. This kind of graph provides a useful visual comparison of site data with a
compliance limit or BTVs. Graphical displays of a data set (e.g., Q-Q plot) should be
used to gain insight knowledge contained in a data set that may not otherwise be clear by
looking at simple test statistics such as t-test, Dixon test statistic, or Shapiro-Wilk (S-W)
test statistic.
6. ProUCL 4.0 can process multiple contaminants (variables) simultaneously and has the
capability of processing data by groups. A valid group column should be included in the
data file.
7. ProUCL 4.0 provides a GOF test for data sets with nondetects. The user can create
additional columns to store extrapolated (estimated) values for nondetects based upon
normal ROS, gamma ROS, and lognormal ROS (robust ROS) methods.
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ProUCL Applications
The methods incorporated in ProUCL 4.0 can be used on data sets with and without BDL and ND
observations. Methods and recommendations as incorporated in ProUCL 4.0 are based upon the results
and findings of the extensive simulation studies as summarized in Singh and Singh (2003), and Singh,
Maichle, and Lee (EPA, 2006). It is anticipated that ProUCL 4.0 will serve as a companion software
package for the following EPA documents:
• Calculating Upper Confidence Limits for Exposure Point Concentrations at Hazardous
Waste Sites (EPA, 2002a), and
• The revised Guidance for Comparing Background and Chemical Concentrations in Soil
for CERCLA Sites (EPA, 2002b).
Methods included in ProUCL 4.0 can be used in various other environmental applications including the
verification of cleanup standards (EPA, 1989), and computation of upper limits needed in groundwater
monitoring applications (EPA, 1992 and EPA, 2004).
In 2002, EPA issued guidance for calculating the UCLs of the unknown population means for
contaminant concentrations at hazardous waste sites. The ProUCL 3.0 software package (EPA, 2004) has
served as a companion software package for the EPA (2002a) guidance document for calculating UCLs of
mean contaminant concentrations at hazardous waste sites. ProUCL 3.0 has several parametric and
nonparametric statistical methods that can be used to compute appropriate UCLs based upon full-
uncensored data sets without any ND observations. ProUCL 4.0 retains the capabilities of ProUCL 3.0,
including goodness-of-fit (GOF) and the UCL computation methods for data sets without any BDL
observations. However, ProUCL 4.0 has the additional capability to perform GOF tests and computing
UCLs and other upper limits based upon data sets with BDL observations.
ProUCL 4.0 defines log-transform (log) as the natural logarithm (In) to the base e. ProUCL 4.0 also
computes the maximum likelihood estimates (MLEsj and the minimum variance unbiased estimates
(MVUEs) of unknown population parameters of normal, lognormal, and gamma distributions. This, of
course, depends upon the underlying data distribution. ProUCL 4.0 computes the (7 - a)100% UCLs of
the unknown population mean, ^i, using 5 parametric and 10 nonparametric methods. It should be pointed
out that ProUCL 4.0 computes the simple summary statistics for detected raw and log-transformed data
for full data sets without NDs, as well as for data sets with BDL observations. It is noted that estimates of
mean and sd for data sets with NDs based upon rigorous statistical methods (e.g., MLE, ROS, K-M
methods) are note provided in the summary statistics. Those estimates and the associated upper limits for
data sets with NDs are provided under the menu options: Background and UCL.
It is emphasized that throughout this User Guide, and in the ProUCL 4.0 software, it is assumed that one
is dealing with a single population. If multiple populations (e.g., background and site data mixed together)
are present, it is recommended to first separate them out (e.g., using appropriate statistical population
partitioning techniques), and then compute appropriate respective 95% UCLs separately for each of the
identified populations. Outliers, if any, should be identified and thoroughly investigated. ProUCL 4.0
provides two commonly used simple classical outlier identification procedures: 1) the Dixon test and 2)
the Rosner test. Outliers distort most parametric statistics (e.g., mean, UCLs, upper prediction limits
(UPLs), test statistics) of interest. 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
those nonparametric statistics (including bootstrap methods), which are based upon higher order statistics
such as UPLs and UTLs. Decisions about the disposition (exclusion or inclusion) of outliers in a data set
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used to estimate the EPC terms or BTVs should be made by all parties involved (e.g., project team, EPA,
local agency, potentially responsible party, etc.) in the decision making process.
The presence of outlying observations also distorts statistics based upon bootstrap re-samples. The use of
higher order values (quantiles) of the distorted statistics for the computation of the UCLs or UPLs based
upon bootstrap t and Hall's bootstrap methods may yield unstable and erratic UCL values. This is
especially true for the upper limits providing higher confidence coefficients such as 95%, 97.5%, or 99%.
Similar behavior of the bootstrap t UCL is observed for data sets having BDL observations. Therefore, the
bootstrap t and Hall's bootstrap methods should be used with caution. It is suggested that the user should
examine various other UCL results and determine if the UCLs based upon the bootstrap t and Hall's
bootstrap methods represent reasonable and reliable UCL values of practical merit. If the results based
upon these two bootstrap methods are much higher than the rest of methods, then this could be an
indication of erratic behavior of those bootstrap UCL values, perhaps distorted by outlying observations.
In case these two bootstrap methods yield erratic and inflated UCLs, the UCL of the mean should be
computed using the adjusted or the approximate gamma UCL computation method for highly skewed
gamma distributed data sets of small sizes. Alternatively, one may use a 97.5% or 99% Chebyshev UCL
to estimate the mean of a highly skewed population. It should be noted that typically, a Chebyshev UCL
may yield conservative and higher values of the UCLs than other methods available in ProUCL 4.0 This
is especially true when data are moderately skewed and sample size is large. In such cases, when the
sample size is large, one may want to use a 95% Chebyshev UCL or a Chebyshev UCL with lower
confidence coefficient such as 92.5% or 90% as estimate of the population mean.
ProUCL Methods
ProUCL 4.0 provides 15 UCL computation methods for full data sets without any BDL observations; 5
are parametric and 10 are nonparametric methods. The nonparametric methods do not depend upon any
assumptions about the data distributions. The five parametric UCL computation methods are:
o Student's t-UCL,
o Approximate gamma UCL using chi-square approximation,
o Adjusted gamma UCL (adjusted for level significance),
o Land's H-UCL, and
o Chebyshev inequality-based UCL (using MVUEs of parameters of a lognormal
distribution).
The 10 nonparametric methods are:
1. The central limit theorem (CLT)-based UCL,
2. Modified-t statistic (adjusted for skewness)-based UCL,
3. Adjusted-CLT (adjusted for skewness)-based UCL,
4. Chebyshev inequality-based UCL (using sample mean and sample standard deviation),
5. Jackknife method-based UCL,
6. UCL based upon standard bootstrap,
7. UCL based upon percentile bootstrap,
8. UCL based upon bias-corrected accelerated (BCA) bootstrap,
9. UCL based upon bootstrap t, and
10. UCL based upon Hall's bootstrap.
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Environmental scientists often encounter trace level concentrations of COPCs when evaluating sample
analytical results. Those low level analytical results cannot be measured accurately, and therefore are
typically reported as less than one or more DL values. However, the practitioners need to obtain reliable
estimates of the population mean, //;, and the population standard deviation, ah and upper limits including
the UCL of the population mass (measure of central tendency) or mean, UPL, and UTL. Several methods
are available and cited in the environmental literature (Helsel (2005), Singh and Nocerino (2002), Millard
and Neerchal (2001)) that can be used to estimate the population mean and variance. However, till to date,
no specific recommendations are available for the use of appropriate methods that can be used to compute
upper limits (e.g., UCLs, UPLs) based upon data sets with BDL observations. Singh, Maichle, and Lee
(EPA, 2006) extensively studied the performance of several parametric and nonparametric UCL
computation methods for data sets with BDL observations. Based upon their results and findings, several
methods to compute upper limits (UCLs, UPLs, and UTLs) needed to estimate the EPC terms and BTVs
have been incorporated in ProUCL 4.0.
In 2002, EPA issued another Guidance for Comparing Background and Chemical Concentrations in Soil
for CERCLA Sites (EPA, 2002b). This EPA (2002b) background guidance document is currently being
revised to include statistical methods that can be used to estimate the BTVs and not-to-exceed values
based upon data sets with and without the BDL observations. In background evaluation studies, BTVs,
compliance limits, or not-to-exceed values often need to be estimated based upon defensible background
data sets. The estimated BTVs or not-to-exceed values are then used for screening the COPCs, to identify
the site AOCs or hot spots, and also to determine if the site concentrations (perhaps after a remediation
activity) are comparable to background concentrations, or are approaching the background level
concentrations. Individual point-by-point site observations (composite samples preferred) are sometimes
compared with those not-to-exceed values or BTVs. It should be pointed out that in practice, it is
preferred to use hypotheses testing approaches to compare site versus background concentrations
provided enough (e.g., at least 8-10 detected observations from each of the two populations) site and
background data are available. Chapter 1 provides practical guidance on the minimum sample size
requirements to estimate and use the BTVs, single and two-sample hypotheses testing approaches to
perform background evaluations and background versus site comparisons. Chapter 1 also briefly
discusses the differences in the definitions and uses of the various upper limits as incorporated in ProUCL
4.0. Detailed discussion of the various methods to estimate the BTVs and other not-to-exceed values for
full-uncensored data sets (Chapter 5) without any nondetect values and for left-censored data sets
(Chapter 6) with nondetect values are given in the revised background guidance document.
ProUCL 4.0 includes statistical methods to compute UCLs of the mean, upper limits to estimate the
BTVs, other not-to-exceed values, and compliance limits based upon data sets with one or more detection
limits. The use of appropriate statistical methods and limits for exposure and risk assessment, and site
versus background comparisons, is based upon several factors:
1. Objective of the study;
2. Environmental medium (e.g., soil, groundwater, sediment, air) of concern;
3. Quantity and quality of the available data;
4. Estimation of a not-to-exceed value or of a mean contaminant concentration;
5. Pre-established or unknown cleanup standards and BTVs; and
6. Sampling distributions (parametric or nonparametric) of the concentration data sets
collected from the site and background areas under investigation.
In background versus site comparison studies, the population parameters of interest are typically
represented by upper threshold limits (e.g., upper percentiles, upper confidence limits of an upper
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percentile, upper prediction limit) of the background data distribution. It should be noted that the upper
threshold values are estimated and represented by upper percentiles and other values from the upper tail
of the background data distribution. These background upper threshold values do not represent measures
of central tendency such as the mean, the median, or their upper confidence limits. These environmental
parameters may include:
• Preliminary remediation goals (PRGs), Compliance Limits,
• Soil screening levels (SSLs),
• Risk-based cleanup (RBC) standards,
• BTVs, compliance limits, or not-to-exceed values, and
• Maximum concentration limit (MCL) or alternative concentration limit (ACL) used in
Groundwater applications.
When the environmental parameters listed above are not known or pre-established, appropriate upper
statistical limits are used to estimate those parameters. The UPL, UTL, and upper percentiles are typically
used to estimate the BTVs, not-to-exceed values, and other parameters listed above. Depending upon the
availability of site data, point-by-point site observations are compared with the estimated (or pre-
established) BTVs and not-to-exceed values. If enough site and background data are available, two-
sample hypotheses testing approaches (preferred method to compare two populations) are used to
compare site concentrations with background concentrations levels. The hypotheses testing methods can
also be used to compare contaminant concentrations of two site AOCs, surface and subsurface
contaminant concentrations, or upgradient versus monitoring well contaminant concentrations.
Background versus Site Comparison Evaluations
The following statistical limits have been incorporated in ProUCL 4.0 to assist in background versus site
comparison evaluations:
Parametric Limits for Full-Uncensored Data Sets without Nondetect Observations
• UPL for a single observation (Normal, Lognormal) not belonging to the original data set
• UPL for next k (k is user specified) or k future observations (Normal, Lognormal)
• UTL, an upper confidence limit of a percentile (Normal, Lognormal)
• Upper percentiles (Normal, Lognormal, and Gamma)
Nonparametric Limits for Full-Uncensored Data Sets without Nondetect Observations
Nonparametric limits are typically based upon order statistics of a data set such as a background or a
reference data set. Depending upon the size of the data set, higher order statistics (maximum, second
largest, third largest, and so on) are used as these upper limits (e.g., UPLs, UTLs). The details of these
methods with sample size requirements can be found in Chapter 5 of the revised Guidance for Comparing
Background and Chemical Concentrations in Soil for CERCLA Sites (EPA, 2002b). It should be, noted
that the following statistics might get distorted by the presence of outliers (if any) in the data set under
study.
• UPL for a single observation not belonging to the original data set
• UTL, an upper confidence limit of a percentile
• Upper percentiles
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• Upper limit based upon interquartile range (IQR)
• Upper limits based upon bootstrap methods
For data sets with BDL observations, the following parametric and nonparametric methods to compute
the upper limits were studied and evaluated by Singh, Maichle, and Lee (EPA, 2006) via Monte Carlo
Simulation Experiments. Depending upon the performances of those methods, only some of the methods
have been incorporated in ProUCL 4.0. Methods (e.g., Delta method, DL method, uniform (0, DL)
generation method) not included in ProUCL 4.0 do not perform well in comparison with other methods.
Note: When the percentage ofNDs in a data set is high (e.g., > 40%-50%), especially when multiple
detection limits might be present, it is hard to reliably perform GOF tests (to determine data distribution)
on those data sets with many NDs. The uncertainty associated with those GOF tests will be high,
especially when the data sets are of small sizes (< 10-20). It should also be noted that the parametric
MLE methods (e.g., for normal and lognormal distributions) often yield unstable estimates of mean and
sd. This is especially true when the number ofnondetects exceeds 40%-50%. In such situations, it is
preferable to use nonparametric (e.g., KM method) methods to compute statistics of interest such as
UCLs, UPLs, and UTLs. Nonparametric methods do not require any distributional assumptions about the
data sets under investigation. Singh, Maichle, and Lee (EPA, 2006) also concluded that the performance
of the KM estimation method is better (in terms of coverage probabilities) than various other parametric
estimation (e.g., MLE, EM, ROS) methods.
Parametric Methods to Compute Upper Limits for Data Sets with Nondetect Observations
• Simple substitution (proxy) methods (0, DL/2, DL)
• MLE method, often known as Cohen's MLE method - single detection limit
• Restricted MLE method - single detection limit - not in ProUCL 4.0
• Expectation maximization (EM) method - single detection limit - not in ProUCL 4.0
• EPA Delta log method - single detection limit - not in ProUCL 4.0
• Regression method on detected data and using slope and intercept of the OLS regression
line as estimates of standard deviation, sd, and mean (not a recommended method)
• Robust ROS (regression on order statistics) on log-transformed data - nondetects
extrapolated (estimated) using robust ROS; mean, sd, UCLs, and other statistics
computed using the detected and extrapolated data in original scale - multiple detection
limits
• Normal ROS - nondetects extrapolated (estimated) using normal distribution, mean, sd,
UCLs, and other statistics computed using the detected and extrapolated data - multiple
detection limits.
• It is noted that the estimated NDs often become negative and even larger than the
detection limits (not a recommended method)
• Gamma ROS - nondetects extrapolated (estimated) using gamma distribution, mean, sd,
UCLs, and other statistics computed using the detected and extrapolated data - multiple
detection limits
Nonparametric Methods to Compute Upper Limits for Data Sets with Nondetect Observations
• Bootstrap Methods
o Percentile Bootstrap on robust ROS
o Percentile Bootstrap
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o BCA Bootstrap
o Bootstrap t
• Jackknife Method
o Jackknife on robust ROS
• Kaplan-Meier (KM) Method
o Bootstrap (percentile, BCA) using KM estimates
o Jackknife using KM estimates
o Chebyshev Method using KM estimates
• Winsorization Method
For uncensored full data sets without any NDs, the performance (in terms of coverage for the mean) of
the various UCL computation methods was evaluated by Singh and Singh (2003). The performance of the
parametric and nonparametric UCL methods based upon data sets with nondetect observations was
studied by Singh, Maichle, and Lee (EPA, 2006). Several of the methods listed above have been
incorporated in ProUCL 4.0 to compute the estimates of EPC terms (95% UCL), and of BTVs (UPLs,
UTLs, upper percentiles). Methods that did not perform well (e.g., poor coverage or unrealistically large
values, infeasible and biased estimates) are not included in ProUCL 4.0. Methods not incorporated in
ProUCL 4.0 are: EPA Delta Log method, Restricted MLE method, and EM method, substitution method
(0, and DL), and Regression method.
Note: It should be noted that for data sets with NDs, the DL/2 substitution method has been incorporated
in ProUCL 4.0 only for historical reasons and also for its current default use. It is well known that the
DL/2 method (with NDs replaced by DL/2) does not perform well (e.g., Singh, Maichle, and Lee (EPA,
2006)) even when the percentage of NDs is only 5%-10%. It is strongly suggested to avoid the use of
DL/2 method for estimation and hypothesis testing approaches used in various environmental
applications. Also, when the % of NDs becomes high (e.g., > 40%-50%), it is suggested to avoid the use
of parametric MLE methods. For data sets with high percentage of NDs (e.g., > 40%), the distributional
assumptions needed to use parametric methods are hard to verify; and those parametric MLE methods
may yield unstable results.
It should also be noted that even though the lognormal distribution and some statistics based upon
lognormal assumption (e.g., Robust ROS, DL/2 method) are available in ProUCL 4.0, ProUCL 4.0 does
not compute MLEs of mean and sd based upon a lognormal distribution. The main reason is that the
estimates need to be computed in the original scale via back-transformation (Shaarawi, 1989, and Singh,
Maichle, and Lee (EPA, 2006)). Those back-transformed estimates often suffer from an unknown amount
of significant bias. Hence, it is also suggested to avoid the use of a lognormal distribution to compute
MLEs of mean and sd, and associated upper limits, especially UCLs based upon those MLEs obtained
using a lognormal distribution.
ProUCL 4.0 recommends the use of an appropriate UCL to estimate the EPC terms. It is desirable that the
user consults with the project team and experts familiar with the site before using those recommendations.
Furthermore, there does not seem to be a general agreement about the use of an upper limit (e.g., UPL,
percentile, or UTL) to estimate not-to-exceed values or BTVs to be used for screening of the COPCs and
in site versus background comparison studies. ProUCL 4.0 can compute both parametric and
nonparametric upper percentiles, UPLs, and UTLs for uncensored and censored data sets. However, no
specific recommendations have been made regarding the use of UPLs, UTLs, or upper percentiles to
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estimate the BTVs, compliance limits, and other related background or reference parameters. However,
the developers of ProUCL 4.0 prefer the use of UPLs or upper percentiles to estimate the background
population parameters (e.g., BTVs, not-to-exceed values) that may be needed to perform point-by-point
site versus background comparisons.
The standard bootstrap and the percentile bootstrap UCL computation methods do not perform well (do
not provide adequate coverage to population mean) for skewed data sets. For skewed distributions, the
bootstrap t and Hall's bootstrap (meant to adjust for skewness) methods do perform better (in terms of
coverage for the population mean) than the other bootstrap methods. However, it has been noted (e.g.,
Efron and Tibshirani (1993), and Singh, Singh, and laci (2002b)) that these two bootstrap methods
sometimes yield erratic and inflated UCL values (orders of magnitude higher than the other UCLs). This
may occur when outliers are present in a data set. Similar behavior of the bootstrap t UCL is observed
based upon data sets with NDs. Therefore, whenever applicable, ProUCL 4.0 provides cautionary
statements regarding the use of bootstrap methods.
ProUCL 4.0 provides several state-of-the-art parametric and nonparametric UCL, UPL, and UTL
computation methods that can be used on uncensored data sets (full data sets) and on data sets with BDL
observations. Some of the methods (e.g., Kaplan-Meier method, ROS methods) incorporated in ProUCL
4.0 are applicable on left-censored data sets having multiple detection limits. The UCLs and other upper
limits computation methods in ProUCL 4.0 cover a wide range of skewed data distributions with and
without the BDLs arising from the environmental applications.
ProUCL 4.0 also has parametric and nonparametric single and two-sample hypotheses testing approaches
required to: compare site location (e.g., mean, median) to a specified cleanup standard; perform site
versus background comparisons; or compare of two or more AOCs. These hypotheses testing methods
can handle both full (uncensored data sets without NDs) and left-censored (with nondetects) data sets.
Specifically, two-sample tests such as t-test, Wilcoxon Mann-Whitney (WMW) Rank Sum test, quantile
test, and Gehan's test are available in ProUCL 4.0 to compare concentrations of two populations.
Single sample parametric (Student's t-test) and nonparametric (sign test, Wilcoxon Signed Rank (WSR)
test, tests for proportions and percentiles) hypotheses testing approaches are also available in ProUCL 4.0.
The single sample hypotheses tests are useful when the environmental parameters such as the clean
standard, action level, or compliance limits are known, and the objective is to compare site concentrations
with those known threshold values. Specifically, a t-test (or a sign test) may be used to verify the
attainment of cleanup levels at an AOC after a remediation activity; and a test for proportion may be used
to verify if the proportion of exceedances of an action level (or a compliance limit) by sample
concentrations collected from the AOC (or a MW) exceeds a certain specified proportion (e.g., 1%, 5%,
10%). As mentioned before, ProUCL 4.0 can perform these hypotheses on data sets with and without
nondetect observations.
Note: It should be noted that as cited in the literature, some of the hypotheses testing approaches (e.g.,
nonparametric two-sample WMW) deal with the single detection limit scenario. If multiple detection
limits are present, all NDs below the largest detection limit need to be considered as NDs (Gilbert, 1987,
andHelsel, 2005). This in turn may reduce the power and increase uncertainty associated with test. As
mentioned before, it is always desirable to supplement the test statistics and test conclusions with
graphical displays such as the multiple Q-Q plots and side-by-side box plots. ProUCL 4.0 can graph box
plots and Q-Q plots for data sets with nondetect observations. Gehan test as available in Pro UCL 4.0
should be used in case multiple detection limits are present. ProUCL 4.0 can draw Q-Qplots and box
plots for data sets with and without nondetect observations.
10
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It should be pointed out that when using two-sample hypotheses approaches (WMWtest, Gehan test, and
quantile test) on data sets with NDs, both samples and variables (e.g., site-As, Back-As) should be
specified as having nondetects. This means, a ND column (0 = ND, and 1 = detect) should be provided
for each variable (here D_site-As, andD_Back-As) to be used in this comparison. If a variable (e.g., site-
As) does not have any nondetects, still a column with label D_site-As should be included in the data set
with all entries = 1 (detected values).
Moreover, in single sample hypotheses tests (e.g., sign test, proportion test) used to compare site
mean/median concentration level with a cleanup standard, Q or compliance limit (e.g., proportion test),
all NDs (if any) should lie below the cleanup standard, Q.
The differences between these tests should be noted and understood. Specifically, 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 compares if the 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., reporting limit, action level). All of these tests have been incorporated in
ProUCL 4.0. Most of the single sample and two-sample hypotheses tests also report associated p-values.
For some of the hypotheses tests (e.g., WMWtest, WSR test, proportion test), large sample approximate
p-values are computed using continuity correction factors.
Graphical Capabilities
ProUCL 4.0 has useful exploratory graphical methods that may be used to visually compare the
concentrations of:
1. A site area of concern (AOC) with an action level. This can be done using a box plot of
site data with action level superimposed on that graph,
2. Two or more populations, including site versus background populations, surface versus
subsurface concentrations, and
3. Two or more AOCs.
The graphical methods include double and multiple quantile-quantile (Q-Q) plots, side-by-side box plots,
and histograms. Whenever possible, it is desirable to supplement statistical test results and statistics with
visual graphical displays of data sets. There is no substitute for graphical displays of a data set as the
visual displays often provide useful information about a data set, which cannot be revealed by simple test
statistics such as t-test, SW test, Rosner test, WMW test. For example, in addition to providing
information about the data distribution, a normal Q-Q plot can also help identify outliers and multiple
populations that might be present in a data set. This kind of information cannot be revealed by simple test
statistics such as a Shapiro-Wilk (SW) test or Rosner's outlier test statistic. Specifically, the SW test may
lead to the conclusion that a mixture data set (representing two or more populations) can be modeled by a
normal (or lognormal) distribution, whereas the occurrence of obvious breaks and jumps in the associated
Q-Q plot may suggest the presence of multiple populations in the mixture data set. It is suggested that the
user should use exploratory tools to gain necessary insight into a data set and the underlying assumptions
(e.g., distributional, single population) that may not be revealed by simple test statistics.
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Note: On a Q-Q plot, observations well separated from the majority of the data may represent potential
outliers, and obvious jumps and breaks of significant magnitude may suggest the presence of observations
from multiple populations in the data set.
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
quite common in many environmental applications. ProUCL 4.0 offers this option for data sets with and
without nondetects. The Group Option provides a powerful tool to perform various 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 an example, the same data set may consist of samples
from the various groups or populations representing site, background, two or more AOCs, surface,
subsurface, monitoring wells. The graphical displays (e.g., box plots, Q-Q plots) and statistics
(computations of background statistics, UCLs, hypotheses testing approaches) of interest can be
computed separately for each group by using this option.
Technical Guide
In addition to this User Guide, a Technical document also accompanies ProUCL 4.0, providing useful
technical details of the graphical and statistical methods as incorporated in ProUCL 4.0. Most of the
mathematical algorithms and formulas (with references) used in the development of ProUCL 4.0 are
summarized in the Technical Guide.
Minimum Hardware Requirements
• Intel Pentium 1.0 GHz
• 45 MB of hard drive space
• 512 MB of memory (RAM)
• CD-ROM drive
• Windows 98 or newer. ProUCL was thoroughly tested on NT-4, Windows 2000, and
Windows XP Operating Systems (limited testing on Windows ME).
Software Requirements
ProUCL 4.0 has been developed in the Microsoft .NET Framework using the C# programming language.
As such, to properly run ProUCL 4.0, the computer using the program must have the .NET Framework
pre-installed. The downloadable .NET files can be found at one of the following two Web sites:
• http://msdn.microsoft.com/netframework/downloads/updates/default.aspx
Note: Download .Net version 1.1
• http://www.microsoft.com/downloads/details.aspx?FamilyId=262D25E3-F589-4842-
8157-034DlE7CF3A3&displavlang=en
The first Web site lists all of the downloadable .NET Framework files, while the second Web site
provides information about the specific file (s) needed to run ProUCL 4.0. Download times are estimated
at 57 minutes for a dialup connection (56K), and 13 minutes on a DSL/Cable connection (256K).
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Installation Instructions
• Download the file SETUP.EXE from the EPA Web site and save to a temporary location.
Note: This download is pending release - beta-testers: See text file on CD.
• Run the SETUP.EXE program. This will create a ProUCL directory and two folders:
1) the USER 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.
Getting Started
The functionality and the use of the methods and options available in ProUCL 4.0 have been illustrated
using Screen Shots of output screen generated by ProUCL 4.0. ProUCL 4.0 uses a pull-down menu
structure, similar to a typical Windows program.
The screen below appears when the program is executed.
Navigation
Panel
4
Main
Window
Log Panel
The screen consists of three main window panels:
• The MAIN WINDOW displays data sheets and outputs from the procedure used.
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• The NAVIGATION PANEL displays the name of data sets and all generated outputs.
o At present, the navigation panel can hold at most 20 outputs. In order to see more
files (data files or generated output files), one can click on Widow Option.
• The LOG PANEL displays transactions in green, warnings in orange, and errors in red.
For an example, when one attempts to run a procedure meant for censored data sets on a
full-uncensored data set, ProUCL 4.0 will print out a warning message in orange in this
panel.
o Should both panels be unnecessary, you can click —J or choose Configure ^-
Panel ON/OFF
The use of this option will give extra space to see and print out the statistics of interest. For an example,
one may want to turn off these panels when multiple variables (e.g., multiple Q-Q plots) are analyzed and
GOF statistics and other statistics may need to be captured for all of the variables.
P?J>roUeL-f .0
Navigation Fsmsr-T'
Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing Background LJCL Window Help
JM
, On/Off
1
7
Name
ft) Worksheet wst
1
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Chapter 1
Guidance on the Use of Statistical Methods and Associated
Minimum Sample Size Requirements
This chapter briefly describes the differences between the various statistical limits (e.g., UCLs, UPLs,
UTLs) often used to estimate the environmental parameters of interest including exposure point
concentration (EPC) terms and background threshold values (BTVs). Suggestions are provided about the
minimum sample size requirements needed to use statistical inferential methods to estimate the
environmental parameters: EPC terms, BTVs and not-to-exceed values, and to compare site data with
background data or with some pre-established reference limits (e.g., preliminary remediation goals
(PRGs), action levels, compliance limits). It is noted that several EPA guidance documents (e.g., EPA
1997, 2002a, 2006) discuss in details about data quality objectives (DQOs) and sample size
determinations based upon those DQOs needed for the various statistical methods used in environmental
applications.
Also, appropriate sample collection methods (e.g., instruments, sample weights, discrete or composite,
analytical methods) depend upon the medium (e.g., soil, sediment, water) under consideration. For an
example, Gerlach and Nocerino (EPA, 2003) describe optimal soil sample (based upon Gy theory)
collection methods. Therefore, the topics of sample size determination based upon DQOs, data validation,
and appropriate sample collection methods for the various environmental media are not considered in
ProUCL 4.0 and its associated Technical Guide. It is assumed that data sets to be used in ProUCL are of
good quality, and whenever possible have been obtained using the guidance provided in various EPA
(2003, 2006) documents. It is the users' responsibility to assure that adequate amount of data have been
collected, and the collected data are of good quality.
Note: In Pro UCL 4.0 and its associated guidance documents, emphasis is given on the practical
applicability and appropriate use of statistical methods needed to address statistical issues arising in risk
management, background versus site evaluation studies, and various other environmental applications.
Specifically, guidance on minimum sample size requirements as provided in this chapter is useful when
data have already been collected, or it is not possible (e.g., due to resource limitations) to collect the
number of samples obtained using DQO processes as described in EPA (2006).
Decisions based upon statistics obtained using data sets of small sizes (e.g., 4 to 6 detected observations)
cannot be considered reliable enough to make a remediation decision that affects human health and the
environment. For an example, a background data set of size 4 to 6 is not large enough to characterize
background population, to compute BTV values, or to perform background versus site comparisons. In
order to perform reliable and meaningful statistical inference (estimation and hypothesis testing), one
should determine the sample sizes that need to be collected from the populations under investigation
using appropriate DQO processes and decision error rates (EPA, 2006). However, in some cases, it may
not be possible (e.g., resource constraints) to collect the same number of samples recommended by the
DQO process. In order to address such cases, minimum sample size requirements for background and site
data sets are described.
The use of an appropriate statistical method depends upon the environmental parameter(s) being
estimated or compared. The measures of central tendency (e.g., means, medians, or their upper confidence
limits (UCLs)) are often used to compare site mean concentrations (e.g., after remediation activity) with a
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cleanup standard, Cs, representing some central tendency measure of a reference area or other known
threshold representing a measure of central tendency. The upper threshold values, such as the compliance
limits (e.g., alternative concentration limit (ACL), maximum concentration limit (MCL)), or not-to-
exceed values, are used when individual point-by-point observations are compared with those not-to-
exceed values or other compliance limit. It should be noted that depending upon whether the
environmental parameters (e.g., BTVs, not-to-exceed value, EPC term, cleanup standards) are known or
unknown, different statistical methods with different data requirements are needed to compare site
concentrations with pre-established (known) or estimated (unknown) cleanup standards and BTVs.
ProUCL 4.0 has been developed to address issues arising in exposure assessment, risk assessment, and
background versus site comparison applications. Several upper limits, and single- and two-sample
hypotheses testing approaches, for both full uncensored and left-censored data sets, are available in
ProUCL 4.0. The details of the statistical and graphical methods included in ProUCL 4.0 can be found in
the ProUCL Technical Guidance. In order to make sure that the methods in ProUCL 4.0 are properly
used, this chapter provides guidance on:
1. analysis of site and background areas and data sets,
2. collection of discrete or composite samples,
3. appropriate use of the various upper limits,
4. guidance regarding minimum sample sizes,
5. point-by-point comparison of site observations with BTVs,
6. use of hypotheses testing approaches,
7. using small data sample sets,
8. use of maximum detected value, and
9. discussion of ProUCL usage for special cases.
1.1 Background Data Sets
The project team familiar with the site should identify and chose a background area. Depending upon the
site activities and the pollutants, the background area can be site-specific or a general reference area. An
appropriate random sample of independent observations should be collected from the background area. A
defensible background data set should represent a "single" background population (e.g., representing
pristine site conditions before any of the industrial site activities) free of contaminating observations such
as outliers. In a background data set, outliers may represent potentially contaminated observations from
impacted site areas under study or possibly from other polluted site(s). This scenario is common when
background samples are obtained from the various onsite areas (e.g., large federal facilities). Outlying
observations should not be included in the estimation (or hypotheses testing procedures) of the BTVs.
The presence of outliers in the background data set will yield distorted estimates of the BTVs and
hypothesis testing statistics. The proper disposition of outliers to include or not include them in the data
set should be decided by the project team.
Decisions based upon distorted statistics can be incorrect, misleading, and expensive. It should be noted
that the objective is to compute background statistics based upon the majority of the data set representing
the dominant background population, and not to accommodate a few low probability outliers that may
also be present in the background data set. A couple of simple classical outlier tests (Dixon and Rosner
tests) are available in ProUCL 4.0. Since these classical tests suffer from masking effects (e.g., extreme
outliers may mask the occurrence of other intermediate outliers), it is suggested that these classical outlier
tests should always be supplemented with graphical displays such as a box plot or a Q-Q plot. The use of
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robust and resistant outlier identification procedures (Singh and Nocerino, 1995, Rousseeuw and Leroy,
1987) is recommended when multiple outliers may be present in a data set. Those methods are beyond the
scope of ProUCL 4.0.
An appropriate background data set of a reasonable size (preferably computed using DQO processes) is
needed to characterize a background area including computation of upper limits (e.g., estimates of BTVs,
not-to-exceed values) based upon background data sets and also to compare site and background data sets
using hypotheses testing approaches. As mentioned before, a small background data set of size 4 to 6 is
not large enough to compute BTVs or to perform background versus site comparisons. At the minimum, a
background sample should have at least 8 to 10 (more observations are preferable) detected observations
to estimate BTVs or to use hypotheses testing approaches.
1.2 Site Data Sets
A defensible data set from a site population (e.g., AOC, EA, RU, group of monitoring wells) should be
representative of the site area under investigation. Depending upon the site 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
site 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 a planning and sampling
design before starting to collect samples from the various site areas under investigation. Specifically, the
availability of an adequate amount of representative site data is required from each of those site sub-
populations defined by sample depths, soil types, and the various other characteristics. For detailed
guidance on soil sample collections, the reader is referred to Gerlach and Nocerino (EPA (2003)).
The site data collection requirements depend upon the objective of the study. Specifically, in background-
versus-site comparisons, site data are needed to perform:
• Individual point-by-point site observation comparisons with pre-established or estimated
BTVs, PRGs, cleanup standards, and not-to-exceed-values. Typically, this approach is
used when only a small number (e.g., < 4 to 6) of detected site observations (preferably
based upon composite samples) are available which need to be compared with BTVs and
not-to-exceed values.
• Single sample hypotheses tests to compare site data with pre-established cleanup
standards, Cs (e.g., representing a measure of central tendency); or with BTVs and not-to-
exceed values (used for tests for proportions and percentiles). The hypotheses testing
approaches are used when enough site data are available. Specifically, when at least 8 to
10 detected (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 the two types (Type 1 and
Type 2) of error rates 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 the background threshold value (e.g., 95th
percentile) by site observations, even when the site and background populations have
comparable distributions. The probabilities of these chance exceedances increase as the
sample size increases.
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Sample Size Probability of Exceedance
1 0.05
2 0.10
5 0.23
8 0.34
10 0.40
12 0.46
64 0.96
• Two-sample hypotheses testing to compare site data distribution with background data
distribution to determine if the site concentrations are comparable to background
concentrations. Adequate amount of data need to be made available from the site as well
as the background populations. It is preferable to collect at least 8 to 10 detected
observations from each of the population under comparison.
1.3 Discrete Samples or Composite Samples?
In a data set (background or site), collected 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.
• If possible, the use of composite site samples is preferred when comparing individual
point-by-point site observations from an area (e.g., area of concern (AOC), remediation
unit (RU), exposure area (EA)) with pre-established or estimated BTV, compliance limit
(CL), or other not-to-exceed value. This comparison approach is useful when few (< 4 to
6) detected site observations are compared with a pre-established or estimated BTV or
other not-to-exceed threshold.
• 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 (e.g., exceeding 8 to 10) detected site observations are available. Details of
the single sample hypothesis approaches are widely available in EPA documents (1989,
1997, and 2006). Selected single sample hypotheses testing procedures are available in
ProUCL 4.0.
• If a two-sample hypotheses testing approach is used to perform site versus background
comparisons, then samples from both of the populations should be either all discrete or
all composite samples. The two-sample hypothesis testing approach is used when many
(e.g., exceeding 8 to 10) site, as well as background, observations are available. For better
and more accurate results with higher statistical power, the availability of more
observations (e.g., exceeding 10-15) from each of the two populations is desirable,
perhaps based upon an appropriate DQO process, as described in an EPA guidance
document (2006).
1.4 Upper Limits and Their Use
The appropriate computation and use of statistical limits depend upon their applications and the
parameters (e.g., EPC term, not-to-exceed value) they are supposed to be estimating. Depending upon the
objective of the study, a pre-specified cleanup standard, Cs, or a risk-based cleanup (RBC) can be viewed
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as to represent: 1) as average contaminant concentration; or 2) a not-to-exceed upper threshold value.
These two threshold values, an average value, jU0, and a not-to-exceed value, A0, represent two
significantly different parameters, and different statistical methods and limits are used to compare the site
data with these two different parameters or threshold values. Statistical limits, such as an upper
confidence limit (UCL) of the population mean, an upper prediction limit (UPL) for an independently
obtained "single" observation, or independently obtained k observations (also called future k
observations, next k observations, or k different observations), upper percentiles, and upper tolerance
limits (UTLs), are often used to estimate the environmental parameters, including the EPC terms,
compliance limits (e.g., ACL, MLC), BTVs, and other not-to-exceed values. Here, UTL95%-95%
represents a 95% confidence limit of the 95th percentile of the distribution of the contaminant under study.
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. Specifically, the differences between UCLs and UPLs
(or upper percentiles), and UCLs and UTLs should be clearly understood and acknowledged. A UCL with
a 95% confidence limit (UCL95) of the mean represents an estimate of the population mean (measure of
the central tendency of a data distribution), whereas a UPL95, a UTL95%-95%, and an upper 95th
percentile represent estimates of a threshold value in the upper tail of the data distribution. Therefore, a
UCL95 should represent a smaller number than an upper percentile or an upper prediction limit. Also,
since a UTL 95%-95% represents a 95% UCL of the upper 95th percentile, a UTL should be > the
corresponding UPL95 and the 95th upper percentile. Typically, it is expected that the numerical values of
these limits should follow the order given as follows:
Sample Mean < UCL95 of Mean < Upper 95th Percentile < UPL95 of a Single Observation < UTL95%-
95%
It should also be pointed out that as the sample size increases, a UCL95 of the mean approaches
(converges to) the population mean, and a UPL95 approaches the 95th percentile. The differences among
the various upper limits are further illustrated in Example 1-1 below. It should be noted that, in some
cases, these limits might not follow the natural 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%. 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 becomes smaller than the sample mean, especially
when the data are mildly skewed to moderately skewed and the sample size is large (e.g., > 50, 100).
Example 1-1: Consider a simple site-specific background data set associated with a Superfund site. The
data set (given in Appendix 5 of the revised Guidance for Comparing Background and Chemical
Concentrations in Soil for CERCLA Sites (EPA, 2002b)) has several inorganic contaminants of potential
concern, including aluminum, arsenic, chromium, iron, and lead. It is noted that iron concentrations
follow a normal distribution. Upper limits for the iron data set are summarized in Table 1 -1. It is noted
that the upper limits do follow the order 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
UPL95for4
Observations
21618
UTL95/95
21149
95% Upper
Percentile
17534
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A 95% UCL (UCL95) of the mean is the most commonly used limit in environmental applications. For an
example, a 95% UCL of mean is used as an estimate of the EPC. A UCL95 should not be used to estimate
a background threshold value (a value in the upper tail of the background data distribution) to be
compared with individual site observations. There are many instances in background evaluations and
background versus site comparison studies, when it is not appropriate to use a 95% UCL. Specifically,
when point-by-point site observations are to be compared with a BTV, then that BTV should be estimated
(or represented) by a limit from the upper tail of the reference set (background) data distribution.
A brief discussion about the differences between the applications and uses of the various statistical limits
is provided below. This will assist a typical user in determining which upper limit (e.g., UCL95 or
UPL95) to use to estimate the parameter of interest (e.g., EPC or BTV).
• A UCL represents an average value that should be compared with a threshold value also
representing an average value (pre-established or estimated), such as a mean cleanup
standard, Cs. For an example, a site 95% UCL exceeding a cleanup value, Cs, may lead to
the conclusion that the cleanup level, Cs, has not been attained by the site area under
investigation. It should be noted that UCLs of means are typically computed based upon
the site data set.
• When site averages (and not individual site observations) are compared with a threshold
value (pre-determined or estimated), such as a PRO or a RBC, or with some other
cleanup standard, Cs, then that threshold should represent an average value, and not a not-
to-exceed threshold value for individual observation comparisons.
• 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 site averages:
with a specified or pre-established cleanup standard (single sample hypothesis), or with
the background population averages (two-sample hypothesis).
• A UPL, an upper percentile, or an 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 individual site observations are compared with those limits. A
site observation for a contaminant exceeding a background UTL or UPL may lead to the
conclusion that the contaminant is a contaminant of potential concern (COPC) to be
included in further risk evaluation and risk management studies.
• When individual point-by-point site observations are compared with a threshold value
(pre-determined or estimated) of a background population or some other threshold and
compliance limit value, such as a PRG, MLC, or ACL, then that threshold value should
represent a not-to-exceed value. Such BTVs or not-to-exceed values are often estimated
by a 95% UPL, UTL 95%-95%, or by an upper percentile. ProUCL 4.0 can be used to
compute any of these upper limits based upon uncensored data sets as well as data sets
with nondetect values.
• As the sample size increases, a UCL approaches the sample mean, and a UPL95
approaches the corresponding 95th upper percentile.
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• It is pointed out that the developers of ProUCL 4.0 prefer the use of a 95% UPL (UPL95)
as an estimate of BTV or a not-to-exceed value. As mentioned before, the option of
comparing individual site observations with a BTV (specified or estimated) should be
used when few (< 4 to 6) detected site observations (preferably composite values) are to
be compared with a BTV.
• When enough (e.g., > 8 to 10) detected site observations are available, it is preferred 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 4.0.
It is re-emphasized that only averages should be compared with the averages or UCLs, and individual site
observations should be compared with UPLs, upper percentiles, or UTLs. For an 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. It is hard to justify comparing a UCL of one population with a UPL of
the other population. Conclusions (e.g., site dirty or site clean) derived by comparing UCLs and UPLs, or
UCLs and upper percentiles as suggested in Wyoming DEQ, Fact Sheet #24 (2005), cannot be considered
fair and reliable. Specifically, the decision error rates associated with such comparisons can be
significantly different from the specified (e.g., Type I error = 0.1, Type II error = 0.1) decision errors.
1.5 Point-by-Point Comparison of Site Observations with BTVs,
Compliance Limits, and Other Threshold Values
Point-by-point observation comparison method is used when a small number (e.g., 4 to 6 locations) of
detected site observations are compared with pre-established or estimated BTVs, screening levels, or
preliminary remediation goals (PRGs). In this case, individual point-by-point site observations (preferably
based upon composite samples from various site locations) are compared with estimated or pre-
established background (e.g., USGS values) values, PRGs, or some other not-to-exceed value. Typically,
a single exceedance of the BTV, PRG, or of a not-to-exceed value by a site (or from a monitoring well)
observation may be considered as an indication of contamination at the site area under investigation. The
conclusion of an exceedance by a site value is some times confirmed by re-sampling (taking a few more
collocated samples) that site location (or a monitoring well) exhibiting contaminant concentration in
excess of the BTV or PRG. If all collocated (or collected during the same time period) sample
observations collected from the same site location (or well) exceed the PRG (or MLC) or a not-to-exceed
value, then it may be concluded that the location (well) requires further investigation (e.g., continuing
treatment and monitoring) and cleanup.
When BTV contaminant concentrations are not known or pre-established, one has to collect, obtain, or
extract a data set of an appropriate size that can be considered as representative of the site related
background. Statistical upper limits are computed using the data set thus obtained, which are used as
estimates of BTVs and not-to-exceed values. It should be noted that in order to compute reasonably
reliable and accurate estimates of BTVs and not-to-exceed values based upon a background (or reference)
data set, enough background observations (minimum of 8 to 10) should be collected, perhaps using an
appropriate DQO process as described in EPA (2006). Typically, background samples are collected from
a comparable general reference area or site-specific areas that are known to be free of contamination due
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to any of the site related activities. Several statistical limits can be used to estimate the BTVs based upon
a defensible data set of an adequate size. A detailed description of the computation and estimation of
BTVs is given in Chapter 5 (for uncensored data sets) and in Chapter 6 for data sets with nondetects of
the revised background guidance document. Once again, the use of this point-by-point comparison
method is recommended when not many (e.g., < 4 to 6) site observations are to be compared with
estimated BTVs or PRGs. An exceedance of the estimated BTV by a site value may be considered as an
indication of the existing or continuing contamination at the site.
Note: When BTVs are not known, it is suggested that at least 8 to 10 (more are preferable) detected
representative background observations be made available to compute reasonably reliable estimates of
BTVs and other not-to-exceed values.
The point-by-point comparison method is also useful when quick turnaround comparisons are required.
Specifically, when the decisions have to be made in real time by a sampling or screening crew, or when
few detected site samples are available, then individual point-by-point site concentrations are compared
either with pre-established PRGs, cleanup goals and standards, or with estimated BTVs and not-to-exceed
values. The crew can use these comparisons to make the following informative decisions:
1. Screen and identify the COPCs,
2. Identify the polluted site AOCs,
3. Continue or stop remediation or excavation at a site AOC or a RU, or
4. Move the cleanup apparatus and crew to the next AOC or RU.
During the screening phase, an exceedance of a compliance limit, action level, a BTV, or a PRO by site
values for a contaminant may declare that contaminant as a COPC. Those COPCs are then included in
future site remediation and risk management studies. During the remediation phase, an exceedance of the
threshold value such as a compliance limit (CL) or a BTV by sample values collected from a site area (or
a monitoring well (MW)) may declare that site area as a polluted AOC, or a hot spot requiring further
sampling and cleanup. This comparison method can also be used to verify if the site concentrations (e.g.,
from the base or side walls of an excavated site area) are approaching or meeting PRO, BTV, or a cleanup
standard after excavation has been conducted at that site area.
If a larger number of detected samples (e.g., greater than 8 tolO) are available from the site locations
representing the site area under investigation (e.g., RU, AOC, EA), then the use of hypotheses testing
approaches (both single sample and two-sample) is preferred. The use of a hypothesis testing approach
will control the error rates more tightly and efficiently than the individual point-by-point site observations
versus BTV comparisons, especially when many site observations are compared with a BTV or a not-to-
exceed value.
Note: In background versus site comparison evaluations, scientists usually prefer the use of hypotheses
testing approaches to point-by-point site observation comparisons with BTVs or not-to-exceed values.
Hypotheses testing approaches require the availability of larger data sets from the populations under
investigation. Both single sample (used when BTVs, not-to-exceed values, compliance limits, or cleanup
standards are known and pre-established) and two-sample (used when BTVs and compliance limits are
unknown) hypotheses testing approaches are available in ProUCL 4.0.
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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 contaminant concentrations of two (e.g., site versus background) or
more (several monitoring wells (MWs)) populations. The uses of hypotheses testing approaches in those
environmental applications are described as follows.
1.6.1 Single Sample Hypotheses - BTVs and Not-to-Exceed Values are Known
(Pre-established)
When pre-established BTVs and not-to-exceed values are used, such as the USGS background values
(Shacklette and Boerngen (1984)), thresholds obtained from similar sites, or pre-established not-to-exceed
values, PRGs, or RBCs, there is no need to extract, establish, or collect a background or reference 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 DQO) or decide (depending upon resources) about
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 area (e.g., RU,
AOC) under investigation. When the number of available detected site samples is less than 4 to 6, one
might perform point-by-point site observation comparisons with a BTV; and when enough detected site
observations (> 8 to 10, more are preferable) are available, it is desirable to use single sample hypothesis
testing approaches.
Depending upon the parameter (e.g., the average value, fi0, or a not-to-exceed value, A0), represented by
the known threshold value, one can use single sample hypothesis tests for population mean (t-test, sign
test) or single sample tests for proportions and percentiles. The details of the single sample hypotheses
testing approaches can be found in EPA (2006) and the Technical Guide for ProUCL 4.0. Several single
sample tests listed as follows are available in ProUCL 4.0.
One-Sample t-Test: This test is used to compare the site mean, ju, with a specified cleanup standard, Cs,
where the cleanup standard, Cs, represents an average threshold value, fi0- The Student's t-test (or a UCL
of mean) is often used (assuming normality of site data or when site sample size is large such as larger
than 30, 50) to determine 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 nondetect observations provided all nondetects (e.g., associated detection limits) fall below
the specified threshold value. Q. These tests are used to compare the site location (e.g., median, mean)
with a specified cleanup standard, Cs, representing a similar location measure.
One-Sample Proportion Test or Percentile Test: When a specified cleanup standard, A0, such as a
PRG or a BTV represents an upper threshold value of a contaminant concentration distribution
(e.g., not-to-exceed value, compliance limit) rather than the mean threshold value, JLIO, of the
contaminant concentration distribution, then a test for proportion or a test for percentile (or
equivalently a UTL 95%-95%) can be used to compare site proportion or site percentile with the
specified threshold or action level, A0. This test can also handle ND observations provided all
NDs are below the compliance limit.
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In order to obtain reasonably reliable estimates and test statistics, an adequate amount of representative
site data (8 to 10 detected observations) is needed to perform the hypotheses tests. As mentioned before,
in case only a few (e.g., < 4 to 6) detected site observations are available, then point-by-point site
concentrations may be compared with the specified action level, A0.
1.6.2 Two-Sample Hypotheses - When 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, a two-sample hypothesis testing approach can
be 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 (collected) from each of the two populations under investigation. Site and background data
requirements (e.g., based upon DQOs) to perform two-sample hypothesis test approaches are described in
EPA (1989b, 2006), Breckenridge and Crockett (1995), and the VSP (2005) software package. 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) into sample size
determination. That is, a larger number (> 10 to 15) of representative background (or site) samples should
be collected from larger background (or site) areas. As mentioned before, every effort should be made to
collect as many samples as determined using DQO processes as described in EPA documents (2006).
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, and they need to be estimated based upon a data set from a background or reference
population. Specifically, two-sample hypotheses testing approaches are used to compare 1) the average
contaminant concentrations of two or more populations such as the background population and the
potentially contaminated site areas, or 2) the proportions of site and background observations exceeding a
pre-established compliance limit, A0 In order to derive reliable conclusions with higher statistical power
based upon hypothesis testing approaches, enough data (e.g., minimum of 8 to 10 samples) should be
available from all of the populations under investigation. It is also desirable to supplement statistical
methods with graphical displays, such as the double Q-Q plots, or side-by-side multiple box plots, as
available in ProUCL 4.0. Several parametric and nonparametric two-sample hypotheses testing
approaches, including Student's t-test, the Wilcoxon-Mann-Whitney (WMW) test, Gehan's test, and
quantile test are included in ProUCL 4.0. Details of those methods are described in the ProUCL 4.0
Technical Guide. It should be noted that the WMW, Gehan, and quantile tests are also available for data
sets with NDs. Gehan's test is specifically meant to be used on data sets with multiple detection limits. It
is also suggested that for best and reliable conclusions, both the WMW and quantile tests should be used
on the same data set. The details of these two tests with examples are given in EPA (1994, 2006).
The samples collected from the two (or more) populations should all be of the same type obtained using
similar analytical methods and apparatus. In other words, the collected site and background samples
should be all 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.
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1.7 Minimum Sample Size Requirements
Due to resource limitations, it may not be possible (nor needed) to sample the entire population (e.g.,
background area, site area, areas of concern, exposure areas) under study. Statistics is used to draw
inference(s) about the populations (clean, dirty) and their known or unknown parameters (e.g.,
comparability of population means, not-to-exceed values, upper percentiles, and spreads) based upon
much smaller data sets (samples) collected from those populations under study. In order to determine and
establish BTVs, not-to-exceed values, or site-specific screening levels, defensible data set(s) of
appropriate size(s) needs to be collected from background areas (e.g., site-specific, general reference or
pristine area, or historical data). 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 size and boundaries based upon all current and future objectives for the data collection. The
size and area of the population (e.g., a remediation unit, area of concern, or an exposure unit) may be
determined based upon the potential land use, and other exposure and risk management objectives and
decisions. Moreover, appropriate effort should be made to properly collect soil samples (e.g., methods
based upon Gy sampling theory), as described in Gerlach and Nocerino (2003).
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 contaminant concentration are comparable; or if the site concentrations exceed the background
threshold concentration level; or if an adequate amount of cleanup and remediation approaching the BTV
or a cleanup level have been performed at polluted areas (e.g., AOC, RU) of the site under study.
In order to perform statistical inference (estimation and hypothesis testing), one needs to determine the
sample sizes that need to be collected from the populations (e.g., site and background) under investigation
using appropriate DQO processes (EPA 2006). However, in some cases, it may not be possible to collect
the same number of samples as determined by using a DQO process. For example, the data might have
already been collected (often is the case in practice) without using a DQO process, or due to resource
constraints, it may not be possible to collect as many samples as determined by using a DQO-based
sample size formula. It is observed that, in practice, the project team and the decision makers may not
collect enough background samples, perhaps due to various resource constraints. However, every effort
should be made to collect at least 8 to 10 (more are desirable) background observations before using
methods as incorporated in ProUCL 4.0. The minimum sample size recommendations as described here
are useful when resources are limited (as often is the case), and it may not be possible to collect as many
background and site (e.g., AOC, EU) samples as computed using DQOs and the sample size
determination formulae given in the EPA (2006). Some minimum sample size requirements are also given
in Fact Sheet #24, prepared by Wyoming Department of Environmental Quality (June 2005).
As mentioned before, the topics of DQO processes and the sample size determination are described in
detail in the EPA (2006) guidance document. Therefore, the sample size determination formulae based
upon DQO processes are not included in ProUCL 4.0 and its Technical Guide. It should be noted that
DQO-based sample size determination routines are available in DataQUEST (EPA, 1997) and VSP
(2005) software packages. Guidance and suggestions are provided on the minimum number of
background and site samples needed to be able to use statistical methods for the computation of upper
limits, and to perform single sample tests, two-sample tests such as t-test, and the Wilcoxon-Mann-
Whitney (WMW) test. The minimum sample size recommendations (requirements) as described here are
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made so that reasonably reliable estimates of EPC terms and BTVs, and defensible values of test statistics
for single or two-sample hypotheses tests (e.g., t-test, WMW test), can be computed.
1.7.1 Minimum Sample Size for Estimation and Point-by-Point Site Observation Comparisons
• Point-by-point observation comparison method is used when a small number (e.g., 4 to 6
locations) of detected site observations are compared with pre-established or estimated
BTVs, screening levels, or PRGs. In this case, individual point-by-point site observations
(preferably based upon composite samples from various site locations) are compared with
estimated or pre-established background (e.g., USGS values) values, PRGs, or some
other not-to-exceed value.
• When BTV contaminant concentrations are not known or pre-established, one has to
collect, obtain, or extract a data set of an appropriate size that can be considered as
representative of the site related background. Statistical upper limits are computed using
the data set thus obtained; which are used as estimates of BTVs and not-to-exceed values.
It should be noted that in order to compute reasonably reliable and accurate estimates of
BTVs and not-to-exceed values based upon a background (or reference) data set, enough
background observations (minimum of 8 to 10) should be collected perhaps using an
appropriate DQO process as described in EPA (2006). Typically, background samples are
collected from a comparable general reference area or a site-specific area.
• When enough (e.g., > 8 to 10) detected site observations are available, it is preferred 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.
1.7.2 Minimum Sample Size Requirements for Hypothesis Testing Approaches
Statistical methods (as in ProUCL 4.0) used to estimate EPC terms, BTVs, PRGs, or to compare the site
contaminant concentration data distribution with the background data distribution can be computed based
upon small site and background data sets (e.g., of sizes 3, 4, 5, or 6). However, those statistics cannot be
considered representative and reliable enough to make important cleanup and remediation decisions. It is
recommended not to use those statistics to draw cleanup and remediation decisions potentially impacting
the human health and the environment. It is suggested that the estimation and hypothesis testing methods
as incorporated in ProUCL 4.0 may not be used on background data sets with fewer than 8 to 10 detected
observations. Also, when using hypotheses testing approaches, it is suggested that the site and
background data be obtained using an appropriate DQO process as described in EPA (2006). In case that
is not possible, it is suggested that the project team at least collect 8 to 10 observations from each of the
populations (e.g., site area, MWs, background area) under investigation.
Site versus background comparisons and computation of the BTVs depend upon many factors, some of
which cannot be controlled. These factors include the site conditions, lack of historical information, site
medium, lack of adequate resources, measurement and analytical errors, and accessibility of the site areas.
Therefore, whenever possible, it is desirable to use more than one statistical method to perform site versus
background comparison. The use of statistical methods should always be supplemented with appropriate
graphical displays.
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1.8 Sample Sizes for Bootstrap Methods
Several parametric and nonparametric (including bootstrap methods) UCL, UPL, and other limits
computation methods for both full-uncensored data sets (without nondetects) and left-censored data sets
with nondetects are available in ProUCL 4.0. It should be noted that bootstrap resampling methods are
useful when not too few (e.g., < 10-15) and not too many (e.g., > 500-1000) detected 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 (with replacement) are drawn from the same data
set. Therefore, in order to obtain bootstrap resamples with 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 10-15. Also, it is not required 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., > 1000) 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. Moreover, bootstrapping a large data set of size greater
than 500 or 1000 will be time consuming.
1.9 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
quite common in many environmental applications. ProUCL 4.0 offers this option for data sets with and
without nondetects. The Group Option provides a powerful tool to perform various 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 an example, the same data set may consist of samples
from the various groups or populations representing site, background, two or more AOCs, surface,
subsurface, monitoring wells. The graphical displays (e.g., box plots, Q-Q plots) and statistics
(computations of background statistics, UCLs, hypotheses testing approaches) of interest can be
computed separately for each group by using this option.
It should be pointed out that it is the users' responsibility to provide adequate amount of detected data to
perform the group operations. For an example, if the user desires to produce a graphical Q-Q plot (using
only detected data) with regression lines displayed, then there should be at least two detected points (to
compute slope, intercept, sd) in the data set. Similarly if the graphs are desired for each of the group
specified by the group ID variable, there should be at least two detected observations in each group
specified by the group variable. ProUCL 4.0 generates a warning message (in orange color) in the lower
panel of the ProUCL 4.0 screen. Specifically, the user should make sure that a variable with nondetects
and categorized by a group variable should have enough detected data in each group to perform the
various methods (e.g., GOF tests, Q-Q plots with regression lines) as incorporated in ProUCL 4.0.
1.10 Use of Maximum Detected Values as Estimates of Upper Limits
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 values (EPA, 1992b). Also, many times in practice, the BTVs and not-to-exceed values are
estimated by the maximum detected value. This section discusses the appropriateness of using the
maximum detected value as estimates of the EPC term, BTVs, or other nor-to-exceed values.
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1.10.1 Use of Maximum Detected Values to Estimate BTVs and Not-to-Exceed Values
It is noted that BTVs and not-to-exceed values represent upper threshold values in 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 maximum detected value. As described earlier, upper limits, such
as UPLs, UTLs, and upper percentiles, are used to estimate the BTVs and not-to-exceed values. It is noted
that a nonparametric UPL or UTL is often estimated by higher order statistics such as the maximum value
or the second largest value (EPA 1992a, RCRA Guidance Addendum). The use of higher order statistics
to estimate the UTLs depends upon the sample size. For an example: 1) 59 to 92 samples, a
nonparametric UTL95%-95 is given by the maximum detected value; 2) 93 to 123 samples, a
nonparametric UTL95%-95 is given by the second largest maximum detected value; and 3) 124 to 152
samples, a UTL95%-95 is given by the third largest detected value in the sample.
Note: Therefore, when a data set does not follow a discernable distribution, the maximum observed value
(or other high order statistics) may be used as an estimate ofBTV or a not-to-exceed value, provided the
maximum value does not represent an outlier or a contaminating observation perhaps representing a hot
location.
1.10.2 Use of Maximum Detected Values 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 values (EPA, 1992b). Specifically, a RAGS document (EPA, 1992) 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)
exceeded the maximum value. ProUCL 4.0 can compute a 95% UCL of mean using several methods
based upon normal, Gamma, lognormal, and non-discernable distributions. In past (e.g., EPA, 1992b),
only two methods were used to estimate the EPC term based upon: 1) Student's t-statistic and a normal
distribution, and 2) Land's H-statistic (1975) and a lognormal model. The use of H-statistic often yields
unstable and unpractically large UCL95 of the mean (Singh, Singh, and laci, 2002). For skewed data sets
of smaller sizes (e.g., < 30, < 50), H-UCL often exceeds the maximum detected value. This is especially
true when extreme high outliers may be present in the data set. Since the use of a lognormal distribution
has been quite common (e.g., suggested as a default model in a RAGS document (EPA, 1992)), the
exceedance of the maximum detected value by H-UCL95 is frequent for many skewed data sets of
smaller sizes (e.g., < 30, < 50). It is also be noted that for highly skewed data sets, the sample mean
indeed can even exceed the upper 90%, 95%, etc., percentiles, and consequently, a 95% UCL of mean can
exceed the maximum observed value of a data set.
All of 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 a highly polluted outlying observation. Obviously, it is not desirable to use a highly polluted
value as an estimate of average exposure (EPC term) for an exposure area. This is especially true when
one is dealing with lognormally distributed data sets of small sizes. As mentioned before, for such highly
skewed data sets that cannot be modeled by a gamma distribution, a 95% UCL of the mean should be
computed using an appropriate distribution-free nonparametric method.
It should be pointed out that the EPC term represents the average exposure contracted by an individual
over an exposure area (EA) during a long period of time; therefore, 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. It is unlikely
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that an individual will visit the location (e.g., in an EA) of the maximum detected value all of the time.
One can argue that the use of this practice results in a conservative (higher) estimate of the EPC term. The
objective is to compute an accurate estimate of the EPC term. Several other methods (instead of H-UCL)
as described in EPA (2002), and included in ProUCL 4.0 (EPA 2007), are available to estimate the EPC
terms. It is unlikely (but possible with outliers) that the UCLs based upon those methods will exceed the
maximum detected value, unless some outliers are present in the data set. ProUCL 4.0 displays a warning
message when the recommended 95% UCL (e.g., Hall's or bootstrap t UCL with outliers) of the mean
exceeds the observed maximum concentration. When a 95% UCL does exceed the maximum observed
value, ProUCIAO recommends the use of an alternative UCL computation method based upon the
Chebyshev inequality. The detailed recommendations (as functions of sample size and skewness) for the
use of those UCLs are summarized in ProUCL 3.0 User Guide (EPA, 2004).
Singh and Singh (2003) studied the performance of the max test (using the maximum observed value as
an estimate of the EPC term) via Monte Carlo simulation experiments. They noted that for skewed data
sets of small sizes (e.g., < 10-20), the max test does not provide the specified 95% coverage to the
population mean, and for larger data sets, it overestimates the EPC term, which may require unnecessary
further remediation. The use of the maximum value as an estimate of the EPC term also ignores most
(except for maximum value) of the information contained in the data set. With the availability of so many
UCL computation methods (15 of them), the developers of ProUCL 4.0 do not recommend using the
maximum observed value as an estimate of the EPC term representing an average exposure by an
individual over an EA. Also, for the distributions considered, the maximum value is not a sufficient
statistic for the unknown population mean.
Note: It is recommended that the maximum observed value NOT be used as an estimate of the EPC term
representing average exposure contracted by an individual over an EA. For the sake of interested users,
ProUCL displays a warning message when the recommended95% UCL (e.g., Hall's bootstrap UCL etc.)
of the mean exceeds the observed maximum concentration. For such scenarios (when a 95% UCL does
exceed the maximum observed value), an alternative 95% UCL computation method is recommended by
ProUCL 4.0.
1.10.3 Samples with Nondetect Observations
Nondetect observations (or less than obvious values) are inevitable in most environmental data sets.
Singh, Maichle, and Lee (EPA, 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 nondetect observations. They concluded that the UCLs obtained using the
substitution methods, including the replacement of nondetects by respective DL/2, do not perform well
even when the percentage of nondetect observations is low, such as 5%-10%. They recommended
avoiding the use of substitution methods to compute UCL95 based upon data sets with nondetect
observations.
1.10.4 Avoid the Use of DL/2 Method to Compute UCL95
Based upon the results of the report by Singh, Maichle, and Lee (EPA, 2006), it is strongly recommended
to avoid the use of the DL/2 method to perform GOF test, and to compute the summary statistics and
various other limits (e.g., UCL, UPL) often used to estimate the EPC terms and BTVs. Until recently, the
DL/2 method has been the most commonly used method to compute the various statistics of interest for
data sets with BDL observations. The main reason of its common use has been the lack of the availability
of other defensible methods and associated programs that can be used to estimate the various
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environmental parameters of interest. Today, several other methods (e.g., KM method, bootstrap
methods) with better performances are available that can be used to compute the various upper limits of
interest. Some of those parametric and nonparametric methods are available in ProUCL 4.0. Even though
the DL/2 method (to compute UCLs, UPLs, and for goodness-of-fit test) has also been incorporated in
ProUCL 4.0, its use is not recommended due to its poor performance. The DL/2 method is included in
ProUCL 4.0 only for historical reasons as it had been the most commonly used and recommended method
until recently (EPA, 2006). Some of the reviewers of ProUCL 4.0 suggested and requested the inclusion
of DL/2 method in ProUCL for comparison purposes.
Note: The DL/2 method has been incorporated in ProUCL 4.0 for historical reasons only. NERL-EPA,
Las Vegas strongly recommends avoiding the use of DL/2 method even when the percentage (%) ofNDs
is as low as 5%-10%. There are other methods available in ProUCL 4.0 that should be used to compute
the various summary statistics and upper limits based upon data sets with multiple detection limits.
1.10.5 Samples with Low Frequency of Detection
When all of the sampled data values are reported as nondetects, the EPC term should also be reported as a
nondetect value, perhaps by the maximum reporting limit (RL) or maximum RL/2. Statistics (e.g.,
UCL95) computed based upon only a few detected values (e.g., < 4 to 6) cannot be considered reliable
enough to estimate the EPC terms having potential impact on the human heath and the environment.
When the number of detected data is small, it is preferable to use simple ad hoc methods rather than using
statistical methods to compute the EPC terms and other upper limits. Specifically, it is suggested that in
cases when the detection frequency is low (e.g., < 4%-5%) and the number of detected observations is
low, the project team and the decision makers together should make a decision on site-specific basis on
how to estimate the average exposure (EPC term) for the contaminant and area under consideration. For
such 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, it is also suggested that when most (e.g., > %95) of the observations for a contaminant lie
below the detection limit(s) or reporting limits (RLs), the sample median or the sample mode (rather than
the sample average which cannot be computed accurately) may be used as an estimate the EPC term. Note
that when the majority of the data are nondetects, the median and the mode will also be a nondetect. The
uncertainty associated with such estimates will be high. It is noted that 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.
Note: In case the number of available detected samples is small (< 5), it is suggested that the project
team decide about the estimation of the EPC term on site-specific basis. For such small data sets with
very few detected values (< 5), the final decision ("policy decision ") on how to estimate the EPC term
should be determined by the project team and decision makers.
1.11 Other Applications of Methods in ProUCL 4.0
In addition to performing background versus site comparisons for CERCLA and RCRA sites, and
estimating the EPC terms in exposure and risk evaluation studies, the statistical methods as incorporated
in ProUCL 4.0 can be used to address other issues dealing with environmental investigations that are
conducted at Superfund or RCRA sites.
30
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1.11.1 Identification of COPCs
Risk assessors and RPMs often use screening levels or BTVs to identify the COPCs during the screening
phase of a cleanup project to be conducted at a contaminated site. The screening for the COPCs is
performed prior to any characterization and remediation activities that may have to be conducted at the
site under investigation. This comparison is performed to screen out those contaminants that may be
present in the site medium of interest at low levels (e.g., at or below the background levels or pre-
established screening levels) and may not pose any threat and concern to human health and the
environment. Those contaminants may be eliminated from all future site investigations, and risk
assessment and risk management studies.
In order to identify the COPCs, point-by-point site observations (preferably composite samples) are
compared with pre-established 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 right there in the site field. The project team should decide about the type of
site samples (discrete or composite) and the number of detected site observations (not more than 4 to 6)
that should be collected and compared with the screening levels or the BTVs. In case BTVs, screening
levels, or not-to-exceed values are not known, the availability of a defensible background or reference
data set of reasonable size (e.g., > 8 to 10, more are preferable) is required to obtain reliable estimates of
BTVs and screening levels. When a reasonable number of detected site observations are available, it is
preferable to use hypotheses testing approaches. The contaminants with concentrations exceeding the
respective screening values or BTVs may be considered as COPCs, whereas contaminants with
concentrations (in all collected samples) lower than the screening value, PRG, or an estimated BTV may
be omitted from all future evaluations including the risk assessment and risk management investigations.
1.11.2 Identification of Non-Compliance Monitoring Wells
In monitoring well (MW) compliance assessment applications, individual (often discrete) contaminant
concentrations from a MW are compared with pre-established ACL, MCL, or an estimated compliance
limit (CL) based upon a group of upgradient wells representing the background population. An
exceedance of the MCL or the BTV by a MW concentration may be considered as an indication of
contamination in that MW. In such 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 contaminants in both the original sample and the re-sample(s) exceed the MCL or BTV,
then that MW may require closer scrutiny, perhaps triggering the remediation remedies as determined by
the project team. If the concentration data from a MW for about 4 to 5 continuous quarters (or another
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. Statistical
methods as described in Chapters 5 and 6 of the revised background guidance document (EPA, 2002b)
can be used to estimate the not-to-exceed values or BTVs based upon background or upgradient wells in
case the ACLs or MCLs are not pre-determined.
1.11.3 Verification of the Attainment of Cleanup Standards, Cs
Hypothesis testing approaches may be used to verify the attainment of the cleanup standard, Cs, at
polluted site areas of concern after conducting remediation and cleanup at the site AOC (EPA, 2006). In
order to properly address this scenario, a site data set of adequate size (minimum of 8 to 10 detected site
observations) needs to be made available from the remediated or excavated areas of the site under
investigation. The sample size should also account for the size of the remediated site area; meaning that
-------
larger site areas should be sampled more (with more observations) to obtain a representative sample of the
site under investigation.
Typically, the null hypothesis of interest is H0: Site Mean, |o,s>=Cs versus the alternative hypothesis, HI:
Site Mean, |o,s < Cs> where the cleanup standard, Cs, is known a priori. The sample size needed to perform
such single sample hypotheses tests can be obtained using the DQO process-based sample size formula as
given in the EPA (2006) documents. In any case, in order to use this test, a minimum of 8 to 10 detected
site samples should be collected. The details of statistical methods used to perform single sample
hypothesis as described above can be found in EPA (2006).
1.11.4 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 (preferably using composite samples representing a site location) are compared
with a pre-established or estimated BTV. Exceedances of the BTV by site observations may be
considered as representing locations with elevated concentrations (hot spots). Chapters 5 and 6 of the
revised background guidance document (EPA, 2002b) describe several methods to estimate the BTVs
based upon full data sets without nondetects and left-censored data sets with nondetect observations.
The rest of the chapters of this User Guide illustrate the use of the various procedures as incorporated in
ProUCL 4.0. Those methods are useful to analyze environmental data sets with and without the nondetect
observations. It is noted that ProUCL 4.0 is the first software package equipped with single sample and
two-sample hypotheses testing approaches that can be used on data sets with nondetect observations.
32
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Chapter 2
Entering and Manipulating Data
2.1 Creating a New Data Set
By simply executing the ProUCL 4.0, a new worksheet is generated and displayed for the user to enter
data.
To create a new worksheet: click Js-i or choose File ^- New
2.2 Opening an Existing Data Set
If your data sets are stored in the ProUCL data format (* .wst), then click x or choose File ^ Open
• If your data sets are stored in the Microsoft Excel format (* .xls), then choose File
Other Files... ^Import Excel... OR File ^ Load Excel Data
Edit Configure Summary Statistics R05 Est NDs Graphs Outiier Tests Goodness -of -Fit Hypothesis Testng Background UCL Window Help
Ni Load Excel Data < (I 0123456789;
Close Export Excel...
Save |
Save As ... i
Print i
Print Preview i
Exit I
2
3
4
5
6
7
Possible Error Messages:
o When you import an Excel file, make sure that you have an empty worksheet. If there
is no empty worksheet, then you must create a new worksheet before importing an
Excel file. Otherwise, there will be an error message in the Log Panel: "[Error]
Worksheet must be empty."
o First open a new worksheet and then import the Excel file.
o Make sure that the file you trying to import is not currently open. Otherwise, there
will be the following warning message in the Log panel:
o "[Information] Unable to open C:\***.xls." Check the validity of this file.
33
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2.3 Input File Format
Note that ProUCL 4.0 does not require that data in each column must end with a nonzero
value (ProUCL 3.0 requires this). Therefore, all zero values (in the beginning, middle, or
end of data columns) are treated as valid zero values as part of the data set.
The program can read Excel files. The user can perform typical Cut, Paste, and Copy
operations.
The first row in all input data files consist of alphanumeric (strings of numbers and
characters) names representing the header row. Those header names may represent
meaningful variable names such as Arsenic, Chromium, Lead, Group-ID, and so on.
o The Group-ID column has the labels for the groups (e.g., Background, AOC1,
AOC2, 1, 2, 3, a, b, c, Site 1, Site 2, and so on) that might be present in the data set.
The alphanumeric strings (e.g., Surface, Sub-surface) can be used to label the various
groups.
o The data file can have multiple variables (columns) with unequal number of
observations.
o Except for the header row and columns representing the group labels, only numerical
values should appear in all other columns.
o All alphanumeric strings and characters (e.g., blank, other characters, and strings),
and all other values (that do not meet the requirements above) in the data file are
treated as missing values.
o Also, a large value denoted by 1E31 (= IxlO31) can be used to represent missing data
values. All entries with this value are ignored from the computations. These values
are counted under number of missing values.
2.4 Number Precision
You may turn "Full Precision" on or off by choosing Configure
On/OFF
Full Precision
r pVorkS.hert.wstI.
[J File Edit Kll Summary Statistics RQS Est. NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing Background IJCL Window Help
°n '
• By leaving "Full Precision" turned off, ProUCL will display numerical values using an
appropriate (default) decimal digit option. However, by turning "Full Precision" off, all
decimal values will be rounded to the nearest thousandths place.
• "Full Precision" on option is specifically useful when one is dealing with data sets
consisting of small numerical values (e.g., < 1) resulting in small values of the various
estimates and test statistics. These values may become so small with several leading zeros
34
-------
(e.g., 0.00007332) after the decimal. In such situations, on may want to use "Full
Precision" option to see nonzero values after the decimal.
Note: For the purpose of this User Guide, unless noted otherwise, all examples have been described
using the "Full Precision " off option. This option prints out results up to 3 significant digits after the
decimal.
2.5 Entering and Changing a Header Name
1. Highlight the column whose header name (variable name) you want to change by clicking either
the column number or the header as shown below.
2.
3.
Right-click and then clicl
0 1 2
Arsenic
1 I 4.5|
2 5.6
3 4.3
4 5.4
5 9.2
c "Header Name"
0 1 2
^^^^^^^^^^^^^^^^^^^^^^E2^l Header Name l^^^H
1 ! 4-b!
2 5.6
3 4.3
4 5.4
5 9.2
Change the Header Name.
^^^^^^^^^^^^1 HI
^^^^^^^^^^^^^^H
Header Name: Arsenic Site 1
OK Cancel
4. Click the OK button to get the following output with the changed variable name.
35
-------
0
Arsenic Site 1
1 i
2
3
4
5
4.5]
5.6
4.3
5.4
9.2
2.6 Saving Files
9 Edit Configure
New
Open ,,,
Load Excel Data
Other Files ... >
Close
Save As ,,,
Print
Print Preview
Exit
• Save option allows the user to save the active window.
• Save As option allows the user to save the active window. This option follows typical
Windows standards, and saves the active window to a file in Excel 95 (or higher) format.
All modified/edited data files, and output screens (excluding graphical displays)
generated by the software can be saved as Excel 95 (or higher) spreadsheet.
2.7 Editing
Click on the Edit menu item to reveal the following drop-down options.
File |gw Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing Background UCL Window Help
I C
Copy Ctri+C L
Na-.igal Paste ctf+v jl
0
1
The following Edit drop-down menu options are available:
• Cut option: similar to a standard Windows Edit option, such as in Excel. It performs
standard edit functions on selected highlighted data (similar to a buffer).
• Copy option: similar to a standard Windows Edit option, such as in Excel. It performs
36
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typical edit functions on selected highlighted data (similar to a buffer).
• Paste option: similar to a standard Windows Edit option, such as in Excel. It performs
typical edit functions of pasting the selected (highlighted) data to the designated
spreadsheet cells or area.
• It should be noted that the Edit option could also be used to Copy Graphs. This topic
(copying and pasting graphs) is illustrated in detail in Chapter 13.
2.8 Handling Nondetect Observations
• ProUCL 4.0 can handle data sets with single and multiple detection limits.
• For a variable with nondetect observations (e.g., arsenic), the detected values, and the
numerical values of the associated detection limits (for less than values) are entered in the
appropriate column associated with that variable.
• Specifically, the data for variables with nondetect values are provided in two columns.
One column consists of the detected numerical values with less than (< DLj) values
entered as the corresponding detection limits (or reporting limits), and the second column
represents their detection status consisting of only 0 (for less than values) and 1 (for
detected values) values. The name of the corresponding variable representing the
detection status should start with d_, or D_ (not case sensitive) and the variable name.
The detection status column with variable name starting with a D_ (or a d_) should have
only two values: 0 for nondetect values, and 1 for detected observations.
• For an example, the header name, D_Arsenic is used for the variable, Arsenic having
nondetect observations. The variable D_Arsenic contains a 1 if the corresponding Arsenic
value represents a detected entry, and contains a 0 if the corresponding entry for variable,
Arsenic, represents a nondetect. An example data set illustrating these points is given as
follows.
37
-------
^taampte.wst
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
jj]
0 1
Arsenic D_Arsenic
4.5 0
5.6 1
4.3 0
5.4 1
9.2 1
6.2 1
6.7 1
5.8 1
8.5 1
5.65 1
5.4 1
5.5 1
5.9 1
5.1 1
5.2 1
4.5 0
6.1 1
6.1 1
6.8 1
5 1
2
Mercury
0.07
0.07
0.11
0.2
0.61
0.12
0.04
0.06
0.88
0.125
0.18
0.21
0.29
0.44
0.12
0.055
0.055
0.21
0.67
0.1
0.8
0.26
0.97
0.05
0.26
3 4
D_Mercury Vanadium
1 16.4
1 16.8
0 17.2
0 18.4
1 15.3
1 30.8
1 29.4
1 13.8
1 18.8
1 17.25
1 17.2
1 16.3
1 16.8
1 17.1
1 10.3
1 15.1
1 24.3
1 18
1 16.8
1 12
1
1
1
1
1
5 6
Zinc Group — ^
89.3 Surface
90.7 Surface
95.5 Surface
113 Surface
266 Surface
80.9 Surface
80.4 Surface
89.2 Surface
182 Surface
80.4 Surface
91.9 Subsurface
112 Subsurface
172 Subsurface
88 Subsurface
80.7 Subsurface
66.3 Subsurface
75 Subsurface
185 Subsurface
184 Subsurface
68.4 Subsurface
^r1
2.9 Caution
• Care should be taken to avoid any misrepresentation of detected and nondetected values.
Specifically, it is advised not to have any missing values (blanks, characters) in the
D_column (detection status column). If a missing value is located in the D_ column (and
not in the associated variable column), the corresponding value in the variable column is
treated as a nondetect, even if this might not have been the intention of the user.
• It is mandatory that the user makes sure that only a 1 or a 0 are entered in the detection
status D_column. If a value other than a 0 or a 1 is entered in the D_ column (the
detection column), results may become unreliable, as the software defaults to any number
other than 0 or 1 as a nondetect value.
• When computing statistics for full data sets without any nondetect values, the user should
select only those variables (from the list of available variables) that contain no nondetect
observations. Specifically, nondetect values found in a column chosen for the summary
statistics (full-uncensored data set) will be treated as a detected value; whatever value
(e.g., detection limit) is entered in that column will be used to compute summary
statistics for a full-uncensored data set without any nondetect values.
• Two-Sample Hypotheses: It should be noted that at present, when using two-sample
hypotheses approaches (WMW test, Gehan test, and quantile test) on data sets with NDs,
38
-------
both samples or variables (e.g., site-As, Back-As) should be specified as having
nondetects, even though one of the variables may not have any ND observations. This
means, a ND column (with 0 = ND, and 1 = detect) should be provided for each variable
(here D_site-As, and D_Back-As) to be used in this comparison. If a variable (e.g., site-
As) does not have any nondetects, still a column with label D_site-As should be included
in the data set with all entries = 1 (detected values).
• The following sample (not from a Superfund site) data set given on the next page
illustrates points related to this option and issues listed above. The data set considered
contains some nondetect measurements for Arsenic and Mercury. It should also be noted
that the Mercury concentrations are used to illustrate the points related to nondetect
observations only. Arsenic and Zinc concentrations are used to illustrate the use of the
group variable, Group (Surface, Subsurface).
• If for mercury, one computes summary statistics (assuming no nondetect values) using
"Full" data set option, then all nondetect values (with "0" entries in D_Mercury column)
will be treated as detected values, and summary statistics will be computed accordingly.
2.10 Summary Statistics for Data Sets with Nondetect Observations
• In order to compute simple summary statistics or to compute other statistics of interest
(e.g., background statistics, GOF test, UCLs, outliers) for variables with nondetect
values, one should choose the nondetect option, "With NDs" from the various available
menu options such as Outliers, Background Statistics, UCLs, Goodness-of-Fit test, Q-Q
plot, Box Plot.
• It should be noted that "summary" statistics for a data set with nondetect observations
represent (at least in ProUCL 4.0) simple summary statistics based upon the data set
without using nondetect observations. All other parametric and nonparametric statistics
and estimates of population mean, variance, percentiles (e.g., MLEs, KM, and ROS
estimates) for variables with nondetect observations are given in other menu options such
as background statistics and UCL. The simple "Summary Statistics/With NDs" option
only provides simple statistics (e.g., % NDs, max ND, Min ND, and Mean of detected
values) based upon detected values. These statistics (e.g., sd of log -transformed detected
values) may help a user to determine the degree of skewness (e.g., mild, moderate, high)
of the data set consisting of detected values. These statistics may also help the user to
choose the most appropriate method (e.g., KM (BCA) UCL or KM (t) UCL) to compute
UCLs, UPLs, and other limits.
• As mentioned before, various other statistics and estimates of interest (e.g., mean, sd,
UCLs, UTLs, MLEs, and KM estimates) for data sets with nondetect observations are
computed in other the menu options (UCLs, Outliers, Background Statistics, GOF tests)
available in ProUCL 4.0.
39
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2.11 Warning Messages and Recommendations for Datasets with
Insufficient Amount of Data
• ProUCL 4.0 provides some warning messages and recommendations for datasets with
insufficient amount of data to calculate meaningful estimates and statistics of interest. For
an example, it is not desirable to compute an estimate the EPC term based upon a data set
of size less than 5, especially when nondetects may be present in the data set. In such
cases, it is suggested to use site specific values (perhaps determined by the Project Team)
to estimate environmental parameters (e.g., EPC term, not-to-exceed background value)
of interest.
• Some examples of datasets with insufficient amount of data include datasets with less
than 4 distinct observations, datasets with only one detected observation, and datasets
consisting of all nondetects.
• Some of the warning messages given by ProUCL 4.0 are shown below.
G eneral B ackground S tattslics f or D ala S ets wih NorrOetects
U ser S elected 0 ptions
From File C:\Doeuments and Settings\narmbya\Desktop\ProUCL-Test\EVlLDATA3.wst
Full Precision OFF
Confidence Coefficient S5%
Coverage 90%
Different or Future K Values 1
Number of Bootstrap Operations 2000
Var
General Statistics
Total Number of Observations 15
Raw Statistics
Minimum 1
Maximum 3
Second Largest 3
First Quartile 1
Median 2
Third Quartile 3
Mean 2
SD 0.845
Coefficient of Variation 0.423
Skewness 0
Warning: T here are ernl^ 3 D is&inct Values in this
-------
,j -^ ene(a| u CL S tat jst jcs f OI p ata s ets with N onfietecte
User Selected Option:
From File CADocurnents and Seltings\narmbya\DesklopSProUCL-Test\EVILDATA3.wst
Full Precision OFF
Confidence Coefficient 35%
Number of Bootstrap Operations 2000
Var_x3
Number of Valid Data
Number of Distinct Detected Data
General Statistics
15
Number of Detected Data 1
Number of Non-Detect Data 14
Percent Non-Detects 93.33K
Warning: Onif orse distinct data value ¥ia$ detected! ProOCL for arej? othef software) shoyld not be used on such a date se$
It is syggiested to use alternative site specific values determined by the Project Team to estimate environmental paraineters fe.@., EPC, BTV3-
The data set fof variable Var_K3 was nol processed!
! General Background Statistics for Data Sets with Non-Detects
User Selected Options
From File C: SD ocuments and S ettings\narmbya\D esktopVPtoU CL-T est\EVI LD AT A3. wst
Full Precision OFF
Confidence Coefficient 95%
Coverage 30%
Different or Future K Values 1
Number of Bootstrap Operations 2000
Vaf_x10
General Statistics
Number of Valid Data 15
NumberofDistinctDetectedData 0
NurnberofDetectedData 0
NumbetofNon-DetectData 15
Warning: AH observations are Non-Detects (MDs], therefore all statistics ar®d esteates shoukl also bs Wsi
S pacifically, sample mean, U CLs, U PLs, and otherstatisticsaiealsoNDs lying below the largest detedkn EmK
T he Proiecl T earn may decide lo use alternative site specific values to estimate environmental parameters (e.g., EPC, BTVT-
T he data set for variable Var_s1Swasnol processed!
41
-------
Number of Valid Data
Number of Distinct Detected Data
Raw Statistics
Minimum Detected
Maximum Detected
Mean of Detected
3D of Detected
Minimum Non-Detect
Maximum Non-Detect
General Statistics
15
2
1
4
2.5
1643
2
5
Note: Data have multiple OLs • Use of KM Method is recommended
For all methods (except KM, DL/2, and ROS Methods),
Observations < Largest ND are treated asNDs
NumberofDetectedData 6
NumberofNon-DetectData 9
Percent Non-D elects 60.00%
Log-transformed Statistics
Minimum Detected 0
Maximum Detected 1.386
Mean of Detected 0.693
SD of Detected 0.759
M inimum N on-D elect 0.693
M aximum N on-D elect 1.609
Number treated as Non-Detect 15
Number treated as Detected 0
SingleDLNon-DetectPercentage 100.00%
Warning: Data set lias orii^ 2 District Detested Vakses.
This niaf not be adequate enough lo compute meaningly) and reliable lest statistics ami estimates.
T he Project T earn may decide io use alternative site specific values to estimate environmental parameters fag., EPC, BTV}
U nless D ata Q uality 0 bjectives (D Q 0 s) have been met, it is suggested to collect additional observations.
The number of detected data may not be adequate enough to perfoim GOF tests, bootstrap, and ROS methods.
Those methods will letuin a WA' value onyour output display
It is necessary to have 4 or more Distinct Values for bootstrap methods.
It is recommended to have 10 to 15 or more observations for accurate and meaningful results and eslroates
2.12 Handling Missing Values
• ProUCL 4.0 can handle missing values within a data set.
• All blanks, alphanumeric strings (except for group variables), or the specific large
number value Ie31 are considered as missing values.
• A group variable (representing two or more groups, populations, AOCs, MWs) can have
alphanumeric values (e.g., MW1, MW2,...).
• ProUCL 4.0 ignores all missing values in all mathematical operations it performs.
Missing values are therefore not treated as being part of a data set.
• Number of Valid Samples or Number of Valid Observations represents the Total Number
of Observations less the Number of Missing Values. If there are missing values, then
number of valid samples = total number of observations.
Valid Samples = Total Number of Observations - Missing Values.
42
-------
• It is important to note, however, that if a missing value not meant (e.g., a blank, or Ie31)
to represent a group category is present in a "Group" variable, ProUCL 4.0 will treat that
blank value (or Ie31 value) as a new group. Any variable that corresponds to this missing
value will be treated as part of a new group and not with any existing groups. It is
therefore very important to check the consistency and validity of all data sets before
performing complex mathematical operations.
• ProUCL 4.0 prints out the number of missing values (if any) associated with each
variable in the data sheet. This information is provided in several output sheets (e.g.,
summary statistics, background statistics, UCLs) generated by ProUCL 4.0.
For further clarification of labeling of missing values, the following example illustrates the terminology
used for the number of valid samples, number of unique and distinct samples on the various output sheets
generated by ProUCL 4.0.
Example: The following example illustrates the notion of Valid Samples, Unique or Distinct Samples,
and Missing Values. The data set also has nondetect values. ProUCL 4.0 computes these numbers and
prints them on the UCLs and background statistics output.
X
2
4
2.3
1.2
w34
l.OE+031
anm
34
23
0.5
0.5
2.3
2.3
2.3
34
73
D x
1
1
1
0
0
0
0
0
1
1
0
0
Valid Samples: Represents the total number of observations (censored and uncensored) excluding the
missing values. If a data set has no missing value, then the total number of data points equals number of
valid samples.
Missing Values: All values not representing a real numerical number are considered as missing values.
Specifically, all alphanumeric values including blanks are considered as missing values. Also unrealistic
big numbers such as 1 .Oe31 are also considered as missing values and are considered as not valid
observations.
Unique or Distinct Samples: The number of unique samples or number of distinct samples represents all
unique (or distinct) detected values. Number of unique or distinct values is computed for detected values
43
-------
only. This number is especially useful when using bootstrap methods. As well known, it is not desirable
and advisable to use bootstrap methods, when the number of unique samples is less than 4-5.
2.13 User Graphic Display Modification
Advanced users are provided two sets of tools to modify graphics displays. A graphics tool bar is
available above the graphics display and the user can right-click on the desired object within the graphics
display, and a drop-down menu will appear. The user can select an item from the drop-down menu list by
clicking on that item. This will allow the user to make desired modifications as available for the selected
menu item. An illustration is given as follows.
2.13.1 Graphics Tool Bar
BUHisto.Group.gst
Histograms for Arsenic, NROS_Arsenic
Arsenic
NROS Arsenic
The user can change fonts, font sizes, vertical and horizontal axis's, select new colors for the various
features and text. All these actions are generally used to modify the appearance of the graphic display.
The user is cautioned that these tools can be unforgiving and may put the user in a situation where the
user cannot go back to the original display. Users are on their own in exploring the robustness of these
tools. Therefore, less experienced users may want to stay away from using these drop-down menu graphic
tools.
2.13.2 Drop-Down Menu Graphics Tools
These tools allow the user to move the mouse to a specific graphic item like an Axis Label or a display
feature. The user then right-clicks their mouse and a drop-down menu will appear. This menu presents the
user with available options for that particular control or graphic object. There is less of chance of making
an unrecoverable error but that risk is always present. As a cautionary note, the user can always delete the
graphics window and redraw the graphical displays by repeating their operations from the datasheet and
menu options available in ProUCL 4.0. An example of a drop-down menu obtained by right-clicking the
mouse on the background area of the graphics display is given as follows.
44
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Histograms for Arsenic, NROS Arsenic
Gallery
Color
Properties...
* Statistical Studies
NROS Arsenic 1 1 1
45
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Chapter 3
Select Variables Screen
3.1 Select Variables Screen
• Variables need to be selected to perform statistical analyses.
• When the user clicks on a drop-down menu for a statistical procedure, the following
window will appear.
Select Variables
Variables
Selected
Name
ID
Count
Mercury
Vanadium
Zinc
Group
30
20
20
20
Name
ID
Count
Group by variable:
OK
Cancel
The Options button is available in certain menus. The use of this option leads to a
different pop-up window.
Multiple variables can be processed simultaneously in ProUCL 4.0. Note that this option
was not available in ProUCL 3.0. ProUCL 4.0 can generate graphs, compute UCLs, and
background statistics simultaneously for all selected variables.
Moreover, if the user wants to perform statistical analysis on a variable (e.g.,
contaminant) by a Group variable, click the arrow below the Group by variable to get a
drop-down list of available variables to select an appropriate group variable. For an
example, a group variable (e.g., Site Area) can have alphanumeric values such as AOC1,
46
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AOC2, AOC3, and Background. Thus in this example, the group variable name, Site
Area, takes four values such as AOC1, AOC2, AOC3, and Background.
• The Group variable is particularly useful when data from two or more samples need to be
compared.
• Any variable can be a group variable. However, for meaningful results, only a variable,
that really represents a group variable (categories) should be selected as a group variable.
• The number of observations in the group variable and the number observations in the
selected variables (to be used in a statistical procedure) should be the same. In the
example below, the variable "Mercury" is not selected because the number of
observations for Mercury is 30; in other words mercury values have not been grouped.
The group variable and each of the selected variables have 20 data values.
Select Variables
Variables
Name
ID
Count
Mercury
Group
30
20
Selected
Name
ID
Arsenic
Vanadium
Zinc
Group by variable:
Arsenic ( Count = 20 )
vlercury ( Count = 30 )
Vanadium ( Count = 20
Zinc (Count = 20
Count
20
20
20
• It is recommended not to assign any missing value such as a "Blank" for the group
variable. If there is a missing value (represented by blanks, strings or 1E31) for a group
variable, ProUCL 4.0 will treat those missing values as a new group. As such, data values
corresponding to the missing Group will be assigned to a new group.
Caution: Once again, care should be taken to avoid misrepresentation and improper use of group
variables. It is recommended not to assign any missing value for the group variable.
47
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More on Group Option
• The Group Option provides a powerful tool to perform various 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 an example, the same data set
may consist of samples from the various groups (populations). The graphical displays
(e.g., box plots, Q-Q plots) and statistics of interest can be computed separately for each
group by using this option.
• In order to use this option, at least one group variable (with alphanumeric values) should
be included in the data set. The various values of the group variable represent different
group categories that may be present in the data set. This can be seen in the example.wst
data set used earlier in Chapter 2.
• At this time, the number of values (representing group membership) in a Group variable
should equal to the number of values in the variable (e.g., Arsenic) of interest that needs
to be partitioned into various groups (e.g., monitoring wells).
• Typically, the data for the various groups (categorized by the group variable) represent
data from the various site areas (e.g., background, AOC1, AOC2, ...), or from monitoring
wells (e.g., MW1, MW2, ...).
3.1.1 Graphs by Groups
• Individual or multiple graphs (Q-Q plots, box plots, and histograms) can be displayed on
a graph by selecting the "Graphs by Groups" option.
• Individual graph for each group (specified by the selected group variable) is produced by
selecting the "Individual Graph" option.
• Multiple graphs (e.g., side-by-side box plots, multiple Q-Q plots on the same graph) are
produced by selecting the "Group Graph" option for a variable categorized by a group
variable. Using this "Group Graph" option, multiple graphs can be displayed for all sub-
groups included in the Group variable. This option is useful when data to be compared
are given in the same column and are classified by the group variable.
• Multiple graphs (e.g., side-by-side box plots, multiple Q-Q plots) for selected variables
are produced by selecting the "Group Graph" option. Using the "Group Graph" option,
multiple graphs can be displayed for all selected variables. This option is useful when
data (e.g., lead) to be compared are given in different columns, perhaps representing
different populations.
Note: It should be noted that it is the users' responsibility to provide adequate amount of detected data to
perform the group operations. For an example, if the user desires to produce a graphical Q-Q plot (using
only detected data) with regression lines displayed, then there should be at least two detected points (to
compute slope, intercept, sd) in the data set. Similarly if the graphs are desired for each of the group
specified by the group ID variable, there should be at least two detected observations in each group
specified by the group variable. ProUCL 4.0 generates a warning message (in orange color) in the lower
panel of the ProUCL 4.0 screen. Specifically, the user should make sure that a variable with nondetects
48
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and categorized by a group variable should have enough detected data in each group to perform the
various methods (e.g., GOF tests, Q-Qplots with regression lines) as incorporated in ProUCL 4.0.
As mentioned before, 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 quite common in many environmental applications.
The usefulness of the group option is illustrated throughout the User Guide using various methods as
incorporated in ProUCL 4.0.
49
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Chapter 4
Summary Statistics
This option is used to compute general summary statistics for all variables in the data file. Summary
statistics can be generated for full data sets without nondetect observations, and for data sets with
nondetect observations. Two Menu options: Full and With NDs are available.
• Full - This option computes summary statistics for all variables in a data set without any
nondetect values.
• With NDs - This option computes simple summary statistics for all variables in a data set
that have nondetect (ND) observations. For variables with ND observations, only simple
summary statistics are computed based upon detected observations only.
o For this option, no attempt is made to compute estimates of population parameters
(e.g., mean, sd, SE) using parametric (e.g., MLE) or nonparametric (e.g., KM,
bootstrap) estimation methods. Those statistics are generated in other estimation
modules (e.g., Background and UCL) of ProUCL 4.0.
Each menu option (Full and With NDs) has two sub-menu options:
• Raw Statistics
• Log-Transformed
• In ProUCL, log-transformation means natural logarithm (In)
• When computing summary statistics for raw data, a message will be displayed for each
variable that contains non-numeric values.
• The Summary Statistics option computes log-transformed data only if all of the data
values for the selected variable(s) are positive real numbers. A message will be displayed
if non-numeric characters, zero, or negative values are found in the column
corresponding to the selected variable.
4.1 Summary Statistics with Full Data Sets
1. Click Summary Statistics >• Full
P? Prp,UCL4..q r [Worksheet.wstj
File Edit Configure
Summary Statistics
RQS Est, NDs Graphs Outiier Tests Goodness-of-Fit Hypothesis Testing Background UCL Window Help
Ray; Statistics
2. Select either Log-Transformed or Raw Statistics option.
50
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3. The Select Variables Screen (see Chapter 3) will appear.
• Select one or more variables from the Select Variables screen.
• If summary statistics are to be computed by a Group variable, then select a group variable
by clicking the arrow below the Group by variable button. This will result in drop-down
list of available variables, and select the proper group variable.
• Click on the OK button to continue or on the Cancel button to cancel the Summary
Statistic option.
Raw Statistics
11 FuLl_Raw_Stats.ost
From File: DAexample.wst
Variable NumObs
Arsenic (subsurface)
Arsenic (surface)
Vanadium (subsurface)
Vanadium (surface)
Zinc (subsurface)
Zinc (surface)
MJ
10
10
10
10
10
10
Minimum
4.5
4.3
10.3
13.8
66.3
80.4
S ummary S tatistics for R aw Full D ata S ets
Maximum
6.8
9.2
24.3
30.8
185
266
Mean
5.56
6.185
16.4
19.53
114.4
116.7
Median
5.45
5.725
16.85
17.23
95.45
90
Variance
0.449
2.5
13.86
33.72
2271
3680
SD MAD/0.67; Skewness
0.67
1.581
3.723
5.807
47.66
60.66
0.667
1.075
1.26
2.669
35.21
13.86
0.344
0.969
0.495
1.46
0.745
2.111
Kurtosis
-0.047
0.295
2.092
0.896
-1.288
4.065
cv
0.121
0.256
0.227
0.297
0.416
0.52
>J
Log-Transformed Statistics
EH Full_Log_Stats.ost
From File: D: \exarnple. wst
Variable NumObs
Arsenic [subsurface)
Arsenic (surface)
Vanadium (subsurface)
Vanadium (surface)
Zinc [subsurface)
Zinc (surface)
NJJ
10
10
10
10
10
10
!
*.
rd
Summary Statistics for Log-Transformed Full Data Sets
Minimum Maximum
1.504
1.459
2.332
2.625
4.194
4.387
1.917
2.219
3.19
3.428
5.22
5.583
Mean
1.709
1.795
2.774
2.938
4.666
4.673
Median Variance
1.696
1.745
2.824
2.846
4.558
4.5
0.0144
0.0591
0.0538
0.0699
0.159
0.162
SD MAD/O.G7E Skewness
0.12
0.243
0.232
0.264
0.399
0.403
0.123
0.176
0.0735
0.157
0.425
0.163
0.0692
0.527
-0.394
1.194
0.452
1.734
Kurtosis
-0.14
-0.153
1.351
0.504
-1.394
2.184
CV
0.0702
0.135
0.0836
0.09
0.0855
0.0862
_LJ
4. The resulting Summary Statistics screen as shown above can be saved as an Excel file. Click
Save from the file menu.
51
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5. On the output screen, the following summary statistics are displayed for each selected variable in
the data file.
NumObs = Number of Observations
Minimum = Minimum value
Maximum = Maximum value
Mean = Sample average value
Median = Median value
Variance = Classical sample variance
SD = Classical sample standard deviation
MAD = Median absolute deviation
MAD/0.675 = Robust estimate of variability, population standard deviation, a
Skewness = Skewness statistic
Kurtosis = Kurtosis statistic
CV = Coefficient of Variation
The details of these summary statistics are described in an EPA (2006) guidance document and also in the
ProUCL Technical Guide (A. Singh and A.K. Singh (EPA, 2007).
4.2 Summary Statistics with NDs
1. Click Summary Statistics ^ With NDs
Navigation Panel
ROSEst. NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing Background UCL Window Help
Rav Statistics
Name
•_y Worksheet wst
2. Select either Log-Transformed or Raw Statistics option.
3. The Select Variables Screen (Chapter 3) will appear.
• Select variable(s) from the list of variables.
• Only those variables that have nondetect values will be shown.
• If summary statistics are to be computed by a Group variable, then select a group variable
by clicking the arrow below the Group by variable button. This will result in a drop-
down list of available variables; then select the proper group variable.
• Click on the OK button to continue or on the Cancel button to cancel the summary
statistics operations.
Note: It should be noted that in ProUCL 4.0, "Summary Statistics "for a data set with nondetect
observations represent simple summary statistics based upon the data set without using nondetect
observations. All other parametric and nonparametric statistics and estimates of population mean,
52
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variance, percentiles (e.g., MLEs, KM, andROS estimates) for variables with nondetect observations are
given in other estimation menu options such as background statistics and UCL. The simple "Summary
Statistics/With NDs " option only provides simple statistics (e.g., % NDs, max ND, Min ND, Mean of
detected values) based upon detected values. These statistics (e.g., sd of log-transformed detected values)
may help a user to determine the degree ofskewness (e.g., mild, moderate, high) of the data set consisting
of detected values. These statistics may also help the user to choose the most appropriate method (e.g.,
KM(BCA) UCL or KM ft) UCL) to compute UCLs, UPLs, and other limits.
Raw Statistics - Data Set with NDs
1WND Raw Stats.ost
From File: D:\example,wst
Summary Statistics for Raw Data Sets with NDs
Variable NumObs NumNDs 2 NDs
Arsenic 17 3 15.00%
Mercury 25 5 16.67%
Raw Statistics using Detected Observations
Maximum Minimum Mean Median SD MAD/0.67E Skewness CV
5 9.2 6.126 5.8 1.15 0.593 1.783 0.188
0.04 0.99 0.312 0.18 0.315 0.163 1.202 1.01
liU
Log-Transformed Statistics - Data Set with NDs
HI WND_Log_Stats.ost
From File: D:Vexamplawst
Summary Statistics for Log-transformed Data Sets with NDs
Statistics using Detected Log-transformed Observations
Variable NumObs NumNDs % NDs Maximum Minimum Mean Median SD MAD/Q.67J Skewness CV
Arsenic 17 3 15.00% 1.609 2.219 1.798 1.758 0.168 1.458 0.0935 0.106
Mercury 25 5 16.67% -3.219 -0.0101 -1.66 -1.715 1.032 0.238 -0.622 1.4
MJ
The Summary Statistics screen shown above can be saved as an Excel file. Click the save
from the file menu.
On the results screen, the following summary statistics are displayed for each selected
variable from the data file.
53
-------
Num Obs = Number of Observations
NumNDs = Number of Nondetects
% NDs = Percentage of Nondetect observations
Minimum = Minimum value
Maximum = Maximum value
Mean = Sample average value
Median = Median value
SD = Classical sample standard deviation
MAD = Median Absolute Deviation
MAD/0.675 = Robust estimate of variability (standard deviation)
CV = Coefficient of Variation
54
-------
55
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Chapter 5
Estimating Nondetects Using ROS Methods
Regression on order statistics (ROS) can be used to extrapolate nondetect observations using a normal,
lognormal, or gamma model. ProUCL 4.0 has three ROS estimation methods that can be used to estimate
or extrapolate nondetect observations. The use of this option generates additional columns consisting of
all extrapolated nondetects and detected observations. These columns are appended to the existing open
spreadsheet. The user should save the updated file if they want to use the generated data for their other
application(s).
1.
Click ROS Est. NDs ^ Gamma ROS
P3 ProUCL 4.0 .- [C:\Narain\ProUCL-pata\Data\Oahu.wstl
M*]JBJMSJJ3J Normal ROS j
0 1 Lognormal ROS [ 3 A 5 67 8
Arsenic - L> Arsenic
1 ! 1! °
^^m
Window Help
9 10
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
Select Variables
Variables
Selected
Name
Count
Name
Count
Arsenic
Group by Variable
! 3
OK
Cancel
• Click on the OK button to continue or on the Cancel button to cancel the option.
56
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Output Screen for ROS Est. NDs (Gamma) Option
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0
.Arsenic
I 1!
1
I 1-7
j 1
| 1
I 2
\ 3.2
; 2
2
2.8
I 2
j 2
I 2
i 2
• 2
0.7
0.9
i 0.5
: 0.5
j 0.9
i 0.5
i 0.7
; 0.6
1.5
1
D_Arsenic
' 0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
1
1
1
1
0
1
1
1
1
2
GROS_Arsertic
1.031 51 31 074651 303
1.1993230BOSS37iO
1.7
1.33761048919618
1.46028213343334
0.957885781528449
3.2
1.08600369175045
1.19098252772635
2.S
1.28363461914326
1.368290930400230
1.44721892249559
1.52178645574469
1.59289577776005
0.7
0.9
0.5
0.5
1.226870494768360
0.5
0.7
O.G
1.5
Note: Columns with similar naming convention are generated for each selected variable and distribution
using this ROS option.
57
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Chapter 6
Graphical Methods (Graph)
Three commonly used graphical displays are available under the Graphs option:
o Box Plot
o Histogram
o Multi-QQ
• The box plots and multiple Q-Q plots can be used for Full data sets without nondetects
and also for data sets with nondetect values.
• Three options are available to draw Q-Q plots with nondetect (ND) observations.
Specifically, Q-Q plots are displayed only for detected values, or with NDs replaced by 1A
DL values, or with NDs replaced by the respective detection limits. The statistics
displayed on a Q-Q plot (mean, sd, slope, intercept) are computed according to the
method used. The NDs are displayed using the smaller font and in red color.
• ProUCL 4.0 can display box plots for data sets with NDs. This kind of graph may not be
very useful when many NDs may be present in a data set.
o A few choices are available to construct box plots for data sets with NDs. For an
example, some texts (e.g., Helsel) display box plots only for the detected
observations. Specifically, all nondetects below the largest detection limit (DL), and
portion of the box plot (if any) below the largest DL are not shown on the box plot. A
horizontal line is displayed at the largest detection limit level.
o ProUCL 4.0 constructs a box plot using all detected and nondetect (using DL values)
values. ProUCL 4.0 shows the full box plot. However, a horizontal line is displayed
at the largest detection limit.
• When multiple variables are selected, one can choose to: 1) produce a multiple graphs on
the same display by choosing the Graph by group variable option, or 2) produce separate
graphs for each selected variable.
• The Graph by group variable option produces side-by-side box plots, or multiple Q-Q
plots, or histograms for the groups of the selected variable representing samples obtained
from multiple populations (groups). These multiple graphs are particularly useful to
perform two (background versus site) or more sample visual comparisons.
o Additionally, Box Plot has an optional feature, which can be used to draw lines at
statistical (e.g., upper limits of background data set) limits computed from one
population on the box plot obtained using the data from another population (a site
area of concern). This type of box plot represents a useful visual comparison of site
data with background threshold values (background upper limits).
o Up to four (4) statistics can be added (drawn) on a box plot. If the user inputs a value
in the value column, the check box in that row will get activated. For example, the
58
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user may want to draw horizontal lines at 80th percentile, 90th percentile, 95th
percentile, or a 95% UPL) on a box plot.
6.1 Box Plot
1. Click Graphs ^ Box Plot
P? ProUCL4.0 - pWoricSheet.wst]
ay File Edit Configure Summary Statistics ROS Est. NDs
olc> '
Outlier Tests Goodness -of-Fit Hypothesis Testing Background LJCL VYindor.' help
Navigation Panel
0
Histogram
Mu!t-QQ
Full ('.v/oNDsj
\Yith fjDs
Name
^WorkSheet.wst 1 '
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select an appropriate variable representing a
group variable.
• When the Group by variable button is clicked, the following window is shown.
Graph by Groups •
f* Individual Graphs
Label
f" Group Graphs
Value
1. r r
2. r r
3. r f
4. r r
r Graphical Display Options'
| *•* Color Gradient
| r For Export (BW Printers)
Cancel
59
-------
• The default option for Graph by Groups is Individual Graphs. This option will
produce one graph for each selected variable. If you want to put all the selected variables
into a single graph, then select the Group Graphs option. This Group Graphs option is
used when multiple graphs categorized by a Group variable have to be produced on the
same graph.
• The default option for Graphical Display Options is Color Gradient. If you want to use
and import graphs in black and white into a document or report, then check the radio
button next to For Export (BW Printers).
• Click on the OK button to continue or on the Cancel button to cancel the Box Plot (or
other selected graphical) option.
Box Plot Output Screen (Single Graph)
Selected options: Label (Background UPL), Value (103.85), Individual Graphs, and Color Gradient.
Box Plol for Zinc
60
-------
Box Plot Output Screen (Group Graphs)
Selected options: Group Graphs and Color Gradient.
Box Plots for X (1). X (2). X (3)
6.2 Histogram
1. Click Graphs ^- Histogram
3? ProUCL 4.0 - [Worksheet. wst]
•g File Edit Configure Summary Statistics ROS Est. NDs [ J Outlier Tests Goodness -of-Fit Hypothesis Testing Background UCL Window Help
E]b&jja]_H]_ml_n]
Navigation Panel
Name |j_
©Worksheet. wst 1
0
1
Box Plot ^
Multi-QQ K
I I
3456789
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When that option button is clicked, the following window will be shown.
61
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B! Graphs (Histograms)
Graph by Groups
f Individual Graphs
'•' Group Graphs
Graphical Display Options
(f Color Gradient
C For Export (BW Printers)
Cancel
The default selection for Graph by Groups is Individual Graphs. This option produces
a histogram (or other graphs) separately for each selected variable. If multiple graphs or
graphs by groups are desired, then check the radio button next to Group Graphs.
The default option for Graphical Display Options is Color Gradient. If you want to use
and import graphs in black and white into a document or report, then check the radio
button next to For Export (BW Printers).
Click on the OK button to continue or on the Cancel button to cancel the Histogram (or
other selected graphical) option.
Histogram Output Screen
Selected options: Group Graphs and Color Gradient.
Histograms for Arsenic (subsurface). Arsenic (surface)
62
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6.3 Multi-QQ
6.3.1 Multi-QQ (Full)
1. Click Graphs ^ Multi-QQ
2. Multi-QQ can be obtained for data sets with (With NDs) and without NDs (Full).
• When that option button is clicked, the following window will be shown.
P? ProUCL 4,0 r |WorJcSheet.vwt]
File Edit Configure Summary Statistics ROS Est. NDs
oi aisle FC|
Navigation Panel
Outlier Tests Goodness-of-Fit Hypothesis Testing Background UCL Window Help
Box Plot > I
Histogram f , IL
Name
£> Worksheet wst
3 . Select either Full or With NDs.
4. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When the Group by variable option button is clicked, the following window will appear.
9 Multi QQ Options
Display Regression Lines
(• Do Not Display
f** Display Regression Lines
Graphical Display Options
f** Color Gradient
r For Export (BW Printers)
OK
Cancel
63
-------
• The default option for Display Regression Lines is Do Not Display. If you want to see
regression lines on graphs, then check the radio button next to Display Regression
Lines.
• The default option for Graphical Display Options is Color Gradient. If you want to see
the graphs in black and white, then check the radio button next to For Export (BW
Printers).
• Click on the OK button to continue or on the Cancel button to cancel the selected Multi-
QQ option.
Note: For Multi-QQ plot option, for both "Full" as well as for data sets "With NDs, " the values along
the horizontal axis represent quantiles of a standardized normal distribution (Normal distribution with
mean 0 and standard deviation 1). Quantiles for other distributions (e.g., Gamma distribution) are used
when using Goodness-of-Fit (GOF) test option.
Output Screen for Multi-QQ (Full)
Selected options: Group Graph, Do Not Display Regression Lines, and Color Gradient.
Multiple Q-Q Plots
for Arsenic (subsurface), Arsenic (surface)
9.90
9.50
930
900
0
a. 40
8.10
™ 7.SO
O
•fi 7.50
I™
.Q 690
0 a
•0 6.60
| 630
O 6.00
5.70
540 fl
5.10 fl
4.30
450
420
390
-1 0 1
Theoretical Quantiles (Standard Normal)
Arsenic (subsurface)
N = 10
Mean = 5 5600
Sd = 0.6703
Slope - 0.7023
Intercept - 5.5SOO
Correlation. R = 0.9351
Arsenic (surface)
Mean = 6. 1850
Sd = 1 .581 1
Slope - 1 .5930
Intercept = 6.1 350
Correlation.!? = 09474
• Arsenic (subsurface) Arsenic (surface)
6.3.2 Multi-QQ (with NDs)
1. Click Graphs >• Multi-QQ
ProUCL 4.0 - [Worksheet. wstj
•P File Edit Configure Summary Statistics ROS Est. NDs [ S Outlier Tests Goodness-of-Fit Hypothesis Testing Background UCL Window Help
Box Plot
Histogram
e| H| ml
64
-------
Select With NDs option by clicking on it.
The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When the Group by variable option button is clicked, the following screen appears.
HMMtttftafi.llIio.ns.
Display Non-Delects
(" Do not Display Non-D elects
<•" Display Non-D elect Values
<~ D isplay 112 N on-D elect Values
Display Regression Lines
f* Do Not Display
f" Display Regression Lines
Graphical Display Options
(* Color Gradient
f" For Export (BW Printers)
OK
Cancel
The default option for Display Regression Lines is Do Not Display. If you want to see
regression lines, then check the radio button next to Display Regression Lines.
The default option for Display Nondetects is Display Nondetect Values.
o Do not Display Nondetects: Selection of this option excludes the NDs detects and
plots only detected values on the associated Q-Q plot. The statistics are computed
using only detected data.
o Display Nondetect Values: Selection of this option treats detection limits as detected
values and plots those detection limits and detected values on the Q-Q plot. The
statistics are computed accordingly.
65
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o Display Vz Nondetect Values: Selection of this option replaces the detection limits
with their half values, and plots half detection limits and detected values on the Q-Q
plot. The statistics are computed accordingly.
• The default option for Graphical Display Options is Color Gradient. If you want to see
the graphs in black and white, then check the radio button next to For Export (BW
Printers).
• Click on the OK button to continue or on the Cancel button to cancel the Multi-QQ
option.
Output Screen for Multi-QQ (without NDs)
Options: Do Not Display Regression Lines, Do not display Nondetects, and Color Gradient.
Q-Q Plot with without NDs for Mercury
Theoretic*) Quantiles (Standard Normal]
66
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Output Screen for Multi-QQ (with NDs)
Options: Do not Display Regression Line, Display Nondetect Values, and Color Gradient.
Q-Q Plot with NDs for Mercury
1.00 H
a
0.90 J
0.30 H
0.70
H
g 0.60 '
1
£ 0-50
a
o
•o
-S
6
0.30 M
0.20
0.10 a B
i a H J
j H
0.00
-2-101 2
Theoretical Quantiles (Standard Normal)
• Mercury
Mercury
Mean - 0.3055
3d -0.2937
Slope -0.271 7
Intercept = 0 3055
Correlation, R = 0.9001
Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red.
The legend size is made smaller for BWprinters.
Output Screen for Multi-QQ (with NDs)
Selected options: Do not Display Regression Lines, Display Vz, Nondetect Values, and Color Gradient.
a -mm
Theoretical Quantiles (Standard Normal)
N-30
Mean = 0.2829
Sd - 0.2962
Slope = 0.2S56
Intercept - 0.2329
CuirelFilion.R-08721
Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red.
67
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Chapter 7
Simple Classical Outlier Tests
Outliers are inevitable in data sets originating from environmental applications. There are many graphical
(Q-Q plots, Box plots), classical (Dixon, Rosner, Welch), and robust methods (biweight, Huber, PROP)
available to identify outliers. It is well known that the classical outlier tests (e.g., Dixon test, Rosen test,
EPA, 2006) suffer from masking (e.g., extreme outliers may mask intermediate outliers) effects. The use
of robust outlier identification procedures is recommended to identify multiple outliers, especially when
dealing with multivariate (having multiple contaminants) data sets. However, those preferred and more
effective robust outlier identification methods are beyond the scope of ProUCL 4.0. Several robust outlier
identification methods (e.g., based upon biweight, Huber, and PROP influence functions) are available in
the Scout software package (EPA, 1999).
The two simple classical outlier tests (often cited in environmental literature): Dixon and Rosner tests are
available in ProUCL 4.0. These tests can be used on data sets with and without nondetect observations.
These tests also require the assumption of normality of the data set without the outliers. It should be noted
that in environmental applications, one of the objectives is to identify high outlying observations that
might be present in the right tail of a data distribution as those observations often represent contaminated
locations of a polluted site. Therefore, for data sets with nondetects, two options are available in ProUCL
4.0 to deal with data sets with outliers. These options are: 1) exclude nondetects and 2) replace NDs by
DL/2 values. These options are used only to identify outliers and not to compute any estimates and limits
used in decision-making process.
It is suggested that these two classical outlier identification procedures be supplemented with graphical
displays such as Q-Q plots, Box and Whisker plot (called box plot), and IQR (= upper quartile, Q3 -
lower quartile, Ql). These graphical displays are available in ProUCL 4.0. Box plots with whiskers are
often used to identify outliers (e.g., EPA, 2006). Typically, a box plot gives a good indication of extreme
(outliers) observations that may present in a data set. The statistics (lower quartile, median, upper quartile,
and IQR) used in the construction of a box plot do not get distorted by outliers. On a box plot,
observations beyond the two whiskers may be considered as candidates for potential outliers.
Q-Q plots are also quite useful to identify outliers in a data set. For an example, on a normal Q-Q plot,
observations that are well separated from the bulk (central part) of the data typically represent potential
outliers needing further investigation. Also, significant and obvious jumps and breaks in a Q-Q plot (for
any distribution) are indications of the presence of more than one population. Data sets exhibiting such
behavior of Q-Q plots should be partitioned out in component sub-populations before estimating an EPC
Term or a background threshold value (BTV). It is strongly recommended that both graphical and formal
outlier identification tests should be used on the same data set to identify potential outliers that may be
present in a data set under study. More details about the construction of graphical displays and outliers
test can be found in the Technical Guide for ProUCL 4.0.
Dixon's Test (Extreme Value Test)
• This test is used to identify statistical outliers when the sample size is less than or equal
to 25.
68
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• This test can be used to identify outliers or extreme values in both the left tail (Case 1)
and the right tail (Case 2) of a data distribution. In environmental data sets, extremes
found in the right tail may represent potentially contaminated site areas needing further
investigation or remediation. The extremes in the left tail may represent ND values.
• This test assumes that the data without the suspected outlier are normally distributed;
therefore, it is necessary to perform a test for normality on the data without the suspected
outlier before applying this test.
• This test may suffer from masking in the presence of multiple outliers. This means that if
more than one outlier is suspected, this test may fail to identify all of the outliers.
Therefore, if you decide to use the Dixon's test for multiple outliers, apply the test to the
least extreme value first. Alternatively, use more effective robust outlier identification
procedures. Those outlier identification procedures will be available in Scout (EPA,
1999) software.
Rosner's Test
• This test can be used to identify and detect up to 10 outliers in data sets of sizes 25 and
higher.
• This test also assumes that the data are normally distributed; therefore, it is necessary to
perform a test for normality before applying this test.
Depending upon the selected variable(s) and the number of observations associated with them, either the
Dixon's test or the Rosner's test will be performed.
NOTE: Throughout this User Guide, and in ProUCL 4.0, it is assumed that the user is dealing with a
single population. If multiple populations are present in a data set, it is recommended to separate them
out using appropriate population partitioning methods and techniques. Appropriate tests and statistics
(e.g., goodness-of-fit tests, 95% UCLs, 95% UPLs) should be computed separately for each of the
identified populations. Also, outliers if any should be identified and thoroughly investigated. The presence
of outliers distorts all statistics including the all of the upper limits (UCLs, UPLs, upper percentiles). The
use of distorted statistics and limits may lead to incorrect conclusions having potential adverse effects on
the human health and the environment. Decisions about the disposition of outliers: inclusion or exclusion
in the data set to be used to compute the UCLs, UPLs, and other statistics should be made by all parties
involved. Statistical methods supplemented with graphical displays (e.g., Q-Q plot and box plots) can
only help identify statistical outliers that may be present in a data set. The project team and experts
familiar with the site should interpret and assign physical meaning and significance to those identified
outliers. The entire project team should be involved in taking decisions about the appropriate disposition
(include or not include) of outliers.
69
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7.1 Outlier Test for Full Data Set
1. Click Outlier Tests ^ Full *• Compute
P3 ProUCL 4.0 - [WorkSheet.wst]
•g File Edit Configure Summary Statistics ROSEst. NDs Graphs EElSljS3 Goodness-of-Fit Hypothesis Testing Background UCL Window Help
I
With NDs
Navigation Panel
[•lame
1
1
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• If at least one of the selected variables has 25 or more observations, then click the option
button for the Rosner Test.
Select Number Outliers (Rosner Test)
- ri x
Number of Outliers for Rosner Test [l
Applicable to Rosrrer's test (N >= 25) Only
Dixon's test (N < 25) only tests for one outlier.
OK
Cancel
7.2
• The default option for the number of suspected outliers is 1. In order to use this test, the
user has to obtain an initial guess about the number of outliers that may be present in the
data set. This can be done by using graphical displays such as a Q-Q plot. On this
graphical Q-Q plot, higher observations that are well separated from the rest of the data
may be considered as potential or suspected outliers.
• Click on the OK button to continue or on the Cancel button to cancel the Outlier Tests.
Outlier Test for Data Set with NDs
Typically, in environmental applications, one is interested in identifying high outliers (perhaps
representing contaminated parts of a site area, hot spots) that might be present in the right tail of the data
70
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distribution. Therefore, one may want to use the same outlier identification procedures (e.g., Dixon test,
Rosner test) that are used on full-uncensored data sets (without any NDs). The processes to perform such
tests using ProUCL 4.0 are described as follows.
1.
Click Outlier Tests ^ With NDs ^ Exclude NDs
P? ProUCL 4.0 - jWorlcSheet.wstJ
File Edit Configure Summary Statistics ROS Est. NDs Graphs gffljjMJ^^S Goodness -of-Fit H>pothesis Testing Background UCL Window Help
Full
Navigation Panel
Name
D 1
Arsenic D_Arsenic
DL/2 Estimates
7
W WorkSheet.wst f| '
Output Screen for Dixon's Outlier Test
Outlier Tests for Selected Variables
User Selected Options
From File WorkSbeetwst
Full Precision OFF
Test for Suspected Outliers with Dixon test 1
Test for Suspected Outliers for Rosner test 1
Dixon's O utlier Test for Aisenic
Nbrrberofdata = 11
10% critical value: 0,517
5% critical value: 0.576
1', critical value: 0.679
1. 3 2 is a Potential Oudier (UpperTail)
Test Statistic: 0.556
fcr 10'i significance level, 3.2 is an outlier.
For 5% sicjnifieance level, 3.2 is not an outlier.
For 1 % sigrtificance level, 3.2 is not an outlier.
2.0.5 is a Potential Outiier (LowerTail)
71
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Output Screen for Rosner's Outlier Test
There are many observations in this data set that may represent potential outliers. The potential outliers
can also be seen in the graphical displays associated with this data set.
Selected Options: Number of Suspected Outliers for the Rosner Test = 4
i Outlier Tests for Selected Variables
User S elected 0 ptions
From File DANarairAProUCL 4.0\DataWoclor 1254.wsl
Full Precision OFF
Test for Suspected Outliers with Dixon test 1
Test for Suspected Outliers for Rosner test 2
Rosner's Outlier Test for Aroclt»1254
Number of data: 44
Number of suspected outliers: 2
Potential
tt Mean sd outlier
1 1531.88 3316.58 19000.00
2 1125.65 1998.60 8300.00
Test Critical Critical
value value (5%) value (1 %)
5.27 3.08 3.43
3.58 3.07 3.41
For 5% significance level, there are 2 Potential Outliers
Therefore, Potential Statistical Outliers are
18000.00,8300.00
For 1% Significance Level, there are 2 Potential Outliers
Therefore, Potential Statistical Outliers are
18000.00,8300.00
Box plot of the Aroclorl254 Data Set
Box Plot for Aroclorl 254
20000.00
18000.00
16000.00
14000.00
W BC'OD.D:
.Q
o
6000.00
4000.00
2000.00
0.00 -
72
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Q-Q Plot of the Aroclorl254 Data Set
Q-Q Plot with NDs for ArocloM 254
ucoo.oo
H
c
•J3 12000.00
(0
5 10000.00
^ aooo.oo
•
^ EO:O oo
6
4000.00
2000.00
0.00
Arock>r1254
Total Number of Data = 53
Number of Non-Detects = 9
Number of Detects = 44
Mean =1271.7646
Sd =3105.5855
Slope = 2117.6299
Intercept = 1271.7646
Correlation, R = 0.6693
a aflaaBaa^jjjjj^jj^^^^iiiriBJydlri
Theoretical Quantiles (Standard Normal)
73
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Chapter 8
Goodness-of-Fit (G.O.F.) Tests
Several goodness-of-fit (G.O.F.) tests for full data sets (without nondetects) and for data sets with NDs
are available in ProUCL 4.0. Details of those tests are described in the ProUCL 4.0 Technical Guide. In
this User Guide, those tests and available options have been illustrated using screen shots generated by
ProUCL 4.0.
Two choices are available for Goodness-of-Fit menu: Full and With NDs.
• Full
o This option is used to analyze full data sets without any nondetect observations.
Throughout this User Guide and in ProUCL 4.0, "Full" represents data sets without
nondetect observations.
o This option tests for normal, gamma, or lognormal distribution of the variable(s)
selected using the Select Variables option.
o G.O.F. Statistics: This option is available for both full data sets and for data sets with
NDs. This option simply generates output log of GOF test statistics and derived
conclusions about the data distributions of all selected variables. This option is also
available for variables categorized by a group variable.
P?iRroUCL .4.0 - fp»rkSheet.v*fstJ
•B File Edit Configure Summary Statistics ROS Est. NDs Graphs Outtier Tests
^jgjjii
Navigation Panel
Hypothesis Testing Background UCL Window Help
Name
0 1 2
Arsenic D Arsenic
_ With NDs > Gamma
3 ""* *" •> Lognormal
G.O.F. Statistics
-^Worksheet wst
1
With NDs
o Analyzes data sets that have both nondetected and detected values.
o Six sub-menu items listed and shown below are available for this option.
1. Exclude NDs: tests for normal, gamma, or lognormal distribution of the selected
variable(s) using only the detected values.
2. ROS Estimates: tests for normal, gamma, or lognormal distribution of the selected
variable(s) using the detected values and the extrapolated values for the nondetects.
o Three ROS methods for normal, lognormal, and gamma distributions are
available. This option is used to estimate or extrapolate the NDs based upon the
specified distribution.
o By using the menu item ROS Est. NDs, ProUCL 4.0 actually generates additional
column(s) of ROS estimated NDs based upon the selected distribution. This
74
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option should be used for variables with NDs. This is further illustrated by a
screen shot given in the following
3. DL/2 Estimates: tests for normal, gamma, or lognormal distribution of the selected
variable(s) using the detected values and the ND values replaced by their respective
DL/2 values. This option is included for historical reasons and also for curious users.
Note: The use of fabricated data obtained using DL/2 (DL or 0 values) values is not
a recommended method. At best, the user may use the fabricated data (e.g., DL/2
option) for exploratory reasons. It is suggested that these substitution methods should
not be used for estimation and for hypotheses testing approaches.
4. G.O.F. Statistics: As for the full data sets, this option simply generates output log of
GOF test statistics and other relevant statistics for data sets with nondetects. The
conclusions about the data distributions for all selected variables are also listed on the
generated output file (Excel-type spreadsheet). This option is also available for
variables categorized by a group variable.
10 File Edit Configure Summary Statistics ROSEst.
ojoj BJ H] mini
Navigation Panel
Name
^ Worksheet wst
0
Arsenic . D.
1 ! 1i
2 1
3 1 7
NDs Graphs Gutter Tests l^^ra
Ful
1 2 3 —
.Arsenic
0
0
1
^^S^^ Hypothesis Testing Background UCL Window
^ s
HHH Normal-ROS Estimates
Gamma-ROS Estimates
Log-ROS Estimates
DL/2 Estimates
G.O.F. Statistics
k
y| Normal i
^ Gamma <
^ Lognormal
I
^
Help
9
• When multiple variables are selected from the Select Variables screen, you can choose
either:
o Group Graph option to produce multiple Q-Q plots for all selected variables in a
single graph. The relevant statistics (e.g., slope, intercept, correlation, test statistic
and critical value) associated with the selected variables are shown on the right panel
(in tan color; see page 77). In order to capture all the graphs and results shown on the
window screen, it is preferable to print the graph using the Landscape option. The use
of this option is recommended when a selected variable has data coming from two or
more groups or populations.
o Group Graph option is particularly useful to generate multiple Q-Q plots for the
groups associated with a selected variable. In order to capture all the graphs and
results shown on the window screen, it is preferable to print the graph using the
Landscape option. The user may also want to turn off the Navigation Panel and Log
Panel.
• Individual Graph option is used to generate individual Q-Q plots and the associated
statistics separately for each of the selected variable, one variable at a time.
o The linear pattern displayed by a Q-Q plot suggests an approximate goodness-of-fit
for the selected distribution.
75
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o The program computes the intercept, slope, and the correlation coefficient for the
linear pattern displayed by the Q-Q plot. A high value of the correlation coefficient
(e.g., > 0.95) is an indication of a good fit for that distribution. This high correlation
should exhibit a definite linear pattern in the Q-Q plot. Specifically, when data are
sparse and correlation is high, the use of correlation statistic to determine data
distribution is not desirable. Note that these statistics are displayed on the Q-Q plot.
o On a Q-Q plot, observations that are well separated from the bulk (central part) of the
data typically represent potential outliers needing further investigation.
o Significant and obvious jumps and breaks in a Q-Q plot (for any distribution) are
indications of the presence of more than one population. Data sets exhibiting such
behavior of Q-Q plots should be partitioned out in component sub-populations before
estimating an EPC term or a background threshold value (BTV). It is strongly
recommended that both graphical and formal goodness-of-fit tests should be used on
the same data set to determine the distribution of the data set under study.
Normality or Lognormality Tests: In addition to informal graphical normal and
lognormal Q-Q plots, a formal Goodness-of-Fit (GOF) test is also available to test the
normality or lognormality of the data set.
o Lilliefors Test: a test typically used for samples of size larger than 50 (> 50). When
the sample size is greater than 50, the program defaults to the Lilliefors test.
However, the Lilliefors test (generalized Kolmogorov Smirnov test) is available for
samples of all sizes. There is no applicable upper limit for sample size for the
Lilliefors test.
o Shapiro and Wilk (SW) Test: a test used for samples of size smaller than or equal to
50 (<= 50). In ProUCL 4.0, the SW test is available only for samples of size 50 or
less. It should be noted that the critical values for SW test are now available for
sample of sizes up to 2000 (Royston, 1982). These values are not as yet available in
ProUCL 4.0. This extension of SW test will be available in Scout (EPA, 1999)
software package.
o It should be noted that sometimes these two tests might lead to different conclusions.
Therefore, the user should exercise caution interpreting the results. Specifically, the
user should the pattern exhibited by the associated Q-Q plot.
GOF test for Gamma Distribution: In addition to the graphical gamma Q-Q plot, two
formal empirical distribution function (EOF) procedures are also available to test the
gamma distribution of a data set. These tests are the Anderson-Darling test and the
Kolmogorov-Smirnov test.
o It is noted that these two tests might lead to different conclusions. Therefore, the user
should exercise caution interpreting the results.
o These two tests may be used for samples of sizes in the range of 4-2500. Also, for
these two tests, the value of the shape parameter, k (k hat) should lie in the interval
[0.01, 100.0]. Consult the ProUCL 4.0 Technical Guide (A. Singh and A.K. Singh
(EPA, 2007)) for a detailed description of gamma distribution and its parameters,
including k. Extrapolation beyond these sample sizes and values of k is not
recommended.
76
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• ProUCL computes the relevant test statistic and the associated critical value, and prints
them on the associated Q-Q plot. On this Q-Q plot, the program informs the user if the
data are gamma, normally, or lognormally distributed.
• Even though, the G.O.F. Statistics option prints out all GOF test statistics for all selected
variables, it is suggested that the user should look at the graphical Q-Q plot displays to
gain extra insight (e.g., outliers, multiple population) into the data set.
Note: It is highly recommended not to skip the use of a graphical Q-Q plot to determine the data
distribution as a Q-Q plot also provides a useful information about the presence of multiple populations
or outliers.
8.1 ROS Estimated (Est.) NDs - Saving Extrapolated NDs
• As mentioned before, for a variable with NDs, ProUCL 4.0 can generate additional
column(s) consisting of detected data and the estimated (extrapolated) values of NDs
using the ROS method assuming a normal, lognormal, or a gamma distribution.
• The user may want to use the resulting full data set (detected and estimated NDs) thus
obtained to compute the statistics of interest such as a bootstrap BCA UCL95 or a gamma
95% upper percentile.
• This option of saving estimated NDs is provided only for experienced users and
researchers. It is expected that the user knows and understands the theory behind these
methods. Therefore, it is suggested that this option be used with care. For an example,
often, the use of a ROS method yields infeasible (e.g., negative, exceeding the DLs)
estimates of NDs, and therefore, the associated estimates of EPC terms and of BTVs may
be biased and not reliable. This is especially true when the data set contains potential
outlier(s).
P?; PrUCL 4.0
•0 File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit
oh&l
Navigation Panel
Name
@WorkSheet.wst
1
2
3
4
5
6
T
•'
&
9
10
11
0
Arsenic
1
1
1.7
1
1
2
3.2
2
2
2.8
2
1 2 3
D_Arsen.ic NRQS_Arsenic j
0 0.123366712938579
0 0.563125541541462
1 1.7
00.903911763702151! j
0 1.24633976124866
0).0994872439069666
1 3.2
0 0.273006451203256
0 0.542296528947428
1 2.8
0 0.773119303054192
11
-------
8.2 Goodness-of-Fit Tests with Full Data Sets
1. Click Goodness-of-Fit ^ Full
RJ ProUCL4,§ -
File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests
Goodness-of-fit
a
ml n
Navigation Panel
Name
Hypothesis Testing Background UCL Window Help
Normal i
__ With NDs >! Gamma I
3^™........,^*-...™,., . ~T Q Q
^ 4-^1 Lognormal I ' ° y
G.O.F. Statistics I
0 Worksheet wst I ' 1 !
2. Select the distribution to be tested: Normal, Lognormal, or Gamma
• To test your variable for normality, click on Normal from the drop-down menu list.
• To test variable for lognormality, click on Lognormal from the drop-down menu list.
• To test your variable for gamma distribution, click on Gamma from the drop-down menu
list.
8.2.1 GOF Tests for Normal and Lognormal Distribution
1. Click Goodness-of-Fit ^- Full ^- Normal or Lognormal
P?jProUCL 4.0 - tWprkSheet-wstJ
•0 File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests
f*q
tJ
Navigation Panel
Goodness-of-fit
I Hypothesis Testing Background UCL Window Help
_ Normal i
With NDs > Gamma 1 —
0
1
Name
G.O.F. Statistics
7
CsJWorkSheet.wst
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When the option button is clicked, the following window will be shown.
78
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• Select Confidence Level -
r 90%
<*• 35%
r 99%
Method
<*' Shapiro Wilk
'-"" Lilliefors
Display Regression Lines
<~ Do Not Display
(• Display Regression Liees
Graphs by Group
(* Individual Graphs
i " Group Graphs
Graphical Display Options
<"* Color Gradient
f" For Export (BW Printers)
OK
Cancel
o The default option for the Confidence Level is 95%.
o The default GOF Method is Shapiro Wilk. If the sample size is greater than 50, the
program automatically uses the Lilliefors test.
o The default method for Display Regression Lines is Do Not Display. If you want to
see regression lines on a Q-Q plot, then check the radio button next to Display
Regression Lines.
o The default option for Graphs by Group is Individual Graphs. If you want to see
the plots for all selected variables on a single graph, then check the radio button next
to Group Graphs.
Note: This option for Graphs by Group is specifically provided when the user wants to display multiple
graphs for a variable by a group variable (e.g., site AOC1, site AOC2, background). This kind of display
represents a useful visual comparison of the values of a variable (e.g., concentrations ofCOPC-Arsenic)
collected from two or more groups (e.g., upgradient wells, monitoring wells, residential wells).
o The default option for Graphical Display Options is Color Gradient. If you want
to see the graphs in black and white to be included in reports for later use, then check
the radio button next to For Export (BW Printers).
79
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• Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Normal Distribution (Full)
Selected Options: Shapiro Wilk, Display Regression Line, and For Export (BW Printers).
Normal Q-Q Plot for Arsenic
9.90
9.60
9.30
9.00
8.70
8.40
8.10
7.80
7.50
7.20
6.90
6.60
6.30
6.00
5.70
5.40
5.10
4.80
4.50
4.20
3.90
1 0 1
Theoretical Quantiles (Standard Normal)
Arsenic
N-20
Mean - 5.8725
Sd - 1.2247
Slope-1.1798
Intercept = 5.8725
Correlation, R 0.9281
Shapiro-Wilk Test
Test Value - 0.868
Critical Val(0.05) - 0.905
Data not Normal
-J- Arsenic
Output Screen for Lognormal Distribution (Full)
Selected options: Shapiro Wilk, Display Regression Lines, and Color Gradient.
> 1.90
0)
I
l
I
Lognormal Q-Q Plot for Arsenic
Theoretical Quantiles (Standard Normal)
Mean = 1.7519
Sd = 0.1917
Slope-0.1917
Intercept = 1.7519
Correlation, R = 0.9636
Shapiro-Wilk Test
Test Statistic = 0.932
Critical ValueiO.05) = 0.905
Data appear Lognormal
80
-------
8.2.2 GOF Tests for Gamma Distribution
1. Click Goodness-of-Fit ^ Full ^ Gamma
P? RroUCL 4.0..- (Worksheet.wsfl
•f File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests
ojo
Navigation Panel
Name
© Worksheetwst
Q 1
Arsenic D_Arsenic
1 I Ti " o
gj Hypothesis Testing Background UCL Window Help
Normal
Lognormal
G.O.F. Statistics
2. The Select Variables Screen (described in Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When the option button is clicked, the following window will be shown.
81
-------
Select Confidence Level
r 90%
ff 95 %
r 39%
Method
•'•' Anderson Darling
f Kol mogorov Stni rnov
Display Regression Lines
C Do Not Display
'• Display Regression- Lines
Graph by Groups
•'• Individual Graphs
1 " Group Graphs
'Graphical Display Options
'"•'" Color Gradient
<~ For Export (BW Printers]
Cancel
o The default option for the Confidence Level is 95%.
o The default GOF method is Anderson Darling.
o The default option for Display Regression Lines is Do Not Display. If you want to
see regression lines on the Gamma Q-Q plot, then check the radio button next to
Display Regression Lines.
o The default option for Graph by Groups is Individual Graphs. If you want to see
the graphs for all the selected variables into a single graph, then check the radio
button next to Group Graphs.
o The default option for Graphical Display Options is Color Gradient. If you want
to see the graphs in black and white, check the radio button next to For Export (BW
Printers).
• Click the OK button to continue or the Cancel button to cancel the option.
• Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
82
-------
Output Screen for Gamma Distribution (Full)
Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient.
Gamma Q-Q Plot for Mercury
Mean = 0.3055
k star = 1.1722
= 1 0468
lntercept = -O.Q111
Correlation, R = 09713
Test Statistic = 0.730
Critical ValueCO.05)-0.7S9
Dots appear Gamma Distributed
Theoretical Quantiles of Gamma Distribution
8.3 Goodness-of-Fit Tests Excluding NDs
1. Click Goodness-of-Fit ^ With NDs ^ Exclude NDs
•0 File Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests
Hypothesis Testing Background UCL Window Help
Navigation Panel
Normal-ROS Estimates
Gamma-ROS Estimates
Log-ROS Estimates
DL/2 Estimates
G.O.F. Stafetcs
i? Worksheet wst
2. Select distribution to be tested: Normal, Gamma, or Lognormal.
• To test for normality, click on Normal from the drop-down menu list.
• To test for lognormality, click on Lognormal from the drop-down menu list.
• To test for gamma distribution, click on Gamma from the drop-down menu list.
83
-------
8.3.1 Normal and Log normal Options
1. Click Goodness-of-Fit ^ With NDs ^ Excluded NDs ^ Normal or Lognormal
P?jRrpUCL 4..0 - Qf prkSheet.wstf
File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests
Goodness-rffit
Full
Hypothesis Testing Background UCL Window Help
Navigation Panel
Name
.^WorkSheet.wst
Wtn.AOs>| BtebdeWs
0 1 2
Arsenic D_Arsenic,
1 T i] o'
2 : l" 0
3 1.7 1
Normal-ROS Estimates
Gamma-ROS Estimates
Log-ROS Estimates
DL/2 Estimates
G.O.F. Statistics
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When the option button (Normal or Lognormal) is clicked, the following window will be
shown.
84
-------
"Select Confidence Level •
r 90%
(f 95%
r 99 %
Method
(• ShapiroWilk
f~ Lilliefofs
Display Regression Lines
<~ Do Not Display
'-*" Display Regression Lines
Graphs by Group
(* Individual Graphs
f*~ Group Graphs
Graphical Display Options
(* Color Gradient
r For Export (EW Printers)
OK
Cancel
o The default option for the Confidence Level is 95%.
o The default GOF Method is Shapiro Wilk. If the sample size is greater than 50, the
program defaults to Lilliefors test.
o The default for Display Regression Lines is Do Not Display. If you want to see
regression lines on the associated Q-Q plot, check the radio button next to Display
Regression Lines.
o The default option for Graphs by Group is Individual Graphs. If you want to see
the plots for all selected variables on a single graph, check the radio button next to
Group Graphs.
Note: This option for Graphs by Group is specifically useful when the user wants to display multiple
graphs for a variable by a group variable (e.g., site AOC1, Site AOC2, background). This kind of display
represents a useful visual comparison of the values of a variable (e.g., concentrations ofCOPC-Arsenic)
collected from two or more groups (e.g., upgradient wells, monitoring wells, and residential wells).
85
-------
o The default option for Graphical Display Option is Color Gradient. If you want to
see the graphs in black and white, check the radio button next to For Export (BW
Printers).
• Click the OK button to continue or the Cancel button to cancel the option.
• Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Normal Distribution (Exclude NDs)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and For Export (BW Printers).
Normal Q-Q Plots (Statistics using Detected Data)
forArsenic (subsurface), Arsenic (surface)
Arsenic (subsurface)
Total Number of Data = 1C
Number treated as ND = 1
DL = 4.5
N = 9
Percent NDs = 10%
Mean = 5.6778
Sd-0.5911
Slope - 0.1431
Intercept = 5.6778
Correlation, R = 0.2265
Shapiro Wilk Test
Test Statistic = 0.927
Critical Value |0.05> = 0.829
Data appear Normal
Arsenic (surface)
Total Number of Data = 10
Number treated as ND = 2
DL - 4.5
N = 8
Percent NDs-20%
Mean-6.6313
Sd = 1.4400
Slope - 0.1952
Intercept = 6.6313
Correlation, R = 0.1262
Shapiro-WilkTest
Test Statistic = 0.807
Critical Value(0.05) - 0.818
Data not Normal
- Arsenic (subsurface)
Theoretical Quantiles (Standard Normal)
-o- Arsenic (surface)
86
-------
Output Result for Lognormal Distribution (Exclude NDs)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.
Lognormal Q-Q Plots (Statistics using Detected Data)
forArsenic (subsurface), Arsenic (surface)
• Arsenic (subsurface)
Theoretical Quantiles (Standard Normal)
Arsenic (surface)
Arsenic (subsurftce)
Total Number ot Data = 10
Number treated as ND = 1
DL -1.5040774
N = 8
Percent NDs > 10%
Uteri = 1.7319
SO-0.1019
Slope - 0.1059
Intercept -1.7319
Cor relation ,R * 0.9726
Shapiro-Wife Test
Test Statistic - 0.938
Critical Val(0.05) = 0.829
Data appear Lognormal
Arsenic (surface)
Total Number of Data = 10
Number treated as ND - 2
DL -1 5040774
N-8
Percent NDs - 20%
Mean = 1.3731
Sd- 0.2018
Slope = 0.1997
Intercept -1.8731
Correlaticn,R = 0.9211
Shapirc-Wilk Test
Test Statistic - 0.335
Critical Val(0 05) = O.S1 8
Data appear Lognormal
8.3.2 Gamma Distribution Option
1. Click Goodness-of-Fit ^ With NDs ^ Excluded NDs ^ Gamma
PP ProUCL 4.0 - [Worksheet.wst]
File Edit Configure Summary Statistics ROS Est. NDs Graphs Cutter Tests
Goodness-of-Rt
Hypothesis Testing Background UCL Window Help
Navigation Panel
Name
£JWorkSheet.wst
Arsenic
D_Arsenic
i"
1.7
Normal
^^
Normal-ROS Estimates
Gamma-ROS Estimates > Lognormal
Log-ROS Estimates
DL/2 Estimates
G.O.F. Statistics
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When the option button (Gamma) is clicked, the following window is shown.
87
-------
•Select Confidence Level
T 90 %
^ 35%
T 99%
Method •
f* Anderson Darling
'"" tolmogorovSmirncw
'Display Regression Lines
r Do Not Display
(• Display Regression Lines
Graph by Groups
f* Individual Graphs
f Group Graphs
Graphical Display'Options
(* Color Gradient
r For Export (EM Printers)
OK
Cancel
o The default option for the Confidence Level is 95%.
o The default GOF test Method is Anderson Darling.
o The default method for Display Regression Lines is Do Not Display. If you
want to see regression lines on the normal Q-Q plot, check the radio button next
to Display Regression Lines.
o The default option for Graph by Groups is Individual Graphs. If you want to
display all selected variables on a single graph, check the radio button next to
Group Graphs.
o The default option for Graphical Display Options is Color Gradient. If you
want to see the graphs in black and white, check the radio button next to For
Export (BW Printers).
• Click the OK button to continue or the Cancel button to cancel the option.
• Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
88
-------
Output Screen for Gamma Distribution (Exclude NDs)
Selected options: Anderson Darling, Do Not Display, Individual Graphs, and For Export (BW Printers).
Gamma Q-Q Plot for Mercury with NDs
Statistics using Detected Data
4)
01 0.50
5
O 0.30
Mercury
Total Number of Data = 30
Number treated as ND - 5
DL - 0.5
N-25
Percent NDs-17%
Mean - 0.3124
kstar = 1.0533
Slope- 1.0294
Intercept- 4.0048
Correlation, R-0.9590
Anderson-Darling Test
Test Statistic = 0.861
Critical V,iliicl0.05l - 0.770
Data not Gamma Distributed
Theoretical Quantiles of Gamma Distribution
--i- Mercury
8.4 Goodness-of-Fit Tests with Log-ROS Estimates
1. Click Goodness-of-Fit ^ With NDs ^ Log-ROS Estimates
3?VMI!K!VV|W9T|lff6NffilRPfj^l
•2 File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests
elel tftlHlml n|
Navigation Panel
Name
1
2
3
0 1
Arsenic D_Arsenic
11 0
1 0
1.7 1
2
3
Q Hypothesis Testing Background UCL Window Help
Full >
H2I<33S3
^mmntmf
1
Exclude NDs >
Normal -ROS Estimates >
Gamma-ROS Estimates *
• ^^^^^•1 Loa-ROS Estimates >
DL/2 Estimates K
G.O.F. Statistics
5 ,
i
Normal
Gamma
Lognormal
2. Select the distribution to be tested: Normal, Lognormal, or Gamma
• To test your variable for normality, click on Normal from the drop-down menu list.
• To test a variable for gamma distribution, click Gamma from drop-down menu list.
• To test your variable for lognormality, click on Lognormal from drop-down menu.
89
-------
8.4.1 Normal or Log normal Distribution (Log-ROS Estimates)
1. Click Goodness-of-Fit ^ With NDs ^ Log-ROS Estimates ^ Normal, Lognormal
P? ProUCL 4,0 - ||ft)rkSheet,wstI
File Edit Configure Summary Statistics ROS Est. NDs Graphs Gutter Tests
h&lBlHlmlEl " ™FJ
Hypothesis Testing Background UCL Window Help
*
Navigation Panel
Name
0 1 2
Arsenic D_Arservic
1 [~~J] °
1 \ 1-7 1
3 "
mK Exdude NDs >
Normal-ROS Estimates >
Gamma-ROS Estimates >
LoQ*OSEslmates
DL/2 Estimates > Gamma
G.O.F. Statistics
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When the option button (Normal or Lognormal) is clicked, the following window will be
shown.
90
-------
'•Select Confidence Level
T 90%
<¥ 95%
r 99%
Method
<• ShapiroWilk
T Lilliefors
'Display Regression Lines
f" Do Not Display
'** Display Regression Lines
- Graphs by Group •
& Individual Graphs
f" Group Graphs
- Graphical Display Options -
f*" Color Gradient
T For Export (BW Printers)
OK
Cancel
o The default option for the Confidence Level is 95%.
o The default GOF test Method is Shapiro Wilk. If the sample size is greater than 50,
the program defaults to use the Lilliefors test.
o The default method for Display Regression Lines is Do Not Display. If you want to
see regression lines on the normal Q-Q plot, check the radio button next to Display
Regression Lines.
o The default option for Graphs by Group is Individual Graphs. If you want to
display all selected variables into a single graph, check the radio button next to
Group Graphs.
o The default option for Graphical Display Options is Color Gradient. If you want
to see the graphs in black and white, check the radio button next to For Export (BW
Printers).
91
-------
• Click the OK button to continue or the Cancel button to cancel the option.
• Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Normal Distribution (Log-ROS Estimates)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and For Export (BW Printers).
Normal Q-Q Plots using Robust ROS Method
for Arsenic, Mercury
5 6.00
1
aj 5.00
O
•O 4.00
-J- Arsenic
1 o 1
Theoretical Quantiles (Standard Normal)
-o Mercury
Arsenic
N = 20
Mean - 5.8307
Sd-1.2799
Slope-1.2517
Intercept - 5.8307
Correlation, R = 0.9421
Shapiro-WilkTest
Tea Value-0.895
Critical Val(0.05| - 0.905
Data not Normal
Mercury
N-30
Mean-0.2767
Sd - 0.2984
Slope - 0.2643
Intercept = 0.2767
Correlation, R - 0.8618
Shapiro-WilkTest
Test Value-0.733
Critical Val(0.05| - 0.927
Data not Normal
Note: The legend size of nondetect values is smaller than that of the detected values.
92
-------
Output Screen for Lognormal Distribution (Log-ROS Estimates)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.
Lognormal Q-Q Plot for Group
Statistics using Robust ROS Method
I17
Arsenic i :ul:i:urrai:e i
Theoretical Ouantiles of Gamma Distribution
Arsenic (surface)
Arsenic (subsurface)
N = 1Q
Mean = 1.7063
Sd = 0.1246
Slope = 0.1308
Intercept = 1 .7068
Correlation, R = 0.9366
Shapiro-VMIk Test
Test Statistic = 0.961
Critical Value(O.OS) = 0.842
Data appear Lognormal
Arsenic (surface)
N = 1Q
Mean = 1.7731
Sd = 0.2678
Slope = 0.2758
Intercept -1.7731
Correlation, R = 0.9682
Shapiro-Will; Test
Test Statistic = 0.930
Critical ValueC0.05)-0.842
Data appear Lognormal
Note: The legend size ofnondetect values is smaller than that of the detected values and is shown in red.
8.4.2 Gamma Distribution (Log-ROS Estimates)
1. Click Goodness-of-Fit ^ With NDs ^ Log-ROS Estimates ^ Gamma
P3 ProUCL 4.0 - [Worksheet. wst]
•g File Edit Configure Summary Statistics ROS Est, NDs Graphs Outiier Tests [ J Hypothesis Testing Background UCL Window Help
Full TT
Navigation Panel
Name
^WorkSheet.wst
Arsenic
1
D_Arsenic
II
l"
1.7
Oj
~0~
Exdude NDs
Normal-ROS Estimates
Gamma-ROS Estimates
it
8
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
93
-------
When the option button (Gamma) is clicked, the following window will be shown.
'Select Confidence Level
T 30 %
<* 95%
r 99 %
-Method
(• Anderson Darling
T" KblmogorovSmirrtov
'Display Regression Lines
r Do Not Display
t**" Display Regression Lines
-Graph by Groups
f* Individual Graphs
<^~ Group Graphs
'Graphical Display Options
f* Color Gradient
r For Export (EM Printers)
OK
Cancel
o The default option for the Confidence Level is 95%.
o The default GOF test Method is Anderson Darling.
o The default method for Display Regression Lines is Do Not Display. If you want to
see regression lines on the normal Q-Q plot, check the radio button next to Display
Regression Lines.
o The default option for Graph by Groups is Individual Graphs. If you want to put
all of the selected variables into a single graph, check the radio button next to Group
Graphs.
o The default option for Graphical Display Options is Color Gradient. If you want
to see the graphs in black and white, check the radio button next to For Export (BW
Printers).
• Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
94
-------
Output Screen for Gamma Distribution (Log-ROS Estimates)
Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient.
Gamma Q-Q Plot for Mercury
Statistics using Robust ROS Method
Mean = 0.2767
K Star = 1.0653
Slope = 1.1071
Intercept = -0.0262
Correlation, R = 0.9579
Anderson-Darling Test
Test Statistic = 1.425
Critical Value(O.OS) = 0.772
Data not Gamma Distributed
0.0053 - 0.0400
00165-0.0500
0.0326 - 0.0550
0.0326 - 0.0550
0.0492-0.0600
0.0607 - 0.0645
0.0765 - 0.0700
0.07B5 - 0.0700
00974-0.0367
0.1107-0.0922
1.1000-0.9900
Theoretical Quantiles of Gamma Distribution
Note: The legend size ofnondetect values is smaller than that of detected values and is shown in red.
8.5 Goodness-of-Fit Tests with DL/2 Estimates
1 Click Goodness-of-Fit ^ With NDs ^ DL/2 Estimates
P? ProUCL 4.0 - [WorkSheet.wst]
•S File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests f
E|
Normal-ROS Estimates >
Gamma-ROS Estimates >
Log-ROS Estimates >
G.O.F. Statistics
UCL Window Help
8 9
2. Select the distribution to be tested: Normal, Gamma, or Lognormal
• To test the variable for normality, click on Normal from the drop-down menu list.
• To test the variable for lognormality, click on Lognormal from the drop-down menu list.
95
-------
• To test your variable for gamma distribution, click on Gamma from the drop-down menu
list.
8.5.1 Normal or Log normal Distribution (DL/2 Estimates)
1. Click Goodness-of-Fit ^ With NDs ^ DL/2 Estimates ^ Normal or Lognormal
B? ' ProUCL 4.0 - {Worksheet, wsJJ
File Edit Configure Summary Statistics ROS Est. NDs Graphs Gutter Tests
Hypothesis Testing Background UCL Window Help
Full
Navigation Panel
Name
-if Worksheet .wst
1
2
3
4
Arsenic D_Arsen-ie
! 1! 0
T o
1.7 1
: 1 0
Exdude NDs >
Normal-ROS Estimates >
Gamma-ROS Estimates > J
Log-ROS Estimates >
I23^S^H
G.O.F. Statistics
Normal
Gamma
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When Normal or Lognormal button is clicked, following window is displayed.
96
-------
'•Select Confidence Level
T 90%
<¥ 95%
r 99%
Method
<• ShapiroWilk
T Lilliefors
'Display Regression Lines
f" Do Not Display
'** Display Regression Lines
- Graphs by Group •
& Individual Graphs
f" Group Graphs
- Graphical Display Options -
f*" Color Gradient
T For Export (BW Printers)
OK
Cancel
o The default option for the Confidence Level is 95%.
o The default Method is Shapiro Wilk. If the sample size is greater than 50, the
program defaults to the Lilliefors test.
o The default method for Display Regression Lines is Do Not Display. If you want to
see regression lines on the normal Q-Q plot, check the radio button next to Display
Regression Lines.
o The default option for Graphs by Group is Individual Graphs. If you want to put
all of the selected variables into a single graph, check the radio button next to Group
Graphs.
o The default option for Graphical Display Options is Color Gradient. If you want
to see the graphs in black and white, check the radio button next to For Export (BW
Printers)
97
-------
• Click the OK button to continue or the Cancel button to cancel the option.
• Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Normal Distribution (DL/2 Estimates)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.
Normal Q-Q Plot for Group
Statistics using DL/2 Method
-* Arsenic (subsurface)
Theoretical Quantiles (Standard Normal)
•j Arsenic (surface)
Arsenic (subsurface)
N-10
Mean = 5.3350
3d = 12180
Slope = 1.1396
Intercept = 5.3350
Correlation, R = 0.8792
Shapiro-VMIk Statistic
Test Value = 0.803
Critical Val(0.05) = 0.842
Data not Normal
Arsenic (surface)
N = 10
Mean = 5.7450
Sd = 2.2592
Slope-2.2938
Intercept = 5.7450
Correlaiion, R = 0.9547
Shapiro-VMIk Statistic
Test Value - 0.909
Critical Val(0.05) = 0.842
Data appear Normal
Note: The legend size ofnondetect values is smaller than that of the detected values and is shown in red.
98
-------
Output Screen for Lognormal Distribution (DL/2 Estimates)
Selected options: Shapiro Wilk, Display Regression Lines, Individual Graphs, and For Export (BW Printers).
0.00
4.20
J1.40
-0.60
-0.80
-1.00
flS -1.40
Normal-ROS Estimates >
Gamma-ROS Estimates >
Log-ROS Estimates >
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
• If graphs have to be produced by using a Group variable, then select a group variable by
clicking the arrow below the Group by variable button. This will result in a drop-down
99
-------
list of available variables. The user should select and click on an appropriate variable
representing a group variable.
• When the Gamma option button is clicked, the following window will be shown.
. in..
E3
r Select Confidence Level
T 90%
P 95 %
r m %
."Method
(•" Anderson Darling
'*"** KolmogGravSmirnew
i Display Regression Lines
r Do Not Display
{•" Display Regression Lines
.-•"Graph by Groups
<•* Individual Graphs
f" Group Graphs
Graphical Display Options
(* Color Gradient
r For Export (BW Printers)
OK
Cancel
The default option for the Confidence Level is 95%.
The default Method is Anderson Darling.
The default method for Display Regression Lines is Do Not Display. If you want to
see regression lines on the normal Q-Q plot, check the radio button next to Display
Regression Lines.
The default option for Graph by Groups is Individual Graphs. If you want to put
all of the selected variables into a single graph, check the radio button next to Group
Graphs.
100
-------
o The default option for Graphical Display Options is Color Gradient. If you want
to see the graphs in black and white, check the radio button next to For Export (BW
Printers).
• Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Gamma Distribution (DL/2 Estimates)
Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient.
Gamma Q-Q Plot for Mercury
Statistics using DL/2 Substitution Method
" 060
O
w 05°
a
o
TJ
£ 040
Mercury
hi = 30
Mean = 0 2829
k star - 1.0875
Slope = 1.0897
Intercept = -0.0220
Correlation, R = 0.9617
Anderson-Darling Test
Test Statistic = 1.126
Critical Value(0.05) = 0.771
Data not Gamma Distributed
Theoretical Quantiles of Gamma Distribution
Note: The legend size ofnondetect values is smaller than that of the detected values and is shown in red.
8.6 Goodness-of-Fit Tests Statistics
1 Click Goodness-of-Fit > With NDs > G.O.F. Statistics
P3 ProUCL 4.0 - [WorkSheet.wst]
File Edit Configure Summary Statistics ROS Est. NDs Graphs Outiier Tests
Hypothesis Testing Background UCL Window Help
Navigation Panel
Name
WorkSheet.wst
Hi GOF_Stats_wND ost
Arsenic
1
D Arsenic
i
i"
1.7
Exdude NDs
Normal-ROS Estimates
Gamma-ROS Estimates
Log-ROS Estimates
DL/2 Estimates
2. The Select Variables Screen (Chapter 3) will appear.
• Select one or more variable(s) from the Select Variables screen.
101
-------
When the option button is clicked, the following window will be shown.
Select Confidence Level
.Select Confidence Level
957,
99%
OK
Cancel
• The default confidence level is 95%.
• Click the OK button to continue or the Cancel button to cancel the option.
Sample Output Screen for G.O.F. Test Statistics
User Selected Options
From File WorkSheetwst
Full Precision OFF
Confidence Coefficient 0.95
Arsenic Date
Statistics (Non-Detects Only)
Statistics (Detects Only)
Statistics (All: NDs treated as DL value)
Statistics (All: NDs treated as DL2 value)
Statistics (Normal ROS Estimated Data)
Statistics (Gamma ROS Estimated Data)
Statistics (Lognormal ROS Estimated Data)
Statistics (Detects Only)
Statistics (NDs-DL)
NumObs
24
Number
13
11
24
24
24
24
24
KHat
2.257
3.538
Num Miss
0
Minimum
0.9
0.5
0.5
0.45
-0.0995
0.5
0.349
KStar
2.002
3.124
Num Valid
24
Maximum
2
3.2
3.2
3.2
3.2
3.2
3.2
TNeta Hat
0.548
0.406
Detects
11
Mean
1.608
1.236
1.438
1.002
0.997
1.263
0.972
Log Mean
-0.0255
0.215
NDs
13
Median
2
0.7
1.25
0.95
0.737
1.213
0.7
Log Stdv
0.694
0.574
% NDs
54.17%
SO
0.517
0.965
0.761
0.699
0.776
0.652
0.718
LogC.V.
-27.26
2.669
102
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Output Screen for the G.O.F. Test Statistics - (continued)
Normal Distri Indian Test Results
Test value Crit. (0.95)
Conclusion with Alpha(0.05)
Shapiro-Wilks (Detects Only)
Lilliefors (Detects Only)
Shapiro-Wilks (NDs = DL)
Lilliefors (NDs-DL)
Shapiro-Wilks (NDs = Db'2)
Lilliefors (NDs .DL^)
Shapiro-Wilks (Normal ROS Estimates)
Lilliefors (Normal ROS Estimates)
0.777 0.85 Data Not Normal
0.273 0.267 Data Not Normal
0.89 0.916 Data Not Normal
0.217 0.181 Data Not Normal
0.701 0.916 Data Not Normal
0.335 0.181 Data Not Normal
0.868 0.916 Data Not Normal
0.17 0.181 Data Appear Normal
Gamma Distri tuition Test ResuHs
Test value Crit. (0.95)
Conclusion with Alpha(0.05)
Anderson-Darling (Detects Only)
Kolmogorov-Smirnov (Detects Only)
Anderson-Darling (NDs - DL)
Kolmogorov-Smirnov (NDs = DL)
Anderson-Darling (NDs = Db'2)
Kolrrwgorov-Smirnov (NDs = DU2)
Anderson-Darling (Gamma ROS Estimates)
Kolmoporoy-Smirnov (Gamma ROS Est.)
0.787 0.738
0.254 0.258 Data appear Approximate Gamma Distribution
0,98 0.75
0.214 0.179 Data Not Gamrra Distributed
1.492 0.751
0.261 0.179 Data Not Gafrn-a Distributed
0.524 0.747
0.127 0.179 Data Appear Gamma Distributed
103
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Chapter 9
Single Sample and Two-Sample Hypotheses
Testing Approaches
This chapter illustrates single sample and two-sample parametric and nonparametric hypotheses testing
approaches as incorporated in ProUCL 4.0. ProUCL 4.0 can perform these hypotheses tests on data sets
with and without nondetect observations. It should be pointed out that, when one wants to use two-sample
hypotheses tests on data sets with NDs, ProUCL 4.0 assumes that samples from both of the groups have
nondetect observations. All this means that, a ND column (with 0 or 1 entries only) needs to be provided
for the variable in each of the two groups. This has to be done even if one of the groups has all detected
entries; in this case the associated ND column will have all entries equal to "1." This will allow the user
to compare two groups (e.g., arsenic in background vs. site samples) with one of the groups having some
NDs and the other group having all detected data.
9.1 Single Sample Hypotheses Tests
In many environmental applications, single sample hypotheses tests are used to compare site data
(provided enough site data are available) with pre-specified cleanup standards or compliance limits.
ProUCL 4.0 contains single sample parametric and nonparametric tests including Student's t-test, sign
test, Wilcoxon Signed Rank (WSR) test, and test for proportion. The single sample hypotheses tests are
useful when the environmental parameters such as the clean standard, action level, or compliance limits
(CLs) are known, and the objective is to compare site concentrations with those known threshold values.
Specifically, a t-test (or a sign test) may be used to verify the attainment of cleanup levels at an AOC after
remediation activity; and a test for proportion may be used to verify if the proportion of exceedances of an
action level (or a compliance limit) by sample concentrations collected from an AOC (or a MW) exceeds
a certain specified proportion (e.g., 1%, 5%,
ProUCL 4.0 can perform these hypotheses on data sets with and without nondetect observations.
However, it should be noted that for single sample hypotheses tests (e.g., sign test, proportion test) used
to compare site mean/median concentration level with a cleanup standard, Cs, or a compliance limit (e.g.,
proportion test), all NDs (if any) should lie below the cleanup standard, Cs. For proper use of these
hypotheses testing approaches, the differences between these tests should be noted and understood.
Specifically, a t-test or a 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 compares if the proportion
of site observations from an AOC exceeding a compliance limit (CL) exceeds a specified proportion, P0
(e.g., 5%, 10%). ProUCL 4.0 has useful graphical methods that may be used to visually compare the
concentrations of a site area of concern (AOC) with an action level. This can be done using a box plot of
site data with action level superimposed on that graph. The details of the various single sample
hypotheses testing approaches can be found in EPA guidance documents (1989, 2006). A brief discussion
of these methods is also given in the ProUCL 4.0 Technical Guide.
104
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9.1.1 Single Sample Hypothesis Testing for Full Data without Nondetects
1. Click Hypothesis Testing^- Single Sample
P? ProUCL 4.0 - [WorkSheet.wstJ
File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit
Navigation Panel
Name
1 Sample-Prop
i " TiS
HjpoSiessTesSng
Single Sample M Full (w/o NOsJ
Background UCL Window Help
t-Test
Two Sample > With NDs >' Proportion
_,,5—t—-8 «™ ' ! -7— sjgn test
Wilcoxon Signed Rank
r
£J WorkSheet.wst |
2. Select Full (w/o NDs) - This option is used for full data sets without nondetects.
• To perform a t-test, click on t-Test from the drop-down menu as shown above.
• To perform a proportion test, click on Proportion from the drop-down menu.
• To run a sign test, click on Sign test from the drop-down menu.
• To run a Wilcoxon Signed Rank test, click on Wilcoxon Signed Rank from the drop-
down menu.
9.1.1.1 Single Sample t-Test
1. Click Hypothesis Testing ^ Single Sample ^ Full (w/o NDs) ^ t-Test
R ProUCL 4.0 - [Worksheet.wst]
File Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness-of-Fit KlCT^^^SB-l Background UCL Window Help
Two Sample ^ With NDs > Proportion
Wilcoxon Signed Rank
Navigation Panel
Name
@ WorkSheet.wst
1
3 1
1Sample-Prop
Tl9i
2. The Select Variables Screen (see page 130) will appear.
• Select variable (variables) from the Select Variables screen.
• When the Options button is clicked, the following window will be shown.
105
-------
II Single, Saiifple t Test pptipns
0.95
I
Confidence Level |
Substantial Difference. S T~
(Used with Test Form 2)
Compliance Limit j
^Select Null Hypothesis Form
***" Mean <= Compliance Limit (Form 1)
f" Mean >= Compliance Limit (Form 2)
Mean >= Compliance Limit + S (Form 2)
Mean = Compliance Limit (2 Sided Alternative)
OK
Cancel
o Specify the Confidence Level; default is 0.95.
o Specify meaningful values for Substantial Difference, S and the Compliance Limit.
The default choice for S is "0."
o Select the form of Null Hypothesis; default is Mean <= Compliance Limit (Form 1).
o Click on OK button to continue or on Cancel button to cancel the test.
106
-------
Output for Single Sample t-Test (Full Data without NDs)
1 Sample-1
Si ngle Sample t-Test
Raw Statistics
N umber of Va I id Samples 9
Number of Distinct Samples 9
Minimum 82.39
Maximum 113.2
Mean 99.38
Median. 103.5
SD 10.41
SE of Mean 3.468
HO: Site Mean = 100
Test Value -0.178
Two Sided Critical Value (0.05) 2.306
P-Value 0.863
Concl usion with Alpha = 0.05
Do Not Reject HO. Conclude Mean = 100
P-Value>Alpha(O.Og
9.1.1.2 Single Sample Proportion Test
1. Click Hypothesis Testing ^ Single Sample ^- Full (w/o NDs) ^- Proportion
P? jPrffUg. $.Qr- fWockSheet,wst j.
«B File Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness-of-Fit
0 1
1Samp!e-Prop
413
5.33S6
HwotteasTesBuo:-
SoSe:Sample M :ft»ft*/ofC>sJ '*
Navigation Panel
Name
Background UCL Window Help
| t-Test j
f Sign test !
Wilcoxon Signed Rank |
v. Worksheet wst
2. The Select Variables Screen will appear.
• Select variable (variables) from the Select Variables screen.
• When the Options button is clicked, the following window will be shown.
107
-------
^'••"••i-"""""--^--?-''--;;^^
3.95
0.3
Confidence Level
Proportion
Action/Compliance Limit
Select Null Hypothesis Form
C* P <= Porportion (Form 1)
P >= Proportion (Form 2)
P = Proportion (2 Side Alternatived)
OK
Cancel
in
o Specify the Confidence level; default is 0.95.
o Specify the Proportion level and a meaningful Action/Compliance Limit.
o Select the form of Null Hypothesis; default is P <= Proportion (Form 1).
o Click on OK button to continue or on Cancel button to cancel the test.
108
-------
Output for Single Sample Proportion Test (Full Data without NDs)
O ne-Sample Proportion Test
Raw Statistics
N umber of Val id Samples B5
Number of Distinct Samples 83
Minimum 0.598
Maximum 7.676
Mean 5.183
Median 5.564
SD 1.588
SE of Mean 0.172
Number of Exceedances 27
Sample Proportion of Exceedances 0.318
HO: Site Proportion <= 0.3 (Form 1)
Large Sample z-Test Val ue 0.237
Critical Value (0.05) 1.645
P-Value 0.406
Conclusion with Alpha =0.06
Do Not Reject H0. ConcIude Site Proportion <= 0.3
P-Vali*e> Alpha (0.05)
9.1.1.3 Single Sample Sign Test
1. Click Hypothesis Testing ^- Single Sample ^- Full (w/o NDs) ^- Sign test
File Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness-of-Rt
^pottffisis: Testog.
Sfrrife'Sflrkfc M
Navigation Panel
Name
D 1
Sigr Test
234
Arsenic D_Arsenic
Background UCL Window Heip
t-Test |
- i—mnir |
Two Sample * With NDs * Proportion j................
Wilcoxon Signed Rank !
^JWorkSheet-wst
2. The Select Variables Screen will appear.
• Select variable (variables) from the Select Variables screen.
• When the Options button is clicked, the following window will be shown.
109
-------
0.95
Confidence Level j
Substantial Difference. S I
[Used with Test Form 2) '
Action/Compliance Limit )
Select Null Hypothesis Form
I*" Median <= Compliance Limit [Form 1)
»*"* Median >= Compliance Limit (Form 2)
C~* Median >= Compliance Limit + S (Form 2)
l*"* Median = Comtpl iartce Li mit (2 Sided Alternative)
OK
Cancel
o Specify the Confidence Level; default choice is 0.95.
o Specify meaningful values for Substantial Difference, S and Action/Compliance
Limit.
o Select the form of Null Hypothesis; default is Median <= Compliance Limit (Form
1).
o Click on OK button to continue or on Cancel button to cancel the test.
110
-------
Output for Single Sample Proportion Test (Full Data without NDs)
Si ngle Sample Sign Test
Raw Statistics
N umber of Val id Samples 10
Number of Distinct Samples 10
Minimum 750
Maximum 1161
Mean 925.7
Median 888
SD 136.7
SE of Mean 43.24
Number Above Limit 3
Number Equal Limit 0
Number Below Limit 7
HO: Site Median >= 1000 (Form 2)
Test Value
Lower Critical Value (0.05)
P-Value
Conclusion with Alpha =0.05
Do Hot Reject HO. Conclude Median >= 1000
P-Value>AJpbaCQ05)
3
1
0.172
9.1.1.4 Single Sample Wilcoxon Signed Rank (WSR) Test
1. Click Hypothesis Testing ^- Single Sample ^- Full (w/o NDs) ^- Wilcoxon Signed Rank
F? ProUCL 4.0 - [WorkShest_a,wst]
File Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness -of-Fit
jrtwethesasTesfcg,
SngteSannle
Navigation Panel
Name
1
2
0
WSR1
! ' 97i
1Wi
1 2 3
WSR2
52B
5.89
4
Arsenic D
1
1
Background UCL Window Help
t-Test
Two Sample > With NDs > Proportion
2.
iy Worksheet wst
i,?WorkSheet_a.wst
The Select Variables Screen will appear.
• Select variable (variables) from the Select Variables screen.
• When the Options button is clicked, the following window will be shown.
Ill
-------
3H Single Sample Wilcoxon Signed Rank Test Options X
Confidence Level ] 095
Substantial Difference. S
(Used with Test Form 2)
Act i on/Com pliance Limit
Select Null Hypothesis Form —
Mean/Median <= Compliance Limit (Form 1)
Mearv'Median >= Compliance Limit (Form 2)
Mearv'Median >= Compliance Limit + S (Form 2)
Meanj'Median = Compliance Limit (2 Sided Alternative)
OK
Cancel
o Specify the Confidence Level; default is 0.95.
o Specify meaningful values for Substantial Difference, S, and Action/Compliance
Limit.
o Select the form of Null Hypothesis; default is Mean/Median <= Compliance Limit
(Form 1).
o Click on OK button to continue or on Cancel button to cancel the test.
112
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Output for Single Sample Proportion Test (Full Data without NDs)
Single Sample Wilcoxon Signed Rank Test
Raw Statistics
Number of Valid Samples 10
Number of Distinct Samples 10
Minimum 75S
Maximum 1161
Mean'925.7
Median SSS
SD 136.7
SE of Mean 4124
Number Above Limit 3
Number Equal Limit 0
Number Below Limit 7
T-plus 11.5
T-minus 43.5
HO: SHe Median <= 1000 (Form 1)
Test Value 11.5
Critical Value (0.05).45
P-Value'O.S47
Conclusion with Alpha = 0,05
Do Not Reject HO. Conclude Mean/Median <= 1000
P-Value > Alpha (0.05)
113
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9.12 Single Sample Hypothesis Testing for Data Sets with Nondetects
Most of the one-sample tests such as the Proportion test and the Sign test on data sets with nondetect
values assume that all nondetect observations lie below the compliance limit (CL) or an action level, A0.
The single sample tests cannot be performed if ND observations exceed the CL or action levels.
1. Click on Hypothesis Testing^- Single Sample
{WprkSheet.wstJ
i File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit
ejLj|m| ml
Hypothesis Testing
Navigation Panel
| Background IICL Window Help
Full [w/o NDs) > t
Name
Wilcoxon Signed Rank
"ir Worksheet wst
2. Select the With NDs option
• To perform a proportion test, click on Proportion from the drop-down menu.
• To perform a sign test, click on Sign test from the drop-down menu.
• To perform a Wilcoxon Signed Rank test, click on Wilcoxon Signed Rank from the drop-
down menu list.
9.1.2.1 Single Proportion Test on Data Sets with NDs
1. Click Hypothesis Testing >• Single Sample >• With NDs ^ Proportion
E? ProUCL 4.0- [C:\Narain\ProUCL-Data\Data\WSREPA (2006).wst]
D o s|a|mj uj
Navigation Panel
Name
-i WoikSheet >ist
1
2
0
V/SR1
9 4
1 544
1 2 3
WSR2
5BB
5S9
KfaBBSSSlffla HwxrthessTesfcrw?
^^E^^S^B
Arsenic D_Arsenic
1 0
1 0
^^^^^^^^
Background UCL Window Help
••npHH^i^^l
"'"* "^ ''" V S>gn test 11
wilcoxop Signed Rarf |
2. The Select Variables screen will appear.
• Select variable (variables) from the Select Variables screen.
• When the Options button is clicked, the following window will be shown.
114
-------
Single Sample Proportion Test Options X
Confidence Level [ 0-95
Proportion
Action/Compliance Limit
0.3
Select Null Hypothesis Form —
<•" P <= Porportion [Form 1)
C' P >= Proportion [Form 2)
C P = Proportion (2 Side Alternatived)
OK
Cancel
o Specify the Confidence Level; default is 0.95.
o Specify meaningful values for Proportion and the Action/Compliance Limit.
o Select the form of Null Hypothesis; default is P <= Proportion (Form 1).
o Click on OK button to continue or on Cancel button to cancel the test.
115
-------
Output for Single Sample Proportion Test (with NDs)
Arsenic
Single Sample Proportion Test
Raw Statistics
Number of Valid Samples 24
N umber of Disti net Samples 10
N umber of Non-Detect Data 13
Number of Detected Data 11
Percent Norv Detects 54.17%
Minimum Non-d«tect 0.9
Maximum Men-detect 2
Minimum Detected 0.5
Maximum Detected 3.2
Mean of Detected Data 1.236
Median of Detected Data 0.7
SD of Detected Data 0.965
N urnber of Exceedances 2
Sample Proportion of Exceedances 0.0833
Some Men- Detect Values Exceed
The User Aefan/Compiance limit
Unable to do Proportion Test wit i such
9.1.2.2 Single Sample Sign Test with NDs
1. Click Hypothesis Testing ^- Single Sample ^- With NDs ^- Sign test
- BrY0rkSljeet.wstJ
> File Edit Configure Summary Statistics ROS Est, NDs Graphs Outiier Tests Goodness-of-Fit n^gy^^^gM Background UCL Window Help
Fp| H^S|rKij5im Full (w/o NDs) t s
Two Sample > K]32E2^M1 Proportion
Navigation Panel
Name
iiJWortcSheet.wst
01234
Sign Test Arsenic D_Arsemc
'T'S 1 0
1 D44 10
-•9
2. The Select Variables screen will appear.
• Select variable (variables) from the Select Variables screen.
• When the Options button is clicked, the following window will be shown.
116
-------
1 Single Sample Sign Test Options
Confidence Level 0.95
Substantial Difference. S
(Used with Test Form 2}
Action/Compliance Limit |
Select Null Hypothesis Form -
(* Median <= Compliance Limit (Form 1)
("" Median >= Compliance Limit [Form 2)
(*" Median >= Compliance Limit + S (Form 2)
Median = Compliance Limit [2 Sided Alternative)
OK
Cancel
o Specify the Confidence Level; default is 0.95.
o Specify meaningful values for Substantial Difference, S and Action/Compliance
Limit.
o Select the form of Null Hypothesis; default is Median <= Compliance Limit (Form
1).
o Click on OK button to continue or on Cancel button to cancel the test.
117
-------
Output for Single Sample Sign Test (Data with Nondetects)
Arsenic
Single Sample Sign Test
Raw Statistics
Number of Valid Samples 24
N umber of Disti net Samples 10
Number of Non-Detect Data 13
N umber of Detected Data 11
Percent Non-Detects 54.17%
Minimum Non-detect 0.9
Maximum Non-detect 2
Minimum Detected 0.5
Maximum Detected 3.2
Mean of Detected Data 1,23€
Median of Detected Data 0.7
SD of Detected Data 0.965
Number Above Limit 0
Number Equal Limit 0
Number Below Limit 24
HO: Site Median <= 5 (Forml)
Test Value 0
Upper Critical Value (0.05) 17
P-Value 1
"oncl its ion with Alpha = 0.06
Do Not Reject HO. Conclude Median <= 5
P-Value> Alpha (005)
9.1.2.3 Single Sample Wilcoxon Signed Rank Test with NDs
1. Click Hypothesis Testing ^ Single Sample ^- With NDs ^- Wilcoxon Signed Rank
P?iPrpUCL 4,0 - [V^orkSheet_a.wstJ
? FileEdit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness-of-Fit I
,.! A 1_J 1 ^±—I ^ — jwo 3amp|e
Navigation Panel
Name
43 Worksheet, wst
OWorkSheet_a.wst
1
2
0 1
V/SR1
! ^
1.044
2 3
WSR2
5.BB
589
4
Arsenic D
1
1
Background UCL Window Help
Full Q../Q »Ds) > i
[2222^^^B Proportion
' T '' V sign test
118
-------
2. The Select Variables Screen will appear.
• Select variable (variables) from the Select Variables screen.
• When the Options button is clicked, the following window will be shown.
BH Single Sample Wilcoxon Signed Rank Test Options
Confidence Level
Substantial Difference. S
(Used with Test Form 2)
Action/Compliance Limit
Select Null Hypothesis Form -
0.55
<* Mean/Median C= Compliance Limit (Form 1)
Mean/Median >= Compliance Limit (Form 2)
Mean/Median >= Compliance Limit + S (Form 2)
Meanj'Median = Compliance Limit [2 Sided Alternative)
OK
Cancel
o Specify the Confidence Level; default is 0.95.
o Specify meaningful values for Substantial Difference, S and Action/Compliance
Limit.
o Select the form of Null Hypothesis; default is Mean/Median <= Compliance Limit
(Form 1).
o Click on OK button to continue or on Cancel button to cancel the test.
119
-------
Output for Single Sample Wilcoxon Signed Rank Test (Data with Nondetects)
Arsenic
Single Sample WiIcoxon Signed Rank Test
Raw Statistics
Number of Valid Samples 24
N umber of Disti net Samples 10
Number of Non-Defect Data 13
N umber of Detected Data 11
Percent Non-Defects 54.17%
Minimum Non-detect 0.9
Maximum Non-detect 2
Minimum Detected 0.5
Maximum Detected 3.2
Mean of Detected Data 1.236
Median of Detected Data 0.7
SD of Detected Data Q.9S5
N umber Above Li mil 0
Number Equal Limit 0
Number Below Limit 24
T-plus 0
T-minus 300
HO: Site Median <=6 (Forml)
Large Sample z-Test Value -4.293
Critical Value (0.05) 1.645
P-Value 1
Corel us ion with Alpha = 0.05
Do Not Reject HO. Conclude Mean/Median <= 6
P-Val we > Alpha (0.05)
Data set con tains multiple Nan-Detect values!
All Observations <2 ate treated as Nan-Detects
120
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9.2 Two-Sample Hypotheses Testing Approaches
In this section, the two-sample hypotheses testing approaches as incorporated in ProUCL 4.0 have been
illustrated. These approaches are used to compare the parameters and distributions of the two populations
(e.g., Background vs. AOC) based upon data sets collected from those populations. Both forms (Form 1
and Form 2, Form 2 with Substantial Difference, S) of two-sample hypothesis testing approaches have
been included in ProUCL 4.0. The methods are available for full data sets as well as for data sets with
below detection limit (BDL) values.
• Full - analyzes data sets consisting of all detected values. The following parametric and
nonparametric tests are available:
o Student's t and Satterthwaite tests to compare the means of two populations (e.g.
Background versus AOC).
o F-test to the check the equality of dispersions of two populations.
o Two-sample nonparametric Wilcoxon-Mann-Whitney (WMW) test. This test is
equivalent to Wilcoxon Rank Sum (WRS) test.
o Quantile test is often used to compare upper tails of two data distributions. This test
is normally performed in parallel with WMW test.
• With NDs - analyzes data sets consisting of both nondetected and detected values. The
following tests are available:
o Wilcoxon-Mann-Whitney test. All observations (including detected values) below the
highest detection limit are treated as ND (less than the highest DL) values.
o Quantile test is used to compare upper tails of two data distributions. This test is
performed in parallel with WMW test.
o Gehan's test, useful when multiple detection limits may be present.
The details of these methods can be found in the ProUCL 4.0 Technical Guide and are also available in
EPA (1997, 2006). It is re-stated that the use of informal graphical displays (e.g., side-by-side box plots,
multiple Q-Q plots) should always accompany the formal hypothesis testing approaches listed above.
This is especially warranted when the data sets may consist of observations from multiple populations
(e.g., mixture samples collected from various onsite locations) and outliers.
Note: As mentioned before, it is pointed out that, when one wants to use two-sample hypotheses tests on
data sets with NDs, ProUCL 4.0 assumes that samples from both of the groups have nondetect
observations. This may not be the case, as data from a polluted site may not have any ND observations.
ProUCL can handle such data sets. However, the user will have to provide a ND column (with 0 or 1
entries only) for the selected variable of each of the two groups. Thus when one of the groups (e.g., site
arsenic) has no ND value, the user supplies an associated ND column with all entries equal to "1. " This
will allow the user to compare two groups (e.g., arsenic in background vs. site samples) with one of the
group having some NDs and the other group having all detected data.
121
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9.2.1 Two-Sample Hypothesis Tests for Full Data
Full - This option is used to analyze data sets consisting of all detected values. The following two-sample
tests are available in ProUCL 4.0.
• Student's t and Satterthwaite tests to compare the means of two populations (e.g.,
Background versus AOC).
• F-test is also available to test the equality of dispersions of two populations.
• Two-sample nonparametric Wilcoxon-Mann-Whitney (WMW) test.
• Two-sample quantile test.
• Student's t-Test
o This test can be used to compare the site mean concentration of a COPC with that of
the background mean concentration provided the populations are normally
distributed. The data sets are given by independent random observations, Xu X2, . . .,
Xn collected from a site, and independent random observations, YI, Y2, . . ., Ym
collected from a background population. The same terminology is used for all other
two-sample tests in ProUCL 4.0.
o Student's t-test also assumes that the spread (variance) of the two populations are
approximately equal.
o The F-test can be used to the check the equality of dispersions of two populations.
• Satterthwaite t-Test
o This test is used to compare the population means of two populations when the
variances or Spreads of those populations may not be equal. As mentioned before, the
F-distribution based test can be used to verify the equality of dispersions of two
populations.
• Test for Equality of two Dispersions (F-test)
o This test is used to determine whether the true underlying variances of two
populations are equal. Usually the F-test is employed as a preliminary test, before
conducting the two-sample t-test for testing the equality of means of two populations.
o 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 highly sensitive to departures from normality.
• Two-Sample Nonparametric WMW Test
o This test is used to determine the approximate equality of the two continuous data
distributions. This test also assumes that the shapes (e.g., as determined by spread,
skewness, and graphical displays) of the two populations are roughly equal. The test
is often used to determine if the measures of central locations of the two populations
122
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are significantly different. Specifically, the test can be used to determine if the site
concentrations exceed the background concentrations.
o The Wilcoxon-Mann-Whitney test does not assume that the data are normally or log-
normally distributed. For large samples (e.g., > 20), the distribution of the WMWtest
statistic can be approximated by a normal distribution.
o This test is used to determine if measurements from one population consistently tend
to be larger (or smaller) than those from the other population.
• Two-Sample Quantile Test
o The nonparametric quantile test does not assume that the data are normally or log-
normally distributed. For large samples (e.g., > 20), the distribution of the quantile
test statistic can be approximated by a normal distribution.
o This test is used in parallel with the WMW test. This test is often used in Background
Test Form 1 to determine if the concentrations from the upper tail of site data
distribution are comparable to (lower than or equal to) that of the background data
distribution. The critical values for this Form 1 test are available in EPA, 1994. The
details of the test are given in EPA (1994, 2006).
Note: The use of the tests listed above is not recommended on log-transformed data sets, especially when
the parameters of interests are the population means. In practice, the cleanup and remediation decisions
have to be made in original scale based upon statistics and estimates computed in the original scale. The
equality of means in log-scale does not necessarily imply the equality of means in the original scale. This
topic is discussed in detail in Chapter 3 of the revised background document (EPA, 2002) for CERCLA
sites (currently under revision).
1.
Click on Hypothesis Testing ^ Two Sample
P? ProJJCLJ,0 -
if File Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness-of-Fit
Navigation Panel
HwatiesSTesIno
Single Sample
*"5T *' b
Background UCL Window Help
Name
tTest
Wilcoxon-Mann-Whitney
Qusntile test
® Worksheet wst
0 Worksheet a.wst
2. Select the Full (w/o NDs) option
• To perform a t-test, click on t Test from the drop-down menu.
• To perform a Wilcoxon-Mann-Whitney, click on Wilcoxon-Mann-Whitney from the
drop-down menu list.
• To perform a quantile test, click on Quantile test from the drop-down menu.
123
-------
9.2.1.1 Two-Sample t-Test without NDs
1. Click Hypothesis Testing ^ Two Sample ^- Full (w/o NDs) ^- t Test
P? ProUCL 4.0 - [C:\Narain\PraUCL-Data\Data\WSR EPA (2006).wsQ.
Fi!e Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit Ja^*E^ijt[^|i^ya!i|| Background UCL Window Help
S,nge Sample
Navigation Panel
Name
-i?','<'ort-Sheet >,st
-V WSR EPA (2006) *st
:>I fiJfwfcNOsS M tTat
0
V/SR1
1 044
1093
2 3 r 4 --r •
WSR2 Arsenic D_Arsemc
5.88 1 0
5.8S 1 0
1.4fi 1.7 1
Wiicoxon^lann-Whitney
Quantiie test
2. The Select Variables screen will appear.
• Select variable (variables) from the Select Variables screen.
• Without Group Variable: This option is used when the data values of the variable
(COPC) for the site and the background are given in separate columns.
• With Group Variable: This option is used when data values of the variable (COPC) for
the site and the background are given in the same column. The values are separated into
different populations (groups) by the values of an associated Group Variable. The group
variable may represent several populations (e.g., several AOCs, MWs). The user can
compare two groups at a time by using this option.
• When using this option, the user should select a group variable by clicking the arrow next
to the Group Var option for a drop-down list of available variable. The user selects an
appropriate (meaningful) variable representing groups such as Background and AOC.
The user is allowed to use letters, numbers, or alphanumeric labels for the group names.
o When the options button is clicked, the following window will be shown.
124
-------
Site vs Background Comparison fx]
Substantial Difference. S c
(Used with Test Form 2)
Confidence Coefficient
<~ 99.9% <~ 97.5%
<~ 99.5% ^ 95%
<~ 99% <~ 90%
<•" AOC <= Background (Form 1)
C AOC >= Background (Form 2}
f AOC >= Background + S (Form 2)
C AOC = Background [2 Sided)
OK Cancel
o Specify a useful Substantial Difference, S value. The default choice is 0.
o Choose the Confidence level. The default choice is 95%.
o Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).
o Click on OK button to continue or on Cancel button to cancel the option.
• Click on the OK button to continue or on the Cancel button to cancel the Site versus
Background Comparison.
125
-------
Output for Two-Sample t-Test (Full Data without NDs)
Raw Statistics
Site
Number of Valid Samples 77
Number of Distinct Samples 57
Minimum
Maximum 1S8.S
Mean
Median 0.43
SD 20.02
SE of Mean 2.281
Background
47
40
0.09 0.22
1.33
0.599
3.915
0.54
0.2B4
0.0414
Sitevs Background Two-Sample t-Test
HO: Mu of Site- Mu of Background <= 0
t-Test Critical
Value t [0.050) P-Value
1.134 1.S57 0.129
1.454 1.SG5 0.075
Method DF
Pooled (Equal Variance) 122
Satterthwaite (Unequal Variance) 76.1
Pooled SD 15.799
Conclusion with Alpha = 0.050
* Student t (Pooled] Test: Do Not Reject HO. Conclude Site <= Background
* Satterthwaite Test: Do Not Reject HO. Conclude Site <= Background
9.2.1.2 Two-Sample Wilcoxon-Mann-Whitney (WMW) Test without NDs
1. Click Hypothesis Testing ^- Two Sample ^- Full (w/o NDs) ^- Wilcoxon-Mann-Whitney
Test
P? ProUCL 4.0 - [V/prkSheet-wstJ
f File Edit Configure Summary Statistics ROS Est. NDs Graphs Outiier Tests Goodness-of-Fit EE5ElMEil^MSal Background UCL Window Help
Single Sample
r» K, t ^ -jt- .|, ^
Navigation Panel
Name
v> Worksheet wst
126
-------
2.
The Select Variables Screen will appear.
Select Variables
Variables
Name
Count
Site 1
Background 2
Site 4
10
15
15
'• Without Group Variable
Background / Ambient
Area of Concern / Site
r With Group Variable
| Variabte
Group Var
Background / Ambient
Area of Concern /Site
Optior^s
OK
Cancel
• Select variable (variables) from the Select Variables screen.
• Without Group Variable: This option is used when the data values of the variable
(COPC) for the site and the background are given in separate columns.
• With Group Variable: This option is used when data values of the variable (COPC) for
the site and the background are given in the same column. The values are separated into
different populations (groups) by the values of an associated Group Variable. When using
this option, the user should select a group variable by clicking the arrow next to the
Group Var option for a drop-down list of available variables.
• ProUCL 4.0 has been written using environmental terminology such as performing
background versus site comparisons. However, all the tests and procedures in ProUCL
4.0 can be used for any other application if used properly. The user selects an appropriate
group variable representing groups such as Background and AOC. For other applications
such as comparing a new treatment drug versus older treatment drug, the group variable
may represent the two groups: old drug group and new drug group. The user is allowed to
use letters, numbers, or alphanumeric labels for the group names.
• When the Options button is clicked, the following window is shown.
127
-------
Substantial Difference. S Q
(Used with Test Form 2)
I ""Confidence Coefficient
99.9%
9.5%
r 97.5%
fS" 95%
r 90%
Select Null Hypothesis Form
ADC <= Background (Form 1)
ADC >= Background (Form 2)
f" AOC >= Background + S (Form 2)
AOC = Background [2 Sided)
OK
Cancel
o Specify a Substantial Difference, S value. The default choice is 0.
o Choose the Confidence level. The default choice is 95%.
o Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).
o Click on OK button to continue or on Cancel button to cancel the selected options.
Click on the OK button to continue or on the Cancel button to cancel the Site versus
Background Comparison.
128
-------
Output for Two-Sample Wilcoxon-Mann-Whitney Test (Full Data)
Area of Concern Data: Site
Background Data: Background
Raw Statist
Number of Valid Samples
Number of Distinct Samples
Minimum
Maximum
Mean
Median
SD
SE of Mean
ics
Site
10
9
15
100
48.7
34.5
33.36
10.55
Background
10
8
23
79
49.5
50.5
16.76
5.3
Wilcoxon-Mann-VAwlney (WMW) Test
HO: Mean/Median of Site or AOC <= Mean/Median of Background
Site Rank Sum W-Stat 97.5
WMW Test U-Stat 42.5
WMW Critical Value (0.050) 72
Approximate P-Value 0.727
Concl its ion with Alpha = 0.05
Do Not Reject HO. Conclude She <= Background
9.2.1.3 Two-Sample Quantile Test for Full Data without NDs
As mentioned before, the quantile test is often used in parallel with the WMW test. Typically, both tests
are performed on the same data set before coming to the conclusion about comparability (or non-
comparability) of the data distributions of the two populations.
1.
Click Hypothesis Testing ^ Two Sample ^ Full (w/o NDs) ^ Quantile test
°? File Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness-of-Fit
Background
1 1 23j
2 36
1
Site
15
15
2 3
Background D_Background
Single Sample
tTest
4 - •» •• With NDs t Wilcoxon-Mann-Whitnev JO
Site D_Site ' "
5 0
10 0
129
-------
2. The Select Variables Screen shown below will appear.
Variables
Name
ID
I Count (? Without Group Variable
Site 1
Background 2
Site 4
10
15
15
» Background / Ambient
» Area of Concern / Site
With Group Variable
| Variable
Group Var
Background / Ambient
Area of Concern / Site
Options
OK
Cancel
• Select variable (variables) from the Select Variables screen.
• Without Group Variable: This option is used when the data values of the variable
(COPC) for the site and the background are given in different columns.
• With Group Variable: This option is used when data values of the variable (COPC) for
the site and the background are given in the same column. The values are separated into
different groups by using the values of the associated Group Variable. When using this
option, the user should select a group variable by clicking the arrow next to the Group
Var option for a drop-down list of available variables. The user selects an appropriate
group variable representing groups such as Background and AOC. The user is allowed to
use letters, numbers, or alphanumeric labels for the group names.
• When the Options button is clicked, the following window will be shown.
Quantile Test Options
Select Confidence Coefficient
r 99% r 97.5%
ff 95% r 90%
OK
Cancel
130
-------
o Choose the Confidence level; the default choice is 95%.
o Click on OK button to continue or on Cancel button to cancel the option.
Click on the OK button to continue or on the Cancel button to cancel the Site versus
Background Comparison.
Output for Two-Sample Quantile Test (Full Data)
Area of Concern D ata: Site
Background Data: Background
Raw Statistics
Site
Number of Valid Samples
Number of Distinct Samples
Minimum
Maximum
Mean
Median
SD
SE of Mean
10
9
15
1CC<
48.7
34.5
33.3S
10.55
Background
10
8
23
79
49.5
50.5
16.76
5.3
Quantile Test
HO: Site Concentration <= Back ground Concentration (Form 1)
Appraxi mate R Val ue [0.043) 4
Approxi mate K Val ue (0.043) 4
Number of Site Observations in 'R' Largest 3
Calculated Alpha 0:0433
C one I us ion with Alpha = 0,013
Do Not Reject H 0. Perform Wi I c oxon-Marm-YVrtitney Ranked Sum Test
131
-------
9.2.2 Two-Sample Hypothesis Testing for Data Sets with Nondetects
1. Click Hypothesis Testing^- Two Sample
P? ProUCL 4,0 - [C;\Narain\ProUCL-Data\Data\{iuantile,wstI
I File Edit Configure Summary Statistics ROS Est. NDs Graphs Outiier Tests Goodness-of-Fit EESSgjljgHsl Background UCL Window Help
r---| Single Sample > (
0
Background
1 - ' 2$
1
Site
15
2 3
background D_Background
3 0
4
Site
Full {w/o NDs) ^
Wilcoxon-Mann-Whitney
Gehan
Quantile Test
5 0
3 0 10 0
2. Select the With NDs option. A list of available tests will appear (shown above).
• To perform a Wilcoxon-Mann-Whitney test, click on Wilcoxon-Mann-Whitney from the
drop-down menu list.
• To perform a Gehan test, click on Gehan from the drop-down menu.
• To perform a quantile test, click on Quantile Test from the drop-down menu.
9.2.2.1 Two-Sample Wilcoxon-Mann-Whitney Test with Nondetects
1. Click Hypothesis Testing ^- Two Sample ^- With NDs ^- Wilcoxon-Mann-Whitney
ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\Quairtile,wstj
File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit |jggg^gl| Background UCL Window Help
| a|BJmj d Single Sample
0
Background
1 ' 21
2 ' 36
1
Ste
15
15
2
Background
3
3
3
^Background
0
0
4
Ste
5
10
5
D_Ste
0
0
Full (w/o NDs)
With NDs H Wlcoxon-Mann-WfhlSiey
Gehan
Quantile Test
132
-------
2. The Select Variables Screen shown below will appear.
Variables
Name
ID
I Count (? Without Group Variable
Site 1
Background 2
Site 4
10
15
15
» Background / Ambient
» Area of Concern / Site
With Group Variable
| Variable
Group Var
Background / Ambient
Area of Concern / Site
Options
OK
Cancel
• Select variable (variables) from the Select Variables screen.
• Without Group Variable: This option is used when the data values of the variable
(COPC) for the site and the background are given in separate columns.
• With Group Variable: This option is used when data values of the variable (COPC) for
the site and the background are given in the same column. The values are separated into
different populations (groups) by the values of an associated Group Variable. When using
this option, the user should select a group variable by clicking the arrow next to the
Group Var option for a drop-down list of available variables. The user selects an
appropriate variable representing groups such as Background and AOC. The user is
allowed to use letters, numbers, or alphanumeric labels for the group names.
• When the Options button is clicked, the following window will be shown.
133
-------
Site vs Background Comparison
Substantial Difference: S
(Used with Test Form 2)
^Confidence Coefficient
0
r 99.9%
r 99.5%
r 99%
r 97.5%
(? 95%
r 90%
Select Null Hypothesis Form
(• AOC <= Background (Form
C AOC >= Background (Form 2)
C AOC >= Background + S (Form 2)
C AOC = Background (2 Sided)
OK
Cancel
o Specify a meaningful Substantial Difference, S value. The default choice is 0.
o Choose the Confidence level. The default choice is 95%.
o Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).
o Click on the OK button to continue or on the Cancel button to cancel the selected
options.
• Click on OK button to continue or on Cancel button to cancel the Site versus Background
Comparison.
134
-------
Output for Two-Sample Wilcoxon-Mann-Whitney Test (with Nondetects)
Area of Concern Data: Site
Background Data: Background
Raw Statistics
Site Background
Number of Valid Samples 15 15
Number of Non-Detect Data 8 12
N u mber of Detect D ata 7 3
Minimum Non-Detect 5 3
Maximum Non-Detect 300 25
Percent Non detects 53.33% 80.00%
Minimum Detected 11 8
Maximum Detected 200 22
Mean of Detected Data 74.43 15
Median of Detected Data 70 15
SD of Detected Data 6S.42 7
WilcQior-Manr-VVhrtney Srtevs Background Test
Allobservations <=300(Max DL}
Wi Icoxon- Man n-Whitney (WMW) Test
HO: Mean/Median of Site or AOC <= Mean/Median of Background
Site Rank Sum W-Stat 232.5
WMWTestU-Stat 112.5
WMW C ritical Va I ue (0.050) 152
Approxi mate P-Va I ue 0.508
Conclusion with Alpha = 0.05
Do Not Reject HO. Conclude Site <= Background
Note: In the WMW test, all observations below the largest detection limit are considered as NDs
(potentially including some detected values) and hence they all receive the same average rank. This
action may reduce the associated power of the WMW test considerably. This in turn may lead to incorrect
conclusion. As mentioned before, all hypotheses testing approaches should be supplemented with
graphical displays such as Q-Q plots and box plots. When multiple detection limits are present, the use of
the Gehan test is preferable.
135
-------
9.2.2.2 Two-Sample Gehan Test for Data Sets with Nondetects
1. Click Hypothesis Testing ^ Two Sample ^- With NDs ^- Gehan
P? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\Quantile.w5t]
•g File Edit Configure Summary Statistics ROS Est, NDs Graphs Gutter Tests Goodness-of
Hypothesis Testing
Background UCL Window Help
mjjpj
Background
Site
Background
15
15
D_Background
Jitn NDs ^ I Wilcoxon-Mann-Whitnev
Site
D Site
Gehan
QuanflleTest
2. The Select Variables Screen will appear.
Variables
Name
ID
I Count (? Without Group Variable
Site 1
Background 2
Site 4
10
15
15
» Background / Ambient
» Area of Concern / Site
With Group Variable
| Variable
Group Var
Background / Ambient
Area of Concern / Site
Options
OK
Cancel
• Select variable (variables) from the Select Variables screen.
• Without Group Variable: This option is used when the data values of the variable
(COPC) for the site and the background are given in separate columns.
• With Group Variable: This option is used when data values of the variable (COPC) for
the site and the background are given in the same column. The values are separated into
different populations (groups) by the values of an associated Group Variable. When using
this option, the user should select a group variable by clicking the arrow next to the
Group Var option for a drop-down list of available variables. The user selects a group
variable representing groups such as Background and AOC.
• When the Options button is clicked, the following window will be shown.
136
-------
Site vs Background Comparison
Substantial Difference: S
(Used with Test Form 2)
^Confidence Coefficient
0
r 99.9%
r 99.5%
r 99%
r 97.5%
(? 95%
r 90%
Select Null Hypothesis Form
(• AOC <= Background (Form
C AOC >= Background (Form 2)
C AOC >= Background + S (Form 2)
C AOC = Background (2 Sided)
OK
Cancel
o Specify a Substantial Difference, S value. The default choice is 0.
o Choose the Confidence level. The default choice is 95%.
o Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).
o Click on OK button to continue or on Cancel button to cancel selected options.
Click on the OK button to continue or on the Cancel button to cancel the Site versus
Background Comparison.
137
-------
Output for Two-Sample Gehan Test (with Nondetects)
Area of Concern Data: Site
Background Date Background
Raw Statistics
Number of Valid Samples
Number of Non-Detect Data
Number of Detect Data
Minimum Non-Detect
Maximum Non-Detect
Percent Non detects
Minimum Detected
Maximum Detected
Mean of Detected Data
Median of Detected Data
SD of Detected Data
Site vs Background Gehan Test
H0: M u of Site or AO C > = M u of background
Gehan z Test Value 1.769
Critical z (0,05) -1.645
P-Value 0.962
Conclusion with Alpha = 0.05
Do Not Reject HO. Conclude Site >= Background
P-Value >= alpha (Q.05)
Site
10
2
8
4
35
20.00%
2
43
23.63
22.5
14.74
Background
10
4
6
4
25
40.00%
1
27
12.17
11
3.S42
9.2.2.3 Two-Sample Quantile Test for Data Sets with Nondetects
Quantile test as described in EPA (1994) has been included in ProUCL 4.0. The detailed power of the test
with many ND values is not well studied. The conclusion of this test should also be supplemented with
graphical displays. The use of the Gehan test is preferred when the data set may consist of many NDs
with multiple detection limits.
138
-------
1. Click Hypothesis Testing ^- Two Sample ^- With NDs ^- Quantile Test
f? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\Quantile.wst]
PI 1 Ch u=l I H i IT! 1 ^Tl Single Sample ^ (
0
1 234 a WffJSt!l^fD Wilcoxon-Mann-Whitney 0
Background Site Background DJtackground Site D_Site Gehan
1 n 15 3 050 K^^X^^^^^H
2 36 15| 3 0 10 0
2. The Select Variables Screen will appear.
^^^^H Select Variables ^^^^H
Variables
Name ID | Count * Without Group Variable
Background 2 IS » I Background / Ambient T
Site 4 15
» Area of Concern / Site
f~ With Group Variable
» | Variable |~~
Group Var
Background / Ambient
Area of Concern / Site
Options
OK Cancel
• Select variable (variables) from the Select Variables screen.
• Without Group Variable: This option is used when the data values of the variable
(COPC) for the site and the background are given in separate columns.
• With Group Variable: This option is used when data values of the variable (COPC) for
the site and the background are given in the same column. The values are separated into
different populations (groups) by the values of an associated Group Variable. When using
this option, the user should select a group variable by clicking the arrow next to the
Group Var option for a drop-down list of available variables. The user selects an
appropriate group variable representing groups such as Background and AOC.
• When the Options button is clicked, the following window will be shown.
139
-------
•JJuarrtile Test.{|jjtiori5.
Select Confidence Coefficient
r 99% r 97.5%
<• 95% C 90%
OK
Cancel
o Choose the Confidence level; the default choice is 95%.
o Click on OK button to continue or on Cancel button to cancel the option.
• Click on the OK button to continue or on the Cancel button to cancel the Site versus
Background Comparison.
Output for Two-Sample Quantile Test (with Nondetects)
Area of Concern Date: Site
Background Data: Background
Raw Statistics
Number of Valid Samples
Number of Non-Detect Data
Number of Detect Data
Minimum Non-Detect
Maximum Non-Detect
Percent Non detects
Minimum Detected
Maximum Detected
Mean of Detected Data
Median of Detected Data
SD of Detected Data
Quantile Test
HO: Site Concentration <= Background Concentration (Form 1)
Approximate R Value (0.05) 4
Approximate K Value (0.05) 4
Number of Site Observations in 'FT Largest 4
Non-Detect Values in the'R' Largest- Cannot eanfjleieQuanljfe Test
Site
15
8
7
5
300
53.33%
11
200
74.43
70
6S.42
Background
15
12
3
3
25
80.00%
8
22
15
15
-
140
-------
141
-------
Chapter 10
Background Statistics
This chapter illustrates the computations of various parametric and nonparametric statistics and upper
limits that can be used as estimates of background threshold values (BTVs) and other not-to-exceed
values. The BTV estimation methods are available for all data sets with and without nondetect (ND)
observations. The details of those methods are given in Chapter 5 (full data sets without NDs) and
Chapter 6 (data sets with NDs) of the revised background document for CERCLA sites (EPA, 2002).
Technical details can also be found in the Technical Guide associated with ProUCL 4.0. For each selected
variable, this option computes various upper limits such as UPLs, UTLs, and upper percentiles to estimate
the background threshold values (BTVs) and other compliance limits that are used in site versus
background evaluations.
As before, two choices for data sets are available to compute background statistics:
• Full - computes background statistics for a Full data set without any NDs.
• With NDs - computes background statistics for a data set with nondetected as wells as
detected values. Multiple detection limits are allowed.
The user specifies the confidence level (probability) associated with each interval estimate. The
reasonable confidence level as incorporated in ProUCL 4.0 represents a number in the interval [0.5, 1),
0.5 inclusive. The default choice is 0.95.
For data sets with and without NDs, ProUCL 4.0 can compute the following statistics that can be used as
estimates of BTVs and not-to-exceed values.
• Parametric and nonparametric upper percentiles.
• Parametric and nonparametric upper prediction limits (UPLs) for a single observation,
future or next k (> 1) observations, mean of next k observations. Here future k, or next k
observations may also represent k observations from another population (e.g., site)
different from the sampled (background) population (used to compute UPLs, UTLs).
• Parametric and nonparametric upper tolerance Limits (UTLs).
• Nonparametric IQR-based upper limits.
142
-------
10.1 Background Statistics for Full Data Sets without Nondetects
1. Click Background ^ Full (w/o NDs) Background Statistics
P? PrpUCL 4.0 - {C:\Narain\ProUCL-Oata\Data\Cadraium,(wstJ
aQ File Edit Configure Summary Statistics ROSEst, NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing |
I LICL Window Help
Navigation Panel [
Name
Q Worksheet wst
i$ Cadmium, vvst
D 1
Pop-ID Cadmium
" T| 0.24
i" 0.26
With NDs Background Statistics
Normal
Gamma
Non-Parametric
AH
2. Select Full Background Statistics.
• To compute the background statistics assuming the normal distribution, click on Normal
from the drop-down menu list.
• To compute the background statistics assuming the gamma distribution, click on Gamma
from the drop-down menu list.
• To compute the background statistics assuming the lognormal distribution, click on
Lognormal from the drop-down menu list.
• To compute the background statistics using distribution-free nonparametric methods,
click on Non-Parametric from the drop-down menu list.
• To compute and see all background statistics available in ProUCL 4.0, click on the All
option from the drop-down menu list.
10.1.1 Normal or Lognormal Distribution
1. Click Background ^- Full (w/o NDs) Background Statistics ^- Normal or Lognormal
P? PrpUCL 4.0 -JC:VNarain\ProUCL-pata\pata\qadroium,yretI
•y File Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing
C'lvl ta|"|rr:| n:
Navigation Panel
Background
RiwnfunSaag'iundStattsticB M
Name
c; WorkSheetwst
^Cadmium.wst
0 1
Pop-ID Cadrriurr
V 0 24
1 026
LICL Window Help
With NDs Background Statistics
Gamma
Lognormal '(12
Non-Parametric
All
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables and select an appropriate group variable.
143
-------
When the option button is clicked, the following window will be shown.
1 Background Statistics Options
Confidence Level
Coverage
!Baa
0.9
Different or Future K Values I 1
Number of Bootstrap Operations
OK
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Coverage coefficient (for a percentile) needed to compute UTLs.
Coverage represents a number in the interval (0.0, 1). The default choice is 0.9.
Remember, a UTL is an upper confidence limit (e.g., with confidence level = 0.95)
for a 90% (e.g., with coverage = 0.90) percentile.
o Specify the Different or Future K Values. The default choice is 1. It is noted that
when K = 1, the resulting interval will be a UPL for a single future (or site)
observations. In the example shown above, a value of K = 1 has been used.
o Specify the Number of Bootstrap Operations (resamples). The default choice is
2000.
o Click on OK button to continue or on Cancel button to cancel this option.
• Click on OK button to continue or on Cancel button to cancel the Background Statistics
Options.
144
-------
Output Screen for Normal Distribution (Full)
Cadmium
Raw Statistics
Number of Valid Samples 33
Number of Unique Samples 28
Minimum 0.094
Maximum 21
Second Largest 6.967
Mean 3.024
FirstQuantile 0.35
Median 2.5
Third Quantile 3.8
SD 3.751
Coefficient of Variation 1.24
Skewness 3.607
Normal Distribution Test
Shapiro Wilfc Test Statistic 0.642
5% Shapiro Wilk Critical Value 0.931
Data not Normal at 5Z Sgnifcanoe Lesel
Normal Distribution Test
Shapiro Wilk Test Statistic 0.642
5% Shapi ro Wi I k C ritical Val ue 0.931
Data not Normal at 5% Significance
Background Statistics Assuming Normal Distribution
90% Pereentile (z) 7.83
95% Percerrtile (z) 9.193
99% Pereentile (z) 11.75
95% UTL with 90% Coverage 9.549
95% UPL (t) 9.472
Note: UPL (or upper perc en tile for gamma distributed
data) represents a preferred estimate of BTV
145
-------
Output Screen for Lognormal Distribution (Full)
Log-Transformed Statistics
Number of Valid Samples 33
Number of Unique Samples 28
Minimum -2.364
Maximum 3.045
Second Largest 1.941
Mean 0.459
First Quantile 1.653
Median 0.316
Third Quantile 0.709
SD 1.308
Lognormal Distribution Test
Shapiro Wilk Test Statistic Q.9Q9
5% Shapiro Wilk Critical Value 0.931
Data not Log normal at 551 Significance Lewei
Background Statistics Assuming Lognormal Distribution
30% Percent! le (z) S.45B
95% Percentile (z) 13.6
99% Percent! le (z) 33.18
95% UPL 15
95% UTL with 90% Coverage 15.4
Background Statistics Assuming Lognormal Distribution
30% Percerttile (z) 8.45S
95% Percentile (z) 13.6
99% Percentile (z) 33.18
95% UPL 15
95% UTL with 90% Coverage 15.4
Some Nonparametric Background Statistics
95% Chebyshev UPL 19.62
95% Bootstrap BCA UTL with 90% Coverage 6.737
95% Percentile Bootstrap UTL with 90% Coverage 6.967
Note: UPL (or upper percentile tor gamma distributed
data) represents a preferred estimate of BTV
146
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10.1.2 Gamma Distribution
1. Click Background ^- Full (w/o NDs) Background Statistics ^- Gamma
P3 ProUCL4.0 - [C:\Narain\ProUCL-Data\Data\Cadmium.wst]
1 File Edit Configure Summary Statistics ROS Est. NDs Graphs Gutter Tests Goodness-of-Fit Hypothesis Testing
UCL Window Help
Navigation Panel I
Name
OWorkSheet.wst
•_;; Cadmium.wst
Pop-ID
0.24
1 0.26
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables, and select a proper group variable.
• When the option button is clicked, the following window will be shown.
j§l Background Statistics Options
Confidence Level
Coverage
Different or Future K Values [ 1
Number of Bootstrap Operations
OK
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Coverage level; a number in interval (0.0, 1). Default choice is 0.9.
o Specify the next K. The default choice is 1.
o Specify the Number of Bootstrap Operations. The default choice is 2000.
o Click on OK button to continue or on Cancel button to cancel the option.
Click on OK button to continue or on Cancel button to cancel the Background Statistics
Options.
147
-------
Output Screen for Gamma Distribution (Full)
Cadmium
Raw Statistics
N umber of Val id Samples 33
Number of Unique Samples 28
Minimum 0.094
Maximum 21
Second Largest 6,967
Mean 3.024
First Quarrtile 0.35
Median 2.5
Third Quantile 3.8
SD 3.751
Gamma Distribution Test
k hat 0.902
Theta hat 3.352
nu hat 59.54
k star 0.84
Theta star 3.598
nu star 55.46
95% Percentile of Chisquare (2k) 5.356
A-D Test Statistic 0.794
5% A-D Critical Value 0.78
K-S Test Statistic 0.143
5% K-S Critical Value 0.158
Data fol low Appr. Gamma Distribution at5X Significance Level
Background Statistics Assuming Gamma Distribution
90% Percentile 7.265
95% Percentile 9.637
99% Percentile 15.22
Nonparametic Background Statistics
95% Chebyshev UPL 19.62
95% BCA Bootstrap UTL with 90% Coverage 6.737
95% Bootstrap (%) UTL with 90% Coverage 6.737
Note: UPL (or upper perceri tile far gamma distributed
data) represents a preferred estimate of BTV
148
-------
10.1.3 Nonparametric Methods
1. Click Background ^- Full (w/o NDs) Background Statistics ^- Non-Parametric
E? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
my File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing
UCL Window Help
jgJjaJ.jBlBl.gil el
Navigation Panel I
Name
•rf Worksheet, wst
O MW89.wst
Normal
With NDs Background Statistics >\ Gamma
Lognormal
^^
All
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables, and select a proper group variable.
• When the option button is clicked, the following window will be shown.
al Background Statistics Options
Confidence Level
- n x
Coverage [ 0.9
Different or Future K Values [ 1
Number of Bootstrap Operations f
OK
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9.
o Specify the next K. The default choice is 1.
o Specify the Number of Bootstrap Operations. The default choice is 2000.
o Click on the OK button to continue or on the Cancel button to cancel the option.
• Click on OK button to continue or on Cancel button to cancel the Background Statistics
Options.
149
-------
Output Screen for Nonparametric Option (Full)
Cadmium
Some Non-Parametric Statistics
Number of Valid Samples 33
Number of Unique Samples 28
Minimum 0.054
Maximum 21
Second Largest 6.967
Mean 3.024
First Quarttile 0.35
Median 2.5
Third Quantile 3.8
SD 3.751
Variance 14.07
Coefficient of Variation 1.24
Skewness 3.607
Mean of Log-Transformed data 0.459
SD of Log-Transformed data 1.308
Data Follow Appr. Gamma Distribution al 5% Spaieaice Lewd)
Non-Parametric Badcpound Statistics
90% Percentile 5.27
95% Percentile 6.468
99% Percentile 16.37
95% UTL with 90% Coverage
Order Statistic 32
Achieved CC 0.969
UTL S.9S7
95% BCA Bootstrap LITL with 90% Coverage 6.737
95% Percentile Bootstrap UTL with 90% Coverage 6.967
95%UPL 11.18
35% Chebyshev UPL 19.62
Upper Limit Based upon IQR 8.975
Note: UPL (or upper per c en tile far gamma distributed
data) represents a preferred estimate at BTV
150
-------
10.1.4 All Statistics Option
1. Click Background ^ Full Background Statistics ^ All
ProllCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-fit Hypothesis Testing |jj Q UCL Window Help
•
With NDs Background Statistics
Navigation Panel
Name
•cJWorkSheet.wst
•3 MW89-WSI
Well ID
1
460
1 527
Gamma
Lognormal
Non-Parametric
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables, and select a proper group variable.
• When the option button is clicked, the following window will be shown.
Background Statistics Options
Confidence Level
Coverage I
Different or Future K Values 1
Number of Bootstrap Operations
OK
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Coverage level; a number in the interval (0.0, 1). Default is 0.9.
o Specify the next K. The default choice is 1.
o Specify the Number of Bootstrap Operations. The default choice is 2000.
o Click on OK button to continue or on Cancel button to cancel the option.
• Click on OK button to continue or on Cancel button to cancel the Background Statistics
Options.
151
-------
Output Screen for All Statistics Option (Full)
Cadmium
General Statistics
Total Number of Samples 33
Number of Unique Samples 28
Raw Statistics
Minimym 0.094
Maximum 21
Second Largest 6.967
First Quantile 0.35
Median 2.5
Third Quantile 3.S
Mean 3.024
SD 3.751
Coefficient of Variation 1.24
Skewness 3.607
Log-Transformed Statistics
Minimum -2.3S4
Maximum 3.045
Second Largest 1.941
First Quantile -1.13
Median 0.916
Third Quantile 1.334
Mean 0.459
SD 1.308
Background Statistics
Normal Distribution Test
Shapiro Wilk Test Statistic 0.642
Shapiro Wilk Critical Value 0.931
Oats not Normal si 5% Sjf ^ei
Assuming Normal Distribution
95V. UTL with 90% Coverage 9.549
95% UPL (t) 9.472
90% Percerttile (z) 7.83
95% Percentile (z) 9.193
99% Percentile (z) 11.75
Lognormal D istribution Test
Shapiro Wilk Test Statistic 0.909
Shapiro Wilk Critical Value 0.931
Data imt Lognccmal al§X Sifwieanee Lwd
Assuming Lognormal Distribution
95% UTL with 90% Coverage 15.4
95% UPL (t) 15
90% Percentile (z) 8.458
95% Percentile (z) 13.6
99% Percentile (z) 33.1S
Gamma Distribution Test
k star 0.84
Theta Star 3.598
mi star 55.46
Data Distribution Tests
Data Follow Appr. Gamma Distribution at 5% Significance Level
A-D Test Statistic 0.794
5% A-D Critical Value 0.78
K-S Test Statistic 0.143
5% K-S Critical Value 0.158
Data follow Appx. Gamma Distribution at 5% Significance Level
Assuming Gamma Distribution
90% Percentile 7.265
95% Percentile 9.637
99% Percentile 15.22
Nonparametric Statistics
90% Percentile 5.27
95% Percentile 6.468
99% Percentile 16.37
95% UTL with 90% Coverage 6.967
95% Percentile Bootstrap UTL with 90% Coverage 6.967
95% BCA Bootstrap UTL with 90% Coverage 6.737
95% UPL 11.18
95% Chebyshev UPL 19.62
Upper Threshold Limit Based upon IQR 8.975
Note: UPL (or upper percentile for gamma distributed data) represents a preferred estimate of BTV
152
-------
10.2 Background Statistics with NDs
1. Click Background ^ With NDs Background Statistics
P? ProUCL 4.0 - [C:\Narain\ProUCL Data\Data\MW89.wst]
•y File Edit Configure Summary Statistics ROSEst, NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing K CT UCL Window Help
Ell's! OlHlffl! D| Full (w/o NDs) Background Statistics >
Navigation Panel
Name |
ijj Worksheet wst
& MW89.wst
1
2
3
0
Well ID
1
Mn
1 460
1!" 527
1
579
2
3
4
5 6 ' — , 8 S—
I
I
Normal
Ga
Loc
NO!
Al
nma
normal
-Parametric
12
2. Select the With NDs Background Statistics option.
• To compute the background statistics assuming the normal distribution, click on Normal
from the drop-down menu list.
• To compute the background statistics assuming the gamma distribution, click on Gamma
from the drop-down menu list.
• To compute the background statistics assuming the lognormal distribution, click on
Lognormal from the drop-down menu list.
• To compute the background statistics using distribution-free methods, click on Non-
Parametric from the drop-down menu list.
• To compute all available background statistics in ProUCL 4.0, click on the All option
from the drop-down menu list.
10.2.1 Normal or Lognormal Distribution
1. Click Background ^- With NDs Background Statistics ^- Normal or Lognormal
t? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
•5 File Edit Configure Summary Statistics ROSEst. NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing ESS J| UCL Window Help
Full (w/o NDs) Background Statistics T~L
Navigation Panel
Name
•3WorkSh.8flt.wst
<•) MW89.wst
1! 527]
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables, and select a proper group variable.
• When the option button is clicked, the following window will be shown.
153
-------
? Background Statistics Options
Confidence Level
Coverage
Different or Future K Values | 1
Number of Bootstrap Operations
OK
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9.
o Specify the next K. The default choice is 1.
o Specify the Number of Boostrap Operations. The default choice is 2000.
o Click on the OK button to continue or on the Cancel button to cancel the option.
Click on OK button to continue or on Cancel button to cancel the Background Statistics
Options.
154
-------
Output Screen for Normal Distribution (with NDs)
Arsenic
Raw Statistics
Total Number of Data 24
Number of Non-Deteet Data 13
Number of Detected Data 11
Minimum Detected 0.5
Maximum Detected 3.2
Percent Non-Detects 54.17%
Minimum Norv-detect 0.3
Maximum Non-detect 2
Mean of Detected Data 1.236
SD of Detected Data 0.965
Normal Distribution Test with Detected Values Oirfy
Shapiro Wilk Test Statistic 0.777
5% Shapiro Wilk Critical Value 0.85
Data not Normal at 5% Significance Level
Normal Distri but ion Test with Delected Values Only
Shapiro Wilk Test Statistic 0.777
5% Shapiro Wilk Critical Value 0.85
Data mot Normal at SX Signtirance Lewd
Background Statistics Assuming Normal Distribution
DU2 Substitution Method
Mean 1.002
SD 0.633
95% UTL 90% Coverage 2.2%
95% UPL (t) 2.224
90% Percentile (z) 1.897
95% Percentile (z) 2.151
93% Percentile (z) 2.627
Note: DU2 is not a recommended melhod.
Maximum Likelihood Estimate (MLE) Method
MLE Method is Not Appliable for This Data
155
-------
Maximum Likelihood Estimate (MLE) Method
MLE Method is Not Appliable for This Data
Kaplan-Meier (KM) Method
Mean 0.949
SD 0.713
SE of Mean 0.165
95% UTL with 30% Coverage 2.27
95% KM U PL (t) 2.196
35% KM Chebyshev UPL 4.121
90% Pereentiie (z) 1.863
95% Percentile (z) 2.122
99% Pereentiie (z) 2.608
Note: UPL(or upperpercentilefor gamma distributed data) represents a
preferred esti mate of BTV For ma Example: KM-UPL may be used
when muIti pie Detection limrts are present
156
-------
Output Screen for Lognormal Distribution (with NDs)
Log Transformed Statistics
Number of Valid Samples 24
Number of Unique Samples 10
Minimum -0.693
Maximum 1.163
Second Largest 1.03
Mean 0.215
First Quantile 0.693
Median 0.203
Third Quantile -0.693
SD 0.574
Lognormal Distribution Test
Shapiro Wilk Test Statistic 0.906
5% Shapiro Wilk Critical Value 0.916
D ata Log normal at 5X Lori
Background Statistics Assuming Lognormal Distribution
90% Percentile (z) 2.587
95% Percent! le (z) 3.186
99% Percentile (z) 4.711
95% LI PL 3.3S3
95% UTL with 90% Coverage 3.59
Some Nonparametric Background Statistics
95% Chebyshev UPL 4.825
95% Bootstrap BCA UTL with 90% Coverage 1.8
95% Percentiie Bootstrap UTL with 90% Coverage 3.04
Note: UPL (or upper per c en tile far gamma distributed
data] represents a preferred estimate of BTV
157
-------
10.2.2 Gamma Distribution
1. Click Background ^- With NDs Background Statistics ^- Gamma
E? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
•B File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-F
t Hypothesis Testing 1
^^^^^^2]
UCI Window Hftp
nlCll Pl|HlrTll Fll Fu" 'w/0 NDs' Background Statistics > 1
Navigation Panel 1
Name |
QWorkShaet.wst
•3 MW89.wst
1
2
3
D
Well ID
1
1
1
1
Mr
460
527
579
2
3
4
5
6
^m^m
^Ki^^Ki^^
Lognormal
Non -Parametric
All 1
2.
The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables and select a proper group variable.
• When the option button is clicked, the following window will be shown.
Background Statistics (Gamma)
Confidence Level
Coverage
OK
0.9
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9.
o Click on the OK button to continue or on the Cancel button to cancel option.
Click on OK button to continue or on Cancel button to cancel the Background Statistics.
158
-------
Output Screen for Gamma Distribution (with NDs)
24
13
11
0.5
3.2
54.17%
0.9
2
1.236
0.9S5
Arsenic
Raw Statistics
Total Number of Data
Number of Non-Deiect Data
Number of Detected Data
Minimum Detected
Maximum Detected
Percent Non-Deteots
Minimum Non-detect
Maximum Non-detect
Mean of Detected Data
SD of Detected Data
Gamma Distribution Test with Detected Values Only
kstar 1.702
Theta star 0.727
rm star 37.44
95% Percentile of Chisquare (2k) 8.503
A-D Test Statistic 0.7S7
5% A-D Critical Value 0.738
K-S Test Statistic 0.254
5% K-S Critical Value 0.258
Data follow Appr. Gamma Distribution at 5% Significance Levd
Background Statistics Assuming Gamma Dtsiribeiiioci
Gamma ROS Statistics with Extrapolated Data
Mean 1.263
Median 1.213
SD O.S52
k Star 3.974
Theta Star 0.318
mi Star 190.8
95% Percenti le of Chisquare (2k) 15.43
90% Percentile 1.861
95% Percentile 2.16
99% Percentile 2.801
Kaplan Merer (KM) Method
Mean 0.949
SD 0.713
SE of Mean 0.165
95% UTL 90% Coverage 2.27
95% KM Chebyshev UPL 4.121
95%KMUPL(t) 2.196
90% Percentile 1.863
95% Percentile (z) 2.122
99% Percentile (z) 2.608
Note: UPL(or upperpercentileforgamma distributed data) represents a
preferred estimate of BTV. For an Example: KM-UPL may be used
when multiple detection limits are present
159
-------
10.2.3 Nonparametric Methods (with NDs)
1. Click Background ^- With NDs Background Statistics ^- Non-Parametric
File Edit Configure Summarv Statistics ROS Est. NDs Graphs Gutter Tests Gtradness-ot-Fit Hypothesis Testing
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables and select a proper group variable.
• When the option button is clicked, the following window will be shown.
Background Statistics (Gamma)
Confidence Level
Coverage
Ifflaa
OK
c.s
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Coverage level; a number in interval (0.0, 1). Default choice is 0.9.
o Click on the OK button to continue or on the Cancel button to cancel the option.
• Click on OK button to continue or on Cancel button to cancel the Background Statistics.
160
-------
Output Screen for Nonparametric Methods (with NDs)
Arsenic
Total Nymberof Data 24
Number of NorvDetect Data 13
Number of Detected Data 1 1
Minimum Detected 0.5
Maximum Detected 3.2
Percent Norr Detects 54.17%
Minimum Non-detect 0.9
Maximum Non-{fetect 2
Mean of Detected Data 1 .236
SD of Detected Data 0.965
Mean of Log-Transformed Detected Data -0.0255
SD of Log-Transformed Detected Data 0.694
Data Follow Appr. Gamma Distribution at 5%. Significance Level
Nonparametric Background Statistics
X UTLwith 30%. towerage
Order Statistic 23
Achieved CC 92.02
UTL 2.8
Largest Non-detect at 0 rtier 22
Kaplan-Meter (KM) Method
Mean 0.949
SD 0.713
Standard Error of Mean 0. 1 65
95% UTL 90% Coverage 2.27
95% KM Chebyshev UPL 4.121
95% KM LI PL (t) 2.1%
90% KM Percentile (z) 1.863
95% KM Percent! le (z) 2.122
99% KM Percentile (z) 2.608
Note: U PL (or upper percent! le for gamma distributed data) represents m
preferred esti mate of BTV. For an Example: KM-UPL may be used
when multi pie detection limits are present
161
-------
10.2.4 All Statistics Option
1. Click Background >• With NDs Background Statistics >• All
(? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
•P File Edit Configure Summary Statistics ROS Est. NDs Graphs Outter Tests Goodness-of-Fit Hypothesis Testing
UCL Window Help
Navigation Panel
Name
<3 Worksheet wst
© MW89.wst
WslllD
Mn
460
579
Full (w/o NDs) Background Statistics f f
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables, and select a proper group variable.
• When the option button is clicked, the following window will be shown.
Background Statistics
Confidence Level 0.95
Coverage 09
Next K
OK
11
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9.
o Specify the Next K. The default choice is 1.
o Click on the OK button to continue or on the Cancel button to cancel the option.
Click on OK button to continue or on Cancel button to cancel the Background Statistics.
162
-------
Output Screen for All Statistics Option (with NDs)
Ar:
General Statistics
N umber of Val id Samples 24
Number of Unique Samples 8
Raw Statistics
Minimum Detected 0.5
Maximum Detected 3.2
Mean of Detected 1.236
SD of Detected 0.965
Minimum Non-Deteet 0.9
Maximum Non-Detect 2
Data with Multiple Detection Limits
Mote: Data have multiple DLs - Use of KM Method ss recommended
For all rrelbods (except KM, DL'2. and ROS Methods).
Observations < Largest ND aie treated 33 NDs
N umber of Detected Data 11
Number of Non-Detect Data 13
Percent Nan-Detects 54.17%
Log-transformed Statistics
Minimum Detected -0.693
Maximum Detected 1.163
Mean of Detected -0.0255
SD of Detected 0.694
Minimum Non-Detect -0.105
Maximum Non-Detect 0.693
Single Detection Limit Scenario
Number treated as Non-Detect with Single DL 22
Number treated as Detected with Single DL 2
Single DL Non-Detect Percentage 91.67*4
Background Statistics
Normal Distribution Test with Detected Values Only Lognormal Distribution Test with Detected Values Only
Shapiro Wilk Test Statistic 0.777 Shapiro Wilk Test Statistic 0.86
5% Shapiro Wilk Critical Value 0.85 5% Shapiro Wilk Critical Value 0.85
Data pot ffenrai at 5% Sifplic^Mse Lw^ Data appear Lognormal at5X Significance Level
Assuming Normal Distribution
DL/2 Substitution Method
Mean
SD
95%UTL 90% Coverage
35% UPL (t)
90% Percentile (z)
95% Percenttle (z)
99% Percentile (z)
Assuming Lognormal Distribution
DL/2 Substitution Method
1.002 Mean (Log Scale) -0.16
0.699 SD (Log Scale) 0.542
2.296 95%UTL 90% Coverage 2.327
2.224 95% UPL (t) 2.2
1.897 90% Percentile (z) 1.707
2.151 95% Percentile (z) 2.079
2.627 99% Percentile (z) 3.009
163
-------
Maximum Likelihood EstimBte(MLE) Method N/A
Log ROS Method
Mean in Original Scale
SD in Original Scale
Mean in Log Scale -0.209
SD in Log Scale 0.571
95%UTL 90% Coverage
95% UPL (t)
90S Percentile (z)
95% Percentile (2)
99% Percentile (z)
2.337
2.202
1.SS6
2.075
3.062
Gamma Distribution Test with Detected Values Only
k star (bias corrected) 1 .702
Theta Star 0727
riu star 37.44
Data Distribution Tests with Delected Values Only
Data follow Appr. Gamma Distribution at 5% Significance Level
A-D Test Statistic 0.787
5% A-D Critical Value 0.73B
K-S Test Statistic D.254
5% K-S Critical Value 0.258
DatafollowAppx. Gamma Distribution at 5% Significance Level
Assuming Gamma Distribution
Gamma ROS Statistics with extrapolated Data
Mean 1.2S3
Median 1.213
SD 0.652
k star 3.974
Thetastar 0.318
Muster 190.8
95% Percentile of Chisquare (2k) 1 5.43
90% Percertble 1.861
95% Percentile 2.16
39% Percentile 2.801
Non para metric Statistics
Kaplan-Meier (KM) Method
Mean 0.949
SD 0.713
SE of Mean 0.165
95% KMUTLwith 90% Coverage 2.27
95% KM Cbebyshev UPL 4.121
95% KM U PL (t) 2.196
90% Percentile (z) 1.863
35% Percentile (z) 2.122
99% Percentile (z) 2.608
Note: U PL (or upper percent! le for gamma distributed dab) represents a preferred estimate of BTV
For an Example: KM-UPL may be used when multiple detection limits are present
: OU2 is not a ree«nme.Mled ntefsl
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165
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Chapter 11
Computing Upper Confidence Limits (UCLs) of Mean
The UCL computation module of ProUCL 4.0 represents an update of the UCL module of ProUCL 3.0.
The detailed theory and formulae used to compute gamma and lognormal statistics are given by Land
(1971, 1975), Gilbert (1987), Singh, Singh, and Engelhardt (1997, 1999), Singh et al. (2002a), Singh et al.
(2002b), and Singh and Singh (2003).
Several parametric and nonparametric UCL computation methods for data sets with NDs have been
incorporated in ProUCL 4.0. Methods such as the Kaplan-Meier (KM) and regression on order statistics
(ROS) methods as incorporated in ProUCL 4.0 can handle multiple detection limits. For details regarding
the distributions and methods available in ProUCL 4.0, refer to the ProUCL 4.0 Technical Guide and
Singh, Maichle, and Lee (USEPA, 2006). Recommendations for the computations of UCLs for data sets
with NDs have been made based upon the findings of the simulation experiments performed by Singh,
Maichle, and Lee (USEPA, 2006).
In ProUCL 4.0, two choices are available to compute UCL statistics:
• Full - Computes UCLs for full data sets without any nondetected values.
• With NDs - Computes UCLs for data sets that have detected as well as BDL
observations. It is pointed out that it is not desirable to use statistical methods as
incorporated in ProUCL 4.0 on data sets consisting of all nondetect values. Discussion
about the detection sampling frequency is provided in Chapter 1 of this User Guide.
Some of the available methods can handle multiple detection limits. The program
provides a message to the user about the use of an appropriate method when multiple
detection limits may be present.
• For full data sets without NDs and also for data sets with NDs, the following options and
choices are available to compute UCLs of the population mean.
o The user specifies the confidence level; a number in the interval [0.5, 1), 0.5
inclusive. The default choice is 0.95.
o The program computes several nonparametric UCLs using the central limit theorem
(CLT), Chebyshev inequality, jackknife, and bootstrap re-sampling methods.
o For the bootstrap method, the user can select the number of bootstrap runs (re-
samples). The default choice for the number of bootstrap runs is 2000.
o The user is responsible for selecting an appropriate choice for the data distribution:
normal, gamma, lognormal, or nonparametric. It is desirable that user determines data
distribution using the Goodness-of-Fit test option prior to using the UCL option. The
UCL option informs the user if data are normal, gamma, lognormal, or a non-
discernable distribution. Program computes statistics depending on the user selection.
o For data sets, which are not normal, one may try the gamma UCL next. The program
will offer you advice if you chose the wrong UCL option.
o For data sets, which are neither normal nor gamma, one may try the lognormal UCL.
The program will offer you advice if you chose the wrong UCL option.
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o Data sets that are not normal, gamma, or lognormal are classified as distribution-free
nonparametric data sets. The user may use nonparametric UCL option for such data
sets. The program will offer you advice if you chose the wrong UCL option.
o The program also provides the All option. By selecting this option, the UCLs are
computed using most of the relevant methods available in ProUCL 4.0. The program
informs the user about the distribution of the underlying data set, and offers advice
regarding the use of an appropriate UCL.
o For lognormal data sets, ProUCL can compute only a 90% or a 95% Land's statistic-
based H-UCL of the mean. For all other methods, ProUCL can compute a UCL for
any confidence coefficient in the interval [0.5,1.0), 0.5 inclusive.
o If you have selected a distribution, then ProUCL will provide a recommended UCL
computation method for 0.95, confidence coefficient. Even though ProUCL can
compute UCLs for confidence coefficients in the interval [0.5, 1.0),
recommendations are provided only for 95% UCL; as EPC term is estimated by a
95% UCL of the mean.
Note: It is recommended that the user identify a few low probability outlying observations that may be
present in the data set. Outliers distort many statistics of interest including summary statistics, data
distributions, test statistics, UCLs, and estimates ofBTVs. Decisions based upon distorted statistics may
be misleading and incorrect. The objective is to compute relevant statistics and estimates based upon the
majority of the data set(s) representing the dominant population(s). Those few low probability outlying
observations require separate attention and investigation. The project team should decide about the
proper disposition (to include or not to include) of outliers before computing the statistics to estimate the
EPC terms and BTVs. In order to determine and compare the improper and unbalanced influence of
outliers on UCLs and background statistics, the project team may want to compute statistics using data
sets with outliers and without outliers.
11.1 UCLs for Full Data Sets
11.1.1 Normal Distribution (Full Data Sets without NDs)
1. Click UCL ^ Full ^ Normal
E? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
•^ File Edit Configure Summary Statistics ROS Est, NDs Graphs Outiier Tests Goodness-of-Fit Hypothesis Testing Background
C|«B| olHlralnl
Navigation Panel
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With NDs t
Gamma
Lognormal
Non -Parametric
All
11
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If needed, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables, and select a proper group variable.
• When the option button is clicked, the following window will be shown.
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Confidence Level
Confidence Level ItBEl
OK
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Click on OK button to continue or on Cancel button to cancel the option.
• Click on OK button to continue or on Cancel button to cancel the UCL computation
option.
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Output Screen for Normal Distribution (Full Data without NDs)
Normal UCL Statistics for Full Data Sets
User Selected Options
From File C:\Narain\ProUCL-Data\Data\Aroclor 1254.wst
Full Precision OFF
Confidence Coefficient 95%
Aroclor_V/ithout_NonDetects
Number of Valid Samples 44
Number of Unique Samples 41
Minimum 0.21
Maximum 19000
Mean 1532
Median 94.5
SD 3355
Variance 11255555
Coefficient of Variation 2.19
Skewness 3.756
Shapiro Wilk Test Statistic 0.526
5% Shapi ro Wi I k C ritical Val ue 0.544
Data not Normal at 5°i Significance Level
§5% UCL [Assuming Normal Distribution)
Student's-t LCL
Studenfs-t UCL
Data appear Log normal (0.05)
M ay want to try Log n ormal UCLs
681.6
2382
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11.1.2 Gamma, Lognormal, Nonparametric, All Statistics Option (Full Data without NDs)
1. Click UCL ^ Full ^- Gamma, Lognormal, Non-Parametric, or All
B3 ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wstj
•B File Edit Configure Summary Statistics ROS Est. NDs Graphs Gutter Tests Goodness-of-Fit Hypothesis Testing Background
o|«&| olHlmlnl
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l:'l Window
With NDs t
Help
Normal
Gamma
Lognormal
Non -Parametric
11
2. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If desired, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables, and select a proper group variable.
• When the option button is clicked, the following window will be shown.
Confidence Level
Number of Bootstrap Operations | 2000
OK
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Number of Bootstrap Operations (runs). Default choice is 2000.
o Click on OK button to continue or on Cancel button to cancel the UCLs option.
• Click on OK button to continue or on Cancel button to cancel the selected UCL
computation option.
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Output Screen for Gamma Distribution (Full)
Gamma UCL Statistics for Full Data Sets
User Selected Options
From File C:\N3rain\PraUCL-Data\Datay\rodQr 1254.wst
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap 0perations 2000
Arodor_With-out_Non Detects
Number of Valid Samples 44
Number of Unique Samples 41
Minimum 0-21
Maximum 18000
Mean 1532
Median 94-5
Standard Deviation 3355
Variance 11255595
k star (bias corrected) 0.247
Theta Star 6208
nystar 21.72
Approximate Chi Square Value (.05) 12.12
Adjusted Level of Significance 0.0445
Adjusted Chi Square Value (.05) 11.88
Anderson-Darling Test Statistic 1.01
Anderson-Darling Critical Value O.SE4
Kolmogorov-Smirnov Test Statistic 0.169
Kolmogorov-Smirnov Critical Value 0.146
Data not Garr-ma distributed at 5% Significance Level
95% UCLs (Adjusted for Skewness)
95% Adjusted-CLT UCL 2670
95% Modified-! UCL 2430
95% Non-Parametric UCLs
95% Bootstrap-t UCL 3010
95% Hall's Bootstrap UCL 5380
95% Gamma UCLsfAssuming Gamma Distribution)
95% Approximate Gamma UCL 2744
95% Approximate Gamma LCL 991
95% Adjusted Gamma UCL 2800
95% Adjusted Gamma LCL 976.E
Data appear Lognormal (0_Q5)
May want to try LognormaJ UCLs
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Output Screen for Lognormal Distribution (Full)
I Log normal UCL Statistics fof Full Data Sets
User Selected Options
From File C:\Narain\ProUCL-Data\Data\Aroclor 1254.wst
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap Operations 2000
Aroclor_W]thout_NonDetects
Number of Valid Samples 44
Number of Unique Samples 41
Minimum of log data -1.561
Maximum of log data 9.852
Mean of log data 4.474
SD of log data 3.163
Variance of log data 10
Shapiro Wilfc Test Statistic Q.94S
Shapiro Wilk 5% Critical Value 0.944
Data appear Lognormal at 5% Significance Level
95% UCL (Assuming- Normal Distribution)
95% Student's-t UCL 2332
ML Estimates Assuming Lognormal Distribution
Mean 13038
SD 1938235
Coefficient of Variation 14S.7
Skewness 3285837
Median 87.7
80% Quantile 1256
S0% Quantile 5050
95% Quantile 15935
99% Quantile 137549
MVU Estimate of Median 78.25
MVU Esti mate of Mean 7626
MVU Estimate of SD 165089
MVU Estimate of Standard Error of Mean 5553
UCLs (Assuming Lognormal Distribution)
95% H-UCL S7513
95% Chebyshev (MVUE) UCL 31832
97.5% Chebyshev (MVUE) UCL 42307
99% Chebyshev (MVUE) UCL 62381
Potential UCL to Use
Use 99% Chebyshev (MVUE) UCL 62881
Recommended UCL exceeds the njajdimwi
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Output for Nonparametric Methods (Full Data without NDs)
i Nonparimetric UCL Stafofcs far Full Data Sete
User Selected Options
From File C:\Narain\ProLICL-Data\Data\Aroclor 1254.wst
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap Operations 2000
Aroclor Without NorDetects
Number of Valid Samples 44
Number of Unique Samples 41
Minimum 0.21
Maximum 19000
Mean 1532
Median 94.5
SD 3355
Variance 11255595
Coefficient of Variation 2.19
Skewness 3.756
Mean of log data 4.474
SD of log data 3.163
N on Parametric UCLs
95%CLTUCL 135.1
95%JackknifeUCL 136.1
95% 5 tandard B ootstrap U CL 134.4
95% Bootstrap-l UCL 144.5
95% Hall's Bootstrap UCL 139.7
35% Percentile B ootstrap U CL 136.1
95% BCA Bootstrap UCL 141.3
95% Chebyshev(M ean, S d) U CL 167.3
97.5% Chebyshev(M ean, S d) U CL 189.7
99% Chebyshev(M ean, 5 d) U CL 233.7
Potential UCL to Use
Use 95% Student's-* UCL 136.1
Or 95% Modified-* UCL 136.8
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Output Screen for All Statistics Option (Full Data without NDs)
! General UCL Statistics for Full Data Seb
User SeJM^ Options
From File C:\Narain\ProUCL-Data\Data\Aroclor 1254.wst
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap Operations 2000
Aroclor_WithoutJ4onDetects
General Statistics
N umber of Val id Samples 44
Raw Statistics
Minimum 0-21
Maximum 19000
Mean 1532
Median, 94.5
SD 3355
Coefficient of Variation 2.19
Skewness 3.756
Number of Unique Samples 41
Log-transformed Statistics
Minimym of Leg Data -1.561
Maximum of Log Data 9.852
Mean of log Data 4.474
SD of log Data 3.163
Relevant UCL Statistics
Normal Distribution Test
Shapiro Wilk Test Statistic
Shapiro Wilk Critical Value
Data eet Normal at 5% StgnBearsee Levd
0.526
0.944
Assuming Normal Distribution
95% Siudent's-t UCL 2382
95% UCLs (Adjusted far Skewness)
95% Adjusted-CLT UCL 2670
95% Modified-t UCL 2430
Gamma Distribution Test
k star (bias corrected) 0.247
ThetaStar 6208
nu star 21.72
Approximate Chi Square Value (.05) 12.12
Adjusted Level of Significance 0.0445
Adjusted Chi Square Value 11 .SB
Anderson-Darling Test Statistic 1.01
Anderson-Darling 5% Critical Value 0.8S4
Kolmogorov-Smirn-ov Test Statistic 0.163
Kdmogorov-Smirriov 5% Critical Value 0.146
Data not Gamma Distributed at 5% S§wieases Lwel
Assuming Gamma Distribution
95% Approximate Gamma UCL 2744
95% .Adjusted Gamma UCL 2SOO
Log normal Distribution Test
Shapiro Wilk Test Statistic 0.948
Shapiro Wilk Critical Value 0.944
Data appear Log normal at 5% Significance Level
Assuming Lognormal Distribution
95%H-UCL B7513
95% Chebyshev (MVUE) UCL 31832
97.5% Chebyshev (MVUE) UCL 42307
99% Chebyshev (MVUE) UCL 62881
Data Distributor Tests
Data appear Lognormal at 5% Significance Level
Non para metnc Statistics
95%CLTUCL 2364
95% Jackknife UCL 2382
35% Standard Bootstrap UCL 2352
95% Bootstrap-t UCL 3035
95% Hall's Bootstrap UCL 5521
95% Percentile Bootstrap UCL 2404
95% BCA Bootstrap UCL 2727
95% Chebyshev(Mean, Sd) UCL 3737
97.5% ChebyshevjMean, Sd) UCL 4690
99% ChebyshevjMean, Sd) UCL 6564
Potential UCLto Use
p mended UCL exceeds ttig raessrswn*
Use 99% Chebyshev (MVUE) UCL 62881
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Note: Once again, it should be noted that the number of valid samples represents the total number of
samples minus (-) the missing values (if any). The number of unique or distinct samples simply represents
number of distinct observations. The information about the number of distinct samples is useful when
using bootstrap methods. Specifically, it is not desirable to use bootstrap methods on data sets with only a
few (< 4-5) distinct values.
11.2 UCL for Data Sets with NDs
1. Click UCL ^ With NDs
ProUCt 4,fl .- [C:\Narai,n\ProU|:L-Oata\l)ata\MW89,)TOtl
ay File Edit Configure Summary Statistics ROS Est. NDs Graphs Qutiier Tests Goodness-of-Fit Hypothesis Testing Background ||H9 Window Heip
FULL > t
Normal &&.
Gamma |1
Lognormal ]
Non-Parametric
Navigation Panel
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4) Worksheet wst
$ MW89.wst
2. Choose the Normal, Gamma, Lognormal, Non-Parametric, or All option.
3. The Select Variables Screen (Chapter 3) will appear.
• Select a variable (variables) from the Select Variables screen.
• If desired, select a group variable by clicking the arrow below the Group by variable to
obtain a drop-down list of available variables, and select a proper group variable. The
selection of this option will compute the relevant statistics separately for each group that
may be present in the data set.
• When the option button is clicked, the following window will be shown.
Confidence Level
Number of Bootstrap Operations
jfig
2000
Cancel
o Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
default choice is 0.95.
o Specify the Number of Bootstrap Operations. The default choice is 2000.
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o Click on OK button to continue or on Cancel button to cancel the UCLs option.
Click on OK button to continue or on Cancel button to cancel the selected UCL
computation option.
Output Screen for Normal Distribution (with NDs)
Normal UCL Statistics for Data Sets with Non-Detects
User Selected Options
From File D:\example.wst
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap Operations 2000
Arsenic
Total Number of Data 20
Number of Non-Detect Data 3
NumberofDetectedData 17
Minimum Detected 5
Maximum Detected 9.2
PercentNon-Detects 15.00%
Minimum Non-detect 4.3
Maximum Non-detect 4.5
MeanofDetectedData 6.126
SD of Detected Data 1.15
- of KM is
For all 17..
is for all
176
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Output Screen for Normal Distribution (with NDs) - Continued
Normal Distribution Test with Detected Values Only
S hapiro Wilk T est S tatistic 0.794
5% S hapiro Wilk Critical Value 0.892
Data not Normal at 5% Significance Level
DL/2 Substitution Method
Mean 5.54
SD 1.779
35% DU2 (t) UCL 6.228
Maximum Likelihood Method
Mean 5.774
SD 1.348
95% MLE (t] UCL 6.295
95% MLE (Tiku) UCL 6.301
Kaplan Meier(KM) Method
Mean 5.958
SD 1.104
Standard Error of Mean 0.255
95% KM (t) UCL 6.398
95% KM (z) UCL 6.376
95% KM (BCA) UCL 6.393
95% KM (Percentile Bootstrap) UCL 6.388
Data do not follow a Discernable Distribution (0.05)
May want to try Non-Parametric UCLs
177
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Output Screen for Gamma Distribution (with NDs)
Gamma UCL Statistics for Data Sets with Non-Detects
User Selected Options
From File DAexample.mt
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap Operations 2000
Arsenic
Total Number of Data 20
NumberofMissingValues 20
Number of Non-Delect Data 3
NumberofDetectedData 17
Minimum Detected 5
Maximum Detected 9.2
Percent Non-Detects 15.00%
Minimum Non-detect 4.3
Maximum Non-detect 4.5
MeanofDetectedData 6.126
MedianofDetectedData 5.8
SD of Detected Data 1.15
k Star of Detected Data 28.99
ThetaStarofDetectedData 0.211
NuStarofDetectedData 985.5
For ail
the is for
178
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Output Screen for Gamma Distribution (with NDs) - Continued
Gamma Distribution Test with Detected Values Only
A-D Test Statistic 1.027
5% A-D Critical Value 0.737
K-S Test Statistic 0.214
5% K-S Critical Value 0.209
Data not Gamma Distributed at 5% Significance Level
Gamma ROS Statistics with Extrapolated Data
Minimum 3.516
Maximum 9.2
Mean 5.804
Median 5.625
SD 1.325
kStar 18.24
ThetaStar 0.318
Nu Star 729.8
95% Percentile of Chisquare (2k) 51.58
AppChi2 668.1
95% Gamma Approximate UCL 6.34
95% G amma Adjusted U CL 6.384
Kaplan Meier(KM) Method
Mean 5.958
SD 1.104
StandardErrorofMean 0.255
95% KM (t) UCL 6.398
95% KM (BCA) UCL 6.455
95% KM (Percentile Bootstrap) UCL 6.405
95% KM (Chebyshev) UCL 7.067
Data do not follow a Discernable Distribution (0 05]
May want to try Nonparametric UCLs
179
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Output Screen for Lognormal Distribution (with NDs)
Lognormal UCL Statistics for Data Sets with Nan-Delects
User Selected Options
From File D: \example.wst
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap Operations 2000
Arsenic
General Statistics
Number of Valid Samples 20
Number of Unique Samples 15
Maximum Detected
Mean of Detected Data
Raw Statistics
9.2
6.126
Number of Detected Data 17
Number of Non-Detect Data 3
Percent Non-Detects 15.00%
Maximum Non-detect 4.5
SD of Detected Data 1.15
Log-T ransf ormed S tatistics
Mean of Detected Data 1.798 SD of Detected Data 0.168
Note: Data have multiple DLs • Use of KM Method is recomrn Number treated as Non-Detect 3
For all methods (except KM, DL/2, i lobust I |OS, and Gamma Number treated as Detected 17
those Observations < Largest ND are treated as NDs Single DL Non-Detect Percentage 15.00%
Lognormal Distribution Test with Detected Values Only
Shapiro Wilk Test Statistic 0.854 5% Shapiro Wilk Critical Value 0.892
at S5J S
Note: Once again, it should be noted that the number of valid samples represents total number of samples
minus (-) the missing values (if any). The number of unique or distinct samples simply represents number
of distinct observations. The information about the number of distinct samples is useful when using
bootstrap methods. Specifically, it is not desirable to use bootstrap methods on data sets with only a few
(< 4-5) distinct values.
180
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Output Screen for Lognormal Distribution (with NDs) - Continued
DL/2 Substitution Method
Mean 5.54
SD 1.779
Mean (in Log Scale) 1.648
SD (in Log Scale) 0.398
Robust ROS Method
Mean 5.831
SD 1.28
Mean (in Log Scale) 1.742
SD (in Log Scale) 0.207
Kaplan Meier (KM) Method
Mean 5.958
SD 1.104
SE of Mean 0.255
95% H-Stat (DL/2) UCL 6.522
95% Percentile Bootstrap UCL
95% BCA Bootstrap UCL
95% KM (t) UCL
95% KM (BCA) UCL
95% KM (% Bootstrap) UCL
95%KM(Chebyshev)UCL
97. 5% KM (Chebyshev) UCL
99%KM(Chebyshev)UCL
6.31 8
6.357
6.398
6.43
6.373
7.067
7. 547
8.49
Potential UCL to Use
Data do not follow a Discernable Distribution (0.05)
May want to try Nonpararnetric UCLs
181
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Output Screen for Nonparametric Methods (with NDs)
Nonparametric UCL Statistics for Data Sets with Non-Detects
User Selected Options
From File D:\exannple.wst
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap Operations 2000
Arsenic
Total Number of Data 20
Number of Non-Detect Data 3
Number of Detected Data 17
Minimum Detected 5
Maximum Detected 9.2
Percent Non-Detects 15.00%
Minimum Non-detect 4.3
Maximum Non-detect 4.5
Mean of Detected Data 6.126
Median of Detected Data 5.8
VarianceofDetectedData 1.323
SD of Detected Data 1.15
CV of Detected Data 0.188
S kewness of D elected D ata 1.783
MeanofDetectedlogdata 1.788
SDofDetectedLogdata 0.168
DI s - of KM is
all DL/2,
Dl is for
182
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Output for Nonparametric Methods (with NDs) - Continued
Nonparametric Test with Detected Values Only
Data do not follow a Discernable Distribution (0 06)
WinsorizationMethod 0.168
Mean 5.738
SD 0.624
95%Winsor(t)UCL 5.985
Kaplan Meier (KM) Method
Mean 5.958
SD 1.104
Standard Error of Mean 0.255
95% KM (t) UCL 6.398
95% KM (z) UCL 6.376
95% KM (BCA) UCL 6.478
95% KM (Percentile Bootstrap) UCL 6.438
95% KM (Chebyshev) U CL 7.067
97.5% KM (Cheby she v) U CL 7.547
99% KM (Cheby she v) UCL 8.49
Potential UCL to Use
95% KM (Chebyshev) UCL 7.067
183
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Output Screen for All Statistics Option (with NDs)
General UCL Statistics foi Data Sets with Non-Detects
User Selected Options
From File D:\example.wst
Full Precision OFF
Confidence Coefficient 95%
Number of Bootstrap Operations 2000
Arsenic
Number of Valid Samples
Number of Unique Samples
Raw Statistics
Minimum Detected
Maximum Detected
Mean of Detected
SD of Detected
Minimum Non-Detect
Maximum Non-Detect
General Statistics
20
15
5
9.2
6.126
1.15
4.3
4.5
Number of Detected Data 17
NumberofNon-DetectData 3
Percent Noil-Detects 15.00%
Log transformed Statistics
MinimumDetected 1.609
Maximum Detected 2.219
Mean of Detected 1.799
SD of Detected 0.169
Minimum Non-Detect 1.459
Maximum Non-Detect 1.504
Number treated as Non-Detect 3
Number treated as Detected 17
Single DL Non-Detect Percentage 15.00%
UCL Statistics
Normal Distribution Test with Detected Values Only
Shapiro Wilk Test Statistic
5% Shapiro Wilk Critical Value
Data not formal a! 5X Significance Level
Lognormal Distribution Test with Detected Values Only
0.7S4 S hapiro Wilk T est S tatistic 0.854
0.892 5% S hapiro Wilk Critical Value 0.892
Data no! Lognorraal at 5?£ Significance Level
Note: Once again, it should be noted that the number of valid samples represents the total number of
samples minus (-) the missing values (if any). The number of unique or distinct samples simply represents
number of distinct observations. The information about the number of distinct samples is useful when
using bootstrap methods. Specifically, it is not desirable to use bootstrap methods on data sets with only a
few (< 4-5) distinct values.
184
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Output Screen for All Statistics Option (with NDs) - Continued
Assuming Normal Distribution
DU2 Substitution Method
Mean
SD
95% DU2 (t] UCL
Maximum Likelihood Estirnate(MLE) Method
Mean
SD
95% MLE (t) UCL
95% MLE (Tiku) UCL
Gamma Distribution Test with Detected Values Only
k star (bias corrected)
Theta Star
nu star
A-D Test Statistic
5% A-D Critical Value
K-S Test Statistic
5% K-S Critical Value
D ala not G Distributed at 5% S
Assuming Gamma Distribution
Garnnna ROS Statistics using Extrapolated Data
Minimum
Maximum
Mean
Median
SD
kstar
Theta star
Nu star
AppChi2
95% Gamma Approximate UCL
95% Adjusted Gamma UCL
5.54
1.779
6.228
5.774
1.348
6.295
6.301
28.99
0.211
985.5
1.027
0.737
0.737
0.209
3.516
9.2
5.804
5.625
1.325
18.24
0.318
729.8
668.1
6.34
6.384
Assuming Lognormal Distribution
DL/2 Substitution Method
Mean
SD
95% H-Stat (DL/2) UCL
Robust ROS Method
Mean in Log Scale
SD in Log Scale
Mean in Original Scale
SD in Original Scale
95% Percentile Bootstrap UCL
95% EGA Bootstrap UCL
N onparametic T est with D elected Values Only
Data do not follow a Discernable Distribution (0.05)
Nonparametric Statistics
Kaplan-Meier (KM (Method
Mean
SD
SE of Mean
95% KM [t] UCL
95% KM (z) UCL
95% KM (jackknife) UCL
95% KM (bootstrap t) UCL
95% KM (BCA) UCL
95% KM (percentile) UCL
95% KM (Chebyshev) UCL
97.5% KM (Chebjishev) UCL
99% KM (Chebyshev) UCL
Potential UCLs to Use
95% KM (Chebyshev)UCL
1.648
0.398
6.522
1.742
0.207
5.831
1.28
6.314
6.37
5.958
1.104
0.255
6.398
6.376
6.387
6.673
6.43
6.398
7.067
7.547
8.49
7.067
185
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Chapter 12
Windows
y File Edit Configure Summary Statistics ROS Est. NDs Graphs Clutter Tests Goodness-of-Fit Hypothesis Testing Background
Cascade
Tile Vertically
Tile Horizontally
CJ Worksheet wst
Click on the Window menu to reveal the drop-down options shown above.
The following Window drop-down menu options are available:
• Cascade option: arranges windows in a cascade format. This is similar to a typical
Windows program option.
• Tile option: resizes each window vertically or horizontally and then displays all open
windows. This is similar to atypical Windows program option.
• The drop-down options list also includes a list of all open windows with a check mark in
front of the active window. Click on any of the windows listed to make that window
active. This is especially useful if you have more than 20 windows open, as the
navigation panel only holds the first 20 windows.
186
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187
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Chapter 13
Help
When the Help menu is clicked, the following window will appear.
B? ProUCL 4.0 - [WorkSheet.wst]
•g File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness -of-Fit Hypothesis Testing Background UCL Window
0|«&| olalml nl
Navigation Panel
Name
©WorkSheet.wst
1
2
0
1
2
3
4
5
6
7
8
O
About ProUCL
Technical Support
Three options are available under Help menu:
• About ProUCL: This option provides a brief description of ProUCL 4.0, and all
improvements made compared to ProUCL 3.0.
• Statistical Help: This option executes an online help directory. This option provides
information about the various algorithms and formulae (with references) used in the
development of ProUCL 4.0. More information on the various topics covered under
Statistical Help is provided below.
• Technical Support: This option will provide contact information for primary technical
support via phone and e-mail.
Statistical Help provides online help notes for the methods and options available in ProUCL 4.0. A screen
(shown below), listing the topics containing help notes, appears after clicking on the Help menu.
51
ProUCL 4.0 Online Help Directory
I i. 1.1 MI li,-l|i:
c/M>,
Suck!
188
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The online help directory is divided into the following submenus:
• Getting Started: This chapter provides basic information on the software, including
software installation to various menu displays.
• Summary Statistics: This chapter provides information and examples on procedures for
simple classical summary statistics for data sets with and without nondetects.
• ROS Estimates of NDs: This chapter briefly describes the estimation (extrapolation) of
nondetects using regression on order statistics (ROS) for normal, lognormal, and gamma
distribution.
• Graphs: This chapter provides information and examples on the graphical displays that
ProUCL 4.0 can produce: box plots, histograms, and Multi Q-Q plots.
• Outlier Tests: This chapter provides information and examples for two classical outlier
tests available in ProUCL 4.0: Dixon's and Rosner's tests for data sets with and without
outliers.
• Goodness-of-Fit: This chapter provides information and examples for several goodness-
of-fit tests available in ProUCL 4.0 for data sets with and without NDs.
• Background Statistics: This chapter provides information and examples for the
computation of Background Statistics needed to estimate the BTVs and not-to-exceed
values. These statistics are sometimes used to compare point-by-point site data (not more
than 4 to 5 site samples) with the BTVs.
• Hypotheses Testing: This chapter provides brief descriptions (with examples) of the
various single sample and two-sample hypotheses testing approaches as incorporated in
ProUCL 4.0.
o Single Sample Hypotheses Testing: This chapter provides brief description and
examples of single sample hypotheses testing approaches that are useful to compare
site concentrations with cleanup standards, compliance limits, or not-to-exceed
limits. The minimum sample size requirements for site data are briefly discussed in
Chapter 1 of this User Guide.
o Two-sample hypotheses are used to compare site data with background data (or
upgradient and downgradient wells) provided enough site and background data are
available. The minimum sample size requirements for site and background data are
briefly discussed in Chapter 1 of this User Guide.
• Upper Confidence Limits (UCLs): This option provides information and examples for the
various UCL computation methods as incorporated in ProUCL 4.0.
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Chapter 14
Handling the Output Screens and Graphs
Copying Graphs
1. Click the graph you want to copy or save in the Navigation Panel. The graphs can be saved using
the copy option.
P? ProUCL 4.0 - [Histo_Group.gst]
•P File Edit Configure Window Help
e| B|B|m| c|
- S X
Navigation Panel
Name
^worksheet.wst
@ example, wst
El Histo_Arsenic (sub...
13 Histo_Arsenic (surf..
Histogram Plot for Group
fr
I Arsenic (subsurface) Arsenic (surface)
Log Panel
LOG: 2:03:21 PM ^Information] Box_Arsenic (4.5 ).gst closed!
LOG: 2:03:21 PM >[Information] Box_Arsenic (4.3).gst closed!
LOG: 2:03:33 PM >[Information] Histogram was generated!
LOG: 2:03:33 PM >[Information] Histogram was generated!
2. Click Edit ^ Copy Graph.
File I I Configure Window Help
3. Once the user has clicked "Copy Graph," the graph is ready to be imported (pasted) into most
Microsoft Office applications (Word, Excel, and PowerPoint have been tested) by clicking
Edit ^ Paste in these Microsoft applications.
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4. It is important to note that the graph cannot be saved as its own file and must be imported into an
application to be saved. This will save the graph, but the overall file attribute and properties will
be that of the application in which the graph was saved in. For example, if the graph was saved
within Microsoft Word, the graph will be saved in a document with a .doc extension.
Printing Graphs
1. Click the graph you want to print in the Navigation Panel.
2. Click File ^ Page Setup.
Edit Configure Window Help
New
Open ...
Load Excel Data
Close
Save
Save As ,,,
Page 5etup
Print
Print Preview
Exit
Check the radio button next to Portrait or Landscape, and click OK. In some cases, with larger
headings and captions, it may be desirable to use the Landscape printing option.
Page Setup
Size: Le
S_ource: AL
r Portrait
^^^
i
;
ter
o Select
ii • f i i
Left: [05
Jop: [05
OK | Cance
•EH
3
3
Bottom: [tl5
Jl
4. Click File ^ Print to print the graph, and File ^ Print Preview to preview the graph before
printing.
191
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Edit Configure Window Help
New
Open ...
Load Excel Data
Close
Save
Save As ,,,
Page 5etup
Print
Print Preview
Exit
Printing Non-graphical Outputs
1. Click the output you want to copy or print in the Navigation Panel.
n= File Edit Configure Window Help
jalMniJjn]
Navigation Panel
Name
& worksheet, wst
£j$ example, wst
S| output.ost
From File: D:\example_w
General UCL Statistics
13 NormQQ_Arsenic.gst
Raw Statistics
Total Number of Data 20
Number of Detected Data 17
Number of Non-Detect Data 3
Detection Percentage 65.00%
Minimum Detected
Maximum Detected
Mean of Detected
SD of Detected
Minimum Non-Detect
Maximum Non-Detect
Log Panel
Log-transformed Statistics
Minimum Detected 1.6094
Maximum Detected 2.2192
Mean of Detected 1.7983
SD of Detected 0.16S2
5.0000
9.2000
6.1265
Minimum Non-Detect
Maximum Non-Detect
1.4586
1.5041
1.1500 N ote: Data have multiple D Ls - U se of KM methods is recommended
- Use of KM methods is recommended
4 3000 For all methods (Except KM, DL/2. and Helsers ROSJ
4 5000 Observations < largest ND are treated as NDs
Number treated as Non-Detect
Number treated as Detected
Single DL Detection Percentage
UCL
3
17
85.00%
LOG: 6:1 7:37 PM >[Information] D:\exarnple.wst loaded!
LOG: 6:1 7:45 PM > [Information] Generated results!
LOG: 6:1 9:02 PM >[Informat!on] Generated results!
LOG: 6:30:50 PM >[Information] Q-Q Plot graph for Arsenic was generated!
192
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2.
Click File ^ Print.
^^J Edit Configure Window Help
New
Open .,.
Load Excel Data
Close
Save
Save As ...
Page Setup
Print
Print Preview
Exit
Saving Output Screens as Excel Files
1. Click on the output you want to save in the Navigation Panel List.
2. Click File ^- Other Files... ^-Export Excel (preserve formatting).
^^J Edit Configure Window Help
New
Open ...
Load Excel Data
Other Files ...
Close
Save As ...
Print
Print Preview
Exit
Export Excel (preserve formatting). . .
193
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3. Enter the desired file name you want to use, and click Save, and save the file in the desired folder
using your browser.
Save As ? X
Save in:
uL^J
My Recent
Documents
Desktop
*w^
My Computer
My Network
Places
@ Desktop _J <3=fcetilT
_J My Computer
*JMy Network Places
File name: II3TTH5BJ3 T| S_ave
Save as type: | Excel Workbook (K.xls) _^J Cancel
194
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195
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Chapter 15
Recommendations to Compute a 95% UCL (Estimate of EPC
Term) of the Population Mean, [Ji, Using Symmetric and
Positively Skewed Full Data Sets without any Nondetects
This chapter describes the recommendations for the computation of a 95% UCL of the unknown
population arithmetic mean, }JL\, of a contaminant data distribution based upon full data sets without any
nondetect observations. These recommendations are based upon the findings of Singh, Singh, and
Engelhardt (1997, 1999); Singh etal. (2002a); Singh, Singh, and laci (2002b); and Singh and Singh
(2003). These recommendations are applicable to full data sets without censoring and nondetect (ND)
observations. Recommendations to compute UCL95 based upon data sets with NDs are summarized in
Chapter 16.
For skewed parametric as well as nonparametric data sets, there is no simple solution to compute a 95%
UCL of the population mean, ^. Contrary to the general conjecture, Singh et al. (2002a); Singh, Singh,
and laci (2002b); and Singh and Singh (2003) noted that the UCLs based upon the skewness adjusted
methods, such as the Johnson's modified t and Chen's adjusted-CLrdo not provide the specified
coverage (e.g., 95 %) to the population mean even for mildly to moderately skewed (e.g., a the sd of log-
transformed data in interval [0.5, 1.0)) data sets for samples of size as large as 100. The coverage of the
population mean by the skewness-adjusted UCLs becomes much smaller than the specified coverage of
0.95 for highly skewed data sets, where skewness is defined as a function of a or a (sd of log-
transformed data).
It is noted that even though, the simulation results for highly skewed data sets of small sizes suggest that
the bootstrap t and Hall's bootstrap methods do approximately provide the adequate coverage to the
population mean, sometimes in practice these two bootstrap methods yield erratic inflated values (orders
of magnitude higher than the other UCL values) when dealing with individual highly skewed data sets of
small sizes. This is especially true when potential outliers may be present in the data set. ProUCL 4.0
provides warning messages whenever the recommendations are made regarding the use the bootstrap t
method or Hall's bootstrap method.
15.1 Normally or Approximately Normally Distributed Data Sets
• For normally distributed data sets, a UCL based upon the Student's t-statistic 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 based upon Student's t-statistic may also be used when the
Sd, sy of the log-transformed data, is less than 0.5, or when the data set approximately
follows a normal distribution. Typically, a data set is approximately normal when the
normal Q-Q plot displays a linear pattern (without outliers and jumps of significant
magnitude) and the resulting correlation coefficient is high (e.g., 0.95 or higher). The
jumps and breaks in a Q-Q plot (even with a high correlation coefficient) suggest the
presence of multiple populations in the data set under study.
196
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• Student's t-UCL may also be used when the data set is symmetric (but possibly not
normally distributed). A measure of symmetry (or skewness) is k3. A value of k3 close
to zero (e.g., if the absolute value of the 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 the histogram of the data set.
15.2 Gamma Distributed Skewed Data Sets
In practice, many skewed data sets can be modeled both by a lognormal distribution and a gamma
distribution, especially when the sample size is smaller than 70-100. As well known, the 95%H-UCL of
the mean based upon a lognormal model often behaves in an erratic manner. Specifically, 95%H-UCL
often results in an unjustifiably large and impractical 95% UCL value when the sample size is small (e.g.,
n <20, 50, ..) and skewness is high. Moreover, it is also observed that a 95% UCL based upon Land H-
statistic becomes even smaller than the sample arithmetic mean. This is especially true for mildly skewed
to moderately skewed data sets of large sizes (e.g., > 50-70). In such cases, a gamma model, G(k, 6), may
be used to compute a reliable 95% UCL of the unknown population mean, }j.\.
• Many skewed data sets follow a lognormal as well as a gamma distribution. It should be
noted that the population means based upon the two models could differ significantly. A
lognormal model based upon a highly skewed (e.g., a > 2.5) data set will have an
unjustifiably large and impractical population mean, pi, and its associated UCL. The
gamma distribution is better suited to model positively skewed environmental data sets.
• 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, one
should compute a 95% UCL based upon a gamma distribution. Use of highly skewed
(e.g., a > 2.5-3.0) lognormal distributions should be avoided. For such highly skewed
lognormally distributed data sets that cannot be modeled by a gamma or an approximate
gamma distribution, nonparametric UCL computation methods based upon the
Chebyshev inequality may be used.
• The five bootstrap methods do not perform better than the two gamma UCL computation
methods. It is noted that the performances (in terms of coverage probabilities) of the
bootstrap t and Hall's bootstrap methods are very similar. Out of the five bootstrap
methods, bootstrap t and Hall's bootstrap methods perform the best (with coverage
probabilities for population mean closer to the nominal level of 0.95). This is especially
true when the skewness is quite high (e.g., k < 0.1) and the sample size is small (e.g., n <
10-15). Whenever the use of Hall's UCL or bootstrap t UCL is recommended, an
informative warning message about their use is also provided.
• Contrary to the conjecture, the bootstrap BCA method does not perform better than the
Hall's method or the bootstrap t method. The coverage for the population mean, }j.\,
provided by the BCA method is much lower than the specified 95% coverage. This is
especially true when the skewness is high (e.g., k<\) and the sample size is small (e.g.,
Singh and Singh (2003), and Singh, Singh, and laci (2002b)).
197
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• From the results presented in Singh, Singh, and laci (2002b), and in Singh and Singh
(2003), it is concluded that for data sets which follow a gamma distribution, a 95% UCL
of the mean should be computed using the adjusted gamma UCL when the shape
parameter, k, is: 0.1 < k < 0.5, and for values of £ > 0.5, a 95% UCL can be computed
using an approximate gamma UCL of the mean, ^.
• For highly skewed gamma distributed data sets with k < 0.1, the bootstrap t UCL or
Hall's bootstrap (Singh and Singh 2003) may be used when the sample size is smaller
than 15, and the adjusted gamma UCL should be used when the sample size starts
approaching or exceeding 15. The small sample size requirement increases as the
skewness increases (that is, as k decreases, the required sample size, n, increases).
• The bootstrap t and Hall's bootstrap methods should be used with caution as these
methods may yield erratic, unreasonably inflated, and unstable UCL values, especially in
the presence of outliers. In case Hall's bootstrap and bootstrap t methods yield inflated
and erratic UCL results, the 95% UCL of the mean should be computed based upon the
adjusted gamma 95% UCL. ProUCL 4.0 prints out a warning message associated with the
recommended use of the UCLs based upon the bootstrap t method or Hall's bootstrap
method. The recommendations for gamma distribution are summarized in Table 15-1.
Table 15-1. Computation of a UCL95 of the Unknown Mean, fih of a Gamma Distribution
k
k >0.5
0.1 < k<0.5
k <0.1
k <0.1
Sample Size, n
For all n
For all n
IK15
n>15
Recommendation
Approximate gamma 95% UCL
Adjusted gamma 95% UCL
95% UCL based upon bootstrap t
or Hall's bootstrap method*
Adjusted gamma 95% UCL if available,
otherwise use approximate gamma 95% UCL
*If bootstrap t or Hall's bootstrap methods yield erratic, inflated, and unstable UCL values (which often
happens when outliers are present), the UCL of the mean should be computed using the adjusted gamma
UCL.
15.3 Lognormally Distributed Skewed Data Sets
For lognormally, LN (ju, a2), distributed data sets, the H-statistic-based//-f/CL provides 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 the skewness is high (e.g., a >
2.0). The use of a lognormal model unjustifiably accommodates large and impractical values of the mean
concentration and its UCLs. The problem associated with the use of a lognormal distribution is that the
population mean, //1; of a lognormal model becomes impractically large for larger values of a, which in
turn results in an inflated H-UCL of the population mean, ^\. Since the population mean of a lognormal
model becomes too large, none of the other methods, except for the H-UCL, provides the specified 95%
coverage for that inflated population mean, n\. This is especially true when the sample size is small and
198
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the skewness is high. For extremely highly skewed data sets (with a> 2.5-3.0) of smaller sizes (e.g., < 70-
100), the use of a lognormal distribution-based H-UCL should be avoided (e.g., see Singh et al. (2002a)
and Singh and Singh (2003)). Therefore, alternative UCL computation methods, such as the use of a
gamma distribution or the use of a UCL based upon nonparametric bootstrap methods or the Chebyshev
inequality-based methods, are desirable. All skewed data sets should first be tested for a gamma
distribution. For lognormally distributed data (that cannot be modeled by gamma distribution), methods
summarized in Table 15-2 may be used to compute a 95% UCL of mean.
ProUCL can compute an H-UCL for samples of sizes up to 1000. For highly skewed lognormally
distributed data sets of smaller sizes, alternative methods to compute a 95% UCL of the population mean,
/j.\, are summarized in Table 15-2. Since skewness is a function of o (or a), the recommendations for the
computation of the UCL of the population mean are also summarized in terms of a and the sample size,
n. Here, a is an MLE of o, and is given by the Sd of log -transformed data. Note that Table 15-2 is
applicable only to the computation of a 95% UCL of the population mean based upon lognormally
distributed data sets without nondetect observations.
Table 15-2. Computation of a 95% UCL of Mean, Hi of a Lognormal Population
/\
a
a < 0.5
0.5 < a <1.0
1.0< a <1.5
1.5< a <2.0
1.5< a <2.0
2.5 < a < 3.0
3.0 < a <3.5
a > 3.5
Sample Size, n
For all n
For all n
n<25
n>25
n<20
20 < n < 50
n>50
n<20
20 < n < 50
50 < n < 70
n>70
n < 30
30 < n < 70
70100
n<15
15 150
For all n
Recommendation
Student's t, modified t, or H-UCL
H-UCL
95% Chebyshev (MVUE) UCL
H-UCL
99% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
99% Chebyshev (MVUE) UCL
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Larger of 99% Chebyshev (MVUE) UCL or
99% Chebyshev (Mean, Sd)
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Hall's bootstrap method*
Larger of 99% Chebyshev (MVUE) UCL or
99% Chebyshev (Mean, Sd)
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Use nonparametric methods*
199
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*If Hall's bootstrap method yields an erratic or unrealistically large UCL value, then the UCL of the mean
may be computed based upon the Chebyshev inequality.
15.4 Data Sets without a Discernable Skewed Distribution - Nonparametric
Methods for Skewed Data Sets
The use of gamma and lognormal distributions as discussed here cover a wide range of skewed data
distributions. For skewed data sets which are neither gamma nor lognormal, one can use a nonparametric
Chebyshev UCL or Hall's bootstrap UCL (for small samples) of the mean to estimate the EPC term.
• For skewed nonparametric data sets with negative and zero values, use a 95% Chebyshev
(Mean, Sd) UCL for the population mean, y.\.
For all other nonparametric data sets with only positive values, the following method may be used to
estimate the EPC term.
• For mildly skewed data sets with a < 0.5, one can use the Student's t-statistic or
modified t-statistic to compute a 95% UCL of mean, ^.
• 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, y.\.
• For nonparametric moderately to highly skewed data sets (e.g., a in the interval (1.0,
2.0]), one may use a 99% Chebyshev (Mean, Sd) UCL or a 97.5% Chebyshev (Mean, Sd)
UCL of the population mean, //1; to obtain an estimate of the EPC term.
• For highly skewed to extremely highly skewed data sets with a in the interval (2.0, 3.0],
one may use Hall's UCL or a 99% Chebyshev (Mean, Sd) UCL to compute the EPC term.
• Extremely skewed nonparametric data sets with a exceeding 3 provide poor coverage.
For such highly skewed data distributions, none of the methods considered provide the
specified 95% coverage for the population mean, ^. The coverage provided by the
methods decrease as 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 less than 0.95) when the sample size is small.
It is noted that the coverage for the population mean provided by Hall's method (and
bootstrap t method) does not increase much as the sample size, n, increases. However, as
the sample size increases, coverage provided by the 99% Chebyshev (Mean, Sd) UCL
method also increases. Therefore, for larger samples, a UCL should be computed based
upon the 99% Chebyshev (Mean, Sd) method. This large sample size requirement
increases as a increases. These recommendations are summarized in Table 15-3.
200
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Table 15-3. Computation of 95% UCL of Mean, fih Based Upon a Skewed Data Set (with all positive values) without a
Discernable Distribution, where G issrfof Log-Transformed Data
/*,
a
o <0.5
0.5 < a < 1.0
1.0< a <2.0
2.0 < a <3.0
3.0 < a <3.5
a > 3.5
Sample Size, n
For all n
For all n
n<50
n>50
n<10
n>10
n < 30
n>30
n< 100
n>100
Recommendation
95% UCL based on Student's t or modified t-statistic
95% Chebyshev (Mean, Sd) UCL
99% Chebyshev (Mean, Sd) UCL
97.5% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
* 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. The results as summarized in Tables 15-1 through 15-3 are summarized in Table 15-4, shown on
the next page.
201
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Table 15-4. Recommended UCL95 Computation Methods for Full-Uncensored Data Sets without Nondetect Observations
Skewness
Sample Size
95% Student t
95% modified t
_i
o
z>
X
s?
m
o>
95% Hall's Bootstrap
95% Chebyshev (MVUE)
97.5% Chebyshev (MVUE)
99% Chebyshev (MVUE)
95% Chebyshev (Mean, Sd)
97.5% Chebyshev (Mean, Sd)
99% Chebyshev (Mean, Sd)
95% Approx. Gamma
95% Bootstrap t
95% Adjusted Gamma
Normal or Approximate Normal (with a < 0.5) Distribution
A\\n
•
•
Gamma Distribution
k <0.1
0.1 < k <0.5
k >0.5
n<-\5
n>-\5
A\\n
A\\n
•
•
•
•
•
•
Lognormal Distribution
a < 0.5
0.5 < a < 1.0
1.0< a <1.5
1.5< o <2.0
2.0 < a <2.5
2.5 < a < 3.0
3.0 < a <3.5
A\\n
A\\n
n<25
n>25
n<20
20 < n < 50
n >50
n<20
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Table 15-4. Recommended UCL95 Computation Methods for Full-Uncensored Data Sets without Nondetect
Observations-Continued
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15.5 Should the Maximum Observed Concentration be Used as an
Estimate of the EPC Term?
This topic has been discussed earlier in Chapter 1. It is included here only for the convenience of the user.
In practice, atypical user tends to use the maximum sample value as an estimate of the EPC term. This is
especially true when the sample size is small or the data are highly skewed. The discussion and
suggestions as described in Chapter 1 apply to both Chapters 15 and 16. Singh and Singh (2003) studied
the max test (using the maximum observed value as an estimate of the EPC term) in their simulation
study. Previous (e.g., RAGS document (EPA, 1992)) use of the maximum observed value has been
recommended as a default option to estimate the EPC term when a 95% UCL (e.g., the H-UCL) exceeded
the maximum value. Only two 95% UCL computation methods, namely the Student's t-UCL and Land's
H-UCL, were used previously to estimate the EPC term (e.g., EPA, 1992). ProUCL 4.0 can compute a
95% UCL of the mean using several methods based upon the normal, gamma, lognormal, and
"nonparametric" distributions. Furthermore, since the EPC term represents the average exposure
contracted by an individual over an exposure area (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.
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Singh and Singh (2003) also noted that for skewed data sets of small sizes (e.g., < 10-20), the max test
does not provide the specified 95% coverage to the population mean, and for larger data sets, it
overestimates the EPC term which may require unnecessary further remediation. For the distributions
considered, the maximum value is not a sufficient statistic for the unknown population mean. The use of
the maximum value as an estimate of the EPC term ignores most (except for the maximum value) of the
information contained in a data set. It is, therefore, not desirable to use the maximum observed value as
an estimate of the EPC term representing average exposure by an individual over an EA.
It is recommended that the maximum observed value NOT be used as an estimate of the EPC term.
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. When a 95%
UCL exceeds the maximum observed value, ProUCL recommends the use of an alternative UCL method
based upon a 97.5% or 99% Chebyshev UCL.
It should also be noted that for highly skewed data sets, the sample mean indeed can even exceed the
upper 90%, or higher, etc., percentiles, and consequently, a 95% UCL of the mean can exceed the
maximum observed value of a data set. This is especially true when one is dealing with lognormally
distributed data sets of small sizes. For such highly skewed data sets which cannot be modeled by a
gamma distribution, a 95% UCL of the mean should be computed using an appropriate nonparametric
method as summarized in Table 15-4.
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205
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Chapter 16
Recommendations to Compute a 95% UCL of the Population
Mean, fJi, Using Data Sets with Nondetects with Multiple
Detection Limits
This chapter summarizes the recommendations based on the simulation experiments conducted by Singh,
Maichle, and Lee (USEPA, 2006) to compare the performances of the UCL computation methods based
upon data sets with BDLs and multiple detection limits (DLs). ProUCL 4.0 suggests the use of
appropriate UCLs based upon the findings of Singh, Maichle, and Lee (USEPA, 2006). For convenience,
the recommended UCL95 computation methods have been tabulated in Table 16 as functions of the
sample size, skewness, and censoring intensity. General observations and recommendations regarding the
difficulties associated with data sets with NDs are described first.
16.1 General Recommendations and Suggestions
• In practice, it is not easy to verify (perform goodness-of-fit) the distribution of a left-
censored data set with NDs. Therefore, emphasis is given on the use of nonparametric
UCL95 computation methods, which can also be used to handle multiple detection limits.
• This is specifically true when the percentage (%) of nondetects exceeds 40%-50%.
• Most of the parametric MLE methods assume that there is only one detection limit. It
should also be noted that the MLEs behave in an unstable manner when the % of NDs
exceeds 40%-50%. Moreover, as mentioned before, it is hard to verify and justify the
conclusion of a GOF test for data sets with nondetects in excess of 40%-50%.
• Therefore, for data sets with many nondetects (> 40%-50%), it is suggested to use
nonparametric methods to estimate the various environmental parameters (BTVs, EPC
terms) of interest and to perform site versus background comparisons.
• In practice, a left-censored data set often has multiple detection limits. For such methods,
the KM method can be used. ProUCL 4.0 provides UCL computation methods that can
be used on data sets with multiple detection limits including the DL/2 method, KM
method, and robust ROS method.
• As mentioned earlier, for reliable and accurate results, it is suggested that the user make
sure that the data set under study represents a single statistical population (e.g.,
background reference area, or an AOC) and not a mixture population (e.g., clean and
contaminated site areas).
• It is recommended to identify all of the potential outliers and study them separately. The
computation of the statistics such as UCL95 and background statistics should be based
upon the majority of the data set representing a single dominant population. Decisions
about the appropriate disposition (include or not include) of outliers should be made by
all interested members of the project team. When in doubt, it is suggested to compute and
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• Simple classical outlier identification methods (Dixon test and Rosner test) are also
available in ProUCL 4.0. More effective robust outlier procedures (e.g., Rousseeuw and
Leroy (1987) and Singh and Nocerino (1995)) are available in Scout (1999).
• In case a data set represents a mixture sample (from two or more populations), one should
partition the mixture sample into component sub-samples (e.g., Singh, Singh, and
Flatman(1994)).
• Avoid the use of transformations (to achieve symmetry) while computing the upper limits
for various environmental applications, as all remediation, cleanup, background
evaluation decisions, and risk assessment decisions have to be made using statistics in the
original scale. Also, it is more accurate and easier to interpret the results computed in the
original scale. The results and statistics computed in the original scale do not suffer from
an unknown amount of transformations bias.
• Specifically, avoid the use of a lognormal model even when the data appear to be
lognormally distributed. Its use often results in incorrect and unrealistic statistics of no
practical purpose or importance or significance. Several variations of estimation methods
(e.g., robust ROS and FP-ROS on log-transformed data, delta lognormal method) on log-
transformed data have been developed and used by the practitioners. This has caused
some confusion among the users of the statistical methods dealing with environmental
data sets. The proper use of a lognormal distribution (e.g., how to properly back-
transform UCL of mean in the log-scale to obtain a UCL of mean in original scale) is not
clear to many users, which in turn may result in the incorrect use and computation of an
estimate (= UCL95) of the population mean.
• The parameter in the transformed space may not be of interest to make cleanup decisions.
The cleanup and remediation decisions are often made in the original raw scale;
therefore, the statistics (e.g., UCL95) computed in transformed space need to be back-
transformed in the original scale. It is not clear to a typical user how to back-transform
results in the log-scale or any other scale obtained using a Box-Cox (BC)-type
transformation to original raw scale. The transformed results often suffer from significant
amount of transformation bias.
• The question now arises-how one should back-transform results from a log-space (or any
other transformed space) to the original space? Unfortunately, no defensible guidance is
available in the environmental literature to address this question. Moreover, the back-
transformation formula will change from transformation to transformation (BC-type
transformations), and the bias introduced by such transformations will remain unknown.
Therefore, in cases when a data set in the "raw" scale cannot be modeled by a parametric
distribution, it is desirable to use nonparametric methods (many available in ProUCL 4.0)
rather than testing or estimating a parameter in the transformed space.
• On page (78) of Helsel (2005), the use of the robust ROS MLE method (Kroll, C.N. and
J.R. Stedinger (1996)) has been suggested to compute summary statistics. In this hybrid
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method, MLEs are computed using log-transformed data. Using the regression model as
given by equation (3-21) of Section 3, the MLEs of the mean (used as intercept) and sd
(used as slope) in the log-scale are used to extrapolate the NDs in the log-scale. Just like
in the robust ROS method, all of the NDs are transformed back in the original scale by
exponentiation. This results in a full data set in the original scale. One may then compute
the mean and sd using the full data set. The estimates thus obtained are called robust ROS
ML estimates (Helsel (2005), and Kroll and Stedinger (1996)). However, the
performance of such a hybrid estimation method is not well known. Moreover, for higher
censoring levels, the MLE methods sometimes behave in an unstable manner, especially
when dealing with moderately skewed to highly skewed data sets (e.g., with a >1.0).
o It should be noted that the performance of this hybrid method is unknown.
o It is not clear why this method is called a robust method.
o The stability of the MLEs obtained using the log-transformed data is doubtful,
especially for higher censoring levels.
o The BCA and (% bootstrap) UCLs based upon this method will fail to provide the
adequate coverage for the population mean for moderately skewed to highly skewed
data sets.
• The DL/2 (t) UCL method does not provide adequate coverage (for any distribution and
sample size) for the population mean, even for censoring levels as low as 5%, 10%, 15%.
This is contrary to the conjecture and assertion (e.g., EPA (2000)) often 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.
This DL/2 (t) UCL method is not recommended by the authors and developers of ProUCL 4.0; it is
included only for comparison or research purposes.
• The KM method is a preferred method as it can handle multiple detection limits.
Moreover, the nonparametric UCL95 methods (KM (BCA), KM (z), KM (%), KM (t))
based upon the KM estimates provide good coverages for the population mean (e.g.,
Helsel (2005) and Singh et al. (2006)).
• For a symmetric distribution (approximate normality), several UCL95 methods provide
good coverage (-95%) for the population mean, including the Winsorization mean,
Cohen's MLE (t), Cohen's MLE (Tiku), KM (z), KM (t), KM (%) and KM (BCA) (e.g.,
Helsel (2005) and Singh et al. (2006)).
16.2 Recommended UCL95 Methods for Normal (Approximate Normal)
Distribution
• Fornormal and approximately normal (e.g., symmetric or with sd, a < 0.5) distribution:
The most appropriate UCL95 computation methods for normal or approximately normal
distributions are the KM (t) or KM (%) methods. For symmetric distributions, both of
these methods perform equally well on left-censored data sets for all censoring levels (%
nondetects) and sample sizes.
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16.3 Recommended UCL95 Methods for Gamma Distribution
• Highly skewed gamma distributions, G(k, 9), with shape parameter, k < 1:
o Use the nonparametric KM (Chebyshev) UCL95 method for censoring levels < 30%.
o Use the nonparametric KM (BCA) UCL95 method for censoring levels in the interval
[30%, 50%).
o Use the nonparametric KM (t) UCL95 method for censoring levels > 50%.
• Moderately skewed gamma distributions, G(k, 9), with shape parameter, K k < 2:
o For censoring level < 10%, use the KM (Chebyshev) UCL95 method.
o For higher censoring levels [10%, 25%), use the KM (BCA) UCL95 method.
o For censoring levels in [25%, 40%), use the KM (%) UCL95 method.
o For censoring levels > 40%, use the KM (t) UCL95 method.
• Mildly skewed gamma distributions, G(k, 9), with k > 2:
o Use the KM (BCA) UCL95 method for lower censoring levels (< 20%).
o For censoring levels in the interval [20%, 40%), use the KM (%) UCL95.
o For censoring > 40%, use the KM (t) UCL95 computation method.
16.4 Recommended UCL95 Methods for Lognormal Distribution
• Mildly skewed data sets with a < 1:
o For censoring levels (< 20%) and sample of sizes less than 50-70, use the KM
(Chebyshev) UCL95.
o For censoring levels (< 20%) and samples of sizes greater than 50-70, use the KM
(BCA)UCL95.
o For censoring levels in the interval [20%, 40%) and all sample sizes, use the KM
(BCA)UCL95.
o For censoring level > 40%, use the KM (%) or KM (t) UCL95 method.
• Data sets with a in the interval (1, 1.5]:
o For censoring levels < 50% and samples of sizes < 40, use the 97.5% KM
(Chebyshev) UCL.
o For censoring levels < 50% , samples of sizes > 40, use 95% KM (Chebyshev) UCL.
o For censoring levels > 50%, use the KM (BCA) UCL95 for samples of all sizes.
• Highly skewed data sets with a in the interval (1.5, 2]:
o For sample sizes < 40, censoring levels <50%, use 99% KM (Chebyshev) UCL.
o For sample sizes > 40, censoring levels < 50%, use 97.5% KM (Chebyshev) UCL.
o For samples of sizes < 40-50 and censoring levels > 50%, use the 97.5% KM
(Chebyshev) UCL.
o For samples of sizes > 40-50, and censoring levels > 50%, use the 95% KM
(Chebyshev) UCL.
• Use a similar pattern for more highly skewed data sets with a > 2.0, 3.0:
o For extremely highly skewed data sets, an appropriate estimate of the EPC term (in
terms of adequate coverage) is given by a UCL based upon the Chebyshev inequality
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and KM estimates. The confidence coefficient to be used will depend upon the
skewness. For highly skewed data sets, a higher (e.g., > 95%) confidence coefficient
may have to be used to estimate the EPC.
o As the skewness increases, the confidence coefficient also increases.
o For such highly skewed distributions (with a > 2.0, 3.0), for lower sample sizes (e.g.,
< 50-60), one may simply use 99% KM (Chebyshev) UCL to estimate the population
mean, EPC term, and other relevant threshold (e.g., UPL, percentiles) values.
o For sample sizes greater than 60, one may use a 97.5% KM (Chebyshev) UCL as an
estimate of the population mean or mass.
16.5 Recommended Nonparametric UCL Methods
• For symmetric or approximately symmetric distribution-free, nonparametric data sets
with a < 0.5: Use the same UCL computation methods as for the data sets coming from
a normal or an approximate normal (symmetric) population. These methods are
summarized above in the normal UCL computation section.
• For skewed distribution-free, nonparametric data sets with a >0.5: Most of the
recommended UCL computation methods for a lognormal distribution, as described
above in the lognormal UCL section, do not assume the lognormality of the data set.
Therefore, the UCL computation methods, as described in the lognormal UCL
computation section, can be used on skewed nonparametric data sets that do not follow
any of the well-known parametric distributions.
The suggested parametric and nonparametric UCL95 computation methods for data sets with nondetect
observations are summarized in Table 16, shown on the next page.
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Table 16. Recommended UCL95 Computation Methods for Left-Censored Data Sets with Nondetect Observations
Skewness
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Table 16. Recommended UCL95 Computation Methods for Left-Censored Data Sets with Nondetect Observations -
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Note: In Table 16, phrase "All n " represents only valid (e.g., n > 3) and recommended (n > 8 to 10)
values of the sample size, n. As mentioned throughout the report, it is not desirable to use statistical
methods on data sets of small sizes (e.g., with n < 8 to 10). However, it should be noted that the sample
size requirements and recommendations (n > 8 to 10) as described in this report are not the limitations of
the methods considered in this report. One of the main reasons for the recommendation that the sample
size should be at least 8 to 10 is that the estimates and UCLs based upon small data sets, especially with
many below detection limit observations (e.g., 30%, 40%, 50%, and more), may not be reliable and
accurate enough to draw conclusions for environmental applications. It should be noted that in order to
be able to use bootstrap re-sampling methods, it is desirable to have a minimum of 10-15 observations
(e.g., n > 10-15). Therefore, the phrase "Alln" in Table 16, should be interpreted as that the sample size,
n, is least 8 to 10. The software, ProUCL 4.0, will provide appropriate warning messages when a user
tries to use a method on data sets of small sizes.
Also, Hall's bootstrap and bootstrap t methods to compute a UCL based upon a full data set (without
nondetects) should be used with caution. These two bootstrap methods may yield erratic and unstable
UCL results, especially, when outliers are present. In such cases, it is desirable to use alternative UCL
212
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methods based upon Chebyshev inequality. ProUCL software provides a warning message for erratic
UCL results based upon Hall's bootstrap t methods.
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Glossary
This glossary defines selected words in this User Guide to describe impractically large UCL values of the
unknown population mean, mi. In practice, the UCLs based upon Land's H-statistic (H-UCL), and
bootstrap methods such as the bootstrap t and Hall's bootstrap methods (especially when outliers are
present) can become impractically large. The UCLs based upon these methods often become larger than
the UCLs based upon all other methods by several orders of magnitude. Such large UCL values are not
achievable as they do not occur in practice. Words like "unstable" and "unrealistic" are used to describe
such impractically large UCL values.
UCL: upper confidence limit of the unknown population mean.
Coverage = Coverage Probability: The coverage probability (e.g., = 0.95) of a UCL of the population
mean represents the confidence coefficient associated with the UCL.
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.
Stable UCL: The UCL of a population mean is a stable UCL if it represents a number of a practical merit,
which also has some physical meaning. That is, a stable UCL represents a realistic number (e.g.,
contaminant 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.
Reliable UCL: This is similar to a stable UCL.
Unstable UCL = Unreliable 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 population mean. It
represents an impractically large value that cannot be achieved in practice. For example, the use of Land's
H-statistic often results in impractically large inflated UCL value. 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 in this User
Guide.
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