4>EPA
United States
Environmental Protection
Agency
       ProUCL Version 5.1
             User Guide
   Statistical Software for Environmental Applications
      for Data Sets with and without Nondetect
                Observations
        RESEARCH AND DEVELOPMENT

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                                                               EPA/600/R-07/041
                                                                 October 2015
                                                                  www.epa.gov
                ProUCL Version  5.1
                         User Guide
     Statistical Software for Environmental Applications

           for Data Sets with and without Nondetect

                            Observations

                               Prepared for:

                            Felicia Barnett, Director
        ORD Site Characterization and Monitoring Technical Support Center (SCMTSC)
                    Superfund and Technology Liaison, Region 4
                         U.S. Environmental Protection Agency
                      61 Forsyth Street SW, Atlanta, GA 30303

                               Prepared by:
                     Anita Singh,  Ph.D. and Robert Maichle

                           Lockheed Martin/SERAS
                               IS&GS-CIVIL
                            2890 Woodbridge Ave
                              Edison NJ 08837
                   U.S. Environmental Protection Agency
                    Office of Research and Development
                        Washington, DC 20460
Notice: Although this work was reviewed by EPA and approved for publication, it may not necessarily reflect official
     Agency policy. Mention of trade names and commercial products does not constitute endorsement or
     recommendation for use.
                                                                     129cmb07

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                                         NOTICE

The United States Environmental Protection Agency (U.S. EPA) through its Office of Research and
Development (ORD) funded and managed the research described  in  ProUCL Technical  Guide and
methods incorporated in the ProUCL software. It has been peer reviewed by the U.S. EPA and approved
for publication. Mention of trade names or commercial products does not constitute endorsement or
recommendation by the U.S. EPA for use.

   •   All versions  of the  ProUCL software including the current version ProUCL 5.1  have been
       developed by Lockheed Martin, IS&GS - CIVIL under the Science, Engineering, Response and
       Analytical contract with the U.S. EPA and is made available through the U.S. EPA Technical
       Support Center (TSC) in Atlanta, Georgia (GA).

   •   Use of any portion of ProUCL that does  not comply with the ProUCL Technical Guide is not
       recommended.

   •   ProUCL contains embedded licensed software. Any modification of the ProUCL source code
       may violate the embedded licensed software agreements and is expressly forbidden.

   •   ProUCL software provided by the U.S. EPA was scanned with McAfee VirusScan version 4.5.1
       SP1 and is certified free of viruses.

With respect to ProUCL distributed software and  documentation, neither the U.S. EPA nor any of their
employees, assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of
any information, apparatus, product, or process disclosed. Furthermore, software and documentation are
supplied "as-is" without guarantee or warranty, expressed or implied, including without limitation, any
warranty of merchantability or fitness for a specific purpose.

ProUCL software is a statistical software package providing statistical methods described in various U.S.
EPA guidance documents. ProUCL does not describe U.S. EPA policies and should not be considered to
represent U.S. EPA policies.

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Minimum Hardware Requirements

ProUCL 5.1 will function but will run slowly and page a lot.

          •   Intel Pentium 1.0 gigahertz (GHz)
          •   45 MB of hard drive space
          •   512 MB of memory (RAM)
          •   CD-ROM drive or internet connection
          •   Windows XP (with SP3), Vista (with SP1 or later), and Windows 7.

ProUCL 5.1 will function but some titles and some Graphical User Interfaces (GUIs) will need to be
scrolled. Definition without color will be marginal.

          •   800 by 600 Pixels
          •   Basic Color is preferred

Preferred Hardware Requirements

          •   1 GHz or faster Processor.
          •   1 gigabyte (GB) of memory (RAM)
          •   1024 by 768 Pixels or greater color display

Software Requirements

ProUCL 5.1 has been developed  in the  Microsoft .NET Framework 4.0 using  the C# programming
language. To properly run ProUCL 5.1 software, the computer using the program must have the .NET
Framework 4.0 pre-installed. The downloadable .NET Framework 4.0 files can be obtained from one of
the following websites:

          •   http://msdn.microsoft.com/netframework/downloads/updates/default.aspx
              http://www.microsoft.com/en-us/download/details.aspx?id=17851
              Quicker site for 32 Bit Operating systems

          •   http://www.microsoft.com/en-us/download/details.aspx?id=24872
              Use this site if you have a 64 Bit operating system

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Installation Instructions when Downloading ProUCL 5.1 from the EPA Web Site

          •   Download the file SETUP.EXE from the EPA Web site and save to a temporary location.

          •   Run the SETUP.EXE program. This will create a ProUCL directory and two folders:
              1) The 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.

Installation Instructions when Copying ProUCL 5.1 from a CD

          •   Create a folder named ProUCL 5.1 on a local hard drive of the machine you wish to
              install ProUCL 5.1.

          •   Extract the zipped file ProUCL.zip to the folder you have just created.

          •   Run ProUCL.exe

Note: If you have extension turned off, the program will show with the name ProUCL in your directory
and have an Icon with the label ProUCL.

Creating a Shortcut for ProUCL 5.1 on Desktop

          •   To create a shortcut of the ProUCL program on your desktop, go  to your ProUCL
              directory and right click on the executable program and send it to desktop. A ProUCL
              icon will be displayed on your desktop. This shortcut will point to the ProUCL directory
              consisting of all files required to execute ProUCL 5.1.

Caution: Because  all files in your ProUCL directory are needed to execute the ProUCL software, one
needs to generate a shortcut using the process described above. Simply dragging the ProUCL executable
file from Window Explorer onto your desktop will not work successfully (an error message will appear)
as all files needed to run the software are not available on your desktop. Your shortcut should point to the
directory path with all required ProUCL files.

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                                      ProUCL 5.1

Software ProUCL version  5.1 (ProUCL 5.1), its earlier versions:  ProUCL version  3.00.01,  4.00.02,
4.00.04, 4.00.05, 4.1.00, 4.1.01, and ProUCL 5.0.00, associated Facts Sheet, User Guides and Technical
Guides (e.g., EPA 2010a, 2010b, 2013a, 2013b) can be downloaded from the following EPA website:

http://www.epa.gov/osp/hstl/tsc/software.htm
http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm

Material for ProUCL webinars offered in March 2011, and relevant literature used in the development of
various ProUCL versions can also be downloaded from the above EPA website.

Contact Information for all Versions of ProUCL

Since 1999, the ProUCL software has been developed under the direction of the Technical Support Center
(TSC). As of November 2007, the direction of the TSC is transferred from Brian Schumacher to Felicia
Barnett.  Therefore, any comments or questions concerning all versions of ProUCL software should be
addressed to:

       Felicia Barnett, Director
       ORD Site Characterization and Monitoring Technical Support Center (SCMTSC)
       Superfund and Technology Liaison, Region 4
       U.S. Environmental Protection Agency
       61 Forsyth Street SW, Atlanta, GA 30303-8960
       barnett.felicia@epa.gov
       (404)562-8659
       Fax: (404) 562-8439

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Getting Started
The look and feel of ProUCL 5.1 is similar to that of ProUCL 5.0; and they share the same names for
modules and drop-down menus. The functionality and the use of the methods and options available in
ProUCL  5.1 have been illustrated using  Screen  shots of output screens  generated by ProUCL 5.1.
ProUCL 5.1 uses a pull-down menu structure, similar to a typical Windows program. For modules where
no changes have been made in ProUCL since 2010 (e.g., Sample Sizes), screen shots as used in ProUCL
5.0 documents  have been  used in ProUCL 5.1  documents. Some of the screen shots generated using
ProUCL 5.1 might have ProUCL 5.0 in their titles as those screen shots have not been re-generated and
replaced. The screen shown below appears when the program is executed.
             Navigation
             Panel
               4
Main
Window
                                                    Log Panel
The above screen consists of three main window panels:

       •  The MAIN WINDOW displays data sheets and outputs results from the procedure used.

       •  The NAVIGATION PANEL displays the name of data sets and all generated outputs.

              o  The navigation panel can hold up to 40 output files. In order to see more files (data
                 files or generated output files), one can click on Widow Option.

              o  In the NAVIGATION PANEL, ProUCL assigns self explanatory names to output
                 files generated using the various modules of ProUCL.  If the same module (e.g.,
                 Time Series Plot) is used many times, ProUCL identifies them by using letters a, b,
                 c,...and so on as shown below.

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                                                     Navigation Panel
                                                  Name
                                                  Well-mp-27jds
                                                  REGRESSES
                                                  Theil-Senjds
                                                  Trend Test.gst
                                                  Time Series .gst
                                                  Time Series_a.gst
                                                  Time Series_b.gst
                                                  Time Series_c.gst
                                                  Mann-Kendall :ds
                                                  Trend Test_a.gst
               o   The user may want to assign names of his choice to these output files when saving
                   them using the "Save" or "Save As" Options.

        •   The LOG PANEL displays transactions in green, warning messages in orange, and errors in
           red. For an example, when one attempts to run a procedure meant for left-censored data sets
           on a full-uncensored data set, ProUCL 5.1 will output a warning in orange in this panel.

               o   Should both panels be unnecessary, you can choose Configure ^- Panel ON/OFF.

The use of this option gives extra space to see and  print out the statistics of interest.  For example, one
may want to turn off these panels when multiple variables (e.g., multiple quantile-quantile  [Q-Q] plots)
are analyzed and goodness-of-fit (GOF) statistics and  other statistics may need to be captured for all of
the selected variables. The following screen was generated using ProUCL 5.0. An identical screen will be
generated using ProUCL 5.1 with title name as ProUCL 5.1-  [WorkSheet.xls].

                                                                             CL 5.0 - [WorkSheetxIs
,; File
Nav
Name
WoikShe
Edit | Stats/Sample Sizes Graphs Statistical Tests Upper Limits/BTVs UCLs/EPCs Windows Help
Configure Display ^
Cut CtrkX
Copy Ctrl+C
Paste Ctrl+V
Header Name


5
6

Full Precision
^] Log Panel
** Navigation Panel
Excel 2003

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                              EXECUTIVE SUMMARY

The  main objective of the  ProUCL software  funded by the United States Environmental Protection
Agency (EPA) is to compute rigorous statistics to help decision makers and project teams in making good
decisions at a polluted site which are cost-effective, and protective of human health and the environment.
The  ProUCL software is based upon the  philosophy that rigorous statistical methods can be used to
compute reliable estimates of population parameters and decision making statistics including: the upper
confidence limit (UCL)  of the mean, the  upper tolerance limit  (UTL), and the upper prediction limit
(UPL) to help decision makers and project teams in making correct decisions. A few commonly used text
book type methods (e.g., Central Limit  Theorem [CLT], Student's t-UCL)  alone cannot address all
scenarios and  situations occurring in environmental  studies. Since many environmental decisions are
based upon a 95 percent (%) UCL (UCL95) of the population mean, it is important to compute UCLs of
practical merit. The  use and  applicability of a statistical method (e.g., student's  t-UCL, CLT-UCL,
adjusted gamma-UCL, Chebyshev UCL, bootstrap-t UCL) depend upon data size,  data skewness, and
data distribution.  ProUCL computes decision  statistics using several  parametric  and nonparametric
methods covering  a wide-range  of data variability, distribution,  skewness, and sample size.  It is
anticipated that the availability of the statistical methods in the ProUCL software covering a wide range
of environmental data  sets  will  help the decision  makers in making more informative and  correct
decisions at Superfund and Resource Conservation and Recovery Act (RCRA) sites.

It is noted  that for moderately skewed to highly skewed environmental data sets, UCLs based on the CLT
and the Student's t-statistic fail to provide the desired coverage (e.g.,  0.95) to the population mean even
when the sample sizes are as large as 100 or more. The sample size requirements associated with the CLT
increases with skewness. It would be incorrect to state that a CLT or Student's statistic based UCLs are
adequate to estimate  Exposure Point Concentrations (EPC) terms based upon skewed data sets.  These
facts have  been described in the published documents (Singh, Singh, and Engelhardt [1997, 1999]; Singh,
Singh, and laci 2002; Singh and Singh 2003; and Singh et al. 2006) summarizing simulation experiments
conducted on positively skewed data sets to evaluate the performances of the various UCL computation
methods. The  use of a parametric lognormal distribution on a lognormally distributed data set yields
unstable impractically large  UCLs values,  especially when the standard deviation (sd) of the log-
transformed  data becomes greater than 1.0 and  the data set is of small size less than (<) 30-50. Many
environmental data sets can be modeled by a gamma as well as  a lognormal distribution. The use of a
gamma distribution on  gamma distributed  data  sets tends to yield UCL values  of practical  merit.
Therefore, the use of gamma distribution based decision statistics such as UCLs, UPLs, and UTLs should
not be dismissed by stating that it is easier to use a lognormal model to compute these upper limits.

The  suggestions made  in ProUCL are based upon  the  extensive  experience  of the  developers in
environmental statistical methods, published environmental literature, and procedures described in many
EPA guidance documents. These suggestions are made to help the users in selecting the most appropriate
UCL to estimate the EPC term which is routinely used in exposure assessment  and  risk management
studies of the USEPA. The suggestions are based upon the findings of many simulation studies described
in Singh, Singh, and Engelhardt (1997, 1999); Singh, Singh, and laci (2002); Singh and Singh (2003); and
Singh et al. (2006).  It should be  pointed out that  a typical simulation study does not (cannot) cover all
real world data sets of various sizes and skewness from all distributions. When deemed  necessary, the
user may want to consult a statistician to select an appropriate upper limit to estimate the EPC term and
other environmental parameters of interest. For an analyte (data set)  with skewness (sd of logged data)
near the end points of the skewness intervals presented in decision tables of Chapter 2  (e.g., Tables 2-9

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through 2-11), the user may select the most appropriate UCL based upon the site conceptual site model
(CSM), expert site knowledge, toxicity of the analyte, and exposure risks associated with that analyte.

The inclusion of outliers in the computation of the various decision statistics tends to yield inflated values
of those decision statistics, which can lead to poor decisions. Often statistics that are computed for a data
set which includes a few outliers tend to be inflated and represent those outliers rather than representing
the main dominant population of interest (e.g., reference area).  Identification of outliers, observations
coming from population(s) other than the main dominant population is suggested, before computing the
decision  statistics needed  to address project objectives. The project team may want to perform the
statistical evaluations twice,  once with outliers and once without outliers. This exercise will help the
project team in computing reliable and defensible decision statistics which are needed to make cleanup
and remediation decisions at polluted sites.

The initial development during 1999-2000 and all subsequent upgrades and enhancements of the ProUCL
software  have been funded  by  U.S. EPA through its Office of Research and Development (ORD).
Initially ProUCL was developed  as a research tool  for U.S. EPA  scientists and researchers of the
Technical Support Center (TSC) and ORD- National Exposure Research Laboratory  (NERL), Las Vegas.
Background evaluations, groundwater (GW) monitoring, exposure and risk management and cleanup
decisions in support of the Comprehensive Environmental Recovery, Compensation, and Liability Act
(CERCLA) and RCRA site projects of the U.S. EPA are often derived based upon test statistics such as
the  Shapiro-Wilk  (S-W)  test,  t-test,  Wilcoxon-Mann-Whitney (WMW) test, analysis of variance
(ANOVA), and Mann-Kendall (MK) test and decision  statistics including UCLs of the mean, UPLs, and
UTLs. To address  the statistical needs of the environmental projects of the  USEPA,  over the years
ProUCL  software has been  upgraded and  enhanced  to include many graphical  tools and statistical
methods described in many EPA guidance documents including: EPA 1989a, 1989b,  1991, 1992a, 1992b,
2000 Multi-Agency Radiation Survey and Site Investigation Manual (MARSSIM), 2002a, 2002b, 2002c,
2006a, 2006b, and 2009.  Several statistically rigorous methods (e.g., for data sets with nondetects [NDs])
not easily available in the existing guidance  documents and in the  environmental  literature are also
available in ProUCL 5.0/ProUCL 5.1.

ProUCL  5.1/ProUCL 5.0 has graphical, estimation, and hypotheses testing methods for uncensored-full
data sets and for left-censored data sets including ND observations with multiple detection limits (DLs) or
reporting limits (RLs). In addition to computing general statistics, ProUCL 5.1 has goodness-of-fit (GOF)
tests for normal, lognormal  and gamma  distributions, and parametric  and nonparametric methods
including bootstrap methods  for skewed data sets for computation of decision making statistics such as
UCLs of the mean (EPA 2002a), percentiles,  UPLs for a pre-specified number of future observations
(e.g., k with k=\, 2, 3,...), UPLs for mean of future k (>1)  observations,  and UTLs  (e.g., EPA 1992b,
2002b, and 2009). Many positively skewed environmental data sets can be modeled by  a lognormal as
well as a gamma model. It is  well-known that for moderately skewed to highly skewed data sets, the use
of a lognormal distribution tends to yield inflated and unrealistically  large values of the decision statistics
especially when the sample size  is small (e.g.,  <20-30). For gamma distributed skewed uncensored and
left-censored data sets, ProUCL software computes decision  statistics including  UCLs, percentiles, UPLs
for future k (>1) observations, UTLs, and upper simultaneous limits (USLs).

For data sets  with NDs,  ProUCL has several estimation methods including  the Kaplan-Meier (KM)
method, regression on order  statistics (ROS) methods  and substitution methods (e.g., replacing NDs by
DL,  DL/2).   ProUCL 5.1 can be used  to compute upper limits which adjust  for  data skewness;
specifically, for skewed  data sets, ProUCL computes upper limits using KM estimates in gamma
(lognormal) UCL and UTL equations provided the detected observations in the left-censored data set
follow a gamma (lognormal)  distribution.  Some poor performing commonly used and cited methods such

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as the DL/2 substitution method and H-statistic based UCL computation method have been retained in
ProUCL 5.1 for historical reasons, and research and comparison purposes.

The  Sample Sizes module of ProUCL can be used to develop data quality objectives (DQOs) based
sampling designs and to perform power evaluations needed to address statistical issues associated with a
variety of site projects. ProUCL provides user-friendly options to enter the desired values for the decision
parameters such as Type I and Type II error rates, and other  DQOs used to determine the minimum
sample sizes needed to address project  objectives. The Sample Sizes module can compute DQO-based
minimum sample sizes  needed:  to  estimate the population mean; to perform single and two-sample
hypotheses testing  approaches; and in acceptance sampling to accept or reject a batch of discrete items
such as a lot of drums containing hazardous waste. Both parametric (e.g., t-test) and nonparametric (e.g.,
Sign test, WMW test, test for proportions) sample size determination methods are available in ProUCL.

ProUCL has exploratory graphical methods for both uncensored data sets and for left-censored data sets
consisting of ND observations. Graphical methods in ProUCL include histograms,  multiple quantile-
quantile (Q-Q) plots, and side-by-side box plots. The use of graphical displays provides additional insight
about the information contained in a data set that may not otherwise be revealed by the use of estimates
(e.g., 95% upper limits)  and test statistics (e.g., two-sample t-test, WMW test). In addition to providing
information about the data distributions  (e.g., normal or gamma), Q-Q plots are also useful in identifying
outliers  and the presence of mixture populations (e.g., data from several populations) potentially present
in a data set. Side-by-side box plots and multiple Q-Q plots are useful to visually compare two or more
data sets,  such as:  site-versus-background concentrations, surface-versus-subsurface concentrations, and
constituent concentrations of several GW monitoring wells (MWs). ProUCL also has a couple of classical
outlier test procedures, such as the Dixon test and the Rosner test which can be used on uncensored data
sets as well as on left-censored data sets  containing ND observations.

ProUCL has parametric and nonparametric single-sample and two-sample hypotheses testing approaches
for uncensored as well as left-censored  data sets.  Single-sample hypotheses tests: Student's t-test, Sign
test,  Wilcoxon Signed  Rank  test,  and the Proportion test are used  to compare  site mean/median
concentrations (or some other threshold  such as an upper percentile) with some average cleanup standard,
Cs (or a not-to-exceed compliance limit,  Ao) to verify the attainment of cleanup levels (EPA, 1989a; 2000,
2006a) at remediated site areas of concern. Single-sample tests  such as the Sign test and Proportion test,
and upper limits including UTLs and UPLs are also used to perform intra-well comparisons. Several two-
sample hypotheses tests as described in EPA guidance documents (e.g., 2002b,  2006b, 2009) are also
available in the ProUCL software.  The two-sample hypotheses testing approaches in ProUCL include:
Student's t-test, WMW test, Gehan test and Tarone-Ware (T-W) test. The two-sample tests are  used to
compare concentrations  of two populations such as site versus background,  surface versus subsurface
soils, and upgradient versus downgradient wells.

The  Oneway ANOVA module in ProUCL has both classical and nonparametric  Kruskal-Wallis (K-W)
tests. Oneway ANOVA  is used to  compare means  (or medians) of multiple  groups such as comparing
mean concentrations of areas of concern and to perform inter-well comparisons.  In GW monitoring
applications, the ordinary least squares (OLS) regression model,  trend tests, and time series plots are used
to identify upwards or downwards  trends potentially present in constituent concentrations identified in
wells over a certain period of time.  The Trend Analysis module performs the M-K trend test and Theil-
Sen (T-S) trend test on data sets with missing values; and generates trend graphs displaying a parametric
OLS regression line and nonparametric T-S trend  line. The Time  Series Plots  option can be used to
compare multiple time-series data sets.

The  use of the  incremental sampling  methodology  (ISM) has been recommended by the  Interstate
Technology and Regulatory Council (ITRC 2012) for  collecting ISM soil samples  to  compute mean

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concentrations of the decision units (DUs) and sampling units (SUs)  requiring characterization and
remediation activities. At many polluted sites, a large amount of discrete onsite and/or offsite background
data are already available which cannot be directly compared with newly collected ISM data. In order to
provide a tool to compare the existing discrete background data with actual field onsite or background
ISM data, a Monte Carlo Background Incremental Sample Simulator (BISS) module was incorporated in
ProUCL 5.0 and retained in ProUCL 5.1 (currently blocked from general use) which may be used on a
large  existing  discrete  background data set.  The  BISS  module simulates incremental  sampling
methodology based equivalent background incremental samples. The  availability of a large discrete
background data set collected from areas with geological conditions comparable to the DU(s) of interest
is a pre-requisite for successful application of this module. For now, the BISS module has been blocked
for use  as this module is awaiting adequate guidance and instructions for its intended use  on discrete
background data sets.

ProUCL software is a user-friendly freeware package providing statistical and graphical tools needed to
address statistical issues described in many U.S. EPA guidance documents. ProUCL 5.0/ProUCL 5.1 can
process many constituents (variables) simultaneously to: perform statistical tests (e.g., ANOVA and trend
test statistics) and compute decision statistics including UCLs of mean, UPLs, and UTLs - a capability
not available in several  commercial software  packages  such as Minitab 16  and NADA for R (Helsel
2013). ProUCL also has the capability of processing data by group variables. Special care has been taken
to  make the software as user friendly as  possible. For example, on the various GOF graphical displays,
output sheets  for GOF tests, OLS and ANOVA, in addition to critical values and/or  p-values, the
conclusion  derived based upon those values is also displayed. ProUCL is easy to use and does not require
any programming skills as needed when using commercial software packages and programs written in R
script.

Methods incorporated in ProUCL have  been tested  and  verified  extensively  by the  developers,
researchers, scientists, and users.   The results  obtained by ProUCL are in agreement with the  results
obtained by using other software packages including Minitab, SASฎ, and programs written in R Script.
ProUCL 5.0/ProUCL 5.1 computes decision statistics (e.g., UPL, UTL) based upon the KM method in a
straight forward manner without flipping the data and re-flipping the computed statistics for left-censored
data sets; these operations are not easy for a typical user to understand  and perform. This can become
unnecessarily tedious when  computing  decision  statistics for multiple  variables/analytes.  Moreover,
unlike survival analysis, it is important to compute an accurate estimate of the sd which is needed to
compute decision making statistics including UPLs and UTLs. For left-censored data sets, ProUCL
computes a KM estimate  of sd directly. These issues are elaborated  by examples discussed in this  User
Guide and in the accompanying ProUCL 5.1 Technical Guide.

ProUCL does not represent a policy software of the government. ProUCL has been developed on limited
resources, and it does provide many statistical methods often used  in environmental applications. The
objective of the freely available user-friendly software,  ProUCL is  to provide statistical and graphical
tools to address environmental issues of environmental site projects for  all users including  those users
who cannot or may not want to program and/or do not have access to commercial software packages.
Some users have criticized  ProUCL  and pointed out  some  deficiencies such as:  it does not  have
geostatistical  methods; it does not perform simulations; and does not offer programming interface for
automation. Due to the limited scope of ProUCL, advanced methods have not been incorporated  in
ProUCL. For methods not available in ProUCL, users can use other statistical software packages such as
SASฎ (available to EPA personnel) and R script to address their computational needs. Contributions from
scientists and researchers to  enhance methods incorporated in ProUCL will be very much appreciated.
Just like other government documents (e.g., U.S. EPA 2009), various versions of ProUCL (2007, 2009,
2011, 2013,  2016) also make some rule-of thumb type  suggestions  (e.g., minimum sample size
10

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requirement of 8-10) based  upon professional judgment and  experience  of the  developers. It is
recommended that the users/project team/agencies make their own determinations about the rule-of-
thumb type suggestions made in ProUCL before applying a statistical method.
                                                                                           11

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                    ACRONYMS and ABBREVIATIONS
 ACL
 A-D, AD
 AL
 AOC
 ANOVA
 Ao
 BC
 BCA
 BD
 BISS
 BTV
 CC,cc
 CERCLA
 CL
 CLT
 COPC
 Cs
 CSM
 Df
 DL
 DL/2 (t)
 DL/2 Estimates
 DOE
 DQOs
 DU
 EA
 EOF
 EM
 EPA
Alternative compliance or concentration limit
Anderson-Darling test
Action limit
Area(s) of concern
Analysis of variance
Not to exceed compliance limit or specified action level
Box-Cox transformation
Bias-corrected accelerated bootstrap method
Binomial distribution
Background Incremental Sample Simulator
Background threshold value
Confidence coefficient
Comprehensive Environmental Recovery, Compensation, and Liability Act
Compliance limit
Central Limit Theorem
Contaminant/constituent of potential concern
Cleanup standards
Conceptual site model
Degrees of freedom
Detection limit
UCL based upon DL/2 method using Student's t-distribution cutoff value
Estimates based upon data set with NDs replaced by 1/2 of the respective detection
limits
Department of Energy
Data quality objectives
Decision unit
Exposure area
Empirical distribution function
Expectation maximization
United States Environmental Protection Agency
12

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EPC
GA
GB
GHz
GROS
GOF, G.O.F.
GUI
GW
HA
Ho
H-UCL
i.i.d.
ISM
ITRC
k, K
K
K,k
khat
k star
KM (%)
KM (Chebyshev)
KM(t)
KM(z)
K-M, KM
K-S, KS
K-W
LCL
LN,/ซ
LCL
LPL
LROS
Exposure point concentration
Georgia
Gigabyte
Gigahertz
Gamma ROS
Goodness-of-fit
Graphical user interface
Groundwater
Alternative hypothesis
Null hypothesis
UCL based upon Land's H-statistic
Independently and identically distributed
Incremental sampling methodology
Interstate Technology & Regulatory Council
Positive integer representing future or next k observations
Shape parameter of a gamma distribution
Number of nondetects in a data set
MLE of the shape parameter of a gamma distribution
Biased corrected MLE of the shape parameter of a gamma distribution
UCL based upon Kaplan-Meier estimates using the percentile bootstrap method
UCL based upon Kaplan-Meier estimates using the Chebyshev inequality
UCL based upon Kaplan-Meier estimates using the Student's t-distribution critical
value
UCL based upon Kaplan-Meier estimates using critical value of a standard normal
distribution
Kaplan-Meier
Kolmogorov-Smirnov
Kruskal Wallis
Lower confidence limit
Lognormal distribution
Lower confidence limit of mean
Lower prediction limit
LogROS; robust ROS
                                                                                        13

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 LTL
 LSL
 M,m
 MARS SIM
 MCL
 MOD
 MDL
 MK, M-K
 ML
 MLE
 n
 N
 MVUE
 MW
 NARPM
 ND, nd, Nd
 NERL
 NRC
 OKG
 OLS
 ORD
 OSRTI
 OU
 PCA
 PDF, pdf
 .pdf
 PRO
 PROP
 /^-values
 QA
 QC
 Q-Q
Lower tolerance limit
Lower simultaneous limit
Applied to incremental sampling:  number in increments in an ISM sample
Multi-Agency Radiation Survey and Site Investigation Manual
Maximum concentration limit, maximum compliance limit
Minimum detectable  difference
Method detection limit
Mann-Kendall
Maximum likelihood
Maximum likelihood estimate
Number of observations/measurements in a sample
Number of observations/measurements in a population
Minimum variance unbiased estimate
Monitoring well
National Association of Remedial Project Managers
Nondetect
National Exposure Research Laboratory
Nuclear Regulatory Commission
Orthogonalized Kettenring Gnanadesikan
Ordinary  least squares
Office of Research and Development
Office of Superfund Remediation and Technology Innovation
Operating unit
Principal  component  analysis
Probability density function
Files in Portable Document Format
Preliminary remediation goals
Proposed influence function
Probability-values
Quality assurance
Quality
Quantile-quantile
14

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R,r
RAGS
RCRA
RL
RMLE
ROS
RPM
RSD
RV
S
SCMTSC
SD, Sd, sd
SE
SND
SNV
SSL
SQL
su
s-w, sw
T-S
TSC
TW, T-W
UCL
UCL95
UPL
U.S. EPA, EPA
UTL
UTL95-95
USGS
USL
vs.
WMW
Applied to incremental sampling: number of replicates of ISM samples
Risk Assessment Guidance for Superfund
Resource Conservation and Recovery Act
Reporting limit
Restricted maximum likelihood estimate
Regression on order statistics
Remedial Project Manager
Relative standard deviation
Random variable
Substantial difference
Site Characterization and Monitoring Technical Support Center
Standard deviation
Standard error
Standard Normal Distribution
Standard Normal Variate
Soil screening levels
Sample quantitation limit
Sampling unit
Shapiro-Wilk
Theil-Sen
Technical Support Center
Tarone-Ware
Upper confidence limit
95% upper confidence limit
Upper prediction limit
United States Environmental Protection Agency
Upper tolerance limit
95% upper tolerance limit with 95% coverage
U.S. Geological Survey
Upper simultaneous limit
Versus
Wilcoxon-Mann-Whitney
                                                                                        15

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 WRS                Wilcoxon Rank Sum
 WSR                Wilcoxon Signed Rank
 Xp                  pth percentile of a distribution
 <                   Less than
 >                   Greater than
 >                   Greater than or equal to
 <                   Less than or equal to
 A                   Greek letter denoting the width of the gray region associated with hypothesis testing
 Z                   Greek letter representing the summation of several mathematical quantities, numbers
 %                  Percent
 a                   Type I error rate
 /?                   Type II error rate
 0                   Scale parameter of the gamma distribution
 Z                   Standard deviation of the log-transformed data
 A                   carat sign over a parameter, indicates that it represents a statistic/estimate computed
                     using the sampled data
16

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                                       GLOSSARY

Anderson-Darling (A-D) test:  The Anderson-Darling test assesses whether known data come from a
specified distribution. In ProUCL the A-D test is used to test the null hypothesis that a sample data set, x\,
..., xn came from a gamma distributed population.

Background Measurements:   Measurements that are  not site-related  or impacted by site  activities.
Background sources can be naturally occurring or anthropogenic (man-made).

Bias:  The systematic  or persistent distortion of a measured value from its true value  (this  can  occur
during sampling design, the sampling process, or laboratory analysis).

Bootstrap  Method: The bootstrap method is a  computer-based method  for  assigning measures of
accuracy to sample estimates. This technique allows estimation of the sample distribution of almost  any
statistic using only very simple methods. Bootstrap methods are generally superior to ANOVA for small
data sets or where sample distributions are non-normal.

Central Limit Theorem (CLT): The central limit theorem states that given a distribution with a mean, u,
and variance, a2, the sampling distribution of the mean approaches a normal distribution with a mean (u)
and a variance o2/N as N, the sample size, increases.

Censored Data Sets: Data sets that contain one or more observations which are nondetects.

Coefficient of Variation  (CV): A dimensionless quantity used to measure the spread of data  relative to
the size of the numbers. For a normal distribution, the coefficient of variation is given by s/xBar. It is also
known as the relative standard deviation (RSD).

Confidence Coefficient  (CC): The confidence coefficient (a number in the  closed  interval [0,  1])
associated with a confidence interval for a population parameter is the probability that the  random interval
constructed from a random  sample (data set) contains the true value of the parameter.  The confidence
coefficient is related to the  significance level of an associated hypothesis test by the equality: level of
significance =  1 - confidence coefficient.

Confidence Interval:  Based upon the sampled data set, a confidence interval for a parameter is a random
interval within which the unknown population parameter, such as the mean, or a future  observation, x0,
falls.

Confidence Limit: The lower or an upper boundary of a confidence interval. For example, the 95% upper
confidence limit (UCL) is given by the upper bound of the associated confidence interval.

Coverage, Coverage Probability: The coverage probability (e.g., = 0.95) of an upper confidence limit
(UCL) of the population mean represents the confidence coefficient associated with the UCL.

Critical Value: The critical value for a hypothesis test is  a threshold  to which the value of the  test
statistic is compared to determine whether or not the null hypothesis is rejected. The critical value for  any
hypothesis test depends on the sample  size, the significance level, a at which the test is  carried out,  and
whether the test is one-sided or two-sided.
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Data Quality Objectives  (DQOs): Qualitative and quantitative  statements derived from the DQO
process that clarify study technical and quality objectives, define the appropriate type of data, and specify
tolerable levels of potential decision errors that will be used as the basis for establishing the quality and
quantity of data needed to support decisions.

Detection Limit: A measure of the capability of an analytical method to distinguish samples that do not
contain a specific analyte from samples that contain  low concentrations of the analyte. It is the lowest
concentration or  amount of the target analyte that can  be determined to be different from zero by a single
measurement at a stated level of probability. Detection limits are analyte and matrix-specific and may be
laboratory-dependent.

Empirical Distribution Function (EOF): In statistics, an empirical distribution function is a cumulative
probability distribution function that concentrates probability \ln at each of the n numbers in a sample.

Estimate: A numerical value computed using a random data set (sample), and is used to guess (estimate)
the population parameter of interest (e.g., mean). For example, a sample mean represents an estimate of
the unknown population mean.

Expectation  Maximization (EM): The EM algorithm is  used to approximate a probability  density
function  (PDF).  EM is typically used  to compute  maximum likelihood estimates  given incomplete
samples.

Exposure Point Concentration (EPC): The constituent concentration within an exposure unit to which
the receptors are exposed. Estimates  of the  EPC represent the concentration term  used  in  exposure
assessment.

Extreme Values: Values that are well-separated from the majority of the  data set coming  from the
far/extreme tails  of the data distribution.

Goodness-of-Fit (GOF): In general, the level of agreement between an observed set of values and a set
wholly or partly derived from a model of the data.

Gray Region: A range of values of the population parameter of interest (such as mean constituent
concentration) within which the consequences of making a decision error are relatively minor.  The gray
region is bounded on one side by the action level.  The width of the gray region is denoted by the Greek
letter delta, A, in this guidance.

H-Statistic: Land's statistic used to compute UCL of mean of a lognormal population

H-UCL: UCL based on Land's H-Statistic.

Hypothesis:  Hypothesis is a statement about the population parameter(s) that may be supported or
rejected by examining the data set collected for this purpose. There are two hypotheses: a null hypothesis,
(Ho), representing a testable presumption (often set up  to be rejected based upon the sampled data), and an
alternative hypothesis (HA), representing the logical opposite of the null hypothesis.

Jackknife Method:  A statistical procedure in which, in its simplest form, estimates are  formed of a
parameter based  on a set of N observations by deleting each observation in turn to obtain, in addition to
the usual estimate based on N observations, N estimates each based on N-l observations.
18

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Kolmogorov-Smirnov (KS) test: The Kolmogorov-Smirnov test is used to decide if a data set comes
from a population with a specific distribution. The Kolmogorov-Smirnov test is based on the empirical
distribution function (EDF). ProUCL uses the KS test to test the null hypothesis if a data set follows a
gamma distribution.

Left-censored Data Set: An observation is left-censored when it is below a certain value (detection limit)
but it is unknown by how much; left-censored observations are also called nondetect (ND) observations.
A data set consisting of left-censored observations is called a left-censored data set.  In environmental
applications trace concentrations of chemicals may indeed be present in an environmental sample (e.g.,
groundwater,  soil, sediment) but cannot be detected and are reported as less than the detection limit of the
analytical instrument or laboratory method used.

Level of Significance (a): The error probability  (also known as false positive error  rate)  tolerated of
falsely rejecting the null hypothesis and accepting the alternative hypothesis.

Lilliefors test: A goodness-of-fit test that tests for normality of large data sets when population mean and
variance are unknown.

Maximum Likelihood Estimates (MLE): MLE is a popular statistical method used to make inferences
about parameters of the underlying probability distribution of a given data set.

Mean: The sum of all the  values of a set of measurements divided by the number of values  in the set; a
measure of central tendency.

Median: The middle value for an ordered set of n values. It is represented by the central value when n is
odd or by the  average of the two most central values when n is even. The median is the 50th percentile.

Minimum Detectable Difference (MOD):  The  MDD is  the smallest difference in means that the
statistical test can resolve. The MDD depends on  sample-to-sample variability, the number of samples,
and the power of the statistical test.

Minimum Variance Unbiased Estimates (MVUE):  A minimum variance unbiased estimator (MVUE or
MVU estimator) is an unbiased estimator of parameters, whose variance is minimized for all values of the
parameters. If an estimator is unbiased, then its mean squared error is equal to its variance.

Nondetect (ND) values: Censored data values. Typically, in environmental applications, concentrations
or measurements that are less than the analytical/instrument method detection limit or reporting limit.

Nonparametric: A term  describing statistical methods that do  not assume a particular population
probability distribution, and are  therefore  valid  for data from any population  with any probability
distribution, which can remain unknown.

Optimum: An interval is optimum if it possesses optimal properties as defined in the statistical literature.
This may mean that  it is the  shortest interval providing  the specified  coverage (e.g., 0.95)  to the
population mean. For example, for normally distributed data sets, the UCL of the population  mean based
upon Student's t distribution is optimum.
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Outlier: Measurements (usually larger or smaller than the majority of the data values in a sample) that
are not representative of the population from which they were drawn. The presence of outliers distorts
most statistics if used in any calculations.

Probability - Values (p-value): In statistical hypothesis testing, the p-value associated with an observed
value,  ^observed of some random variable T used as a test statistic is the probability that, given that the null
hypothesis is true, T will assume a value as or more unfavorable to the null hypothesis  as the observed
value Observed. The null hypothesis is rejected for all levels of significance, a greater than or equal to the p-
value.

Parameter: A parameter is an unknown or known constant associated with the distribution used to model
the population.

Parametric: A term describing  statistical methods that  assume a probability distribution  such  as a
normal, lognormal, or a gamma distribution.

Population: The total collection of N objects, media, or people to be studied and from which a sample is
to be drawn. It is the totality of items  or units under consideration.

Prediction Interval: The interval (based  upon historical data,  background data) within which a newly
and independently  obtained (often  labeled  as a  future observation)  site  observation (e.g.,  onsite,
compliance well) of the predicted variable (e.g., lead) falls  with  a given probability (or confidence
coefficient).

Probability of Type II (2) Error (P): The probability, referred to as  (3 (beta), that the null hypothesis will
not be rejected when in fact it is false (false negative).

Probability of  Type I (1) Error = Level of Significance (a):  The  probability, referred to as a (alpha),
that the null hypothesis will be rejected when in fact it is true (false positive).

pth Percentile  or pth Quantile:  The specific value,  Xp of a distribution that partitions a data set of
measurements in such a way that the  p percent (a number between 0  and 100) of the measurements fall at
or below this value, and (100-p) percent of the measurements exceed  this value, Xp.

Quality Assurance  (QA):  An  integrated   system   of  management  activities  involving  planning,
implementation, assessment, reporting, and quality improvement to ensure that a process, item, or service
is of the type and quality needed and expected by the client.

Quality Assurance Project Plan: A formal document describing, in comprehensive detail, the necessary
QA, quality control (QC), and other technical activities that  must be implemented to  ensure that the
results of the work performed will satisfy the stated performance criteria.

Quantile Plot:  A graph that displays the entire distribution of a data set, ranging from the lowest to the
highest value. The vertical axis represents the measured concentrations, and the horizontal axis is used to
plot the percentiles/quantiles of the distribution.

Range: The numerical difference between the minimum and maximum of a set of values.
20

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Regression on Order Statistics (ROS): A regression line is fit to the normal scores of the order statistics
for the  uncensored observations and is used  to fill in values  imputed from the straight line for the
observations below the detection limit.

Resampling: The repeated process of obtaining representative samples  and/or measurements of a
population of interest.

Reliable UCL: see Stable UCL.

Robustness:  Robustness is used  to compare statistical tests.  A  robust test is the one with good
performance (that  is  not unduly affected by outliers and underlying assumptions) for  a wide variety of
data distributions.

Resistant Estimate:  A test/estimate which is not affected by outliers is called a resistant test/estimate

Sample: Represents a random sample (data set) obtained from the population of interest (e.g., a site area,
a reference area, or  a monitoring well). The sample is supposed to be a representative sample of the
population under study. The sample is used to draw inferences about the population parameter(s).

Shapiro-Wilk (SW) test: Shapiro-Wilk test is a goodness-of-fit test that tests the null hypothesis that a
sample data set, x\,..., xป came from a normally distributed population.

Skewness: A measure  of asymmetry  of  the distribution of the  parameter under  study (e.g.,  lead
concentrations). It  can also be measured in terms of the standard deviation of log-transformed data. The
greater the standard deviation, the greater is the skewness.

Stable UCL: The  UCL of a population mean is a stable UCL if it represents a number of practical merit
(e.g., a realistic value which can actually occur at a site), which also has some physical meaning. That is,
a stable UCL represents  a realistic number (e.g., constituent  concentration)  that can occur in  practice.
Also,  a stable UCL  provides the specified (at least approximately, as much as possible,  as  close as
possible to the specified value) coverage (e.g., -0.95) to  the population mean.

Standard Deviation (sd, sd, SD):  A  measure of variation (or  spread) from an average value of the
sample data values.

Standard Error (SE): A measure of an estimate's variability (or precision).  The greater the  standard
error in relation to the size of the estimate, the  less reliable is the estimate. Standard errors are needed to
construct confidence  intervals for the parameters of interests such as the population mean and population
percentiles.

Substitution Method: The substitution method is a method for handling NDs in a data set, where the ND
is replaced by a defined value such as 0, DL/2 or DL prior to statistical calculations or graphical analyses.
This method  has  been  included  in  ProUCL  5.1  for historical  comparative  purposes  but  is  not
recommended for use. The bias introduced by applying the substitution method cannot be quantified
with any certainty. ProUCL 5.1 will provide a warning when this option is chosen.

Uncensored Data  Set: A data set without any censored  (nondetects) observations.
                                                                                              21

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Unreliable UCL, Unstable UCL, Unrealistic  UCL: The  UCL of a population mean is unstable,
unrealistic, or unreliable if it is orders of magnitude higher than the other UCLs of a population mean. It
represents an unpractically large value that cannot be achieved in practice. For example, the use of Land's
H-statistic often results in an impractically large inflated UCL  value. Some other UCLs, such as the
bootstrap-t UCL and Hall's UCL, can be inflated by outliers resulting in an impractically large and
unstable value. All such impractically large  UCL values are called unstable, unrealistic,  unreliable, or
inflated UCLs.

Upper Confidence Limit (UCL): The upper boundary (or limit) of a confidence interval of a parameter
of interest such as the population mean.

Upper Prediction Limit (UPL):  The upper boundary of a prediction interval for an independently
obtained observation (or an independent future observation).

Upper Tolerance Limit  (UTL): A confidence  limit on a percentile of the population  rather than a
confidence limit on the mean. For example, a 95% one-sided UTL for 95% coverage represents the value
below which 95% of the population values are expected to  fall with 95 % confidence. In other words, a
95% UTL with coverage coefficient 95% represents a 95% UCL for the 95th percentile.

Upper Simultaneous Limit (USL): The upper boundary of the largest value.

xBar: arithmetic average of computed using the sampled data values
22

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                             ACKNOWLEDGEMENTS

We wish to express our gratitude and thanks to our friends and colleagues who have contributed during
the development of past versions of ProUCL and to all of the many people who reviewed, tested, and
gave helpful suggestions throughout the development of the ProUCL software package.  We wish to
especially acknowledge EPA scientists including  Deana Crumbling, Nancy Rios-Jafolla, Tim Frederick,
Dr. Maliha Nash,  Kira Lynch, and Marc Stiffleman; James Durant of ATSDR, Dr. Steve Roberts of
University of Florida, Dr. Elise A. Striz of the National Regulatory Commission (NRC), and Drs. Phillip
Goodrum and John Samuelian of Integral Consulting Inc. for testing and reviewing ProUCL 5.0 and its
associated guidance documents, and for providing helpful comments and suggestions. We also wish to
thank Dr. D. Beal of Leidos for reviewing ProUCL 5.0.

Special thanks go  to Ms. Donna Getty and Mr. Richard Leuser of Lockheed Martin for providing a
thorough technical and editorial review of ProUCL  5.1 and also ProUCL 5.0 User Guide and Technical
Guide.  A special  note of thanks is due to Ms. Felicia Barnett of EPA ORD Site Characterization and
Monitoring Technical  Support Center (SCMTSC), without  whose assistance the  development of the
ProUCL 5.1 software and associated guidance documents would not have been possible.

Finally,  we wish to dedicate the ProUCL 5.1 (and ProUCL 5.0) software package to our friend and
colleague, John M. Nocerino who had contributed significantly in the development of ProUCL and Scout
software packages.
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                               Table of Contents


NOTICE	1

Minimum Hardware Requirements	2
Software Requirements	2

Installation Instructions when Downloading ProUCL 5.1 from the EPA Web Site	3

ProUCL5.1	4

Contact Information for all Versions of ProUCL	4
EXECUTIVE SUM MARY	7

GLOSSARY	17

ACKNOWLEDGEMENTS	23
Table of  Contents	24

INTRODUCTION OVERVIEW OF ProUCL VERSION 5.1 SOFTWARE	29
       The Need for ProUCL Software	34
       ProUCL 5.1 Capabilities	37
       ProUCL 5.1 Technical Guide	44
Chapter  1 Guidance on the Use of Statistical Methods in ProUCL Software	45
       1.1    Background Data Sets	45
       1.2    Site Data Sets	46
       1.3    Discrete Samples or Composite Samples?	47
       1.4    Upper Limits and Their Use	48
       1.5    Point-by-Point Comparison of Site Observations with BTVs, Compliance Limits and
       Other Threshold Values	50
       1.6    Hypothesis Testing Approaches and Their Use	51
             1.6.1  Single  Sample Hypotheses (Pre-established BTVs and Not-to-Exceed Values are
                   Known)	51
             1.6.2  Two-Sample Hypotheses (BTVs and Not-to-Exceed Values are Unknown)	52
       1.7    Minimum Sample Size Requirements and Power Evaluations	53
             1.7.1  Why a data set of minimum size, n = 8-10?	54
             1.7.2  Sample Sizes for Bootstrap Methods	55
       1.8    Statistical Analyses by a Group ID	56
       1.9    Statistical Analyses for Many Constituents/Variables	56
       1.10   Use of Maximum Detected Value as Estimates of Upper Limits	56
             1.10.1 Use of Maximum Detected Value to Estimate BTVs and Not-to-Exceed Values
                    	57
             1.10.2 Use of Maximum Detected Value to Estimate EPC Terms	57
       1.11   Samples with Nondetect Observations	58
             1.11.1 Avoid the Use of the DL/2 Substitution Method to Compute UCL95	58
             1.11.2 ProUCL Does Not Distinguish between Detection Limits, Reporting limits, or
                   Method Detection Limits	59
       1.12   Samples with Low Frequency of Detection	59
             1.13.1 Identification of COPCs	60
             1.13.2 Identification of Non-Compliance Monitoring Wells	60
24

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              1.13.3  Verification of the Attainment of Cleanup Standards, Cs	60
              1.13.4  Using BTVs (Upper Limits) to Identify Hot Spots	61
       1.14    Some General Issues, Suggestions and Recommendations made by ProUCL	61
              1.14.1  Handling of Field Duplicates	61
              1.14.2  ProUCL Recommendation about ROS Method and Substitution (DL/2) Method
                     	61
              1.14.3  Unhandled Exceptions and Crashes in ProUCL	61
       1.15    The Unofficial User Guide to ProUCL4 (Helsel and Gilroy 2012)	62
       1.16    Box and Whisker Plots	69

Chapter 2 Entering and Manipulating Data	74
       2.1     Creating a New Data Set	74
       2.2     Opening an Existing Data Set	74
       2.3     Input File Format	75
       2.4     Number Precision	76
       2.6     Saving Files	78
       2.7     Editing 79
       2.8     Handling Nondetect Observations and Generating Files with Nondetects	79
       2.9     Caution 80
       2.10    Summary Statistics for Data Sets with Nondetect Observations	81
       2.11    Warning Messages and Recommendations for Data Sets with an Insufficient Amount of
       Data   82
       2.12    Handling Missing Values	84
       2.13    User Graphic Display Modification	86
              2.13.1  Graphics Tool Bar	86
              2.13.2  Drop-Down Menu Graphics Tools	86
Chapter 3 Select Variables Screen	88
       3.1     Select Variables Screen	88
              3.1.1   Graphs by Groups	90
Chapter 4 General Statistics	93
       4.1     General Statistics for Full Data Sets without NDs	93
       4.2     General Statistics with NDs	95
Chapter 5 Imputing Nondetects  Using ROS Methods	97

Chapter 6 Graphical Methods (Graph)	99
       6.1     Box Plot	101
       6.2     Histogram	106
       6.3     Q-Q Plots	107
       6.4     Multiple Q-Q Plots	109
              6.4.1   Multiple Q-Q plots (Uncensored data sets)	109
       6.5     Multiple Box Plots	110
              6.5.1   Multiple Box plots (Uncensored data sets)	110
Chapter 7 Classical Outlier Tests	112
       7.1     Outlier Test for Full Data Set	113
       7.2     Outlier Test for Data Sets with NDs	114

Chapter 8 Goodness-of-Fit (GOF)  Tests for Uncensored and Left-Censored Data Sets.119
       8.1     Goodness-of-Fit test in ProUCL	119
       8.2     Goodness-of-Fit Tests for Uncensored Full Data Sets	122
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             8.2.1   GOF Tests for Normal and Lognormal Distribution	123
             8.2.2   GOF Tests for Gamma Distribution	125
       8.3    Goodness-of-Fit Tests Excluding NDs	127
             8.3.1   Normal and Lognormal Options	127
             8.3.2   Gamma Distribution Option	131
       8.4    Goodness-of-Fit Tests with ROS Methods	133
             8.4.1   Normal or Lognormal Distribution (Log-ROS Estimates)	133
             8.4.2   Gamma Distribution (Gamma-ROS Estimates)	135
       8.5    Goodness-of-Fit Tests with DL/2 Estimates	137
             8.5.1Normal or Lognormal Distribution (DL/2 Estimates)	137
       8.6    Goodness-of-Fit Test Statistics	137
Chapter 9  Single-Sample and Two-Sample Hypotheses Testing Approaches	141
       9.1    Single-Sample Hypotheses Tests	141
             9.1.1   Single-Sample Hypothesis Testing for Full Data without Nondetects	142
                    9.1.1.1 Single-Sample t-Test	143
                    9.1.1.2 Single-Sample Proportion Test	144
                    9.1.1.3 Single-Sample Sign Test	146
                    9.1.1.4 Single-Sample Wilcoxon Signed Rank (WSR) Test	149
             9.1.2   Single-Sample Hypothesis Testing for Data Sets with Nondetects	151
                    9.1.2.1 Single Proportion Test on Data Sets with NDs	151
                    9.1.2.2 Single-Sample Sign Test with NDs	154
                    9.1.2.3 Single-Sample Wilcoxon SignedRank TestwithNDs	155
       9.2    Two-Sample Hypotheses Testing Approaches	157
             9.2.1   Two-Sample Hypothesis Tests for Full Data	158
                    9.2.1.1 Two-Sample t-Test without NDs	160
                    9.2.1.2 Two-Sample Wilcoxon-Mann-Whitney (WMW) Test without NDs
                           	163
             9.2.2   Two-Sample Hypothesis Testing for Data Sets with Nondetects	165
                    9.2.2.1 Two-Sample Wilcoxon-Mann-Whitney Test with Nondetects	166
                    9.2.2.2 Two-Sample Gehan Test for Data Sets with Nondetects	168
                    9.2.2.3 Two-Sample Tarone-Ware Test for Data Sets with Nondetects.. Ill

Chapter 10 Computing Upper Limits to Estimate Background Threshold Values Based
       Upon Full Uncensored Data Sets and Left-Censored Data Sets with Nondetects 175
       10.1   Background Statistics for Full Data Sets without Nondetects	176
             10.1.1  Normal or Lognormal Distribution	177
             10.1.2  Gamma Distribution	179
             10.1.3  Nonparametric Methods	182
             10.1.4  All Statistics Option	184
       10.2   Background Statistics with NDs	186
             10.2.1  Normal or Lognormal Distribution	187
             10.2.2  Gamma Distribution	190
             10.2.3  Nonparametric Methods (with NDs)	193
             10.2.4  All Statistics Option	194

Chapter 11  Computing Upper Confidence Limits (UCLs) of Mean Based Upon Full-
       Uncensored Data Sets and Left-Censored Data Sets with Nondetects	200
       11.1   UCLs for Full (w/o NDs) Data Sets	202
             11.1.1  Normal Distribution (Full Data Sets without NDs)	202
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             11.1.2 Gamma, Lognormal, Nonparametric, All Statistics Option (Full Data without
                   NDs)	204
       11.2   UCL for Left-Censored Data Sets with NDs	208
Chapter 12 Sample Sizes Based Upon User Specified Data Quality Objectives (DQOs)
       and Power Assessment	212
       12.1   Estimation of Mean	214
       12.2   Sample Sizes for Single-Sample Hypothesis Tests	215
             12.2.1 Sample Size for Single-Sample t-Test	215
             12.2.2 Sample Size for Single-Sample Proportion Test	216
             12.2.3 Sample Size for Single-Sample Sign Test	217
             12.2.4 Sample Size for Single-Sample Wilcoxon Signed Rank Test	219
       12.3   Sample Sizes for Two-Sample Hypothesis Tests	220
             12.3.1 Sample Size for Two-Sample t-Test	220
             12.3.2 Sample Size for Two-Sample Wilcoxon Mann-Whitney Test	221
       12.4   Sample Sizes for Acceptance Sampling	223

Chapter 13 Analysis of Variance	224
       13.1   Classical Oneway ANOVA	224
       13.2   Nonparametric ANOVA	226

Chapter 14 Ordinary Least Squares of Regression and  Trend Analysis	228
       14.1   Simple Linear Regression	228
       14.2   Mann-Kendall Test	232
       14.3   Theil - Sen Test	235
       14.4   Time Series Plots	238
Chapter 15 Background  Incremental  Sample Simulator (BISS)  Simulating BISS Data
       from a Large Discrete Background Data	244

Chapter 16 Windows	246

Chapter 17 Handling the Output Screens and Graphs	247
       17.1   Copying and Saving Graphs	247
       17.2   Printing Graphs	248
       17.3   Making Changes in Graphs  using Tools and Properties	250
       17.4   Printing Non-graphical Outputs	250
       17.5   Saving Output Screens as Excel Files	251
Chapter 18 Summary and Recommendations to Compute a 95% UCL for Full
       Uncensored and Left-Censored Data Sets with NDs	253
       18.1   Computing UCL95s of the Mean Based Upon Uncensored Full Data Sets	253
       18.2   Computing UCLs Based Upon Left-Censored Data Sets with Nondetects	254

REFERENCES	256
                                                                                  27

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28

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                                   INTRODUCTION


            OVERVIEW OF ProUCL VERSION 5.1 SOFTWARE

The main objective of the ProUCL software funded by the U.S.EPA is to compute rigorous decision
statistics to help the decision makers in making reliable decisions which are cost-effective, and protective
of human health and the environment.  The development of ProUCL software is based upon  the
philosophy that rigorous statistical methods can be used to compute representative estimates of population
parameters (e.g., site mean, background percentiles) and accurate decision making statistics (including the
upper confidence limit  [UCL] of the mean, upper tolerance limit [UTL],  and upper prediction limit
[UPL]) which  will assist  decision makers and project  teams in making sound decisions. The use and
applicability of a statistical method (e.g., student's t-UCL, Central Limit Theorem (CLT)-UCL, adjusted
gamma-UCL, Chebyshev UCL, bootstrap-t UCL)  depend upon data size, data variability, data skewness,
and data  distribution.  ProUCL computes decision statistics  using several parametric and nonparametric
methods  covering a wide-range of data variability, skewness, and sample size. A couple of text book
methods  described in most of the statistical text books (e.g., Hogg and Craig,  1995) based upon the
Student's  t-statistic and the CLT alone cannot address all scenarios and situations commonly occurring in
environmental  studies. It is incorrect to assume that Student's t-statistic and/or CLT based UCLs of mean
will provide the desired coverage  (e.g., 0.95) to the population mean irrespective of the skewness of the
data set/population under consideration. These issues have been discussed in detail in Chapters 2 and 4 of
the accompanying ProUCL 5.1 Technical Guide.  Several examples are provided in the Technical Guide
which elaborate on these issues.

The use  of  a  parametric  lognormal distribution  on  a  lognormally distributed data set tends to yield
unstable impractically large UCL  values, especially when the standard deviation of the log-transformed
data is greater than 1.0 and the data set is of small size such  as less than 30-50 (Hardin and Gilbert 1993;
Singh, Singh, and Engelhardt 1997). Many environmental data sets can be modeled by a gamma as well
as a lognormal distribution. Generally, the use of a gamma distribution on gamma distributed data sets
yields UCL values of practical merit (Singh,  Singh,  and  laci  2002). Therefore, the use of gamma
distribution based decision statistics such as UCLs, UPL, and UTLs cannot be dismissed just because it is
easier to use a  lognormal model to compute these upper limits. The two distributions do not behave in a
similar manner.  The  advantages  of computing  the gamma distribution-based  decision  statistics  are
discussed in Chapters  2 through 5 of the ProUCL Technical Guide.

Since many environmental decisions are  made based upon a 95% UCL of the population mean,  it is
important to compute  reliable UCLs and other decision making statistics of practical merit. In an effort to
compute  stable UCLs of the population  mean  and other decision  making  statistics, in addition  to
computing the  Student's t statistic  and the  CLT based statistics (e.g., UCLs, UPLs), significant effort has
been made to  incorporate rigorous  statistical methods for  computing UCLs (and other limits) in the
ProUCL  software, covering a wide-range of data skewness and sample sizes (e.g., Singh, Singh, and
Engelhardt,  1997; Singh,  Singh,  and laci,  2002; and  Singh, Singh, 2003). It is anticipated that  the
availability of the statistical methods in the  ProUCL  software, which can be applied to a wide range of
environmental  data sets, will help decision makers  in making more informative, practical and sound
decisions.

It is noted that even for skewed data sets, practitioners tend  to use the CLT or  Student's t-statistic based
UCLs of mean for "large" sample sizes of 25-30 (rule-of-thumb to use CLT). However, this  rule-of-
thumb does not apply for moderately to highly skewed data  sets, specifically when a (standard deviation
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of the log-transformed data) starts exceeding 1. The large sample size requirement associated with the use
of the CLT depends upon the skewness of the data distribution under consideration. The  large sample
requirement associated with  CLT for the  sample mean to follow an approximate normal distribution
increases with the data skewness; and for highly skewed data sets, even samples of size greater than
(>)100 may not be large enough for the sample mean to follow an approximate normal distribution.  For
moderately skewed to highly  skewed environmental data sets, as expected, UCLs based on the CLT and
the  Student's t-statistic fail to  provide the desired coverage of the population mean even when the sample
sizes are as large as 100 or more. These facts have been verified in the published simulation experiments
conducted on positively skewed data sets (e.g., Singh, Singh, and Engelhardt, 1997; Singh, Singh, and
laci, 2002; and Singh and Singh, 2003); some graphs showing the simulation results are provided in
Appendix B of the ProUCL 5.1 Technical Guide.

The initial development and  all subsequent upgrades  and  enhancements of the ProUCL software have
been funded by the U.S. EPA through its Office of Research and Development (ORD).  Initially ProUCL
was developed as a research tool for scientists and researchers of the Technical Support Center and ORD-
NERL, Las Vegas.  During  1999-2001, the initial intent and objectives  of developing  the  ProUCL
software (Version 1.0 and Version 2.0) were to provide a statistical research tool for EPA scientists which
can be used to compute theoretically sound 95% upper confidence limits (UCL95s) of the mean routinely
used in exposure assessment, risk management and cleanup  decisions made at various CERCLA and
RCRA sites (EPA 1992a, 2002a). During 2002, the peer-reviewed ProUCL version 2.1 (with Chebyshev
inequality  based UCLs) was released for public  use. Several researchers have  developed  rigorous
parametric and nonparametric statistical methods (e.g., Johnson 1978; Grice and Bain 1980; Efron [1981
1982]; Efron  and Tibshirani  1993; Hall [1988,  1992]; Sutton 1993; Chen 1995; Singh, Singh,  and
Engelhardt 1997; Singh, Singh, and laci 2002] to compute upper limits (e.g., UCLs) which adjust for data
skewness.  Since Student's t-UCL, CLT-UCL, and percentile bootstrap UCL fail to provide the desired
coverage to the population mean of skewed distributions, several parametric (e.g., gamma distribution
based) and nonparametric (e.g., bias-corrected accelerated [BCA]  bootstrap  and bootstrap-t, Chebyshev
UCL) UCL computation methods which adjust for data skewness were incorporated in ProUCL versions
3.0  and 3.00.02 during 2003-2004. ProUCL version 3.00.02 also had graphical Q-Q plots and GOF tests
for  normal, lognormal, and gamma distributions; capabilities to statistically analyze multiple variables
simultaneously were also incorporated in ProUCL 3.00.02 (EPA 2004).

It is important to compute decision statistics (e.g., UCLs, UTLs) which are  cost-effective and protective
of human  health and the  environment (balancing between Type  I and Type  II errors), therefore,  one
cannot dismiss the use of the better [better than t-UCL, CLT-UCL, ROS and KM percentile bootstrap
UCL, KM-UCL (t)] performing UCL  computation methods including gamma UCLs  and the various
bootstrap UCLs which adjust for data skewness. During 2004-2007, ProUCL was upgraded to versions
4.00.02, and 4.00.04. These  upgrades included exploratory graphical  (e.g., Q-Q plots, box plots)  and
statistical (e.g., maximum likelihood estimation [MLE], KM, and ROS) methods for left-censored data
sets consisting of nondetect  (NDs) observations  with  multiple DLs or RLs. For uncensored and left-
censored data sets, these upgrades provide statistical methods to compute upper limits: percentiles, UPLs
and UTLs needed to  estimate site-specific background level constituent concentrations or background
threshold values (BTVs). To address statistical needs of background evaluation projects (e.g., EPA 2000,
2002b), several single-sample and two-sample hypotheses testing approaches were also included in these
ProUCL upgrades.

During 2008-2010, ProUCL was upgraded to ProUCL 4.00.05. The upgraded ProUCL was enhanced by
including methods to compute gamma distribution based UPLs and UTLs (Krishnamoorthy, Mathew, and
Mukherjee 2008). The Sample Size module to compute DQOs-based minimum sample sizes, needed to
30

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address statistical issues associated with environmental projects (e.g., EPA 2000,2002c, 2006a, 2006b),
was also incorporated in ProUCL 4.00.05.

During 2009-2011, ProUCL 4.00.05 was upgraded to  ProUCL 4.1 and 4.1.01. ProUCL 4.1 (2010) and
4.1.01 (2011) retain all capabilities of the previous versions of ProUCL software. Two new modules:
Oneway ANOVA and Trend Analysis were included in ProUCL 4.1. The Oneway ANOVA module has
both parametric and nonparametric  ANOVA  tests to perform inter-well  comparisons. The Trend
Analysis module can be used to determine potential upward or downward trends present in constituent
concentrations identified in GW monitoring wells (MWs).  The Trend Analysis module  can compute
Mann-Kendall (MK) and  Theil-Sen  (T-S) trend statistics to determine upward  or downward trends
potentially present in analyte concentrations.  ProUCL 4.1 also has the OLS Regression module. In
ProUCL 4.1,  some modifications were made in decision tables which are  used  to make suggestions
regarding the use  of UCL95 for estimating EPCs.  Specifically, based upon experience, developers of
ProUCL re-iterated that the use of a lognormal distribution  for estimating EPCs and BTVs  should be
avoided, as the use of the lognormal distribution tends to yield unrealistic and  unstable values of decision
making statistics including UCLs, UPLs, and UTLs.  This is especially true when the sample size is <20-
30 and the data  set is moderately to highly skewed. During March  2011, webinars  were  presented
describing the capabilities and use  of the methods available in ProUCL 4.1,  which can be downloaded
from the EPA ProUCL website.

ProUCL version 5.0.00 (EPA 2013, 2014) represents  an  upgrade of ProUCL 4.1.01 (EPA June  2011)
which represents an upgrade of ProUCL 4.1.00 (EPA 2010). For uncensored and left-censored data sets,
ProUCL 5.0.00 (ProUCL 5.0) contains all statistical and graphical methods  that were available in the
previous versions of the ProUCL software package except for some poor performing and restricted  (e.g.,
can be used only  when a single detection limit is present) estimation methods such as the  MLE and
winsorization methods  for left-censored data sets. ProUCL has  GOF tests for normal,  lognormal, and
gamma distributions for uncensored and left-censored data sets with NDs. ProUCL 5.0 has the extended
version of the Shapiro-Wilk (S-W) test to perform normal  and lognormal GOF tests for data sets of sizes
up to 2000 (Royston [1982, 1982a]).  In addition to normal and lognormal distribution- based decision
statistics, ProUCL software computes UCLs, UPLs, and UTLs based upon the gamma distribution.

Several enhancements were made in the UCLs/EPCs and Upper Limits/BTVs modules of the ProUCL
5.0 software. A new statistic, an upper simultaneous limit (USL) (Singh and Nocerino 2002; Wilks 1963)
has been incorporated in the Upper Limits/BTVs module  of ProUCL 5.0 for data sets consisting of NDs
with multiple DLs. A two-sample hypothesis test, the Tarone-Ware  (T-W; Tarone  and Ware,  1978) test
has also been incorporated in ProUCL 5.0. Nonparametric tolerance limits have been enhanced, and for
specific values of confidence coefficients, coverage probability, and sample size, ProUCL 5.0 outputs the
confidence coefficient  (CC) actually  achieved by a UTL. The Trend Analysis and OLS Regression
modules can handle missing events when computing  trend test statistics and generating trend graphs.
Some new methods using  KM estimates  in gamma  (and lognormal) distribution-based UCL,  UPL, and
UTL equations have been incorporated to compute the  decision statistics for data sets consisting of
nondetect observations. To facilitate the computation of UCLs from ISM based samples (ITRC 2012); the
minimum sample size requirement has been lowered to  3, so that one can compute the UCL95 based upon
ISM data sets of sizes >3.

All known bugs,  typographical errors, and discrepancies found by the developers and  users of the
ProUCL software package were addressed in ProUCL  version 5.0.00.  Specifically, a discrepancy found
in the estimate of mean based upon the KM method was fixed in ProUCL 5.0. Some changes were made
in the decision logic used  in the Goodness of Fit and UCLs/EPCs modules. In practice, based upon a

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given data set, it is well known that the two statistical tests (e.g., T-S and OLS trend tests) can lead to
different conclusions. To streamline the  decision logic associated with the computation of the various
UCLs, the decision tables in ProUCL 5.0 were updated. Specifically, for each distribution if at least one
of the two GOF tests (e.g., Shapiro-Wilk or Lilliefors test for normality) determines that the hypothesized
distribution holds, then ProUCL concludes that the data set follows the hypothesized distribution, and
decision statistics are computed accordingly.  Additionally, for gamma distributed data sets, ProUCL 5.0
suggests the use of the: adjusted gamma UCL  for samples of sizes < 50 (instead of 40 suggested in
previous versions); and approximate gamma UCL for samples of sizes >50.

Also, for samples of larger sizes (e.g., with n > 100) and small values of the gamma shape parameter, k
(e.g., k < 0.1), significant discrepancies were found in the critical values of the two gamma GOF test
statistics (Anderson-Darling [A-D] and Kolmogorov Smirnov [K-S] tests) obtained using the two gamma
deviate generation algorithms: Whitaker (1974) and Marsaglia and Tsang (2000). For values of k < 0.2,
the critical values of the two gamma GOF tests: A-D and K-S tests have been updated using the currently
available more accurate gamma deviate generation algorithm due to Marsaglia and Tsang's (2000); more
details about the implementation of their algorithm can be  found in Kroese,  Taimre, and  Botev (2011).
For values of the shape parameter, k=0.025, 0.05, 0.1, and 0.2, the critical value tables for these two tests
were updated by incorporating the newly generated critical  values for the three significance levels: 0.05,
0.1, and 0.01. The updated tables are provided in Appendix A of the ProUCL 5.0/ProUCL 5.1 Technical
Guide. It should be noted that for k=0.2, the older and  the newly generated critical values are in general
agreement; therefore, critical values for k=0.2 were not replaced in tables summarized in Appendix A of
the ProUCL Technical Guide.

ProUCL 5.0 also has a new Background Incremental Sample Simulator (BISS) module (temporarily
blocked for general public use) which can be used on a large existing discrete background data set to
simulate background incremental samples. The availability of a large discrete data set collected from
areas with geological formations and conditions comparable to the DUs (background or onsite) of interest
is a requirement for successful application of this module. The simulated BISS data can be compared with
the actual field ISM (ITRC 2012) data collected from the various DUs using other modules of ProUCL
5.0. The values of the BISS data are not directly available to users; however, the simulated  BISS data can
be accessed by the  various modules of ProUCL 5.0 to  perform  desired  statistical  evaluations.  For
example, the simulated  background  BISS data can be merged  with the actual field ISM data after
comparing the two data sets using a two-sample t-test; the simulated BISS or the merged data can be used
to compute a UCL of the mean or a UTL.

Note: The ISM methodology used to develop the  BISS module is a relatively new approach; methods
incorporated in this BISS module requires further investigation. For now, the BISS module has been
blocked for use in ProUCL 5.0/ProUCL 5.1 as this module is awaiting adequate guidance and instructions
for its intended use on discrete background data sets.

ProUCL 5.0  is a user-friendly freeware  package providing statistical and graphical tools  needed to
address statistical issues described in several EPA guidance documents. Considerable effort was made to
provide a detailed technical guide to help practitioners understand the  statistical methods needed to
address the statistical needs of their environmental projects.  ProUCL generates detailed output sheets and
graphical  displays  for each method which  can be  used  to educate students learning environmental
statistical methods. Like previous versions, ProUCL 5.0 can process many variables simultaneously to
compute various tests (e.g., ANOVA and trend test statistics) and decision statistics including UCL of the
mean,  UPLs,  and UTLs,  a capability not available in other software packages such as Minitab  16 and
NADA for R (Helsel 2013).  Without the availability of this option, the user has to compute decision and
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test statistics for one variable at a time which becomes cumbersome when dealing with a large number of
variables. ProUCL 5.0 also has the capability of processing data by groups. ProUCL 5.0 is easy to use; it
does not require any programming skills as needed when using programs written in R Script.

Deficiencies Identified in  ProUCL 5.0:  For  ProUCL to be compatible with Microsoft Office 8  and
provide Excel-compatible Spreadsheet functionality (e.g., ability to input/output *.xlsx files), ProUCL 5.0
used FarPoint Spread 5 for .NET; and for graphics,  ProUCL 5.0 used the development software package,
ChartFx 7.  The look and feel of ProUCL 5.0 is quite different from its previous versions; all main menu
options were re-arranged. However, the use of  upgraded development softwares resulted in some
problems. Specifically, it takes an unacceptably long time to save large ProUCL 5.0 generated output files
using FarPoint Spread 5. Also the use of ChartFx 7 caused some problems in properly labeling axes for
histograms. Additionally some unhandled exceptions and crashes were  noted by users. The unhandled
exceptions were mainly noted for "bad" data sets including data sets not following ProUCL input format;
data sets with not enough observations; and data sets with not enough detects.

ProUCL 5.1: ProUCL  5.1 represents an upgrade of ProUCL 5.0 to address deficiencies  identified in
ProUCL 5.0. ProUCL 5.1 retains all capabilities  of ProUCL  5.0 as described above. All modules in
ProUCL 5.1, and their look and feel is the same as in ProUCL 5.0. In this document, any statement made
about the capabilities of ProUCL 5.0 also apply to  ProUCL version 5.1; and  to save  time, not all screen
shots  used in ProUCL  5.0 manuals have  been replaced in the ProUCL 5.1  User Guide and Technical
Guide. Upgrades in ProUCL 5.1 (not available in earlier versions) have been labeled  as New in ProUCL
5.1 in this document.

All known bugs, crashes, and unhandled exceptions (e.g., on bad data sets)  found in ProUCL 5.0 have
been  addressed in ProUCL  5.1. In ProUCL 5.1,  some enhancements have been made in the  Trend
Analysis option of the Statistical  Test module of ProUCL 5.1. ProUCL  5.1 computes  and outputs
residuals for the non-parametric T-S trend line which may be helpful to compute a prediction band around
the T-S trend line. In addition to generating Q-Q plots based upon detected observations, the Goodness of
Fit Tests option of the  Statistical Tests module of ProUCL 5.1 generates censored probability plots for
data sets with NDs.  Some changes have been made in the  decision table used to  make suggestions for
UCL  selection  based upon  a gamma distribution. New licensing agreements were  obtained for the
development softwares: FarPoint and ChartFx. Due to deficiencies present in the development software,
ProUCL 5.1 generated large output files still take a long time to be saved. However, there is a quick work
around to  this problem, instead of saving the output  sheet using ProUCL, one  can copy the output
spreadsheet and save the copied output sheet using Excel. This operation can be carried out instantly.
Also, ChartFx 7.0 has  some deficiencies, and labeling along the  x-axis on a histogram is still not as
desirable as one would like it to be.  Some tools have been added in ProUCL 5.1,  and relevant statistics
(e.g.,  start point, midpoint, and end point) of a histogram bar can be displayed by hovering the cursor on
that bar.

Software ProUCL version 5.1, its earlier versions: ProUCL version 3.00.02, 4.00.02,  4.00.04,  4.1.00,
4.1.01 and ProUCL 5.0, associated Facts Sheet,  User Guides and Technical Guides (e.g., EPA [2004,
2007, 2009a, 2009b, 2010a, 2010b, 2013a, 2013b]) can be downloaded from the EPA website:

http://www.epa.gov/osp/hstl/tsc/software.htm
http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm
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The Need for ProUCL Software

EPA guidance documents (e.g.,  EPA [1989a, 1989b, 1992a, 1992b, 1994,  1996, 2000, 2002a, 2002b,
2002c, 2006a, 2006b,  2009a, and 2009b]) describe statistical methods including: DQOs-based sample
size  determination procedures,  methods to compute decision  statistics: UCL95,  UPL,  and UTLs,
parametric  and nonparametric hypotheses testing approaches, Oneway ANOVA, OLS regression, and
trend  determination approaches. Specifically, EPA guidance documents  (2000, 2002c, 2006a, 2006b)
describe DQOs-based  parametric and nonparametric minimum sample size determination procedures
needed: to  compute decision statistics (e.g., UCL95); to perform site versus background comparisons
(e.g., t-test, proportion test, WMW test); and to determine the number of discrete items (e.g., drums filled
with hazardous material) that need to be sampled to meet the DQOs (e.g., specified proportion, p0 of
defective items, allowable error margin in an estimate of mean).  Statistical methods are used to compute
test statistics (e.g., S-W test, t-test, WMW test, T-S trend statistic) and decision statistics (e.g., 95% UCL,
95% UPL,  UTL95-95) needed to address statistical issues  associated  with CERCLA and RCRA site
projects. For example,  exposure and risk management and cleanup decisions in support of EPA projects
are often made based upon the mean concentrations of the contaminants/constituents of potential concern
(COPCs). Site-specific BTVs are used in site versus background evaluation studies.  A UCL95  is used to
estimate the EPC terms (EPA 1992a, 2002a); and upper limits such as upper percentiles, UPLs, or UTLs
are used to  estimate BTVs or not-to-exceed values (EPA 1992b, 2002b, and 2009). The estimated BTVs
are used to address several objectives:  to  identify the COPCs; to identify the site areas of concern
(AOCs); to perform intra-well comparisons to identify MWs not meeting specified standards; and to
compare onsite constituent concentrations with site-specific background level constituent concentrations.
Oneway ANOVA is used to perform inter-well comparisons and OLS regression and trend tests are often
used to determine potential trends present in constituent concentrations identified in GW monitoring wells
(MWs). Most of the methods described in this paragraph are available in the ProUCL 5.1 (ProUCL 5.0)
software package.

It is noted that not much guidance is available in the guidance documents cited above to compute rigorous
UCLs, UPLs, and UTLs for moderately to highly skewed uncensored and left-censored  data  sets
containing NDs with multiple DLs, a common occurrence in environmental data sets.  Several parametric
and nonparametric methods are available in the statistical literature (Singh, Singh, and Engelhardt 1997;
Singh, Singh, and laci 2002; Krishnamoorthy et al. 2008; Singh,  Maichle, and Lee, 2006) to compute
UCLs and other upper limits which adjust for data skewness. During the years, as new methods became
available to address statistical issues related to environmental projects, those methods were incorporated
in ProUCL software so that environmental scientists and decision makers can make  more accurate and
informed decisions. Until 2006,  not much guidance  was provided on how to compute UCL95s of the
mean and other upper limits (e.g., UPLs and UTLs) based upon data sets containing  NDs with multiple
DLs.  For data sets with NDs, Singh, Maichle, and Lee (2006) conducted an extensive simulation study to
compare the performances of the various estimation methods (in terms of bias in the mean estimate) and
UCL  computation methods  (in terms of coverage provided by a UCL).  They demonstrated that the
nonparametric KM method performs well in terms of bias in estimates of mean.  They also concluded that
UCLs computed using the Student's t-statistic and percentile bootstrap method using the KM estimates do
not provide the desired coverage to the population mean of skewed data sets. They demonstrated that
depending upon sample size and data skewness, UCLs computed using KM estimates,  the BCA bootstrap
method (mildly skewed data sets), the bootstrap-t method, and the Chebyshev inequality (moderately to
highly skewed data sets) provide better coverage (closer to the specified 95% coverage) to the population
mean than other UCL computation methods. Based upon their findings, during 2006-2007, several  UCL
and other upper limits computation methods based upon KM and ROS estimates were  incorporated in the
ProUCL 4.0 software.  It is noted that since the inclusion of the KM method in ProUCL 4.0 (2007), the
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use of the KM method based upper limits has become popular in many environmental applications to
estimate EPC terms and BTVs. The KM method is also described in the latest version of the unified
RCRA guidance document (U.S. EPA 2009).

It is not easy to justify distributional assumptions of data sets consisting of both detects and NDs with
multiple DLs. Therefore, based upon the published literature and experience, parametric UCL (and other
upper limits)  computation  methods  such as the MLE method (Cohen 1991) and  the  expectation
maximization  (EM) method (Gleit  1985)  for normal and  lognormal distributions were not  included
ProUCL 5.0 (and ProUCL 5.1) even though these methods were available in earlier versions  of ProUCL.
Additionally, the winsorization method (Gilbert 1987) available in an earlier version of ProUCL has also
been excluded from ProUCL 5.0  (ProUCL 5.1) due to its poor performance.  During 2015, some
researchers (e.g., from New Mexico State University, Las Cruces, NM) suggested that the EM method
performs better than some of the methods available in ProUCL 5.0,  especially the gamma ROS  (GROS)
method; a method which can be used on left-censored data sets with multiple DLs.  The literature has
articles dealing with MLE and EM methods for data sets with a single censoring point (DL). Further
research needs to be conducted on methods for computing reliable estimates of the mean, sd, and upper
limits based upon parametric MLE and EM methods for data sets with NDs and multiple DLs. As always,
it is the desire of the developers of ProUCL to incorporate the best available methods  in ProUCL. The
developers of ProUCL welcome/encourage other researchers to share their findings about the  EM method
showing that EM method performs better than methods already available in ProUCL 5.0/ProUCL 5.1 for
data sets with single/multiple censoring  points. The developers  of ProUCL have been  enhancing the
ProUCL  software with better performing methods as those methods  become available. Efforts will be
made to incorporate contributed code (with acknowledgement) for superior methods in future versions of
ProUCL. ProUCL software is also used for teaching environmental statistics courses therefore, in addition
to  statistical and graphical methods  routinely used to address statistical needs of environmental  projects,
some poor performing methods such as the substitution DL/2 method and Land's (1975)  H-statistic based
UCL computation  method  have been retained in ProUCL version  5.1 for research  and  comparison
purposes.

Methods incorporated in ProUCL 5.1 and in its earlier versions have been tested and verified extensively
by the developers, researchers, scientists,  and users. Specifically, the results obtained by ProUCL 5.1 are
in  agreement with the results  obtained by using other software packages including Minitab, SASฎ, and
programs available in R-Script (not all methods are available in these  software packages). Additionally,
like ProUCL 5.0, ProUCL 5.1 outputs several intermediate results (e.g., khat and biased corrected kstar
estimates of the gamma shape parameter,  k, and critical values (e.g., tolerance factor, K, used to  compute
UTLs;  critical value, d2max,  used  to compute USL) needed to compute  decision statistics of interest,
which may help interested users to verify  statistical results computed by the ProUCL software. Whenever
applicable, ProUCL provides warning messages and based upon professional experience and findings of
simulation studies, makes suggestions to help a typical user in selecting the most appropriate  decision
statistic (e.g., UCL).

Note: The availability of intermediate results and critical values can be used to compute  lower limits and
two-sided intervals which are not as  yet available in the ProUCL software.

For left-censored data sets, ProUCL 5.1 computes decision statistics (e.g., UCL, UPL,  and UTL) based
upon KM estimates  computed in a straight forward manner without flipping the data and re-flipping the
decision statistics; these operations are not easy for a typical user  to understand and perform and can
become quite tedious  when  multiple analytes  need to be  processed.  Moreover,  in  environmental
applications it is important to compute accurate estimates of sd which are needed to compute  decision
making statistics including UPLs and UTLs.  Decision statistics (UPL, UTL) based upon a KM estimate
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of the of sd and computed using indirect methods can be different from the statistics computed using an
estimate of sd obtained using the KM method directly, especially when one is dealing with a skewed data
set  or  when using a log-transformation.  These issues  are elaborated by  examples discussed in the
accompanying ProUCL 5.1 Technical Guide.

For uncensored data sets, researchers (e.g., Johnson 1978; Chen  1995; Efron and Tibshirani 1993; Hall
[1988,  1992], and additional references found in Chapters 2 and 3) developed parametric (e.g., gamma
distribution based) and nonparametric (bootstrap-t and Hall's bootstrap method, modified-t) methods for
computation of decision statistics which adjust for data skewness. For uncensored positively skewed data
sets, Singh, Singh, and laci (2002) and Singh and Singh (2003) performed simulation experiments to
compare the performances (in terms of coverage probabilities) of the various UCL computation methods
described in the literature. They demonstrated that for skewed data sets, UCLs based upon Student's  t
statistic, central  limit theorem  (CLT), and  percentile  bootstrap method  tend to underestimate the
population mean (EPC).  It is reasonable to state that the findings of the simulation studies performed on
uncensored skewed data sets comparing the performances of the various UCL computation methods can
be extended to skewed left-censored data  sets. Based upon the findings of those studies performed on
uncensored data sets and also using the findings summarized in Singh, Maichle, and Lee (2006), it was
concluded that t-statistic,  CLT, and the  percentile bootstrap method based UCLs  computed using KM
estimates (and  also ROS estimates) underestimate the population mean of moderately skewed to highly
skewed data sets.  Interested users may  want  to  verify these  statements by performing simulation
experiments or other forms of rigorous testing. Like uncensored skewed data sets, for left-censored data
sets, ProUCL 5.1 offers several  parametric and  nonparametric methods for computing UCLs and other
limits which adjust for data skewness.

Due to the lack of research and methods, in earlier versions of the ProUCL software  (e.g., ProUCL
4.00.02, ProUCL 4.0), KM estimates were  used in the normal distribution based equations for computing
the various upper limits for left-censored data sets. However, normal distribution based upper limits (e.g.,
t-UCL) using KM estimates (or any other estimates such as ROS estimates) fail to provide the specified
coverage (e.g., 0.95) of the parameters (e.g., mean, percentiles) of populations with skewed distributions
(Singh, Singh,  laci 2002; Johnson 1978; Chen 1995).  For skewed data sets, ProUCL 5.0/ProUCL 5.1
computes  UCLs applying KM  estimates  in UCL equations  for skewed data  sets (e.g.,  gamma and
lognormal); therefore, some changes have  been made in the decision tables of ProUCL 5.0/ProUCL 5.1
for computing UCL95s.  Also, the nonparametric UCL computation methods (e.g., percentile bootstrap)
do not provide the desired coverage to the population means of skewed distributions (e.g., Hall [1988,
1992],  Efron and  Tibshirani, 1993). For example,  the use of t-UCL or the percentile bootstrap UCL
method on robust ROS estimates or on KM estimates underestimates the population mean for moderately
skewed to highly skewed data sets. Chapters 3 and 5 of the ProUCL Technical Guide describe parametric
and nonparametric KM methods for computing upper limits (and available in ProUCL 5.0/ ProUCL 5.1)
which adjust for data skewness.

The KM method yields good estimates of the population mean and std (Singh, Maichle, and Lee 2006);
however upper limits computed using the KM or ROS estimates in normal equations or in the percentile
bootstrap method do  not  account for skewness  present in the data set. Appropriate UCL  computation
methods which account for data skewness should be used on KM or ROS estimates. For left-censored
data sets, ProUCL 5.0/ProUCL  5.1 compute upper limits using KM estimates in gamma (lognormal)
UCL, UPL, and  UTL equations (e.g.,  also suggested in U.S. EPA  2009)  provided the  detected
observations in the left-censored data set follow a gamma (lognormal) distribution.

Recently, the use of the  ISM  methodology has been recommended (ITRC  2012) for  collecting soil
samples with the purpose of estimating mean concentrations of DUs requiring analysis of human and
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ecological risk and exposure. ProUCL can be used to compute UCLs based upon ISM data as described
and recommended in the ITRC ISM Technical and Regulatory Guide (2012). At many sites, large
amounts of discrete background data are already available which are not directly comparable to the actual
field ISM data (onsite or background). To compare the existing discrete background data with field ISM
data, the BISS module (blocked for general use in ProUCL version 5.1 awaiting guidance and instructions
for its intended use) of ProUCL 5.1  can be used on a large (e.g., consisting of at least 30 observations)
existing discrete background data set. The BISS module simulates the incremental sampling methodology
based equivalent incremental  background samples; and each simulated BISS sample represents  an
estimate of the mean of the population represented by the discrete background data set. The availability of
a large discrete background data set collected from areas with geological conditions comparable to the
DU(s) of interest (onsite DUs) is a requirement for successful application of this module. The user cannot
see the  simulated BISS data; however, the simulated BISS data can be accessed by other modules of
ProUCL 5.0 (ProUCL 5.1) for performing desired statistical evaluations. For  example, the simulated
BISS data can be merged with the actual field ISM background data after comparing the two data sets
using a two-sample t-test. The actual field ISM  or the merged ISM and BISS  data can be accessed  by
modules of ProUCL to compute a UCL of the mean or a UTL.

ProUCL 5.1 Capabilities

Assumptions: Like most statistical methods, statistical methods for computing upper limits (e.g., UCLs,
UPLs, UTLs) are also based upon certain assumptions including the availability of a randomly collected
data set consisting  of independently  and identically distributed (i.i.d) observations representing the
population (e.g., site area, reference area) under investigation. A UCL of the mean (of a population) and
BTV estimates  (UPL, UTL)  should  be computed using  a randomly collected  (simple random  or
systematic  random)  data set  representing a  single statistical population (e.g.,  site population  or
background population). When multiple populations (e.g., background and site  data mixed together) are
present  in a data set, the  recommendation is to separate them first by using the population partitioning
techniques (e.g., Singh, Singh, and Flatman 1994) prior to computing the appropriate decision statistics
(e.g., 95% UCLs).  Regardless of how the  populations are separated, decision  statistics should  be
computed separately  for  each identified population. The topic  of population  partitioning and the
extraction of a valid site-specific background  data set from a broader mixture  data set potentially
consisting of both onsite  and offsite  data are beyond the scope of ProUCL 5.0/ProUCL 5.1. Parametric
estimation and hypotheses testing methods (e.g.,  t-test, UCLs, UTLs) are based upon distributional (e.g.,
normal  distribution, gamma) assumptions. ProUCL includes GOF tests for determining  if a data  set
follows a normal, a gamma, or a lognormal distribution.

Multiple Constituents/Variables:  Environmental  scientists need to evaluate many constituents in  their
decision making processes including exposure and risk assessment, background evaluations, and site
versus background comparisons. ProUCL can process multiple constituents/variables simultaneously in a
user-friendly manner; an option not available in other freeware or commercial software packages  such as
NADA for R (Helsel 2013). This option is very useful when one has to process many variables/analytes
and compute decision statistics (e.g., UCLs, UPLs, and  UTLs) and/or test statistics (e.g., ANOVA test,
trend test) for those variables/analytes.

Analysis by a Group Variable: ProUCL also has the  capability  of processing  data by groups. A valid
group column should be included in the data file. The analyses of data categorized by a group ID variable
such as: 1) Surface  versus  (vs.)  Subsurface; 2) AOC1 vs. AOC2;  3) Site vs.  Background;  and  4)
Upgradient vs. Downgradient MWs are common  in many environmental applications. ProUCL offers this
option for data sets with and without nondetects.  The Group option provides a way to perform statistical
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tests and methods including graphical displays separately for each of the group (samples from different
populations) that may be present in a data set. For example, the same  data set may consist of analytical
data from multiple groups or populations representing site, background, two or more AOCs, surface soil,
subsurface soil, and GW. By  using this option,  the  graphical displays  (e.g.,  box plots, Q-Q  plots,
histograms) and statistics (including computation of background statistics, UCLs, ANOVA test, trend test
and OLS regression statistics) can be easily computed separately for each group in the data set.

Exploratory Graphical  Displays  for Uncensored  and  Left-Censored Data  Sets: Graphical methods
included in the  Graphs module of ProUCL include: Q-Q plots (data in  same column), multiple  Q-Q plots
(data in different columns), box plots, multiple  box plots (data in different columns), and histograms.
These graphs can also be generated for data sets containing ND observations.   Additionally, the OLS
Regression and Trend Analysis module can be used  to generate graphs displaying parametric OLS
regression lines with confidence and prediction intervals around the regression and nonparametric Theil-
Sen trend lines. The Trend Analysis module can generate trend graphs for data sets without a sampling
event variable,  and also generates time series graphs for data sets with a sampling event (time) variable.
Like ProUCL 5.0, ProUCL 5.1 accepts only numerical values for the event variable. Graphical displays of
a data set are useful for gaining added insight regarding a data set that may not otherwise be clear by
looking at test statistics  such as T-S test or MK statistics. Unlike test statistics (e.g., t-test, MK test, AD
test) and decision statistics (e.g., UCL, UTL), graphical displays do not get influenced by outliers and ND
observations. It is  suggested that the final decisions be made based upon  statistical results as  well  as
graphical displays.

Side-by-side box plots or multiple Q-Q plots are useful to graphically  compare concentrations of two or
more groups  (e.g.,  several monitoring wells). The GOF module of ProUCL generates Q-Q plots for
normal, gamma, and lognormal distributions based upon uncensored  as well as left-censored data sets
with NDs. All  relevant  information such as  the test statistics,  critical  values  and probability-values (p-
values), when available are  also displayed on the GOF Q-Q plots.  In  addition to providing information
about the data distribution, a normal Q-Q plot in  the original raw scale also helps to identify outliers and
multiple populations that may be present in a data set. On a Q-Q plot, observations well-separated from
the majority of the data may represent potential outliers coming  from a population different from the main
dominant population  (e.g.,  background population). In a Q-Q plot,  jumps  and breaks of significant
magnitude suggest the presence of observations coming from multiple  populations (onsite and offsite
areas).  ProUCL can also be used to display box plots with horizontal lines displayed/superimposed at
pre-specified  compliance limits (CLs) or computed upper limits (e.g., UPL, UTL). This kind of graph
provides a visual comparison of site  data with compliance limits and/or  BTV estimates.

Outlier Tests:  ProUCL also provides a couple of classical outlier test procedures (EPA 2006b, 2009), the
Dixon test and the Rosner test. The  details of these outlier tests are described in Chapter 7. These outlier
tests often suffer from "masking effects" in the  presence of multiple  outliers.  It is suggested that the
classical outlier procedures should always be accompanied by graphical displays including box plots and
Q-Q plots. Description and  use of the robust and masking-resistant outlier procedures (Rousseeuw and
Leroy 1987;  Singh and Nocerino  1995) are beyond the scope  of ProUCL  5.1.  Interested users are
encouraged to try the Scout 2008 software package (EPA 2009d) for robust outlier identification methods
especially  when  dealing  with multivariate  data  sets  consisting  of  observations  for  several
variables/analytes/constituents.

Outliers  represent observations coming from populations different from the main dominant population
represented by the majority of the data set. Outliers distort most statistics (e.g., mean, UCLs, UPLs, test
statistics) of interest.   Therefore, it is desirable to compute decisions  statistics based upon data sets
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representing the main population and not to compute distorted statistics by accommodating a few low
probability outliers  (e.g., by using a lognormal distribution).  Moreover, it should be noted that even
though outliers might have minimal influence  on hypotheses  testing statistics based upon ranks (e.g.,
WMW test), outliers  do distort several  nonparametric  statistics including bootstrap methods such as
bootstrap-t and Hall's bootstrap UCLs and other nonparametric UPLs and UTLs computed using higher
order statistics.

Goodness-of-Fit Tests: In addition to computing simple summary statistics for data sets with and without
NDs, ProUCL 5.1  includes GOF tests  for normal, lognormal and  gamma distributions.  To test  for
normality (lognormality) of a data set, ProUCL includes  the Lilliefors test and the extended S-W test for
samples of sizes up to 2000 (Royston 1982, 1982a). For the gamma distribution, two GOF tests: the A-D
test (Anderson and Darling 1954) and K-S test (Schneider 1978) are available in ProUCL. For samples of
larger sizes (e.g., with n > 100) and  small values of the gamma shape parameter, k (e.g., k < 0.1),
significant discrepancies were found in the critical values of the two gamma GOF test statistics (A-D and
K-S tests) obtained using the two gamma deviate generation algorithms: Whitaker (1974) and Marsaglia
and Tsang (2000). In ProUCL 5.0 (and ProUCL 5.1), for values of k < 0.2, the critical values of the two
gamma GOF tests:  A-D and K-S tests have been updated using the currently available  more efficient
gamma  deviate generation algorithm due to  Marsaglia and  Tsang's (2000); more  details  about the
implementation of their algorithm can be  found in Kroese, Taimre,  and Botev (2011). For these two GOF
and values of the shape parameter, k=0.025, 0.05, 0.1, and 0.2, critical value tables have been updated by
incorporating the newly generated critical values for three levels of significance: 0.05, 0.1, and 0.01. The
updated tables are provided in Appendix A of the ProUCL Technical  Guide.  It was noted that for k=0.2,
the older (generated in 2002) and the newly generated critical values are in general agreement; therefore,
critical values for k=0.2 were not replaced in tables summarized in Appendix A.

ProUCL  also generates GOF Q-Q plots  for normal, lognormal, and  gamma distributions displaying all
relevant  statistics including GOF test statistics.  GOF tests for data sets with and  without NDs  are
described in Chapters 2 and 3 of the ProUCL Technical Guide. For data sets containing NDs, it is not
easy to verify the distributional assumptions correctly, especially  when the data set consists  of a large
percentage of NDs with multiple DLs and NDs exceeding some detected values. Historically, decisions
about distributions of data sets with NDs are based upon GOF test statistics computed using the data
obtained: without NDs; replacing NDs by 0, DL, or DL/2; using imputed NDs based upon a ROS (e.g.,
lognormal ROS) method. For data sets with NDs, ProUCL 5.1 can perform GOF tests using the methods
listed above. ProUCL 5.1 can also generate censored probability plots (Q-Q plots) which are  very similar
to Q-Q plots generated using detected data. Using the Imputed NDs using ROS Methods option of the
Stats/Sample  Sizes module of ProUCL  5.0,  additional  columns can be generated  for storing imputed
(estimated) values for NDs based upon normal ROS, gamma ROS, and lognormal ROS (also known as
robust ROS) methods.

Sample Size Determination and Power Evaluation:  The Sample Sizes module in ProUCL can be used to
develop DQO-based sampling designs needed to address statistical issues associated with environmental
projects.  ProUCL 5.1 provides  user-friendly options  for  entering the  desired/pre-specified values  for
decision parameters (e.g., Type I and Type II error rates) and other DQOs used to determine minimum
sample  sizes  for statistical applications including:  estimation of the mean, single  and  two-sample
hypothesis testing approaches,  and acceptance sampling for discrete items  (e.g., drums containing
hazardous waste). Both parametric (e.g.,  t-test) and nonparametric (e.g., Sign test, WRS test) sample size
determination  methods as described in  EPA (2000,  2002c,  2006a, 2006b)  guidance documents are
available in ProUCL 5.1. ProUCL also has the sample size determination option for acceptance sampling
of lots of discrete objects such as a lot (batch, set) of drums containing hazardous waste (e.g., RCRA
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applications, EPA 2002c). When the sample size for an application (e.g., verification of cleanup level) is
not computed using the DQOs-based sampling design process, the Sample Size module can be used to
assess the power of the test statistic used in retrospect. The mathematical details of the Sample Sizes
module are given in Chapter 8 of the ProUCL Technical Guide.

Bootstrap Methods: Bootstrap methods are computer intensive nonparametric methods which can be used
to compute decision statistics of interest when a data set does not follow a known distribution, or when it
is difficult to  analytically  derive the distributions of statistics of interest. It is well-known that  for
moderately skewed to highly  skewed data sets, UCLs based upon standard bootstrap and the percentile
bootstrap methods do  not  perform well (e.g., Efron [1981,  1982]; Efron and Tibshirani  1993; Hall
[1988,1992]; Singh, Singh,  and laci 2002; Singh and  Singh 2003,  Singh, Maichle and Lee 2006) as  the
interval  estimates based  upon these bootstrap  methods fail to provide  the specified  coverage  to  the
population mean (e.g., UCL95 does not provide adequate 95% coverage of population mean). For skewed
data sets, Efron and Tibshirani (1993) and Hall (1988, 1992) considered other bootstrap methods such as
the BCA, bootstrap-t and  Hall's bootstrap methods. For skewed data sets, bootstrap-t and Hall's bootstrap
(meant to adjust for skewness) methods perform better (e.g., in terms  of coverage for the population
mean) than the other bootstrap methods. However, it has been noted (e.g., Efron and Tibshirani  1993,
Singh, Singh, and  laci 2002) that these two bootstrap methods tend to yield  erratic and inflated UCL
values (orders of magnitude higher than  other UCLs)  in the presence of outliers. Similar behavior of the
bootstrap-t UCL and Hall's bootstrap UCL  methods is observed for data sets consisting of NDs and
outliers. For nonparametric uncensored and left-censored  data sets with NDs, depending upon data
variability and  skewness, ProUCL recommends the use of BCA  bootstrap, bootstrap-t, or  Chebyshev
inequality based methods  for computing decision statistics. Due to the reasons described above, whenever
applicable, ProUCL 5.0/ProUCL 5.1 provides cautionary notes and warning messages regarding the  use
of bootstrap-t and Halls bootstrap UCL methods.

Hypotheses Testing Approaches: ProUCL  software has both single-sample (e.g., Student's t-test, sign
test, proportion test, WSR test) and two-sample (Student's t-test, WMW test, Gehan test, and T-W test)
parametric and nonparametric hypotheses testing approaches. Hypotheses testing approaches in ProUCL
can handle both full-uncensored data sets and left-censored data sets with NDs. Most of the hypotheses
tests also report associated p-values. For some  hypotheses tests (e.g., WMW test, WSR test, proportion
test), large sample /"-values  based upon  the  normal approximation  are computed  using continuity
correction factors.   The mathematical details of the various single-sample and two-sample hypotheses
testing approaches are described in Chapter 6 the ProUCL Technical Guide.

•   Single-Sample  Tests:  Parametric (Student's t-test) and nonparametric (Sign test, WSR test, tests  for
    proportions and percentiles) hypotheses testing approaches  are available in ProUCL. Single-sample
    hypotheses tests are used when environmental parameters such as the cleanup standard, action level,
    or compliance limits are known, and the objective  is to compare site concentrations with those known
    threshold values. A t-test (or a sign test) may be used to verify the attainment of cleanup levels in an
    AOC after a remediation activity has taken place or a test for proportion may be  used to verify if the
    proportion of exceedances of an action level (Ao or a CL) by sample observations collected from an
    AOC (or a MW) exceeds a certain specified proportion (e.g., 1%, 5%, 10%).

    The differences between these tests should be noted and understood.  A t-test or a Wilcoxon Signed
    Rank (WSR) test are used to compare  the measures of location and central tendencies  (e.g., mean,
    median) of a site area  (e.g., AOC)  to a cleanup standard,  Cs, or action  level  also representing a
    measure of central tendency  (e.g.,  mean,  median);  whereas, a proportion test determines if  the
    proportion of site observations from  an AOC exceeding a compliance limit (CL) exceeds a specified
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    proportion, Po (e.g., 5%, 10%). The percentile test compares a specified percentile (e.g., 95th) of the
    site data to a pre-specified upper threshold (e.g., action level).

•   Two-Sample Tests: Hypotheses tests (Student's t-test, WMW test, Gehan test, T-W test) are used to
    perform site versus background comparisons, compare concentrations of two or more AOCs, or to
    compare  concentrations of  GW collected  from MWs.  As cited  in  the  literature,  some of the
    hypotheses testing approaches (e.g., nonparametric two-sample WMW)  deal with a single detection
    limit scenario. When  using the WMW test on a data set  with multiple  detection  limits,  all
    observations (detects and  NDs) below the  largest detection limit  need to be considered as NDs
    (Gilbert 1987). This in turn tends to reduce the power and increase  uncertainty associated with test.
    As mentioned before, it is always desirable to supplement the test statistics and conclusions with
    graphical  displays such as multiple Q-Q plots and side-by-side box plots. The Gehan test or T-We
    (new in ProUCL 5.1) should be used in cases where multiple detection limits are present.

Note about Quantile Test: For smaller data sets, the  Quantile test as  described in U.S. EPA documents
(U.S.  EPA [1994,  2006b]; Hollander and Wolfe, 1999) is available in ProUCL  4.1 (see ProUCL 4.1
Technical Guide).  In the past, some users incorrectly used this test for  larger data sets.  Due to lack of
resources, this test has not been expanded for data sets of all sizes. Therefore, to avoid confusion and its
misuse for larger data sets, the Quantile test was not included in ProUCL  5.0 and ProUCL 5.1.

Computation of Upper Limits including UCLs. UPLs. UTLs. and USLs: ProUCL software has parametric
and nonparametric methods including bootstrap and  Chebyshev inequality based  methods to compute
decision making statistics such as UCLs of the mean (EPA 2002a), percentiles, UPLs for future k (>1)
observations,  UTLs (U.S. EPA  [1992b and 2009]) and  upper simultaneous limits (USLs) (Singh and
Nocerino [1995, 2002]) based upon uncensored full data sets and left-censored data sets containing NDs
with multiple DLs. Methods incorporated in ProUCL cover a wide range  of skewed  data distributions
with and without NDs. In addition to normal and lognormal distributions based upper limits, ProUCL 5.0
can compute parametric UCLs, percentiles, UPLs for future k (>1) observations, UTLs, and USLs based
upon gamma distributed data  sets. For data sets with NDs, ProUCL  has  several estimation methods
including the  Kaplan-Meier (KM) method (1958), ROS methods (Helsel 2005) and substitution methods
such as replacing NDs with the DL or DL/2 (Gilbert 1987; U.S. EPA 2006b).  Substitution method and
other poor performing methods (e.g., H-UCL for lognormal  distribution)  have been retained, as requested
by U.S. EPA  scientists, in ProUCL 5.0/ProUCL 5.1 for research and comparison purposes. One may not
interpret the  availability of these poor performing methods  in ProUCL as recommended methods by
ProUCL or by the U.SEP A for computing decision statistics.

Computation  of UCLs Based upon Uncensored Data Sets without NDs: Parametric UCL computation
methods in ProUCL for uncensored data sets include:  Student's t-UCL, Approximate gamma UCL (using
chi-square approximation), Adjusted gamma UCL (adjusted for level  significance), Land's H-UCL, and
Chebyshev inequality-based UCL (using minimum variance unbiased estimates (MVUEs) of parameters
of a lognormal distribution).  Nonparametric  UCL computation methods  for data sets without NDs
include: CLT-based UCL, Modified-t-statistic-based UCL (adjusted for  skewness), Adjusted-CLT-based
UCL  (adjusted for skewness),  Chebyshev inequality-based  UCL (using  sample mean and standard
deviation),  Jackknife  method-based  UCL, UCL  based upon standard bootstrap, UCL based  upon
percentile bootstrap, UCL based upon BCA bootstrap, UCL  based upon bootstrap-t,  and UCL based upon
Hall's  bootstrap method.  The  details  of UCL computation methods for uncensored data sets are
summarized in Chapter 2 of the ProUCL Technical Guide.
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Computations of  UPLs, UTLs. and  USLs Based  upon Uncensored  Data Sets without NDs:  For
uncensored data sets without NDs, ProUCL can compute parametric percentiles, UPLs for k (k>l) future
observations, UPLs for mean of k (>1) future  observations,  UTLs, and USLs based upon the normal,
gamma, and lognormal distributions. Nonparametric upper limits are typically based upon order statistics
of a data set. Depending upon the size of the data  set, the  higher order statistics (maximum, second
largest, third largest, and so on) are used to compute these upper limits (e.g., UTLs). Depending upon the
sample size, specified  CC and coverage probability,  ProUCL 5.1 outputs the actual CC achieved by a
nonparametric UTL. The details of the parametric and nonparametric computation methods for UPLs,
UTLs, and USLs are described in Chapter 3 of the ProUCL Technical Guide.

Computation of UCLs. UPLs. UTLs. and USLs  Based upon Left-Censored Data Sets with NDs: For data
sets with NDs, ProUCL computes UCLs, UPLs, UTLs, and USLs based upon the mean and sd computed
using lognormal ROS (LROS, robust ROS), Gamma ROS (GROS), KM,  and DL/2 substitution methods.
To  adjust for skewness in non-normally distributed data sets, ProUCL uses bootstrap methods and
Chebyshev inequality  when computing UCLs  and other limits using  estimates  of the mean  and sd
obtained using the methods (details in Chapters  4 and 5) listed above. ProUCL 5.1  (new in ProUCL 5.0)
uses parametric methods on KM (and ROS) estimates, provided detected observations in the left-censored
data set follow a parametric distribution. For example, if the detected data follow a gamma distribution,
ProUCL uses KM estimates in gamma distribution-based equations when computing UCLs, UTLs, and
other upper limits. When detected data do  not follow  a discernible distribution, depending upon size and
skewness of detected data, ProUCL recommends the  use of Kaplan-Meier (1958) estimates in bootstrap
methods and the Chebyshev  inequality for  computing nonparametric decision statistics (e.g., UCL95,
UPL,  UTL) of interest. ProUCL computes KM estimates directly using  left-censored data  sets without
flipping data and requiring re-flipping of decision statistics.  The KM method incorporated in ProUCL
computes both sd and standard error (SE)  of the mean. As mentioned earlier, for historical reasons and
for comparison  and research purposes, the  DL/2  substitution method and H-UCL based upon  LROS
method  have been retained in ProUCL 5.0/ProUCL  5.1.  The inclusion of the substitution and  LROS
methods in ProUCL should not be inferred as an endorsement of those methods by ProUCL software and
its developers. The details of the UCL computation methods for data sets with NDs are given in Chapter 4
and the detail description of the various other upper limits: UPLs, UTLs, and USLs for data sets with NDs
are given in Chapter 5 of the ProUCL Technical  Guide.

Oneway ANOVA. OLS Regression and Trend  Analysis:  The Oneway ANOVA  module has both
classical and nonparametric K-W ANOVA  tests as described in EPA guidance documents (e.g., EPA
[2006b,  2009]).  Oneway ANOVA is used to compare means (or medians) of multiple  groups such as
comparing mean concentrations  of several  areas of  concern  or performing  inter-well comparisons of
COPC concentrations  at several MWs.   The  OLS Regression option computes  the  classical OLS
regression line and generates graphs displaying the  OLS line, confidence bands  and prediction bands
around the regression line. All statistics of interest including slope, intercept, and correlation coefficient
are displayed on the OLS line graph. The Trend Analysis module has two nonparametric trend tests: the
M-K trend test and T-S trend test. Using this  option, one can generate trend graphs and time-series graphs
displaying a T-S trend line and all other statistics of interest with associated />-values. In addition to slope
and intercept, the  T-S  test in ProUCL 5.1  computes and  outputs residuals  based upon the  computed
nonparametric T-S line.

In GW  monitoring applications, OLS regression,  trend tests, and time  series plots  are often used to
identify trends (e.g., upwards, downwards) in constituent concentrations of GW monitoring wells over a
certain period of time (U.S. EPA 2009). The  details of Oneway ANOVA are given in Chapter 9 and OLS
regression line and Trend tests methods are described in Chapter 10 of the ProUCL Technical Guide.
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BISS Module: At many sites, a large amount of discrete onsite and background data are already available
which are not directly comparable to actual field ISM data. In order to provide a tool to compare the
existing discrete data with ISM data, the BISS module of ProUCL 5.0 may be used on a large existing
discrete data set. The ISM methodology used to develop the BISS module is a relatively new approach;
methods incorporated in this BISS module require further investigation. For now, the BISS module has
been blocked for use in ProUCL 5.0/ProUCL 5.1 as this module is  awaiting adequate guidance for its
intended use on discrete background data sets.

Recommendations and Suggestions in ProUCL: Until 2006, not much guidance was available on how to
compute a UCL95  of the mean and other upper limits (e.g., UPLs and UTLs) for skewed left-censored
data sets containing NDs  with multiple DLs, a common occurrence in environmental data sets.  For
uncensored positively skewed data sets, Singh, Singh, and laci (2002) and Singh and Singh (2003)
performed  extensive simulation experiments to compare the  performances  (in terms  of coverage
probabilities) of several UCL computation  methods  described in  the statistical and environmental
literature.  They noted that the optimal  choice of a decision statistic (e.g., UCL95) depends upon the
sample  size,  data distribution and  data skewness.  They incorporated the results of their findings in
ProUCL 3.1 and higher versions to select the most appropriate UCL to estimate the EPC term.

For data sets with NDs, Singh, Maichle,  and Lee (2006) conducted a similar simulation study to compare
the performances of the various estimation methods (in terms of bias in the mean estimate); and some
UCL computation methods (in terms of coverage provided by a UCL). They demonstrated that the KM
estimation method performs well in terms of bias in estimates of the mean; and for skewed data sets, the t-
statistic, CLT, and the percentile bootstrap method based UCLs  computed using KM estimates (and ROS
estimates) underestimate the population mean. From these findings summarized in Singh,  Singh, and laci
(2002) and Singh, Maichle, and Lee (2006), it is natural to state and assume the findings of the simulation
studies  performed  on  uncensored  skewed  data  sets  comparing performances  of the  various UCL
computation methods can be extended to skewed left-censored data sets.

Like uncensored data sets without NDs, for data sets with NDs,  there is no one single best UCL (and
other upper limits such as  UTL, UPL) which can be used to estimate an EPC (and background threshold
values) for all data sets of varying sizes, distribution, and skewness.  The optimal choice of a decision
statistic depends upon the size, distribution, and skewness of detected observations.

For data sets with and without NDs, ProUCL computes decision statistics including UCLs, UPLs, and
UTLs using several parametric and nonparametric methods covering a wide-range of  sample size, data
variability and skewness. Using the results and findings summarized in the literature cited above, and
based upon the sample size, data distribution, and data skewness,  modules of ProUCL make suggestions
about using the most appropriate decision statistic(s) to estimate population parameter(s) of interest (e.g.,
EPC). The suggestions made in ProUCL are based upon the extensive  professional applied and theoretical
experience of the  developers in environmental statistical  methods, published literature, results  of
simulation studies conducted by the developers of ProUCL and procedures described in many U.S. EPA
guidance documents. These suggestions are made to help the users in  selecting the most appropriate UCL
to estimate an EPC which is routinely used in exposure assessment and risk management studies  of the
U.S. EPA. It  should be pointed out that a typical simulation study cannot cover all data sets of various
sizes and skewness from all types of distributions. For an analyte  (data set) with skewness (sd of logged
data) near the end points of the skewness intervals described in decision tables of Chapter 2 (e.g., Tables
2-9 through 2-11) of the ProUCL Technical Guide, the user/project team may select the most appropriate
UCL based  upon the site CSM, expert site  knowledge, toxicity of  the analyte, and  exposure risks
associated with that analyte. The project team should make the final decision regarding using or not using
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the suggestions/recommendations made by ProUCL. If deemed necessary, the project team may want to
consult a statistician.

Even though, ProUCL  software has been developed using limited government funding, ProUCL 5.1
provides many statistical and graphical methods described in U.S. EPA documents for data sets with and
without NDs. However, one may not compare the availability of methods in ProUCL 5.1 with methods
available in the commercial software packages such as SASฎ and Minitab 16. For example, trend tests
correcting for seasonal/spatial variations and geostatistical  methods are  not available in the ProUCL
software. For those methods, the user is referred to commercial software packages such  as SASฎ. As
mentioned earlier, is the developers of ProUCL recommended supplementing test  results  (e.g., two-
sample test) with graphical displays (e.g., Q-Q plots, side-by-side box plots) especially when data sets
contain  NDs and outliers. With the inclusion of the BISS, Oneway ANOVA, OLS Regression Trend
and the  user-friendly DQOs based Sample Size modules, ProUCL represents a comprehensive software
package equipped with statistical methods and graphical tools  needed to address many environmental
sampling and statistical needs as described in the various CERCLA (U.S. EPA  1989a,  1992a, 2002a,
2002b, 2006a, 2006b), MARSSIM (U.S. EPA 2000), and RCRA (U.S. EPA 1989b, 1992b, 2002c, 2009)
guidance documents.

Finally, the users of ProUCL are cautioned about the use of methods and suggestions described in some
recent  environmental literature. For example,  many decision statistics (e.g., UCLs,  UPLs, UTLs,)
computed using the methods  (e.g., percentile bootstrap,  statistics  using  KM estimates and t-critical
values)  described  in Helsel (2005, 2012) will fail to provide the  desired coverage  for environmental
parameters of interest (mean, upper percentile) of moderately skewed to highly skewed populations and
conclusions derived based upon those decisions statistics may lead to  incorrect conclusions which may
not be cost-effective or protective of human health and the environment.

Note about ProUCL  5.1: ProUCL 5.1  represents an upgrade  of ProUCL 5.0 to address deficiencies
identified in ProUCL 5.0.  ProUCL 5.1 retains all capabilities  of ProUCL 5.0 as described  above. All
modules in ProUCL 5.1, and their look and feel is the same as in ProUCL 5.0. In this document, any
statement made about the capabilities of ProUCL 5.0 also apply to ProUCL version 5.1; and to save time,
not all screen shots used in ProUCL 5.0 manuals  have been replaced in the ProUCL 5.1 User Guide and
Technical Guide. Upgrades in ProUCL 5.1 (not available in earlier versions) have been labeled as New in
ProUCL 5.1 in this document.

ProUCL 5.1 Technical Guide

In addition to this  User Guide, a Technical Guide also accompanies the ProUCL 5.1 software, providing
details  of using the statistical and graphical  methods incorporated in ProUCL 5.1. Most of the
mathematical algorithms and formulae  (with references) used in the development of ProUCL 5.1 are
described in the associated Technical Guide.
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                                        Chapter 1


                Guidance on the Use of Statistical  Methods
                                 in ProUCL Software

Decisions based upon statistics computed using discrete data sets of small  sizes (e.g., < 6) cannot be
considered reliable enough to make decisions that affect human health and the environment. For example,
a background data set of size < 6 is not large enough to characterize a background population, compute
BTV  estimates,  or  to perform background  versus  site comparisons. Several U.S. EPA  guidance
documents (e.g., EPA 2000, 2006a, 2006b) detail DQOs and minimum sample size requirements needed
to address statistical issues associated with different environmental applications.  In order to  obtain
reliable statistical results, an adequate amount of data  should be collected using project-specified DQOs
(i.e., CC, decision error rates). The Sample Sizes module of ProUCL computes minimum sample sizes
based on DQOs specified by the user and described in  many guidance documents.  In some cases, it may
not be possible (e.g., due to resource constraints) to collect the calculated number of samples needed to
meet the project-specific DQOs. Under these circumstances  one can use the Sample Sizes module to
assess the power of the test statistic resulting from the  reduced number of samples which were collected.
Based upon professional experience, the developers of ProUCL 4 software  and its  later versions have
been making  some rule-of-thumb suggestions regarding minimum sample size requirements needed to
perform statistical evaluations such as: estimation of environmental parameters of interest (i.e., EPCs and
BTVs), comparing site data with background data or with some pre-established screening levels (e.g.,
action levels  [ALs],  compliance limits [CLs]). Those rule-of thumb suggestions are described later in
Section  1.7 of this chapter.  It is noted that those minimum sample requirements have been adopted by
some other guidance  documents including the RCRA Guidance Document (EPA 2009).

This chapter also describes the differences between the various statistical upper limits including upper
confidence limits (UCLs) of the mean, upper prediction limits (UPLs) for future observations, and upper
tolerance intervals (UTLs) often used to estimate the environmental parameters of interest including EPC
terms and BTVs.  The use of a statistical method depends upon the environmental parameter(s) being
estimated or compared. The measures of central tendency (e.g., means, medians, or their UCLs) are used
to compare site mean concentrations with a cleanup standard, Cs, also representing some central tendency
measure of a  reference area or some other known threshold representing a measure of central tendency.
The upper threshold values, such as the CLs, alternative concentration limits  (ACL), or not-to-exceed
values, are used when individual point-by-point  observations are compared with those threshold values.
Depending upon whether the environmental parameters (e.g., BTVs, not-to-exceed value, or EPC term)
are  known or unknown, different  statistical methods with different  data requirements are needed to
compare  site concentrations with pre-established (known) or estimated (unknown) standards and BTVs.
Several upper limits, and single and two sample  hypotheses testing approaches, for both full-uncensored
and left-censored data sets are available in the ProUCL software package for performing the comparisons
described above.

1.1    Background Data Sets

Based upon  the  CSM and regional  and expert knowledge  about the site, the project team selects
background or reference areas. Depending upon the site activities and the pollutants, the background area
can be  site-specific or  a  general  reference  area  with conditions  comparable to the site  before
contamination due to site related activities.  An appropriate random sample of independent observations


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(i.i.d) should be collected from the background area. A  defensible background data set represents a
"single" environmental population possibly without any outliers. In a background data set, in addition to
reporting and/or laboratory errors, statistical outliers may also be present.  A few  elevated statistical
outliers present in a background data set may actually  represent potentially contaminated locations
belonging to an impacted site area and/or possibly from other sources; those elevated outliers may not be
coming from the background population under evaluation. Since the presence of outliers in a data set
tends to yield distorted (poor and misleading) values of the decision making statistics (e.g., UCLs, UPLs
and UTLs), elevated outliers should not be included in background data sets and  estimation of BTVs.
The objective here is to  compute background statistics based upon  a data set which represents the main
background population,  and does not accommodate the few low probability high outliers (e.g., coming
from extreme tails of the data distribution) that may also be present in the sampled data. The occurrence
of elevated outliers is common when background samples are collected from various onsite areas (e.g.,
large Federal Facilities). The proper disposition of outliers, to include or not include them in statistical
computations, should be decided by the project team. The project team may want to compute decision
statistics with and without the  outliers to evaluate  the influence  of outliers on the decision  making
statistics.

A couple of classical outlier tests (Dixon and Rosner tests) are available in ProUCL. Since both
of these classical tests suffer from masking  effects (e.g., some extreme outliers  may mask  the
occurrence  of other intermediate outliers),  it is suggested that these classical outlier tests be
supplemented with graphical displays such as a box plot and a Q-Q plot on a raw scale.  The use
of exploratory graphical displays helps in  determining the number of outliers potentially present
in a data set. The use of graphical displays also helps in identifying extreme high outliers as well
as intermediate and mild outliers. The use of robust and masking-resistant outlier identification
procedures  (Singh and Nocerino,  1995, Rousseeuw and Leroy, 1987) is recommended when
multiple outliers are present in a data set.  Those methods are beyond the scope of ProUCL 5.1.
However, several robust outlier identification methods are available in the Scout 2008 version
1.0 software package (EPA 2009d,

An appropriate background data set of a reasonable size (preferably computed using the DQOs processes)
is needed for the data set to be representative of background conditions and to compute upper limits (e.g.,
estimates of BTVs) and  compare site and background data sets  using hypotheses testing approaches. A
background data set should have a minimum of 10 observations, however more observations is preferable.

1.2    Site Data Sets

A data set collected from a site population (e.g., AOC, exposure area [EA], DU, group of MWs) should
be representative of the  population under investigation. Depending upon the areas under investigation,
different soil depths and soil types may be considered as representing different statistical populations. In
such cases, background versus site comparisons  may have to be conducted separately for each of those
sub-populations (e.g., surface and sub-surface layers of an AOC, clay and sandy site areas). These issues,
such as comparing depths  and  soil types,  should  also be considered  in the planning stages when
developing sampling designs. Specifically, the availability of an adequate amount of representative data is
required from each of those site  sub-populations/strata defined by  sample depths,  soil types, and other
characteristics.

Site data collection requirements depend upon the objective(s) of the study. Specifically, in background
versus site comparisons,  site data are needed to perform:
46

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       point-by-point onsite comparisons with pre-established ALs or estimated BTVs. Typically, this
       approach is used when only a small number (e.g., < 6) of onsite observations are compared with a
       BTV or some other not-to-exceed value. If many onsite values need to be compared with a BTV,
       the recommended upper limit to use is the UTL or upper simultaneous limit (USL) to control the
       false positive error rate  (Type I Error Rate). More details can  be found in Chapter 3 of the
       Technical Guide. Alternatively, one can use hypothesis testing approaches (Chapter 6 of ProUCL
       Technical Guide) provided enough observations (at least 10, more  are preferred) are available.

       single-sample hypotheses tests to compare site data with a pre-established cleanup standards, Cs
       (e.g., representing  a measure of central tendency); proportion test to compare site  proportion of
       exceedances  of an AL with a pre-specified allowable proportion, Po. These hypotheses testing
       approaches are used on site data when enough site observations are available. Specifically, when
       at least 10 (more are desirable) site observations are available; it  is preferable to use hypotheses
       testing approaches  to  compare  site  observations with specified threshold values. The use  of
       hypotheses testing approaches can control both types of error rates (Type 1 and Type 2) more
       efficiently than the point-by-point individual observation comparisons. This is especially true as
       the number of point-by-point comparisons increases.  This issue  is illustrated by the following
       table summarizing the probabilities of exceedances (false positive error rate) of a BTV  (e.g., 95th
       percentile) by onsite  observations,  even  when the site and  background  populations have
       comparable distributions. The probabilities of these  chance exceedances increase  as the site
       sample size increases.
Sample Size
1
2
5
8
10
12
64
Probability of
Exceedance
0.05
0.10
0.23
0.34
0.40
0.46
0.96
    •  two-sample hypotheses tests to compare site data distribution with background data distribution
       to determine if the site concentrations are comparable to background concentrations. An adequate
       amount of data needs to be made available from the site as well as the background populations. It
       is preferable to collect at least 10 observations from each population under comparison.

Notes: From a mathematical point of view, one can perform hypothesis tests on data sets consisting of
only 3-4 data values; however, the reliability of the test statistics (and the conclusions derived) thus
obtained is questionable. In these situations it is suggested to supplement the test statistics decisions with
graphical displays.

1.3    Discrete Samples or Composite Samples?

ProUCL  can be used for discrete sample data sets,  as well as on composite sample data sets. However, in
a data set (background or site), samples should be either all discrete  or all composite. In general, both
discrete and composite site samples may be used  for individual point-by-point site comparisons with a
threshold value, and for single and two-sample hypotheses testing applications.
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    •   When using a single-sample hypothesis testing approach, site data can be obtained by collecting
       all discrete or all composite samples. The hypothesis testing approach is used when many (> 10)
       site observations are available. Details of the single-sample hypothesis approaches are widely
       available in EPA guidance documents  (MARSSIM 2000, EPA 1989a, 2006b). Several single-
       sample  hypotheses testing procedures available in ProUCL are described in Chapter 6 of the
       ProUCL 5.1 Technical Guide.

    •   If a  two-sample hypothesis  testing  approach is used  to perform site  versus background
       comparisons, then samples from both of the populations should be either all discrete samples, or
       all composite samples. The two-sample hypothesis testing approaches are used when many (e.g.,
       at least  10) site, as well as background, observations are available. For better results with higher
       statistical power, the availability of more observations perhaps based upon an appropriate DQOs
       process (EPA 2006a) is desirable.  Several two-sample hypotheses tests available in ProUCL 5.1
       are described in Chapter 6 of the ProUCL 5.1 Technical Guide.

1.4    Upper Limits and Their Use

The computation and use of statistical limits depend upon their applications and the parameters (e.g., EPC
term,  BTVs)  they are  supposed  to be  estimating. Depending upon the objective of the  study, a pre-
specified cleanup  standard, Cs, can be viewed as representing: 1) an average (or median) constituent
concentration, //o; or 2) a not-to-exceed upper threshold concentration value, Ao.  These two threshold
values, /uo, and A0, represent two  significantly different parameters, and different statistical methods and
limits are used to compare the site data with these two very different threshold values. Statistical limits,
such as a UCL of the population mean, a UPL for an independently obtained "single" observation, or
independently obtained "k" observations  (also called future k observations, next k observations, or k
different  observations), upper percentiles, and UTLs are often  used to estimate  the environmental
parameters: EPC (fio) and a BTV (Ao).  A new upper limit, USL was included in ProUCL 5.0 which may
be used to estimate a BTV based upon a well-established background data set representing a single
statistical population without any outliers.

It is important to understand and note the differences between the uses and numerical values of these
statistical limits so that they can  be properly used. The differences between UCLs and UPLs (or upper
percentiles), and UCLs and UTLs should be clearly understood. A  UCL with a 95% confidence  limit
(UCL95) of the mean represents  an estimate of the  population mean (measure of the  central tendency),
whereas a UPL95,  a UTL95%-95% (UTL95-95), and an upper 95th percentile represent estimates of a
threshold from  the upper tail of the population distribution such as the 95th percentile.   Here, UPL95
represents a 95% upper prediction limit, and UTL95-95 represents a 95% confidence limit of the 95th
percentile. For mildly skewed to moderately skewed  data sets, the numerical values of these limits tend to
follow the order given as follows.

Sample Mean < UCL95 of Mean < Upper 95th Percentile < UPL95 of a Single Observation < UTL95-95

Example 1-1. Consider a real data set collected from a Superfund site. The data set has several inorganic
COPCs, including aluminum (Al), arsenic (As), chromium (Cr), iron (Fe), lead  (Pb), manganese (Mn),
thallium (Tl)  and vanadium (V). Iron concentrations follow a normal distribution. This data set has  been
used in several  examples throughout the two ProUCL guidance documents (Technical Guide and  User
Guide), therefore it is provided as follows.
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Aluminum
6280
3830
3900
5130
9310
15300
9730
7840
10400
16200
6350
10700
15400
12500
2850
9040
2700
1710
3430
6790
11600
4110
7230
4610
Arsenic
1.3
1.2
2
1.2
3.2
5.9
2.3
1.9
2.9
3.7
1.8
2.3
2.4
2.2
1.1
3.7
1.1
1
1.5
2.6
2.4
1.1
2.1
0.66
Chromium
8.7
8.1
11
5.1
12
20
12
11
13
20
9.8
14
17
15
8.4
14
4.5
3
4
11
16.4
7.6
35.5
6.1
Iron
4600
4330
13000
4300
11300
18700
10000
8900
12400
18200
7340
10900
14400
11800
4090
15300
6030
3060
4470
9230




Lead
16
6.4
4.9
8.3
18
14
12
8.7
11
12
14
14
19
21
16
25
20
11
6.3
13
98.5
53.3
109
8.3
Manganese
39
30
10
92
530
140
440
130
120
70
60
110
340
85
41
66
21
8.6
19
140
72.5
27.2
118
22.5
Thallium
0.0835
0.068
0.155
0.0665
0.071
0.427
0.352
0.228
0.068
0.456
0.067
0.0695
0.07
0.214
0.0665
0.4355
0.0675
0.066
0.067
0.068
0.13
0.068
0.095
0.07
Vanadium
12
8.4
11
9
22
32
19
17
21
32
15
21
28
25
8
24
11
7.2
8.1
16




Several upper limits for iron are summarized as follows, and it be seen that they follow the order (in
magnitude) as described above.

            Table 1-1. Computation of Upper Limits for Iron (Normally Distributed)

Mean
9618

Median
9615

Min
3060

Max
18700

UCL95
11478
UPL95 for a
Single
Observation
18145

UPL95 for 4
Observations
21618

UTL95-95
21149
95%
Upper
Percentile
17534
For highly skewed data sets, these limits may not follow the order described above. This is especially true
when the upper limits are computed based upon a lognormal distribution (Singh, Singh, and Engelhardt
1997).  It is well known that a lognormal distribution based H-UCL95 (Land's UCL95)  often yields
unstable and impractically large UCL values. An H-UCL95 often becomes larger than UPL95 and even
larger than a UTL 95%-95% and the largest sample value. This is especially true when dealing with
skewed data sets of smaller sizes.  Moreover, it should also be noted that in some cases,  a H-UCL95
becomes smaller than the sample mean, especially when the data are mildly skewed and the sample size is
large (e.g., > 50, 100).
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There is a great deal of confusion about the appropriate use of these upper limits. A brief discussion about
the differences between the applications and uses of the statistical limits described above is provided as
follows.

•   A UCL represents an  average value that is compared with a threshold value also representing an
    average value  (pre-established or  estimated), such as a mean Cs. For example, a site 95% UCL
    exceeding a Cs, may lead to the conclusion that the cleanup standard, Cs has not been attained by the
    average site area concentration. It should also be noted that UCLs of means are typically computed
    from the site data set.

•   A UCL represents a "collective" measure of central tendency, and it is not appropriate to compare
    individual site  observations with a UCL. Depending upon data availability, single or two-sample
    hypotheses testing approaches are used to compare a site average or a site median with a specified or
    pre-established cleanup  standard  (single-sample hypothesis), or with the background population
    average or median (two-sample hypothesis).

•   A UPL, an upper percentile, or a UTL represents an upper limit to be used for point-by-point
    individual site observation comparisons. UPLs and UTLs are computed based upon background data
    sets,  and point-by-point onsite observations are  compared with those limits.  A site observation
    exceeding a background UTL may  lead to the conclusion that the constituent is present at the site at
    levels greater than the background concentrations level.

•   When enough  (e.g.,  at least 10) site observations are  available, it is  preferable to use hypotheses
    testing  approaches. Specifically, single-sample  hypotheses testing  (comparing  site to a specified
    threshold) approaches should be used to perform site versus a known threshold comparison; and two-
    sample hypotheses testing (provided enough background data are also available) approaches should
    be used to perform site versus background comparison. Several parametric and nonparametric single
    and two-sample hypotheses testing approaches are available in ProUCL 5.0/ProUCL 5.1.

It is re-emphasized that only averages  should be compared with averages or UCLs, and individual site
observations  should be compared with  UPLs,  upper percentiles, UTLs, or USLs. For example, the
comparison of a 95% UCL of one population (e.g., site) with a 90% or 95% upper percentile of another
population  (e.g., background) cannot be considered fair and reasonable as these limits (e.g., UCL  and
UPL) estimate and represent different parameters.

1.5   Point-by-Point  Comparison of  Site  Observations with  BTVs, Compliance
       Limits and Other Threshold Values

The point-by-point observation comparison method  is used when a small number (e.g., < 6) of site
observations  are compared with pre-established or estimated BTVs, screening levels, or preliminary
remediation goals (PRGs).  Typically, a  single exceedance of the BTV by an onsite (or a monitoring well)
observation may be considered an  indication of the presence of contamination at  the site area under
investigation. The conclusion of an exceedance  by a site value is  sometimes confirmed by re-sampling
(taking a few more collocated samples) at the site location (or a monitoring well) exhibiting constituent
concentrations in excess of the BTV. If all collocated sample  observations (or all  sample observations
collected during the same time period) from the same site location (or well)  exceed the BTV or PRO, then
it may be concluded that the location (well) requires further investigation (e.g.,  continuing treatment and
monitoring) and possibly cleanup.
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When BTV constituent concentrations are not known or pre-established, one has to collect or extract a
background data set of an appropriate size that can be considered representative of the site background.
Statistical upper limits are computed using the background data set thus  obtained, which are used as
estimates of BTVs. To compute reasonably reliable  estimates of BTVs, a minimum of 10 background
observations should be collected, perhaps using an appropriate DQOs process as described in EPA (2000,
2006a).  Several statistical limits listed above are used to estimate BTVs based upon a defensible (free of
outliers, representing the background population) background data set of an adequate size.

The point-by-point comparison method is also useful when quick turnaround comparisons are required in
real time. Specifically, when decisions have to be made in real time by a sampling/screening crew, or
when only a few site samples are available, then individual point-by-point  site concentrations  are
compared either with pre-established cleanup goals or with estimated BTVs. The sampling crew can  use
these comparisons to:  1)  screen and identify the COPCs, 2) identify the potentially polluted  site AOCs, or
3) continue or stop remediation or excavation at an onsite area of concern.

If a larger number of samples (e.g., >10) are available from the AOC, then the use of hypotheses testing
approaches (both single-sample and a two-sample) is preferred. The use  of hypothesis testing approaches
tends  to control the  error rates  more  tightly and efficiently than the individual point-by-point  site
comparisons.

1.6     Hypothesis Testing Approaches and Their Use

Both single-sample and two-sample hypotheses testing approaches are used to make cleanup decisions at
polluted sites, and also to compare constituent concentrations of two (e.g., site versus background) or
more populations (e.g., MWs).

1.6.1   Single Sample Hypotheses (Pre-established BTVs and Not-to-Exceed Values are
       Known)

When pre-established BTVs  are used such as the U.S. Geological Survey (USGS) background values
(Shacklette and Boerngen 1984), or thresholds obtained from similar sites, there is no need to extract,
establish, or collect a background data set. When the BTVs and cleanup standards are known, one-sample
hypotheses are used to compare site  data (provided enough site data are available) with known and pre-
established threshold values. It is suggested that the project team determine (e.g., using DQOs) or decide
(depending upon resources) the number of site observations that should be  collected and compared with
the "pre-established" standards before coming  to a conclusion about the status (clean or polluted) of the
site AOCs. As mentioned earlier, when the number of available site samples  is < 6, one might perform
point-by-point site observation comparisons with a BTV; and when enough  site observations (at least 10)
are available, it is desirable  to use  single-sample hypothesis testing approaches. Depending upon  the
parameter (juo, Ao), represented by the known threshold value, one can use single-sample hypotheses tests
for population  mean or median  (t-test, sign test),  or  use single-sample tests for  proportions and
percentiles. The details of the  single-sample hypotheses testing approaches can be found in EPA (2006b)
guidance document and in Chapter 6 of ProUCL Technical Guide.

One-Sample t-Test: This test is used to compare the site mean, /u, with some specified cleanup  standard,
Cs, where the Cs represents an average threshold value, /no. The Student's t-test (or a UCL of the mean) is
used (assuming normality of site data set or when sample  size  is large, such as larger than 30, 50) to
verify the attainment of cleanup levels at a polluted site after some remediation activities.
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One-Sample Sign Test or Wilcoxon Signed Rank (WSR) Test: These tests are nonparametric tests and can
also handle ND observations, provided the detection limits of all NDs fall below the specified threshold
value, Cs. These tests are used to compare the site location (e.g., median, mean) with some specified Cs
representing a similar location measure.

One-Sample Proportion Test or Percentile Test: When a specified cleanup standard, Ao, such as a PRG or
a BTV represents an upper threshold value of a constituent concentration distribution rather than the mean
threshold value, juo, then a test for proportion or a test for percentile (equivalently UTL 95-95 UTL 95-90)
may be used to compare site proportion (or site percentile) with the  specified threshold or action level, Ao.

1.6.2   Two-Sample Hypotheses (BTVs and Not-to-Exceed Values are Unknown)

When BTVs, not-to-exceed values,  and other cleanup  standards are  not  available, then site  data are
compared directly with the background data. In such cases, two-sample  hypothesis testing approaches are
used to  perform site versus background comparisons. Note that this approach can be used to  compare
concentrations of any two populations including two different site areas or two different monitoring wells
(MWs). In order to use and perform a two-sample hypothesis testing approach, enough data should be
available from each of the two populations. Site and background data requirements (e.g., based upon
DQOs) for performing two-sample hypothesis test approaches are described in EPA (2000, 2002b, 2006a,
2006b) and also in Chapter 6 of the ProUCL 5.1 Technical Guide. While collecting site and background
data, for better representation of populations under investigation, one may also want to account for the
size of the background area (and site area for site samples) in sample size determination. That is, a larger
number (>15-20) of representative  background (and  site)  samples should be  collected  from  larger
background (and site) areas; every effort should be made to collect as many samples as determined by the
DQOs-based sample sizes.

The two-sample  (or more) hypotheses approaches are used when the site parameters (e.g., mean,  shape,
distribution) are  being compared with the background parameters (e.g., mean, shape, distribution). The
two-sample hypotheses testing approach is also used when the cleanup standards or  screening levels are
not known  a priori.  Specifically,  in  environmental applications,  two-sample  hypotheses testing
approaches are used to compare average or median constituent concentrations of two or more populations.
To derive reliable conclusions with higher statistical power based upon  hypothesis testing approaches, an
adequate amount of data (e.g., minimum of 10 samples) should  be collected from all of the populations
under investigation.

The two-sample hypotheses testing approaches incorporated in ProUCL  5.1 are listed  as follows:

    1.   Student t-test (with equal and unequal variances) - Parametric test assumes normality
    2.  Wilcoxon-Mann-Whitney (WMW) test - Nonparametric  test handles data with NDs with one DL
        - assumes two populations have comparable shapes and variability
    3.   Gehan test -  Nonparametric test handles data sets with NDs and multiple  DLs -  assumes
        comparable shapes and variability
    4.   Tarone-Ware (T-W) test - Nonparametric test handles data sets with NDs  and multiple  DLs  -
        assumes comparable shapes and variability

The Gehan and T-W tests are meant to be used on left-censored data sets with multiple DLs.  For best
results, the samples collected from the two (or more) populations should all be of the same type  obtained
using similar analytical methods and apparatus; the collected site and background samples should all be
discrete or all composite (obtained using the same design and pattern), and be collected from the same
52

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medium (soil) at similar depths (e.g., all surface samples or all subsurface samples) and time (e.g., during
the same quarter in groundwater applications) using comparable (preferably same) analytical methods.
Good sample collection methods and  sampling strategies are given in EPA  (1996, 2003) guidance
documents.

Note: ProUCL  5.1  (and previous versions) has been developed using limited government funding.
ProUCL  5.1 is equipped with statistical and graphical methods needed to address many environmental
sampling and statistical issues as described in the various CERCLA, MARSSIM, and RCRA documents
cited earlier.  However, one may not compare the availability of methods in ProUCL 5.1  with methods
incorporated in commercial software packages such as SASฎ and Minitab 16. Not all methods available in
the statistical literature are available in ProUCL.

1.7   Minimum Sample Size Requirements and Power Evaluations

Due to resource limitations, it  is not be possible (nor needed) to sample the  entire population (e.g.,
background  area, site area, AOCs, EAs) under study. Statistics is used to  draw inference(s) about the
populations (clean, dirty) and their known or unknown statistical parameters (e.g., mean, variance, upper
threshold values) based upon much smaller data sets (samples) collected  from those populations. To
determine and establish BTVs and site specific screening  levels,  defensible data set(s) of appropriate
size(s) representing the background population (e.g., site-specific, general  reference area, or historical
data)  need to be collected.  The  project  team and  site experts should decide what represents  a  site
population  and  what represents  a  background population.  The  project team  should  determine  the
population area and boundaries based upon all current and intended future uses, and the objectives of data
collection.  Using the collected  site and  background data  sets, statistical methods  supplemented with
graphical displays are used to perform site versus background  comparisons. The test results and statistics
obtained  by performing such site versus background comparisons are used to determine if the site  and
background  level constituent concentrations are comparable; or if the site concentrations exceed the
background threshold concentration level; or if an adequate amount of remediation approaching the BTV
or some cleanup  level has been performed at polluted site AOCs.

To perform these statistical tests, determine the number of samples that need to be collected from the
populations  (e.g., site and background) under investigation  using appropriate  DQOs processes (EPA
2000, 2006a, 2006b). ProUCL has the Sample Sizes module which can be used to develop DQOs based
sampling designs needed to address  statistical issues  associated with polluted  sites projects. ProUCL
provides user-friendly options to enter the desired/pre-specified values of decision parameters (e.g., Type
I and Type II error rates) to determine minimum sample sizes for the selected statistical applications
including: estimation of mean,  single and  two-sample hypothesis testing  approaches, and acceptance
sampling. Sample size determination methods are available for the sampling  of continuous characteristics
(e.g., lead or Radium 226), as well as for attributes (e.g., proportion of occurrences exceeding a specified
threshold). Both parametric (e.g., t-tests) and nonparametric (e.g.,  Sign test, test for proportions, WRS
test) sample size determination methods are available in ProUCL 5.1  and in its earlier versions (e.g.,
ProUCL  4.1). ProUCL also has sample size determination  methods for acceptance sampling of lots of
discrete objects such as a batch of drums containing hazardous waste (e.g., RCRA applications, U.S. EPA
2002c).

However, due to budgetary or logistical constraints,  it may not be possible to collect the same  number of
samples as determined by applying a DQO process. For example, the data might have already been
collected (as  often is the case) without using a DQO process, or due to resource constraints,  it may not
have been possible to collect as many samples as determined by using a DQO-based sample size formula.
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In practice, the project team and the decision makers tend not to collect enough background samples. It is
suggested to collect at least 10 background  observations  before using statistical methods to perform
background evaluations based upon  data collected using discrete samples. The minimum sample size
recommendations described here are useful when resources are  limited, and it may not be possible to
collect as many background and site samples as computed using DQOs based sample size determination
formulae. In case data are collected without using a DQO process, the Sample Sizes module can be used
to assess the power of the test statistic in retrospect. Specifically, one can use the standard deviation of the
computed test statistic (EPA 2006b) and compute the sample size needed to meet the desired DQOs. If the
computed sample size is greater than the size of the data set used, the project team may want to collect
additional samples to meet the desired DQOs.

Note: From a mathematical point of view, the statistical methods incorporated in ProUCL and described
in this guidance document for estimating EPC terms and BTVs, and comparing site versus background
concentrations can be performed on  small site and background data sets (e.g., of sizes as small as 3).
However,  those statistics  may not  be considered representative and reliable enough to make important
cleanup and remediation  decisions which will  potentially  impact human  health and the environment.
ProUCL provides messages when  the number of detects is <4-5, and  suggests collecting at least 8-10
observations.  Based  upon professional judgment,  as a rule-of-thumb, ProUCL  guidance documents
recommend collecting a minimum of 10 observations when data sets of a size determined by a DQOs
process (EPA 2006) cannot be collected. This however, should  not be interpreted as the general
recommendation and every effort should be made to collect DQOs based number of samples. Some recent
guidance documents  (e.g., EPA 2009) have also adopted  this rule-of-thumb  and suggest collecting a
minimum  of about 8-10 samples in the circumstance that data cannot be collected using a DQO-based
process. However, the project team needs to make  these determinations based upon their comfort level
and knowledge of site conditions.

    •   To allow users to compute  decision  statistics using data  from ISM (ITRC,  2012)  samples,
        ProUCL 5.1 will  compute  decision statistics (e.g.,  UCLs, UPLs, UTLs) based upon samples of
        sizes as small as 3. The user is  referred to the ITRC ISM Technical Regulatory Guide (2012) to
        determine which UCL (e.g., Student's t-UCL or Chebyshev UCL) should be used to estimate the
        EPC term.


1.7.1   Why a data set of minimum size, n = 8-10?

Typically, the computation of parametric upper limits  (UPL, UTL, UCL) depends upon three values: the
sample mean, sample variability (standard deviation) and a critical value. A critical value depends upon
sample size, data distribution, and confidence level.  For samples of small size (< 8-10), the critical values
are  large and unstable, and upper limits (e.g., UTLs, UCLs) based upon a data set  with fewer than 8-10
observations are mainly driven by those critical values. The differences in the corresponding  critical
values tend to stabilize when the sample size becomes larger than 8-10  (see tables below, where degrees
of freedom [df] = sample size - 1). This is one of the reasons ProUCL guidance  documents suggest a
minimum data set size of 10 when  the number of observations determined from sample-size calculations
based upon EPA DQO process exceed the logistical/financial/temporal/constraints of a project.  For
samples of sizes 2-11, 95% critical values used to compute upper limits (UCLs, UPLs, UTLs, and USLs)
based upon a normal distribution are summarized in the subsequent tables. In general, a similar pattern is
followed for critical values used in the computation of upper limits based upon other distributions.
For the normal distribution, Student's t-critical values are used to compute UCLs  and UPLs which are
summarized as follows.
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                             Table of Critical Values oft-Statistic
                                   .HI
                                               Upper-la:] probability p

                                          .O5-      .025      .ill      .01
3.U7S (
l.XSb
l.n*S
1.5^3
1.47o
1 .440
1.415
1.397
1.383
1.372
-..314
1.420
1.35*
I 132
1.015
.443
.845
.800
.833
.812
12.71
4.30.5
3.1H2
2.77a
2.571
2.447
1365
2.31*0
2.262
2.228
1x84
4.K44
<.4o'2
2.444
2.757
2.612
2.517
2.449
2.3*8
2.359
31.ซ
f.>*5
4.541
1.747
3J65
3.14.?
2 .9W
2.4%
2.421
2.?64
                             to

One can see that once the sample size starts exceeding 9-10 (df= 8, 9), the difference between the critical
values starts  stabilizing. For example,  for upper tail probability (= level of significance)  of 0.05, the
difference between critical values for df= 9 and df=\0 is only 0.021, where as the difference between
critical values for df= 4 and 5 is 0.117;  similar patterns are noted for other levels of significance. For the
normal distribution,  critical  values used to  compute UTL90-95,  UTL95-95, USL90, and USL95 are
described as follows. One can see that once the sample size starts exceeding 9-10, the difference between
the critical values starts decreasing significantly.
                                UTL90-95   UTL95-95
USL90
USL95
3
4
5
6
1
8
9
10
11
6.155
4.162
3.407
3.006
2.755
2.582
2.454
2.355
2.275
7.656
5.144
4.203
3.708
3.399
3.187
3.031
2.911
2.815
1.148
1.425
1.602
1.729
1.828
1.909
1.977
2.036
2.088
1.153
1.462
1.671
1.822
1.938
2.032
2.11
2.176
2.234
Note: Nonparametric upper limits (UPLs, UTLs, and USLs) are computed using higher order statistics of
a data set.   To achieve the desired confidence coefficient, samples of sizes much greater than 10 are
required. For details, refer to Chapter 3.  It should be noted that critical values of USLs are significantly
lower than critical values  for UTLs. Critical values associated with UTLs decrease as the sample size
increases. Since, as the sample size increases the maximum of the data set also increases, and critical
values associated with USLs increase with the sample size.

1.7.2  Sample Sizes for Bootstrap Methods

Several nonparametric methods including bootstrap methods for computing UCL, UTL, and other limits
for both full-uncensored data sets and left-censored data sets with NDs are available in ProUCL 5.1.
Bootstrap resampling methods are useful when not too few  (e.g., < 15-20) and not too many (e.g., > 500-
1000) observations are available. For bootstrap methods (e.g., percentile method, BCA bootstrap method,
bootstrap-t method), a large number (e.g., 1000, 2000) of bootstrap resamples are drawn with replacement
from the same data set. Therefore, to obtain bootstrap resamples with at least some distinct values (so that
statistics can be computed from each resample), it is suggested that a bootstrap method should not be used
when dealing with small data sets of sizes less than 15-20. Also, it is not necessary to bootstrap a large
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data set of size greater than 500 or 1000; that is when a data set of a large size (e.g., > 500) is available,
there is no need to obtain bootstrap resamples to compute statistics of interest (e.g., UCLs). One can
simply use a statistical method on the original large data set.

Note: Rules-of-thumb about minimum sample size requirements described in this section are based upon
professional experience of the developers. ProUCL software is not a policy software. It is recommended
that the users/project teams/agencies make  determinations about the minimum number of observations
and minimum number of detects that should be present in a data set before using a statistical method.

1.8     Statistical  Analyses by a Group ID

The analyses of data categorized by a group ID variable such as: 1) Surface vs. Subsurface;  2) AOC1 vs.
AOC2;  3) Site vs. Background; and 4)  Upgradient vs. Downgradient monitoring  wells are common in
environmental applications. ProUCL 5.1 offers this option for data  sets  with and without NDs.  The
Group  Option provides  a tool for performing separate  statistical tests and for  generating  separate
graphical displays for each member/category of the group (samples from different populations) that may
be present in a data set. The graphical displays (e.g., box plots, quantile-quantile plots) and statistics (e.g.,
background  statistics, UCLs, hypotheses tests)  of interest can be computed separately for each group by
using this option. Moreover, using the  Group Option, graphical  methods can display multiple graphs
(e.g., Q-Q plots) on the same graph providing graphical comparison of multiple groups.

It  should be pointed  out that it is the user's responsibility to  provide an adequate amount of data to
perform the group operations. For example, if the user desires to produce a graphical Q-Q plot (e.g., using
only detected data) with regression lines displayed, then there should be at least two detected data values
(to compute slope, intercept, sd) in the data  set. Similarly, if the graphs are desired for each group
specified by the group ID variable, there should be at least two observations in each group specified by
the group variable. When ProUCL data  requirements are not met, ProUCL does not perform  any
computations, and generates  a warning  message (colored orange) in the lower Log Panel of the output
screen of ProUCL 5.1.

1.9     Statistical  Analyses for Many Constituents/Variables

ProUCL software can process multiple analytes/variables simultaneously in a user-friendly  manner This
option  is useful when one has to process multiple variables and compute decision  statistics (e.g., UCLs,
UPLs, and UTLs) and test statistics (e.g., ANOVA test, trend test)  for multiple variables. It is the user's
responsibility to make sure that each selected variable has an adequate amount of data  so that ProUCL
can perform the selected statistical method correctly. ProUCL displays warning messages when a selected
variable does not have enough data needed to perform the selected statistical method.

1.10    Use of Maximum Detected Value as Estimates of Upper Limits

Some practitioners use the maximum detected value as an estimate of the EPC term. This  is especially
true when the sample size is small  such  as < 5, or when a UCL95 exceeds the maximum detected values
(EPA 1992a). Also, many times in practice, the BTVs and not-to-exceed values are estimated by the
maximum detected value (e.g., nonparametric UTLs, USLs).
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1.10.1 Use of Maximum Detected Value to Estimate BTVs and Not-to-Exceed Values

BTVs and not-to-exceed values represent upper threshold values from the upper tail of a data distribution;
therefore, depending upon the data distribution and sample size, the BTVs and other not-to-exceed values
may be estimated by the largest or the second largest detected value. A nonparametric UPL, UTL, and
USL are often estimated by higher order statistics such as the maximum value or the second largest value
(EPA 1992b, 2009, Hahn and Meeker  1991). The use of higher order statistics to estimate the UTLs
depends upon the sample size. For data sets of size: 1) 59 to 92 observations, a nonparametric UTL95-95
is given by the maximum detected value; 2) 93 to 123 observations, a nonparametric UTL95-95 is given
by the second largest maximum detected value; and 3) 124 to 152 observations, a UTL95-95 is given by
the third largest detected value in the sample, and so on.

1.10.2 Use of Maximum Detected Value to Estimate EPC Terms

Some practitioners tend to use the maximum detected value as an estimate of the EPC term.  This is
especially true when the sample size is small such as <  5, or when a UCL95 exceeds the maximum
detected value. Specifically, the EPA (1992a)  document suggests the use of the maximum detected value
as a default value to estimate the EPC term when a 95% UCL (e.g., the H-UCL) exceeds the maximum
value in a data set. ProUCL computes 95% UCLs of the mean using several methods based upon normal,
gamma, lognormal, and non-discernible distributions. In the past, a lognormal distribution was used as the
default distribution to model positively skewed environmental data sets. Additionally, only  two methods
were used to estimate the EPC term based upon: 1) normal distribution and Student's t-statistic, and 2)
lognormal distribution and Land's H-statistic  (Land 1971, 1975). The use of the H-statistic often yields
unstable and impractically large UCL95 of the mean (Singh, Singh, and Engelhardt 1997; Singh, Singh,
and laci 2002). For highly skewed data sets  of smaller sizes (< 30, < 50), H-UCL often exceeds the
maximum detected value. Since the use of a lognormal distribution has been quite common (suggested as
a default model in the risk assessment guidance for Superfund [RAGS] document [EPA 1992a]), the
exceedance of the maximum value by an H-UCL95 is frequent for many skewed data sets of smaller sizes
(e.g., < 30, < 50). These occurrences result in the possibility of using the maximum detected value as an
estimate of the  EPC term.

It should be pointed out that  in some cases,  the maximum observed value  actually might represent an
impacted location.  Obviously, it is not  desirable to use an observation  potentially  representing an
impacted location to estimate the EPC for an AOC.  The EPC term represents the average exposure
contracted by an individual over an EA during a long period of time; the EPC term should be estimated
by using an average value (such as an appropriate 95% UCL of the mean) and not by the maximum
observed concentration. One needs to compute an average  exposure and not the maximum exposure.
Singh and Singh (2003) studied the performance of the max test (using the maximum observed value to
estimate the EPC) via Monte Carlo simulation experiments. They noted that for skewed data sets of small
sizes (e.g., < 10-20), even 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 lead to unnecessary further
remediation.

Several methods, some of which are described in EPA (2002a) and other EPA documents, are available in
versions of ProUCL (i.e., ProUCL 3.00.02 [EPA 2004], ProUCL 4.0 [U.S. EPA 2007],  ProUCL 4.00.05
[EPA 2009, 2010], ProUCL 4.1  [EPA  2011]) for estimating the EPC  terms. For data sets with NDs,
ProUCL 5.0 (and ProUCL 5.1) has some new UCL (and other limits) computation methods which  were
not available in earlier versions of ProUCL. It is unlikely that the UCLs based upon those methods will
exceed the maximum detected value, unless some outliers are present in the data set.
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1.10.2.1      Chebyshev Inequality Based UCL95

ProUCL 5.1 (and its earlier versions) displays a warning message when the suggested 95% UCL (e.g.,
Hall's or bootstrap-t UCL with outliers) of the mean exceeds the detected maximum concentration. When
a 95% UCL does exceed the maximum observed value, ProUCL suggests the use of an alternative UCL
computation method based upon the Chebyshev inequality. One may use a 97.5% or 99% Chebyshev
UCL to estimate the mean  of a highly skewed population.  The use of the Chebyshev inequality to
compute UCLs tends to yield more conservative (but stable) UCLs than other  methods available in
ProUCL software. In such cases, when the sample size is large (and other UCL methods such as the
bootstrap-t method yield unrealistically high values due to presence of outliers), one may want to use a
95% Chebyshev UCL or  a  Chebyshev UCL with a lower confidence  coefficient such as 90%  as an
estimate of the population mean, especially when the sample size is large  (e.g., >100, 150). The details (as
functions of sample size and  skewness) for the use of those UCLs are summarized  in various versions of
ProUCL Technical Guides (EPA 2004, 2007, 2009, 2010d, 2011, 2013a).

Notes: Using the maximum observed value to estimate the EPC term representing the average exposure
contracted by an individual over an EA is not recommended. For the sake of interested users, ProUCL
displays a warning message when the recommended 95% UCL (e.g., Hall's bootstrap UCL) of the mean
exceeds the observed maximum concentration. For such scenarios (when  a 95% UCL does exceed the
maximum observed value), an alternative UCL computation method based  upon Chebyshev inequality is
suggested by the ProUCL software.

1.11   Samples with Nondetect Observations

ND observations are inevitable in most environmental data sets. Singh, Maichle, and Lee (2006) studied
the performances (in terms  of coverages)  of the various UCL95 computation methods including the
simple substitution methods (such as the DL/2 and DL methods) for data sets with ND observations. They
concluded that the UCLs obtained using the substitution methods, including the replacement of NDs by
DL/2; do not perform well even when the percentage of ND observations is low, such as less than 5% to
10%. They recommended avoiding the  use of substitution methods for computing UCL95 based upon
data sets with ND observations.

1.11.1 Avoid the Use of the DL/2 Substitution Method to Compute UCL95

Based upon the results of the report by Singh, Maichle, and Lee (2006), it is recommended to avoid the
use of the  DL/2 substitution method when performing a GOF test, and when computing the summary
statistics and various other limits (e.g., UCL, UPL, UTLs) often used  to estimate the  EPC terms and
BTVs. Until recently, the substitution method has been the most commonly used method for computing
various statistics of interest for data sets which include NDs. The main reason for this has been the lack of
the availability of the other rigorous methods and associated software programs that  can be used to
estimate the various  environmental  parameters of interest.  Today, several  methods (e.g., using KM
estimates)  with better performance, including the Chebyshev inequality and bootstrap methods, are
available for computing the upper limits of interest. Several of those parametric  and nonparametric
methods are available in ProUCL 4.0 and higher versions.  The DL/2 method is included in ProUCL for
historical reasons as it had been the most commonly used and  recommended method until recently (EPA
2006b).  EPA scientists and several reviewers of the  ProUCL  software had suggested and requested the
inclusion of the DL/2 substitution method in ProUCL for comparison and research purposes.
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Notes: Even though the  DL/2 substitution method has been incorporated in ProUCL, its use is  not
recommended due to its poor performance. The DL/2 substitution method has been retained in ProUCL
5.1 for historical and comparison purposes. NERL-EPA, Las Vegas strongly recommends avoiding the
use of this method even when the percentage of NDs is as low as 5% to 10%.

1.11.2 ProUCL Does Not  Distinguish  between Detection  Limits,  Reporting  limits,  or
       Method Detection Limits

ProUCL  5.1 (and all  previous versions) does  not make distinctions between method  detection limits
(MDLs),  adjusted MDLs, sample quantitation limits  (SQLs), reporting limits  (RLs), or DLs.  Multiple
DLs (or  RLs) in ProUCL mean different  values of the detection limits. It is user's responsibility to
understand the differences between these  limits and  use  appropriate  values (e.g., DLs) for nondetect
values below which the laboratory cannot reliably detect/measure the presence of the analyte in collected
samples  (e.g., soil  samples). A data set consisting  of values less than the  DLs (or  MDLs,  RLs) is
considered a left-censored data set.  ProUCL uses statistical methods available  in the statistical literature
for left-censored data sets for computing statistics of interest including mean, sd, UCL, and estimates of
BTVs.

The user determines which  qualifiers (e.g., J,  U, UJ) will be considered as nondetects. Typically, all
values with U or UJ qualifiers are considered as nondetect values. It is the user's responsibility to enter a
value which can be  used to represent a ND value.  For NDs, the user enters the associated DLs or RLs
(and not zeros or half of the detection limits). An indicator column/variable, D_x taking a value, 0, for all
nondetects and a value,  1, for all  detects  is assigned to each variable, x, with NDs.  It is the user's
responsibility to supply the numerical values for NDs  (should be entered as reported DLs) not qualifiers
(e.g., J, U, B, UJ). For example, for thallium with nondetect values, the user creates an associated column
labeled as  D_thallium to tell the software that the data set will have nondetect  values. This  column,
D_thallium consists  of only zeros (0) and ones (1); zeros are used for all values reported as NDs and ones
are used for all values reported as detects.

1.12  Samples with  Low Frequency of Detection

When all of the sampled values are  reported as NDs, the EPC term and other statistical limits should also
be reported as a ND value, perhaps by the  maximum  RL or the  maximum RL/2. The project team will
need to make this determination. Statistics (e.g., UCL95) based upon only a few detected values  (e.g., <
4) cannot be considered reliable enough to  estimate EPCs which can have a potential impact on human
health and the environment. When the number of detected values is small, it is preferable to use ad  hoc
methods  rather than using statistical methods to compute EPCs and other upper limits.  Specifically, for
data sets  consisting  of < 4 detects and for small data sets (e.g., size <  10) with low detection frequency
(e.g.,  < 10%), the project team and the  decision makers should decide, on  a site-specific basis,  how to
estimate the average exposure (EPC) for the constituent and area under consideration.  For data sets with
low detection frequencies, other measures such as the median or mode represent better estimates (with
lesser uncertainty) of the population measure of central tendency.

Additionally, when most (e.g., > 95%) of the observations for a constituent lie below the DLs, the sample
median or the sample mode (rather than the  sample average) may be used as an estimate of the  EPC. Note
that when the majority of the data are NDs, the median and the mode may also be represented by a ND
value. The uncertainty associated with such estimates will be high. The statistical properties, such as the
bias, accuracy, and precision of such estimates, would remain unknown. In order to be able to compute
defensible estimates, it is always desirable to collect more samples.
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1.13   Some Other Applications of Methods in ProUCL 5.1

In addition to performing background versus site comparisons for CERCLA and RCRA sites, performing
trend evaluations based upon time-series data sets, and estimating EPCs in exposure and risk evaluation
studies, the statistical methods in ProUCL can be used to address other issues dealing with environmental
investigations that are conducted at Superfund or RCRA sites.

1.13.1 Identification of COPCs

Risk assessors and remedial project managers (RPMs) often use screening levels or BTVs to identify
COPCs during the screening phase of a cleanup project at a contaminated site. The screening for COPCs
is performed prior to any characterization and remediation activities that are conducted at the  site. This
comparison is performed to screen out those constituents that may be present in the site  medium  of
interest at low levels (e.g., at or below the background levels or some pre-established screening levels)
and may not pose any threat and concern to human health and the environment. Those constituents may
be eliminated from all future site investigations, and risk assessment and risk management studies.

To identify the COPCs, point-by-point site observations  are compared with some pre-established soil
screening  levels (SSL)  or  estimated  BTVs. This  is especially true  when the comparisons of site
concentrations with screening levels or BTVs are conducted in real time  by the sampling or cleanup crew
onsite. The project team should decide the type of site samples (discrete or composite) and the number of
site observations that should be collected and compared with the screening levels or the BTVs. In case
BTVs  or screening  levels are  not known, the availability of a defensible  site-specific background  or
reference  data  set  of reasonable  size (e.g., at least  10)  is required for computing reliable and
representative estimates of BTVs and screening levels. The constituents with concentrations exceeding
the respective  screening values or BTVs may be considered COPCs,  whereas  constituents  with
concentrations (e.g., in  all collected samples) lower than the screening values or  BTVs may be omitted
from all future evaluations.

1.13.2 Identification of Non-Compliance Monitoring Wells

In MW compliance assessment applications, individual (often discrete) constituent concentrations from a
MW are compared with some pre-established limits  such as an  ACL or a maximum concentration limit
(MCL). An exceedance of the MCL or  the BTV (e.g., estimated by a UTL95-95 or a UPL95) by a MW
concentration may be considered an indication of contamination in that MW. For individual concentration
comparisons, the presence of contamination (determined by an exceedance) may have to be confirmed by
re-sampling from that MW. If concentrations of constituents in the original sample  and re-sample(s)
exceed the MCL or BTV, then that MW may require further scrutiny, perhaps  triggering remediation
activities.  If the concentration data from a MW for 4 to 5  continuous quarters (or some other designated
time period determined by the project team) are below the MCL or BTV level, then that MW may be
considered as complying with (achieving) the pre-established or estimated standards.

1.13.3 Verification of the Attainment of Cleanup Standards, Cs

Hypothesis testing approaches are used  to verify the attainment of the cleanup standard, Cs, at site AOCs
after conducting remediation and cleanup at those site AOCs (EPA  1989a, 1994). In order to assess the
attainment of cleanup levels, a representative  data set of adequate size perhaps obtained using  the DQO
process (or a minimum of  10  observations should be collected) needs to be made available  from the
remediated/excavated areas  of the  site under investigation. The  sample  size should also account for the
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size of the remediated site areas: meaning that larger site areas should be sampled more (with more
observations) to obtain a representative sample of the remediated areas under investigation. Typically, the
null hypothesis of interest is Ho: Site Mean, /& > Cs versus the alternative hypothesis, Hi: Site Mean, /4 <
Cs, where the cleanup standard, Cs, is known a priori.

1.13.4 Using BTVs (Upper Limits) to Identify Hot Spots

The use of upper limits (e.g., UTLs) to identify hot spot(s) has also been mentioned in the Guidance for
Comparing Background and Chemical Concentrations in Soil for CERCLA Sites (EPA 2002b). Point-by-
point site observations are compared with a pre-established or estimated BTV. Exceedances of the BTV
by site observations may represent impacted locations with elevated concentrations (hot spots).

1.14   Some   General  Issues,  Suggestions  and  Recommendations  made  by
       ProUCL

Some general issues regarding the handling of multiple DLs by ProUCL and recommendations made
about various substitution and ROS methods for data sets with NDs are described in  the  following
sections.

1.14.1 Handling of Field Duplicates

ProUCL does not pre-process field duplicates. The project team determines how field duplicates will be
handled and pre-processes the data accordingly. For an example,  if the  project  team decides to use
average  values for field duplicates, then averages need to be computed and field duplicates need to be
replaced by their respective average values.  It is the user's responsibility to feed in appropriate values
(e.g., averages,  maximum) for field duplicates.   The user is  advised to refer to the  appropriate  EPA
guidance documents related to collection and use of field duplicates for more information.

1.14.2 ProUCL Recommendation about ROS Method and Substitution (DL/2) Method

For data sets with NDs, ProUCL can compute point estimates of population mean and standard deviation
using the KM and ROS methods (and also using the DL/2 substitution method). The substitution method
has been retained in ProUCL for historical and  research purposes. ProUCL uses Chebyshev inequality,
bootstrap methods,  and normal, gamma, and lognormal distribution based equations on KM  (or ROS)
estimates to compute upper limits  (e.g.,  UCLs, UTLs).  The simulation study conducted by Singh,
Maichle and Lee (2006)  demonstrated that the KM method yields accurate estimates of the population
mean. They also demonstrated that for moderately skewed to highly  skewed data sets, UCLs based upon
KM estimates and BCA bootstrap (mild skewness), KM estimates and Chebyshev inequality (moderate to
high skewness), and KM estimates and bootstrap-t method  (moderate to high skewness) yield better  (in
terms of coverage probability) estimates of EPCs than other UCL methods based  upon the Student's t-
statistic on KM estimates, percentile bootstrap method on KM or ROS estimates.

1.14.3 Unhandled Exceptions and Crashes in ProUCL

A typical   statistical  software, especially developed under  limited  resources may not be able  to
accommodate data  sets with all kinds of deficiencies such as all missing values  for  a variable, or  all
nondetect values for a variable. An inappropriate/insufficient data set can occur in various  forms and not
all of them can be addressed in a scientific program like ProUCL. Specifically, from a programming point
of view, it can be quite burdensome on the programmer to address all potential deficiencies that can occur
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in a data set. ProUCL 5.1 addresses many data deficiencies and produces warming messages.  All data
deficiencies causing  unhandled  exceptions  which were identified by  users have been  addressed in
ProUCL 5.1. However, when ProUCL yields an unhandled exception or crashes, it is highly likely that
there is something wrong with the data set; the user is advised to review the input data set to make sure
that the data set follows ProUCL data and formatting requirements.

1.15   The Unofficial  User  Guide to  ProUCL4 (Helsel and Gilroy 2012)

Several ProUCL 4.1  users sent inquiries  about the validity of the comments made about the ProUCL
software in the Unofficial User Guide to ProUCL4 (Helsel and Gilroy, 2012) and in the Practical Stats
webinar, "ProUCL v4: The Unofficial User Guide," presented by Dr. Helsel on October 15, 2012 (Helsel
2012a). Their inquiries led us to review comments made about the ProUCL4 software and  its associated
guidance documents (EPA 2007, 2009a, 2009b,  2010c, 2010d, and 2011) in the "The Unofficial Users
Guide to ProUCL4" and in the webinar, "ProUCL v4:  The Unofficial User Guide". These two documents
collectively are referred to as the Unofficial ProUCLv4 User Guide in this ProUCL document. The pdf
document describing the  material presented  in the Practical Stats  Webinar (Helsel  2012a)  was
downloaded from the http://www.practicalstats.com website.

In the "ProUCL v4: The Unofficial User Guide", comments have been made about the software and its
guidance documents, therefore,  it  is appropriate to  address those comments in the  present ProUCL
guidance document. It is necessary to provide the detailed response to assure that: 1) rigorous statistical
methods are used to  compute decision making statistics; and 2) the methods incorporated in ProUCL
software are not misrepresented  and misinterpreted.  Some general responses and comments about the
material presented in the webinar and in the Unofficial User Guide to ProUCLv4 are described as follows.
Specific comments and responses  are also considered in the respective chapters of  ProUCL 5.1  (and
ProUCL 5.0) guidance documents.

Note: It is noted that the Kindle version of "ProUCL v4: Unofficial User Guide" is no longer available on
Amazon. Several incorrect theoretical statements and statements misrepresenting ProUCL 4 were made in
that Unofficial User Guide; therefore, a brief response to some of those  statements has been retained in
ProUCL 5.1 guidance documents.

ProUCL is a freeware software package which has been developed under limited government funding to
address statistical issues associated with various environmental site projects.  Not all statistical methods
(e.g., Levene test) described in the statistical literature have been incorporated in ProUCL. One should not
compare ProUCL with commercial  software packages which are expensive and not as user-friendly as the
ProUCL software when addressing  environmental statistical issues. The existing and some new statistical
methods based upon the research conducted by ORD-NERL, EPA Las Vegas during  the last couple of
decades have been incorporated in  ProUCL to address the statistical needs of various environmental site
projects and research studies.  Some of those new methods may not be available in  text books, in the
library of programs written in R-script, and in commercial software packages. However, those methods
are described  in detail in the cited  published literature and also in the ProUCL Technical  Guides (e.g.,
EPA 2007, 2009a,  2009b,  2010c,  2010d, and 2011).  Even though  for uncensored data sets, programs
which compute gamma distribution based UCLs and  UPLs are available in R  Script, programs which
compute  a  95% UCL of mean based upon a gamma distribution on KM estimates are  not as easily
available.

•  In the Unofficial ProUCL v4 User Guide, several statements have been made  about percentiles. There
   are several ways to compute  percentiles. Percentiles computed by ProUCL may or may not be
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identical (don't have to be) to percentiles computed by NADA for R (Helsel 2013) or described in
Helsel and Gilroy (2012). To address users' requests, ProUCL 4.1  (2011)  and its higher versions
compute percentiles that are comparable to the percentiles computed by Excel  2003 and higher
versions.

The literature search suggests that there are a total of nine (9) known types of percentiles, i.e., 9
different methods of calculating percentiles in statistics literature (Hyndman and Fan, 1996). The R
programming language (R Core Team 2012) computes percentiles using those 9 methods using the
following statement in R

    Quantile (x, p, type=k) where p = percentile, k = integer between 1-9

ProUCL computes percentiles using Type 7; Minitab 16 and SPSS compute percentiles using Type 6.
It is simply a matter of choice, as there is no 'best' type to use. Many software packages use one type
for calculating a percentile, and another for generating a box plot (Hyndman and Fan 1996).

An incorrect statement "By definition, the sample mean has a 50% chance of being below the true
population mean" has been made in Helsel and Gilroy (2012) and also in Helsel (2012a). The above
statement is not correct for means of skewed distributions (e.g., lognormal or gamma)  commonly
occurring in environmental applications.  Since Helsel (2012) prefers to use a lognormal distribution,
the incorrectness of the above statement has been illustrated using  a lognormal distribution.  The
mean and median of a lognormal distribution (details in Section 2.3.2 of Chapter 2 of ProUCL 5.1
Technical Guide) are given by:

                       mean =jul = exp(ju + 0.5<72); and median =M = exp(//)

From the above equations, it is clear that the mean of a lognormal distribution is always greater than
the median for all positive values of a (sd of log-transformed variable). Actually the mean is greater
than the pth percentile when a >2zp. For example, when p = 0.80, zp = 0.845, and mean of a
lognormal distribution, y.\ exceeds XQ.SO, the 80th percentile when a > 1.69. In other  words, when a >
1.69 the  lognormal mean  will exceed  the 80th percentile of a lognormal  distribution. Here zp
represents the/?* percentile of the standard normal distribution (SND)  with mean 0 and variance 1.

To demonstrate the incorrectness  of the above statement, a small simulation study was conducted.
The distribution of sample  means based upon samples of size  100 were generated from lognormal
distributions  with n =4, and  varying skewness.  The experiment was performed 10,000 times to
generate the distributions  of sample means.  The probabilities  of  sample means  less than the
population means were computed. The following results are noted.
      Table 1-2. Probabilities p(x 
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       The probabilities summarized in the above table demonstrate that the statement about the mean
       made in Helsel and Gilroy (2012) is incorrect.

    Graphical Methods:  Graphical methods are available in ProUCL as exploratory tools which can be
    generated for both uncensored and left-censored data sets. Exploratory graphical methods are used to
    understand possible  patterns present in data sets and not to compute statistics used  in the decision
    making process. The Unofficial ProUCL Guide makes several comments about box plots and Q-Q
    plots incorporated in ProUCL.  The Unofficial ProUCL Guide states that all graphs with NDs  are
    incorrect. These statements are misleading  and incorrect. The intent of the graphical methods in
    ProUCL is exploratory  for the  purpose of gaining information (e.g., outliers, multiple populations,
    data distribution, patterns, and skewness) about a data set. Based upon the data displayed (ProUCL
    displays a message [e.g., as a sub-title] in this regard) on those graphs, all statistics shown on those
    graphs generated by ProUCL are correct.

    Box Plots: In statistical  literature, one can find several ways to generate box plots. The practitioners
    may have their own preferences to use one method over the other.  All box plot methods including the
    one in ProUCL convey  the same information about the data set (outliers, mean, median, symmetry,
    skewness). ProUCL uses a couple of development tools such as FarPoint spread (for Excel type input
    and output operations) and ChartFx (for graphical displays); and ProUCL generates box plots using
    the built-in box plot feature in ChartFx.  For all practical and exploratory purposes, box plots in
    ProUCL are equally good (if  not  better) as those available in the various commercial  software
    packages, for examining data distribution  (skewed or symmetric), identifying outliers, and comparing
    multiple groups (main objectives of box plots in ProUCL).

       o   As mentioned earlier, it is a matter of choice of using percentiles/quartiles to  construct a box
           plot. There is no 'best' method for constructing a box plot. Many software packages use one
           method (out of 9 as specified above) for calculating a percentile, and another for constructing
           a box plot  (Hyndman and Fan 1996).

    Q-Q plots:  All Q-Q plots incorporated in ProUCL are correct and of high quality. In addition to
    identifying outliers, Q-Q plots are also used to assess data distributions. Multiple Q-Q  plots are useful
    for performing point-by-point comparisons of grouped data sets, unlike box plots based upon the five-
    point summary statistics.  ProUCL has Q-Q  plots for normal,  lognormal, and gamma distributions -
    not all of these graphical capabilities are directly available in other software packages  such as NADA
    for R (Helsel 2013).  ProUCL offers several exploratory options for generating Q-Q plots for data sets
    with NDs. Only detected outlying observations may require additional investigation;  therefore, from
    an exploratory point of view, ProUCL can generate Q-Q plots excluding all NDs  (and other options).
    Under this scenario  there is no need to retain place holders (computing plotting positions used to
    impute  NDs) as the  objective is not to impute NDs. To impute NDs, ProUCL  uses ROS  methods
    (Gamma ROS and log ROS) requiring place holders; and ProUCL computes plotting positions for all
    detects  and NDs to generate a proper regression model  which is  used to  impute  NDs. Also  for
    comparison purposes, ProUCL can be used to generate Q-Q plots on data sets obtained by replacing
    NDs by their respective DLs or DL/2s. In these cases, no NDs are  imputed, and there is no need to
    retain placeholders for NDs. On these Q-Q plots, ProUCL displays some relevant statistics which are
    computed based upon the data displayed on those graphs.

    Helsel (2012a) states that the Summary  Statistics module does not display  KM estimates and that
    statistics based upon logged data are useless. Typically, estimates computed after processing the data
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do not represent summary statistics. Therefore, KM  and ROS estimates  are not displayed in the
Summary Statistics module. These statistics are available in several other modules including the
UCL and BTV modules. At the request of several users, summary statistics are computed based upon
logged data. It is believed that the mean, median, or  standard deviation of logged data do provide
useful information about data skewness and data variability.

To test for the equality of variances, the F-test, as incorporated in ProUCL, performs fairly well and
the  inclusion of the Levene's (1960) test will not add any new capability to the ProUCL software.
Therefore, taking budget constraints into consideration, Levene's test has not been incorporated in the
ProUCL software.

    o  Although it makes sense to first determine if the two variances are equal or unequal, this is
       not a requirement to perform a t-test. The t-distribution based confidence interval or test for
       /Hi  - /Li2 based on the pooled sample variance does not perform better than the approximate
       confidence  intervals based upon Satterthwaite's test.  Hence  testing for the  equality  of
       variances is not required to perform a two-sample t-test. The use of Welch-Satterthwaite's or
       Cochran's method is recommended in all situations (see Hayes 2005).

Helsel (2012a) suggests that imputed NDs should not be made available to the users. The developers
of ProUCL and other researchers like to have access to  imputed NDs. As a researcher, for exploratory
purposes only, one  may want to have  access to imputed NDs to be used in exploratory advanced
methods such as multivariate methods  including data mining, cluster and principal component
analyses. It is noted that one cannot easily perform exploratory methods on multivariate data sets with
NDs. The  availability of imputed NDs makes it possible for researchers and scientists to identify
potential patterns present in complex multivariate data by using data mining exploratory methods on
those multivariate data sets with  NDs. Additional discussion on this topic is considered in Chapter 4
of the ProUCL 5.1 Technical Guide.

    o  The statements summarized above should not be misinterpreted. One may not use parametric
       hypothesis tests such as  a t-test or a classical ANOVA on data sets consisting of imputed
       NDs. These methods require further investigation as the decision errors associated with such
       methods remain unqualified. There are other methods such as the Gehan and T-W tests in
       ProUCL  5.0/ProUCL 5.1 which are better suited to perform two-sample hypothesis  tests
       using data sets with multiple detection limits.

Outliers:  Helsel  (2012a) and Helsel and Gilroy (2012) make several comments about outliers. The
philosophy (with input from EPA  scientists) of the  developers of ProUCL about the outliers in
environmental applications is that those outliers  (unless they represent typographical errors)  may
potentially represent impacted (site related or otherwise) locations or monitoring wells; and therefore
may require further investigation. Moreover, decision  statistics such as a UCL95 based upon a data
set  with outliers gets inflated  and tends to represent those outliers  instead of representing the
population average. Therefore,  a few  low probability outliers coming from the tails  of the  data
distribution may  not be included in the computation  of the  decision making upper limits (UCLs,
UTLs), as those upper limits get distorted by outliers and tend not to represent the parameters they are
supposed to estimate.

    o  The presence  of outliers  in a data set tends to  destroy the normality of the data set. In other
       words, a data set with outliers can seldom (may be when outliers are mild, lying around the
       border of the central and tail parts of a normal distribution) follow a normal distribution.
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           There are modern robust and resistant outlier identification methods (e.g., Rousseeuw and
           Leroy 1987; Singh and Nocerino 1995) which are better suited to identify outliers present in a
           data set; several of those robust outlier identification methods are available in the Scout 2008
           version 1.0 (EPA 2009) software package.

       o   For both  Rosner and Dixon tests, it is the data set (also called the main body of the data set)
           obtained  after removing the outliers (and not the data set with outliers) that needs to follow a
           normal distribution (Barnett and Lewis 1994). Outliers are not known in advance. ProUCL
           has normal Q-Q plots which can be used to get an idea about  potential outliers (or mixture
           populations) present in a data set. However, since a lognormal  model tends to accommodate
           outliers, a data set with outliers can follow  a lognormal distribution; this does not imply that
           the outlier which may  actually represent an impacted/unusual location does not exist! In
           environmental applications, outlier tests should be performed on raw data sets, as the cleanup
           decisions need to be made based upon values in the  raw scale  and not in log-scale or some
           other transformed space. More discussion  about outliers can be found in Chapter 7  of the
           ProUCL 5.1 Technical Guide.

    In Helsel (2012a), it is stated, "Mathematically, the lognormal is simpler and easier to interpret than
    the gamma (opinion)." We do agree with the opinion that the lognormal is simpler and easier to use
    but  the  log-transformation  is  often  misunderstood  and  hence  incorrectly used and  interpreted.
    Numerous examples (e.g., Example 2-1 and 2-2, Chapter 2 of ProUCL Technical Guide) are provided
    in the ProUCL guidance documents illustrating the advantages of the using a gamma distribution.

    It is further stated in  Helsel  (2012a) that ProUCL prefers the gamma  distribution because  it
    downplays outliers as compared to the lognormal. This argument can be turned around - in other
    words, one can say that the lognormal is preferred  by practitioners who want to inflate the effect of
    the outlier.  Setting this argument aside, we prefer the gamma distribution as it does not transform the
    variable so the results are  in the  same scale as the  collected data set. As mentioned earlier, log-
    transformation does appear to be simpler but problems arise when practitioners are not aware of the
    pitfalls  (e.g.,  Singh and Ananda 2002; Singh,  Singh, and  laci  2002) associated with the  use of
    lognormal distribution.

    Helsel (2012a) and Helsel and Gilroy (2012) state that "lognormal and gamma are similar, so usually
    if one is considered possible, so is the other."  This  is another incorrect and misleading statement;
    there are significant differences in the two distributions and in their mathematical properties.  Based
    upon the  extensive experience in environmental statistics and published literature, for skewed data
    sets that follow both lognormal  and gamma distributions, the developers favor the use of the gamma
    distribution over the lognormal distribution. The  use of the gamma distribution based decision
    statistics is preferred to estimate the environmental parameters (mean, upper percentile). A lognormal
    model tends to hide contamination by accommodating outliers and multiple populations whereas a
    gamma distribution tends not to accommodate contamination (elevated values) as can be seen in
    Examples 2-1 and 2-2 of Chapter  2 of the ProUCL 5.1 Technical Guide. The use of the lognormal
    distribution on a data set with outliers tends to yield inflated and distorted estimates which may not be
    protective of human health and the environment; this is especially true  for skewed data sets of small
    of sizes <20-30; the sample size requirement increases with skewness.

    o   In the context of computing a UCL95 of mean, Helsel and Gilroy (2012) and Helsel (2012a) state
       that GROS and LROS methods  are probably never better than the KM method.  It should be
       noted that these three estimation methods compute estimates of mean and standard deviation and
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       not the upper limits used to estimate EPCs and BTVs. The use of the KM method does yield good
       estimates of the mean and standard deviation as noted by Singh, Maichle, and Lee (2006). The
       problem of estimating mean and standard deviation for data sets with nondetects has been studied
       by many researchers as described in Chapter 4 of the ProUCL 5.1 Technical Guide. Computing
       good estimates of mean and sd based upon left-censored data sets  addresses only half of the
       problem. The main issue is  to compute decision statistics (UCL, UPL, UTL) which  account for
       uncertainty and data skewness inherently present in environmental data sets.

    o   Realizing that for skewed data sets, Student's t-UCL, CLT-UCL, and standard and percentile
       bootstrap UCLs do not provide the specified coverage to the population mean for uncensored data
       sets, many researchers (e.g., Johnson 1978; Chen  1995; Efron and Tibshirani 1993;  Hall [1988,
       1992]; Grice and Bain 1980; Singh, Singh, and Engelhardt 1997;  Singh, Singh,  and laci  2002)
       developed parametric (e.g., gamma) and nonparametric (e.g.,  bootstrap-t and Hall's bootstrap
       method, modified-t, and Chebyshev inequality) methods for computing confidence intervals and
       upper limits which adjust for data skewness.  One cannot ignore the work and  findings of the
       researchers cited  above, and assume that Student's t-statistic based upper limits or percentile
       bootstrap method based upper  limits can be used for all data  sets with varying skewness and
       sample sizes.

    o   Analytically, it is not feasible to compare the various estimation and  UCL computation methods
       for  skewed data  sets  containing  ND  observations.   Instead, researchers  use  simulation
       experiments to learn about the distributions and performances of the various statistics (e.g., KM-t-
       UCL, KM-percentile bootstrap  UCL, KM-bootstrap-t UCL,  KM-Gamma UCL). Based upon the
       suggestions  made in published literature and findings summarized in Singh,  Maichle, and Lee
       (2006), it is  reasonable to state  and assume that the findings of the simulation studies performed
       on uncensored skewed data sets comparing  the performances of the various UCL computation
       methods can be extended to  skewed left-censored data sets.

    o   Like uncensored  skewed data  sets,  for  left-censored data  sets, ProUCL 5.0/ProUCL 5.1 has
       several parametric and nonparametric methods to  compute UCLs and other limits which adjust
       for data skewness.  Specifically,  ProUCL uses  KM estimates  in  gamma  equations; in  the
       bootstrap-t method,  and in  the  Chebyshev inequality to compute upper limits for left-censored
       skewed data sets.

•   Helsel (2012a) states that ProUCL 4 is based upon presuppositions. It is emphasized that ProUCL
    does not make any suppositions in advance.  Due to the poor performance of a lognormal model, as
    demonstrated in  the literature and illustrated via  examples throughout ProUCL guidance documents,
    the use of a gamma distribution is preferred when a data set can be modeled by a lognormal model
    and a gamma model. To provide the desired coverage  (as close as possible) for the population mean,
    in earlier versions of ProUCL  (version 3.0), in lieu  of H-UCL, the use  of Chebyshev UCL was
    suggested for moderately and highly skewed data sets.   In later (3.00.02 and higher)  versions  of
    ProUCL, depending upon skewness and  sample size, for gamma distributed data sets, the use of the
    gamma distribution was  suggested for computing the UCL of the mean.

Upper limits (e.g., UCLs, UPLs, UTLs) computed using the Student's t statistic and percentile bootstrap
method (Helsel 2012, NADAfor R, 2013) often fail to provide the desired coverage (e.g., 95% confidence
coefficient) to the parameters (mean, percentile)  of most of the skewed environmental populations. It is
suggested that the practitioners compute the decision making statistics (e.g., UCLs,  UTLs)  by taking: data
                                                                                           67

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distribution; data set size; and data skewness into consideration. For uncensored and left-censored data
sets, several such upper limits computation methods are available in ProUCL 5.1 and its earlier versions.

Contrary to the statements made in Helsel and Gilroy (2012), ProUCL software does not favor statistics
which yield higher (e.g., nonparametric Chebyshev  UCL) or lower (e.g., preferring the use of a gamma
distribution to using a lognormal distribution) estimates of the environmental parameters (e.g., EPC and
BTVs). The main objectives of the ProUCL software funded by the  U.S. EPA is to compute rigorous
decision statistics to  help the decision makers and project teams in making sound decisions which are
cost-effective and protective of human health and the environment.

Cautionary Note: Practitioners and scientists are cautioned about:  1) the suggestions made about the
computations of upper limits described in some recent environmental literature such as the NADA books
(Helsel [2005, 2012]); and 2) the misleading comments made about the ProUCL software  in the training
courses offered by Practical Stats during 2012 and 2013. Unfortunately, comments about ProUCL made
by Practical Stats during their training courses lack professionalism and theoretical accuracy. It is noted
that NADA packages in R and Minitab (2013) developed by Practical Stats do not offer methods which
can be used to compute reliable or accurate decision statistics  for skewed data sets. Decision statistics
(e.g., UCLs, UTLs, UPLs) computed using the methods (e.g., UCLs computed using percentile bootstrap,
and KM and LROS estimates and t-critical values) described in the  NADA books and incorporated  in
NADA packages do not take data distribution and data skewness into consideration. The use of statistics
suggested in NADA books  and in  Practical Stats  training sessions  often fail to  provide  the  desired
specified coverage to environmental parameters of interest  for moderately skewed to highly  skewed
populations. Conclusions derived based upon those  statistics may lead to incorrect conclusions which
may not be cost-effective  or protective of human health and the environment.

Page 75 (Helsel [20121): One of the reviewers of the ProUCL 5.0 software drew our attention to the
following incorrect statement made on page 75 of Helsel (2012):

"If there is only 1 reporting limit, the result is that the mean is identical to a substitution of the reporting
limit for censored observations."

An example of a left-censored data set containing ND observations with one reporting limit of 20 which
illustrates this issue is described as follows.
Y        D_y
20        0
20        0
20        0
7         1
58        1
92        1
100       1
72        1
11        1
27        1

The mean and standard deviation based upon the KM and two substitution methods: DL/2  and DL are
summarized as follows:
68

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Kaplan-Meier (KM) Statistics
Mean          39.4
SD            35.56

DL Substitution method (replacing censored values by the reporting limit)
Mean          42.7
SD            34.77

DL/2 Substitution method (replacing NDs by the reporting limit)
Mean          39.7
SD            37.19

The above example illustrates that the KM mean (when only 1 detection limit is present) is not actually
identical to the mean estimate obtained using the  substitution, DL (RL) method. The statement made in
Helsel's text (and also incorrectly made in his presentations such as the one made at the U.S. EPA 2007
National Association of Regional Project Managers (NARPM) Annual Conference:
http://www.ttemidev.com/narpm2007Admin/conference/) holds only when all observations reported
as detects are greater than the single reporting limit, which is not always true for environmental data sets
consisting of analytical concentrations.

1.16   Box and Whisker Plots

At the request of ProUCL users, a brief description of box plots (also known as box and whisker plots) as
developed by Tukey  (Hoaglin, Mosteller  and Tukey  1991)  is  provided  in this  section.   A box  and
whiskers plot represents a useful and convenient exploratory tool  and  provides a quick five-point
summary of a data set. In  statistical literature, one can find several  ways to generate box plots.  The
practitioners may have their own preferences  to use one  method over the other. Box plots are well
documented in the statistical literature and description of box plots can be easily obtained by surfing the
net. Therefore, the detailed description about the generation of box  plots is not provided  in ProUCL
guidance documents.  ProUCL also generates box plots for data set with NDs. Since box plots are used
for exploratory purposes to identify outliers and also to compare concentrations of two or more groups, it
does not really matter how NDs are  displayed on those box plots.  ProUCL generates box plots using
detection limits and draws  a horizontal line at the  highest detection  limit. Users can draw up to  four
horizontal lines at other levels (e.g., a screening  level, a BTV, or an average) of their choice.

All box plot methods, including the one in ProUCL, represent five-point summary graphs including: the
lowest and  the highest data values, median (50th percentile=second quartile, Q2), 25th percentile (lower
quartile, Ql), and 75th percentile (upper quartile, Q3).  A box and whisker plot also provides  information
about the degree of dispersion (interquartile range (IQR) = Q3-Ql=length/height of the box in a box plot),
the degree  of skewness (suggested by the length of the whiskers)  and unusual data  values known as
outliers. Specifically,  ProUCL (and other software  packages) use the following to generate a box  and
whisker plot.

    •    Q1= 25th percentile, Q2= 50th (median), and Q3 = 75th percentile
    •    Interquartile range= IQR = Q3-Q1 (the length/height of the box in a box plot)
    •    Lower whisker starts at Q1 and the upper whisker starts at Q3.
    •    Lower whisker extends up to the lowest observation or (Ql - 1.5 * IQR) whichever is higher
    •    Upper whisker extends up to the highest observation or (Q3 + 1.5* IQR) whichever is lower
                                                                                             69

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    •   Horizontal bars (also known as fences) are drawn at the end of whiskers
    •   Guidance in statistical  literature suggests that observations lying outside the fences (above the
        upper bar and below the lower bar) are considered potential outliers

    An example box plot generated by ProUCL is shown in the following graph.
                                                Box Plot for Lead
    Box Plot with Fences and Outlier

It should be pointed out that the use of box plots in different scales (e.g., raw-scale and log-scale) may
lead to different conclusions about outliers. Below is an example illustrating this issue.

Example 1-2. Consider an actual data set consisting of copper concentrations collected a Superfund Site.
The data set is: 0.83, 0.87, 0.9, 1, 1, 2, 2, 2.18, 2.73, 5, 7, 15, 22, 46, 87.6, 92.2, 740, and 2960. Box plots
using data in the raw-scale and log-scale are shown in Figures 1-1 and  1-2.
70

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                                           Box Plot for Copper
Figure 1-1. Box Plot of Raw Data in Original Scale

Based upon the last bullet point of the description of box plots described above, from Figure  1-1, it is
concluded that two observations 740 and 2960 in the raw scale represent outliers.
                                         Box Plot for In(copper)
Figure 1-2. Box Plot of Data in Log-Scale

However, based upon the last bullet point about box plots, from Figure  1-2, it is  concluded that two
observations  740 and  2960 in  the log-scale do not represent  outliers. The log-transformation has
accommodated the two outliers. This is one of the reasons ProUCL guidance suggests avoiding the use of
                                                              O          OO           O
log-transformed data. The use of a log-transformation tends to hide/accommodate outliers/contamination.
                                                                                              71

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Note: ProUCL uses a couple of development tools such as FarPoint spread (for Excel type input and
output operations) and ChartFx (for graphical displays). ProUCL generates box plots using the built-in
box plot feature in ChartFx. The programmer has no control over computing various statistics (e.g., Ql,
Q2, Q3, IQR) using ChartFx. So  box plots generated by  ProUCL can differ slightly from box plots
generated by other programs (e.g., Excel). However, for all practical and exploratory purposes, box plots
in ProUCL are equally good (if not better) as available in the various commercial software packages for
investigating data distribution  (skewed or  symmetric),  identifying outliers,  and  comparing multiple
groups (main objectives of box plots).

Precision in Box Plots: Box plots generated using ChartFx round values to  the  nearest integer. For
increased precision of graphical displays (all graphical displays generated by ProUCL), the user can use
the process described as follows.

Position your  cursor on the graph and right-click, a popup menu will appear.  Position the cursor on
Properties and right-click; a windows form labeled Properties will appear. There are three choice at the
top: General,  Series and Y-Axis.  Position the e cursor over the Y-Axis choice and left-click. You can
change  the number of decimals to increase  the precision, change the step to  increase or  decrease the
number Y-Axis values displayed and/or change the direction of the label.  To  show values on the plot
itself, position your cursor on the graph and right-click; a pop-up menu will appear. Position the cursor on
Point Labels and right-click. There are other options available in this pop-up menu including changing
font sizes.
72

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73

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                                         Chapter 2

                         Entering and Manipulating Data
2.1    Creating a New Data Set

By executing ProUCL 5.1, the following file options will appear (the title will show ProUCL 5.1 instead
ofProUCL5.0):
  File   Edit  Stats/Sample Sizes  BIS Simulator  Graphs  Statistical Tests  Upper Limits/BTVs   UCLs/EPCs   Windows  Help

     Navigation Panel
  Name
By choosing the File ^- New option, a new worksheet shown below will appear. The user enters variable
names and data following the ProUCL input file format requirements described in Section 2.3.
                                                                                   VorkSheet
   File  Edit  Stats/Sample Sizes  Graphs  Statistical Tests  Upper Limits/BTVs   UCLs/EPCs  Windows  Help
Navigation Panel
Name
Worksheet jds




o

1

2

3

4

5

6

7

8

9

10

11

T
2
3


2.2    Opening an Existing Data Set

The user can open an existing worksheet (*.xls, *.xlsx, *.wst, and *.ost) by choosing the File ^- Open
Single File Sheet option. The following drop down menu will appear:
     File   Edit   Stats/Sample Sizes   ISM Simulator   Graphs    Statistical Tests   Upper Limits/BTVs

         New
         Open Single File Sheet
         Exit
         Open Excel File with Multiple Sheets
                 Opens First Sheet in an Excel File or an Output or Older ProUCL (,WST) File
74

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t i, > Anita Singh > Desktop > ProUCL 5,0 K
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] Basic Calc for 20 example 08Feb13 5/25/2013 6:38 PM
] Adv Calc for 50 example 08Feb13 5/25/2013 6:37 PM
] DU4 data-bkgd metals-drom deana 5/16/2013 7:07 AM
proucl-review-comments-7-2013 8/9/20139:41 PM
Data-ProUCL 5.0 8/9/201 3 8:35 AM

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Excel Files (.xlsx)
Worksheet files (*.wst)





















1 output files (*-ost)
Choose a file by high lighting the type of file such as .xls as shown above.  This option can also be used
to read in a *.wst worksheet and *.ost output sheet generated by earlier versions (e.g., ProUCL 4.1 and
older) of ProUCL.

By choosing the File  ^-  Excel Multiple  Sheets option, the user can open an Excel file consisting of
multiple sheets. Each sheet will be opened as a separate file to be processed individually by ProUCL 5.1

Caution:  If you are editing a file (e.g., an excel file using Excel), make sure to close the file  before
importing the file into ProUCL using the file open option.

2.3   Input File Format

       •   The program can  read Excel  files. The user can perform  typical  Cut, Paste, and Copy
           operations available under the Edit Menu Option as shown below.
a^ File
Nav
Name
PCA-NDs
SuperFun
Edit Stats/Sample Sizes ISM Graphs Statistical Test:
Configure Display >
Cut Ctrl+X
Copy Ctrl+C
Paste Ctrl+V
Header Name
Full Precision
w1 Log Panel
*' Navigation Panel
Excel 2003

                                                                                            75

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       •   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 holds the labels for the groups (e.g., Background, AOC1, AOC2,
               1, 2, 3, a, b, c, Site 1, Site 2) that might be present in the data set.  Alphanumeric strings
              (e.g., Surface, Sub-surface) can be used to label the  various groups. Most of the modules
              of ProUCL can process data by a group variable.

           o  The  data file  can have   multiple variables  (columns) with  unequal numbers  of
              observations. Most of the modules of ProUCL can process data by a group variable.

           o  Except for the header row  and columns representing  the group labels, only numerical
              values should appear in all other rows.

           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 and are omitted from statistical evaluations.

           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 the number of missing values.

2.4    Number Precision

       •   The  user may turn  "Full Precision" on or off by choosing Configure ^- Full Precision
           On/OFF

       •   By leaving "Full Precision" turned off, ProUCL will display numerical values using an
           appropriate (default) decimal digit option; and by turning "Full Precision" off,  all decimal
           values will be rounded to the nearest thousandths place.

       •   The "Full Precision" on option is specifically useful when 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 (e.g., 0.00007332)
           after the decimal. In such situations, one may want to use the "Full Precision" on option to
           see nonzero values after the decimal.

Note: For the purpose of this User Guide, unless  noted  otherwise, all examples have used the "Full
Precision " OFF option. This option prints out results up to 3 significant digits after the decimal.
76

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2.5    Entering and Changing a Header Name
                                        Stats/Sample Size
                                       Configure Display
                                       Cut Ctrl+X
                                       Copy Ctrl + C
                                       Paste Ctrl+V
                                       Header Name
1.
The user can change variable names (Header Name) using the following process. Highlight the
column whose header name  (variable name) you want to change by clicking either the column
number or the header as shown below.
                                      Arsenic
                                                 1
                                          4.5!
                                          "5.6
                                          4.3
                                          5.4
       Right-click and then click Header Name.
0
1 2
I Header Name I

2 5.6
3
4
4.3
5.4



       Change the Header Name.
                             Header Name
                               Header Name:    [Arsenic Site 1
                                      OK
                                           Cancel
                                                                                         77

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4.      Click the OK button to get the following output with the changed variable name.
2.6     Saving Files
              File  Edit   Stats/Sample Sizes   Graphs   Statistical Tests   Upper Limits/BTVs   UCLs/EPCs
                  New
                  Open Single File Sheet
                  Open Excel File with Multiple Sheets
                  Close
                  Save
                  Save As...
                  Print
                  Print Preview
                  Exit
        The Save option allows the user to save the active window in Excel 2003 or Excel 2007.
        The Save As option also allows the user to  save the active window. This option follows typical
        Windows standards, and saves the  active window to  a file  in .xls  or .xlsx format.  All
        modified/edited data files, and output screens (excluding graphical displays) generated by the
        software can be saved as .xls or .xlsx files.
78

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2.7    Editing

Click on the Edit menu item to reveal the following drop-down options.
                                    Edit   Stats/Sample Sizes
                                        Configure Display

                                        Cut Ctrl+X
                                        Copy Ctrl+C
                                        Paste Ctrl+V
                                        Header Name
       •   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 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.

2.8    Handling Nondetect Observations and Generating Files with Nondetects

       •   Several modules of ProUCL (e.g., Statistical Tests,  Upper limits/BTVs, UCLs/EPCs) handle
           data sets containing ND observations with single and multiple DLs.

       •   The user informs the program about the status of a variable consisting of NDs. For a variable
           with ND 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. No qualifiers or flags (e.g., J, B, U, UJ, X) should be entered in
           data files with ND observations.

       •   Data for variables  with ND values  are provided in two columns. One  column  consists of
           numerical  values of detected  observations  and  numerical  values  of detection  limits  (or
           reporting limits) associated with observations reported as  NDs; and the  second column
           represents their detection status consisting  of only 0 (for ND 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
           ND values, and 1 for detected observations.

       •   For example, the header name,  D_Arsenic is used for the variable, Arsenic having ND
           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 represents a ND entry.
           If this format is not followed, the program will not recognize that the data set has NDs. An


                                                                                             79

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           example data set illustrating these points is given as follows. ProUCL does not distinguish
           between lowercase and uppercase letters.
B! D:\example.wst QOIXl

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
nil
0 1 2
Arsenic D_Arsenic Mercury
4.5 0 0.07
5.6 1 0.07
	 4.3] o| 0.11
5.4 TT~ 0.2
9.2 1 0.61
6.2 1 0.12
6.7 1 0.04
5.8 1 0.06
8.5 0.99
5.65 0.125
5.4 0.18
5.5 | 0.21
5.9 I 0.29
5.1 TT~ 0.44
5.2 1 0.12
4.5 0| 0.055
6.1 T| 0.055
6.1 1 0.21
6.8 Tl 0.67
si i 0.1
0.8
0.26
0.97
0.05
0.26
3
D_Mercury
1
1
0
0
1
1
1
1
f
T~
1
1
1
1
T~
1
1
T~
T"
1
1
1
4
Vanadium
16.4
1E.8
17.2
19.4
15.3
30.8
29.4
13.8
18.9
17.25
17.2
16.3
16.8
17.1
10.3
15.1
24.3
18
16.9
12

1
Tl
5
Zinc
89.3
90.7
95.5
113
266
80.9
80.4
89.2
182
80.4
91.9
G nrj
Group
Surface
Surface
Surface
Surface
Surface
Surface
Surface
Surface
Surface
Surface
Subsurfac
112 Subsurfac
172 Subsurfac
99 Subsurfac
90.7 Subsurfac
66.3 Subsurfac
175 Subsurface
185
184
68.4


Subsurface
Subsurface
Subsurface



- ^
2.9    Caution
       •   Care should be taken to avoid any misrepresentation of detected  and nondetected values.
           Specifically,  do not include  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
           ND, 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 (such as qualifiers) 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 ND value.

       •   When computing statistics for full uncensored data sets without any ND values,  the user
           should select only those  variables (from the list of available variables) that contain no ND
           observations. Specifically, ND 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 ND values.

       •   It is mandatory that the header name of a nondetect column associated with a variable such as
           XYZ should be D_XYZ (or d_Xyz). No other characters or blanks are allowed. However, the
           header (column) names are not case sensitive. If the nondetect column is not labeled properly,
           methods to handle nondetect data will not be activated and shown.
80

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       •   Two-Sample Hypotheses: When using two-sample hypotheses tests (WMW test, Gehan test,
           and T-W test) on data sets with NDs, both samples or variables  (e.g., site-As,  Back-As)
           should be specified as having NDs, even though one of the variables may not have any ND
           observations. This means that 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 NDs, still a column with label D_site-As  should be
           included in the data set with all entries = 1 (detected values).

       •   The sample data set given on the previous page illustrates points related to this option and
           issues listed above. The data set contains some ND measurements for arsenic and mercury. It
           should be noted that mercury concentrations are used to illustrate the points related to ND
           observations; 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 ND values) using "Full" data
           set option, then all ND 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

       •   To compute statistics of interest (e.g., background statistics, GOF test, UCLs, WMW test) for
           variables with ND values, one should choose the ND option, With NDs, from the available
           menu  options such as Stats/Sample  Sizes, Graphs, Statistical Tests, Upper Limits/BTVs,
           and UCLs/EPCs.

       •   The NDs option of these modules gets activated only when your data set contains NDs.

       •   For data sets with NDs, the Stats/Sample Sizes module  of ProUCL 5.0 computes summary
           statistics and other general statistics such as the KM mean and KM standard deviation based
           upon raw as well as log-transformed data.
•a File Edit
Navigation F
Name
Worksheet jds
Well 10jds
WMW-with NDs:xl
Stats/Sample Sizes Graphs Statistical Tests
General Statistics K
Imputed NDs using ROS Methods >
DQOs Based Sample Sizes >
3
2
3
4
—
5
7
—
8
17

Upper Limits/BTVs UCLs/EPCs Windows Help
Full (w/o NDs) >
With NDs >
0
1
D
0
0
1


5



6



7



           The General Statistics/With NDs option also provides simple statistics (e.g., % NDs, Max
           detect, Min detect, Mean) based upon detected values. The statistics computed in log-scale
           (e.g.,  sd of log-transformed detected values) may help  a user to determine the degree of
           skewness (e.g., mild,  moderate,  high)  of  a data set based  upon detected values. These
           statistics may also help the user to choose the most appropriate method (e.g., KM bootstrap-t
           UCL or KM percentile bootstrap UCL) to  compute UCLs, UPLs, and other limits  used to
           compute decision statistics.
                                                                                            81

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        •   All other parametric and nonparametric statistics and estimates of population mean, variance,
           percentiles (e.g., KM, and ROS estimates) for variables with ND observations are provided in
           other menu options such as Upper Limits/BTVs and UCLs/EPCs.

2.11   Warning Messages and Recommendations for Data Sets with an
        Insufficient Amount of Data

        •   ProUCL  provides warning messages and recommendations for data sets with an insufficient
           amount of data for calculating meaningful estimates and statistics of interest.  For example, it
           is not desirable to compute an estimate of the EPC term based upon a discrete (as opposed to
           composite or ISM)  data set of size less than 5, especially when NDs  are also present in the
           data set.

        •   However, to accommodate the computation  of UCLs  and other limits based  upon ISM data
           sets, ProUCL 5.0 allows users to compute UCLs, UPLs, and UTLs based upon data sets of
           sizes as small as 3. The user is advised to follow the guidance  provided in  the ITRC ISM
           Technical Regulatory Guidance Document (2012)  to select an appropriate UCL95 to estimate
           the EPC term.  Due to lower variability in ISM data, the minimum sample size requirements
           for  statistical  methods  used on ISM  data  are lower  than the minimum  sample size
           requirements for statistical methods used on discrete  data sets.

        •   It is suggested that for data sets composed of observations resulting from discrete  sampling,
           at least 10 observations should be collected to compute UCLs and various other limits.

        •   Some examples of data sets with insufficient  amount of data include data sets with less than 3
           distinct observations, data sets with only one  detected observation, and data sets consisting of
           all nondetects.

        •   Some of the warning messages generated by ProUCL 5.0 are shown as follows.

                                    UCL Statistics for Uncensored Full Data Sets
                  User Selected Options
               Date/Time of Computation  3/13/2013 9:28:43 PM
                        From Rle  Not-enough-data-setjds
                      Full Precision  OFF
                 Confidence Coefficient  95%
            Number of Bootstrap Operations 2000
                                            General Statistics
                           Total Number of Observations   2                    Number of Distinct Observations   2
                                                                  Number of Hissing Observations   D
                                      Minimum   2                                 Mean   4.5
                                      Maximum   7                                Median   4.5
                                   Warning: This data set only has 2 observations!
                         Data set is too small to compute reliable and meaningful statistics and estimates!
                                   The data set for variable x was not processed!


                     It is suggested to collect at least 8 to 10 observations before using these statistical methods!
                 If possible, compute and collect Data Quality Objectives (DQO) based sample size and analytical results.
82

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                                       UCL Statistics (or Data Sets with Nor, Detects
            User Selected Options
       Date/Time of Computation    3/13/2013 9:27:39 PM
                      From Rle    Not-enough-data-setjds
                                OFF
                                95%
                                2000
               Full Precision
       Confidence Coefficient
Number of Bootstrap Operations
                                                     General Statistics
                           Total Number of Observations
                                    Number of Detects
                             Number of Distinct Detects
                                      Minimum Detect
                                     Maximum Detect
                                     Variance Detects
                                       Mean Detects
                                      Median Detects
                                    Skewness Detects
                               Mean of Logged Detects

2
2
10
13
4.5


N



                                                    11.5
                                                    11.5
                                                   N/A
                                                    2/34
Number of Distinct Observations
       Number of Non-Detects
Number of Distinct Non-Detects
         Minimum Non-Detect
        Maximum Non-Detect
         Percent Non-Detects
                 SD Detects
                 CV Detects
             Kurtosis Detects
        SD of Logged Detects
                                       Warning: Data set has only 2 Detected Values.
                        This is not enough to compute meaningful or reliable statistics and estimates.
                                                                                                                 5
                                                                                                                 71.43%
                                                                                                                 2.121
                             0.184
                            N/A
                             0.186
                                             Normal GOF Test on Detects Only
                                           Not Enough Data to Perform GOF Tes
       User Selected Options
                     From Rle
                 Full Precision   OFF
         Confidence Coefficient   9K
                    Coverage
Different or Future K Observations
  Number of Bootstrap Operations
                            Background Statistics for Data Sets with Nor Detects

                            Not-enough-data-set_ajtls
                               95%
                               2000
                          Total Number of Observations
                        Number of Distinct Observations
                                   Number of Detects
                            Number of Distinct Detects
                                     Minimum Detect
                                     Maximum Detect
                                   Variance Detected
                                     Mean Detected
                        Mean of Detected Logged Data
                                                  General Statistics
                                                    7
                                                    6
                                                    0
                                                    0
                                                   N/A
                                                   N/A
                                                   N/A
                                                   N/A
                                                   N/A
                                                                                       Number of Missing Observations
       Number of Non-Detects
Number of Distinct Non-Detects
         Minimum Non-Detect
        Maximum Non-Detect
         Percent Non-Detects
               SD Detected
  SD of Detected Logged Data
                               13
                              100%
                               N/A
                               N/A
        Warning: All observations are Non-Detects (NDs). therefore all statistics and estimates should also be NDs!
      Specifically, sample mean. UCLs. UPU. and other statistics are also NDs lying below the largest detection limit!
The Project Team may decide to use alternative site specific values to estimate environmental parameters (e.g.. EPC. BTV).

                                      The data set for variable yy was not processed!
                                                                                                                                             83

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2.12  Handling Missing Values

       •   The modules (e.g., Stats, GOF, UCLs, BTVs, Regression, Trend tests) of ProUCL 5.0 can
           handle missing values within a data set. Appropriate messages are displayed when deemed
           necessary.

       •   All blanks, alphanumeric strings (except for group variables), or the specific large value Ie31
           are considered as missing values.

       •   A  group  variable  (representing  two  or more groups,  populations,  MWs) can  have
           alphanumeric values (e.g., MW01, MW02, AOC1, AOC2).

       •   ProUCL ignores  all missing values in all  statistical evaluations 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 minus the Number of Missing Values. If there are no missing values, then
           number of valid samples = total number of observations.

              Valid Samples = Total Number of Observations - Missing Values.

       •   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 5.1/ProUCL 5.0 will
           treat that blank value (or Ie31 value) as a new group. All variables and values  that correspond
           to this missing value will be treated as part of a new group and not with any existing groups.
           It  is therefore important to check the consistency and validity of all  data sets before
           performing statistical evaluations.

       •   ProUCL prints out the number of missing  values (if any) and the number of reported values
           (excluding the missing values)  associated with each  variable in  the data sheet.  This
           information  is provided  in several output  sheets (e.g.,  General  statistics, BTVs, UCLs,
           Outliers, OLS, Trend Tests) generated by ProUCL 5.1.

       •   Number of missing values in Regression: The  OLS module also handles  the number of
           missing values in the two columns (X  and Y) representing  independent (X) and dependent
           (Y) variables. ProUCL  provides warning messages for bad  data sets (e.g., all  identical
           values) when statistics of interest cannot be computed. However, a bad/extreme data set can
           occur in numerous different ways, and ProUCL may not cover all of those extreme/bad data
           sets. In such cases, ProUCL may still yield an error message. The user needs to review and
           fix his data set before performing regression or trend analysis again.

For further clarification of labeling missing values, the following example illustrates the terminology used
for the number of valid  samples and of unique and distinct  samples on output sheets generated by the
ProUCL software.

Example: The following example illustrates the notion of Valid  Samples, Unique or Distinct Samples,
and Missing Values. The data set  also has ND  values. ProUCL 5.0 computes these numbers and prints
them on the UCLs and background statistics output.
84

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               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
1
1
1
1
1
Valid  Samples: Represents  the  total number  of observations  (censored and uncensored  inclusive)
excluding the missing values. In this case the number of valid samples = 9. 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 treated as missing values.
Specifically, all alphanumeric values including blanks are considered to be missing values. Big numbers
such as 1.0e31  are also treated as missing values and are considered as not valid observations. In the
example above the number of missing values = 4.

Unique or Distinct Samples: The number of unique samples or number of distinct samples represents all
unique (or distinct) detected and nondetected values. This is computed separately for detects and NDs.
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 small. In the example above
total number of unique or distinct samples = 8, number of distinct detects = 6, and number of distinct NDs
(with different detection limits) = 2.
                                           G eneral S tatistks
                      Total Number of Observations   13


                             Number of Detects   10
                        Number of Distinct Detects   6
                               Minimum Detect   2
                              Maximum Detect   73
                              Variance Detects  555.5
                               Number of Distinct Observations   8
                               Number of Missing Observations   4
                                     Number of Non-Detects   3
                               Number of Distinct Non-Detects   2
                                       M inimum N on-D elect   0.5
                                      M aximum N on-D etect   1.2
                                       Percent N on-D elects   23.08%
                                                                                                 85

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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; 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 modifications as available for the selected menu
item. An illustration is given as follows.

2.13.1 Graphics Tool Bar
                                        for
                            : 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 not want to use these drop-down menu graphic tools.

2.13.2 Drop-Down Menu Graphics Tools

Graphs can be modified by using the options shown on the two graphs displayed below. 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.  For example, the user can change colors, title name,
axes labels, font size, and re-size the graphs. There is less 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.  A couple of examples of a drop-down menu obtained by right-clicking the mouse on the
background area of the graphics display are given as follows.
86

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   ! Histo_Group.gst
  at m
         Histograms for Arsenic, NROS_Arsenic
   O
2

0
                                                         Properties...
                                                     51  Statistical Studies
                                                              -  n x
mj File Edit Slau>'Sample Suet BIS Simulator Graphs Statistical Tens Upper Ltmits/BTVs UCLtlPCs Windows Help
                                       Box Plot for Na
Ben PW Ful.g*
BoซPto!FUl.a.9ซ
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                                                                            87

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                                         Chapter 3
                              Select Variables Screen
3.1    Select Variables Screen

       •   The Select Variable screen is associated with all modules of ProUCL.

       •   Variables need to be selected to perform statistical analyses.

       •   When the user clicks on a drop-down menu for a statistical procedure (e.g., UCLs/EPCs), the
           following window will appear.
                        Available Variables
Selected Variables
                         Name
                                   ID
Aluminum
Chromium
Iron
Manganese
Thallium
Vanadium
Benzo lajpynene   S
Naphthalene    5
Benzo[a)pyrene ... 10
                                                       Name
                                                       Arsenic
                                                       Lead
           ID
           1
           4
                                                     Select Group Column (Optional)
           The Options button is available in certain menus. The use of this option leads to another pop-
           up window such as shown below. This window provides the options associated with the
           selected statistical method (e.g., BTVs, OLS Regression).
                                              Enter BTV level

                                                     Confidence Level

                                                          Coverage     0.95

                                        Different or Future K Observations    1

                                         Number of Bootstrap Operations   2000
88

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                                   Display Intervals

                                      Confidence Level

                                          0.95
                                 0 Display Regression Table

                                 EH Display Diagnostics
                               Graphics Options

                                 0 Display XY Plot

                                 	XT Plot Title
                                   Classical Regression  |

                                 0 Display Confidence Interval

                                 0 Display Prediction Interval
ProUCL can process  multiple variables simultaneously.  ProUCL  software  can generate
graphs, and  compute  UCLs, and background statistics  simultaneously  for all  selected
variables shown in the right panel of the screen shot displayed on the previous page.

If the user wants to perform statistical analysis on a variable (e.g., manganese) by a Group
variable, click the arrow below the Select Group Column (Optional) to get a drop-down list
of available variables from which to select an appropriate group variable. For example, a
group variable (e.g., Well ID) can have alphanumeric values such as MW8, MW9, and MW1.
Thus in this example, the group variable name, Well ID, takes 3 values: MW1, MW8, and
MW9. The selected statistical method (e.g., GOF test) performs computations on data sets for
all the groups associated with the selected group variable (e.g., Well ID)
Available Variables Selected Variables












Name ID
Well ID 0
MW-ID 2
Manganese 3
MW-89 5
GW-Mn-89 6
MW9 B
MN9 9
MN-99 11
index 14

< >

ป

ซ







Name ID
Mn-GW 1







< >










Select Group Column (Optional)
I Options |
OK Cmcd |

                                                                                    89

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               The Group variable is 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
                      Available Variables
Selected Variables
Name
Areenic
Iron
Lead
Thallium
Vanadium
Benzoiajpyrene
Naphthalene
Benzo[a)pyTene ..
ID
1
3
4
6
7
8
9
10
                                                        Name
                                                        Aluminum
                                                        Chromium
                                                        Manganese
            ID
            0
            2
            5
                                                      Select Group Column (Optional)
                                             Options
                                                         OK
                                                                    Cancel
           •   As mentioned earlier, one should not 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 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.

           •   The Group Option is a useful tool for performing statistical tests and methods (including
               graphical displays) separately for each of the group (samples from different populations)
               that may be present in a data set. For example, the same data set may consist of samples
               from multiple 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.

           Notes: Once again, care should be taken to avoid misrepresentation and  improper use  of
           group variables. Do not assign any form of a missing value for the group variable.

3.1.1   Graphs by Groups

The following options are available to generate graphs by groups.
90

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                   Graphs    Statistical Tests    U|
                        Box Plot

                        Multiple Box Plots

                        Histogram

                        Q-Q Plots

                        Multiple Q-Q Plots
Individual or multiple graphs (Q-Q plots, box plots, and histograms) can be displayed on
a graph by selecting the Group Column (Optional) option shown as follows.





Available Variables
Name ID
CuLT=1 1
Zn 2
ZnLT=1 3
Zone 4
Zone-Basin 5
Zn-Basin 10
Zone-Allu 13
Zn-Allu 14
nip-Zn-allu-7M) 17

<• >


ป
ซ
i
| Options |

Selected Variables
Name ID
Cu 0
Select Group Column (Option,
Zone (Count = 118) v
OK | CaiceJ |



J)


An individual graph for each group (specified by the selected group variable) is produced
by selecting the Individual  Graph  option; and 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 as shown below.  Using the Group Graph option, multiple graphs are
displayed for all sub-groups included in the Group variable. This option is used when
data are given in the same column and are classified by a group variable.
                                                                              91

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                               Graphs by Groups
                                O Individual Graphs

                                             Label

                                1   a   \2
                                2   a   \2
                                3   a   \2
                                4   a
• Group Graphs


        Value
           •   Multiple graphs for selected variables are produced by selecting options: Multiple Box
               Plots or Multiple Q-Q Plots. Using the Group  Graph option, multiple graphs for all
               selected variables are shown on the  same graphical display. This option is useful when
               data (e.g., site lead and background lead) to be compared are given in different columns.

Notes: It should be noted that it is the users' responsibility to provide an adequate amount of detected data
to perform the group operations. For 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, and sd) in the data set. Similarly, if graphs are desired for each group specified
by a Group ID variable, there should be at least two detected observations in each group specified by the
Group ID  variable.  ProUCL displays a warning message (in orange)  in the  lower Log Panel of the
ProUCL screen when not enough data are available to perform a statistical or graphical operation.
92

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                                         Chapter 4


                                   General Statistics


The General Statistics option is available under the Stats/Sample Sizes module of ProUCL 5.0. This
option is used to compute general  statistics including simple summary  statistics (e.g., mean, standard
deviation) for all selected variables. In addition to simple summary statistics, several other statistics are
computed for full uncensored data  sets (Full  w/o NDs), and for data sets with nondetect (with NDs)
observations (e.g., estimates based upon the KM method). Two Menu options: Full w/o NDs and With
NDs are available.

           •   Full (w/o NDs): This option computes general statistics for all selected variables.

           •   With NDs: This option computes general statistics including the KM method based mean
               and standard deviations for all selected variables with ND  observations.

Each menu option (Full (w/o NDs) and With NDs) has two sub-menu options:

           •   Raw Statistics

           •   Log-Transformed

When  computing general statistics  for raw data,  a message  will be displayed for each variable that
contains  non-numeric  values. The  General Statistics  option  computes  log-transformed (natural log)
statistics  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 a selected variable.

4.1    General Statistics for Full Data Sets without NDs

1.      Click General Statistics ^  Full (w/o NDs)
        i;J  File  Edit  Stats/Sample Sizes  Graphs  Statistical Tests  Upper Limrts/BTVs  UCLs/EPCs Windows  Help
           Navigation F
                     General Statistics
                                              Full (w/o NDs)
                                                              Raw Statistics
                                                                                         10
         Name

         Work Sheet jds
         Well 1(bds
Imputed NDs using ROS Methods

DQOs Based Sample Sizes
                                              With NDs
                                                              Log-Transformed
2.      Select either Log-Transformed or Raw Statistics option.

3.      The Select Variables screen (see Chapter 3) will appear.

           •   Select one or more variables from the Select Variables screen.

           •   If statistics are to be computed by a Group variable, then select a group variable  by
               clicking the arrow below the  Select Group Column (Optional) button. This will result
               in drop-down list of available  variables, and select a proper group variable.
                                                                                              93

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                                                          Select Variables
                              Available Variables
                                   Selected Variables
                                                                            Select Group Column (Optional)
                                                                             sp-Jength  (Count = 150}
                                                                             sp-width  (Count = 150}
                                                                             pit -length  (Count = 150}
                                                                             pt-width  (Count = 150}
               •     Click  on the  OK  button to continue or on  the Cancel  button to cancel the  General
                     Statistics option.
              User Selected Options
                        From File  FULLIRIS-ndsjds
                     Full Precision  OFF
                                                         Raw Statistics
       From Rle: FULLIRIS-ndsjds
                                                   i Statistics for Uncensored Data Sets
                        NumObs  # Missing
               sp-length (1)   50
               sp-length (2)   50
               sp-length (3)   50
4.3
4.9
4.9
5.8
7
7.9
Mean
5.006
5.936
6.588
 SD
0.352
0.516
0.636
 SEM
0.0498
0.073
O.OB99
                                                                                          MAD/0.675  Skewness   Kurtosis
                                                                                                                         CV
                                                Percentiles for Uncensored Data Sets
            Variable      NumObs # Missing   10%ile
               sp-length (1)   50        0        4.59
               sp-length (2)   50        0        5.38
               sp-length (3)  ~50        0        5lT
        4.7
        5.5
                                                   6.1
       25'jle(Q1)   50''jle(Q2)   75%ile(Q3)
        4.8         5          5.2
        5.6         5.9         6.3
        6.225       6.5         6.9
0.297
0.519
0.593
                                 SOMIe
                                 5.32
                                 6.4
                                 7.2
0.12
0.105
0.118
-0.253     0.0704
-0.533     0.087
0.0329    0.0965
                     5.41
                     6.7
                     7.61
                                           5.61      5.751
                                           6.755     6.951
                                           77      7.802
94

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                                         Log-Transformed Statistics
                 User Selected Options
                          From File  FULLIRIS-ndsjds
                        Full Precision  OFF
           From Rle: FULLIRIS-ndsjds
                                        Summary Statistics for Uncensored Log-Transformed Data Sets
               Variable     NumObs ft Missing Minimum Maximum   Mean    Variance     SD
                  sp-tengthfl)   50      0       1.459    1.758    1.608     0.00437     0.0705
                  sp4ength{2!   50      0       1.583    1.346    1.777     0.00761     0.0872
                  sp-length{3)   50      0       1.589    2.067    1.881     0.00343     0.0371
                                                                         MAD/0.675 Skewness Kiirtosis    CV
                                                                          0.0605     -0.0553   -0.231     0.0438
                                                                          0.0873     -0.0852   -0.463     0.0431
                                                                          0.0885     -0.136     0.432    0.0516
                                       PercentiEes for Uncensored Log-Transformed Data Sets

               Variable     NumObs ปMissing  KTSIe   20'
ed NDs using ROS Methods >
Based Sample Sizes >
J • *' 4
3 5 8
4 7 17
Full (w/o NDs) t
1 With NDs t
J „
567
Raw Statistics
Log-Transformed
1 0
0 1
8 3 10 11
2.
Select either Log-Transformed or Raw Statistics option.
                                                                                                                95

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3.       The Select Variables screen (Chapter 3) will appear.


         •   Select variable(s) from the list of variables.


         •   Only those variables that have ND values will be shown.  The user should make sure that the
             variables with NDs are defined properly including the column showing the detection status of
             the various observations.


         •   If statistics are to be computed by a Group variable,  then select a group variable by clicking
             the arrow  below the Select Group Column (Optional) button. This will result in a  drop-
             down list of available variables. Select a proper group variable.


         •   Click on the OK button to continue or on the Cancel button to cancel the summary statistics
             operations.


                                    Raw Statistics - Data Set with NDs

             User Selected Options
                      From File   Zn-alluvial-fan-datajds
                    Full Precision   OFF

        From Rle: Zn-alluvial-fan-datajds

                           Summary Statistics for Censored Data Set (with NDs) using Kaplan  Meier Method

            Variable     NumObs  ซ Missing  Num Ds NumNDs    f. NDs     Min ND    Max ND   KM Mean  KM Var   KM SD   KM CV
             Cu {alluvial fan)  65      3     48      17    26.155,        1       20         3.608   13.08     3.616    1.002
            Cu (basin trough)  49      1     35      14    28.57,1        1       15         4.362   21.64     4.651    1.066

                                Summary Statistics for Raw Data Sets using Detected Data Only

            Variable     NumObs  fl Missing  Minimum Maximum    Mean     Median      Var      SD   MAD/0 675 Skewness   CV
             Cu {alluvial fan)  4S      3      1       20      4.146       2       16.04       4.005    1483     2.256    0.966
            Cu (basintraugh)  35      1      1       23      5.229       3       27.18       5.214    2.965     1.878    0.997

                                   Percentiles using all Detects (Ds) and Non Detects (NDs)

            Variable     NumObs  S Missing  lOSIe   20%ile   25%ile(Qt)  50Sle{Q2)  75%ile(Q3)   SOSIe   90Xile   95%ile   99Xile
             Cu {alluvial fan)  65      3      1       2       2         3       5          7     10       15.2     20
            Cu (basin trough)  43      1      1       2       2         48          9.4    12.4      15      20.12


             The Summary Statistics screen shown above can be  saved as an Excel 2003 or 2007 file.
             Click Save from the file menu.
96

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                                        Chapter 5

                 Imputing Nondetects Using ROS Methods
The imputing of NDs using regression on order statistics (ROS) methods option is available under the
Stats/Sample Sizes module of ProUCL 5.1. This option is provided for advanced users who want to use
the detected and imputed NDs data for exploratory and data mining purposes on multivariate data sets.
For exploratory methods, such as principal component analysis (PCA), cluster, and discriminant analysis
to gain additional insight into potential structures and patterns present in a multivariate (more than one
variable) data set, one may want to use imputed values in graphical displays (line graphs,  scatter plots,
boxplots etc.)  and in the analyses.  To derive conclusions based upon  multivariate data sets  with
nondetects, the developers  suggest the use of the KM method based covariance or correlation matrix to
perform PCA and regression  analysis. These methods  are beyond the scope of the  ProUCL software
which deals only with univariate methods. The details of computing an Orthogonalized Kettenring and
Gnanadesikan (OKG) positive definite KM matrix can be found in Maronna, Martin, and Yohai (2006)
and in Scout 2008 Version 1.0 guidance documents (2009) which can be downloaded from the EPA Site
(http://archive.epa.gov/esd/archive-scout/web/html/). One may not use ROS imputed data to  perform
parametric  statistical  tests  such  as t-test  and ANOVA  test without further investigation. These issues
require  further  research to evaluate  decision errors associated with conclusions  derived using  such
methods.

The ROS methods can be used to impute  ND observations using a  normal, lognormal, or gamma model.
ProUCL has three ROS estimation methods that can be used to impute ND observations. The use of this
option generates additional columns consisting of all imputed NDs and detected  observations. These
columns are appended to the existing open spreadsheet file. The user should save the updated file if they
want to  use the imputed data for their other application(s) such as PCA or discriminant analysis. It is not
easy to perform multivariate statistical methods on data  sets with NDs. The availability of imputed NDs
in a data file helps the  advanced users  who want to use exploratory methods on data sets with ND
observations. Like other statistical methods in ProUCL, NDs can  also be imputed by a group variable.
One can impute NDs using the following steps.
1.
Click Imputed NDs using ROS Methods ^ Lognormal ROS
                                                                      rrOULL D.U - IWMW-WIm INUS.XI5J
   File  Edit
           Stats/Sample Size;  Graphs   Statistical Tests  Upper Limits/BTVs   UCLs/EPCs  Windows  Help
Navigation F
Name
Worksheet ads
Well "IQjds
General Statistics >
Imputed NDs using ROS Methods t
DQOs Based Sample Sizes t
2
IWMWvth NDsjds I

3
4
Normal ROS
Gamma ROS
Loqnormal ROS
8 L 	 : 	

5




6




7

S

9

10


11




2.
The Select Variables screen (Chapter 3) will appear.

   •   Select one or more variable(s) from the Select Variables screen; NDs can be imputed
       using a group variable as shown in the  following screen shot.
                                                                                           97

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                                           Select Variables

                            Available Variables            Selected Variables
                            Name
                            Cu
ID
0
Name
Zn
ID
1
                                                      Select Group Column (Optional)
           Click on the OK button to continue or on the Cancel button to cancel the option.

                    Output Screen for ROS Est. NDs (Lognormal ROS) Option
0
Cu
1
1
3
3
5
1
4~
4
2
2
1
2
5
11
1
2
2
2
2
20
2
2
3
3
2D~
TO"
7
5
1
Zn
10
9

5
18
10
12
10
11
11
19
8
3
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
20
2
Zone
Alluvial Fan
Alluvial Fan
Alluvial Fan
.Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
.Alluvial Fan
Alluvial Fan
.Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
.Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
.Alluvial Fan
Alluvial Fan
Alluvial Fan
Alluvial Fan
3
D_Qj
0
0
1
1
1
r
1
1
1
1
1
1
0
1
0
1
1
1
1
0
1
1
1
1
0
B~
1
1
4 5
D_Zh Ln ROS_Zn (alluvial fan)
0 2.12437734466611
1 I 3
1.0DOOQOE-rf>31
1 5
1 18
0 2.7045642735474
f| 12
1 10
1 11
1 11
1 19
1 I 8
0 2.49463676836713
0 3.1603475071042
0 3.55392730586941
1 10
1 10
1 10
1 10
0 3.92469067412296
1 10
0 4.26969100939485
1 10
0 4.60094330444612
1 10
0 4.9229S559179133
f ~ 10"
1 10
1 20
6
LnROS_Zn [basin trough)
20
10
60
20
12
fT
3.48713118440742
14
4.98477186220711
17
1.87132713438924
11
5
12
4
3
6
3
15
13
4
20
8T
w
60
40
30"
40
17
Notes: For grouped data, ProUCL generates a separate column for each group in the data set as shown in
the above table. Columns with a similar naming convention are generated for each selected variable and
distribution using the ROS option.
98

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                                     Chapter 6

                        Graphical Methods (Graph)

The graphical  methods described here  are used as exploratory tools to get some idea about data
distributions (e.g., skewed, symmetric), potential outliers and/or multiple populations present in a data
set. The following graphical methods are available under the Graphs option of ProUCL 5.1
                                      Box Plot           >

                                      Multiple Box Plots    >

                                      Histogram         ป

                                      Q-Q Plots          ^

                                      Multiple Q-Q Plots   >
       All graphical displays listed above can be generated using uncensored full data sets (Full w/o
       NDs) as well as left-censored data sets with nondetect (With NDs) observations. On box plot
       graphs for data  sets with NDs,  a horizontal line is also  displayed at the highest RL/DL
       associated with ND observations.

       0-0 Plots and Histograms: Q-Q plots and histograms can be generated individually as well as
       by using a Group variable. Graphs generated using the Group Graphs option shown below is
       useful when data for selected variable(s) are given in the same  column  (stacked data)
       categorized by a Group ID.
                                     Options_QCLPIot_NoN Ds
                            Graphs by Groups

                                Individual Graphs     • Group Graphs
                                          OK
Cancel
       For data sets with NDs, three options described below are available to draw Q-Q plots and
       histograms.  Specifically, these graphs are displayed only for detected values, or with NDs
       replaced by  1/2DL values,  or with NDs  replaced by the  respective  DLs. The  statistics
       displayed on a Q-Q plot (mean, sd, slope, intercept) are computed according to the selected
       method. On  Q-Q plots, ND values are displayed using a different symbol. The exploratory Q-
       Q plots described here do not require any placeholders for NDs. These graphs are used only
       to determine the distribution of detected values and to identify potential outliers  and/or
       multiple populations present  in a data set. On histograms, the user can change the number of
       bins (more bins, less bins) used to generate histograms.
                                                                                        99

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                                       Graphs by Groups

                                        O Individual Graphs
                                                       • Group Graphs
                                       Select How to Handle Nondelect Values

                                        >:•:< Use Reported Detection Limit

                                        O Use Detection Unit Divided by 2.0

                                        O Do not Display Nondetects
           Do not Display Nondetects: Selection of this option excludes all NDs from a graphical
           method (Q-Q plots and histograms) and plots only detected values. The statistics shown on
           Q-Q plots are computed only using the detected data.

           Use Reported  Detection Limit: Selection of this option treats DLs/RLs as detected values
           associated  with the ND values. The  graphs  are  generated using  the  numerical values of
           detection limits and statistics displayed on Q-Q plots are computed accordingly.

           Use Detection Limit Divided by 2.0: Selection of this option replaces the DLs with their half
           values.  All Q-Q plots  and histograms  are generated  using  the half  detection limits  and
           detected values. The statistics displayed on Q-Q plots are computed accordingly.

           For data sets in different columns,  one can use the Multiple  Q-Q  Plots option. By default,
           this option will display multiple Q-Q plots for all selected variables on  the same graph. One
           can also generate multiple Q-Q plots by using a group variable.

           Box Plot: Like  Q-Q plots, box plots can also be generated by a Group variable. This option is
           useful when all data are listed in the same column (stacked data) categorized by a Group ID
           variable. On box plots with NDs, a horizontal line is  displayed at the highest detection limit
           level. ProUCL  5.1 constructs a box plot using all detected and nondetected (using associated
           DL values) values. A horizontal line is displayed at the highest detection limit. Box Plots are
           generated using ChartFx, a software used in the development of ProUCL 5.1.

           Multiple Box Plots: For data in different columns, one can use  the Multiple Box Plots option
           to display  multiple  box plots for all  selected variables  on the  same graph. One can also
           generate multiple box plots by using a group variable.

           Box Plots have an optional feature, which can be used to draw up to four (4) horizontal lines
           at pre-established screening levels or at statistical limits (e.g., upper limits) computed using a
           background data set. This option can be used when box plots are generated using onsite data
           and one may be interested in comparing onsite data with background threshold values and/or
           pre-established screening levels. This type of box plot represents a visual comparison of site
           data with background threshold values and/or other action levels. Up to four (4) values can be
           displayed on a  box plot as shown below. If the user inputs a value in the value column, the
           check box in that row will get activated. For example,  the user  may want to display horizontal
           lines  at a  background  UTL95-95 or  some  pre-established  action  level(s) on box  plots
           generated using AOCs data.
100

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6.1     Box Plot

A brief description of the method used to generate Tukey's box plot (also known as box and whisker plot)
is described first.

Box Plot (Box and Whiskers  Plot):   A  box plot (box and whiskers plot) represents a  convenient
exploratory tool and provides a quick five-point summary of a data set. In statistical literature, one can
find several ways to generate box plots. The practitioners may  have their own preferences  to use one
method over the other. Box plots are well documented in the statistical literature and a description of box
plots can be easily obtained by  surfing the net. Therefore, a detailed description about the generation of
box plots is not provided in ProUCL guidance documents.

All box plot methods including the one in ProUCL represent five-point summary graphs including:  the
lowest and the highest  data values, median (50th percentile=second quartile, Q2), 25th percentile (lower
quartile, Ql), and 75th percentile (upper quartile, Q3). A box and whisker plot also provides information
about the degree of dispersion (interquartile range (IQR) = Q3-Ql=length/height of the box in a box plot),
the degree of skewness (suggested by the length of the whiskers) and unusual data values  known  as
outliers. Specifically, ProUCL (and various other software packages) use the following to generate a box
and whisker plot.

    •   Q1= 25th percentile, Q2= 50th (median), and Q3  = 75th percentile
    •   Interquartile range= IQR = Q3-Q1 (the height of the box in a box plot)
    •   Lower whisker starts at Q1 and the upper whisker starts at Q3.
    •   Lower whisker extends up to the lowest observation or (Ql - 1.5 * IQR) whichever is higher
    •   Upper whisker extends up to the highest observation or (Q3 + 1.5* IQR) whichever is  lower
    •   Horizontal bars (also known as fences) are drawn at the end of whiskers
    •   Observations lying outside the fences (above the upper bar and below the lower bar) represent
        potential outliers

ProUCL uses a couple of development tools such as FarPoint spread (for Excel type input  and output
operations) and ChartFx (for graphical displays). ProUCL generates box plots using the built-in box plot
feature in ChartFx. The programmer has no control over computing the statistics (e.g., Ql, Q2, Q3, IQR)
using ChartFx. Boxplots generated by  ProUCL can slightly differ from box plots generated by other
programs (e.g., Excel). However, for all practical and exploratory purposes, box  plots in ProUCL  are
equally good (if not better) than those available in  the various commercial software packages  for
exploring data distribution (skewed or symmetric), identifying outliers,  and comparing multiple  groups
                                                                                            101

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(main objectives of box plots in ProUCL). More details about Box Plots can be found in Section 1.16 of
Chapter 1 of this document.
1.
Click Graphs >• Box Plot
Graphs
Statistical Tests Upper Limits/BTVs UCLs/E
Box Plot >
Multiple Box Plots >
Full (w/o NDs)
With NDs
                                Histogram         I

                                Q-Q Plots
                                Multiple Q-Q Plots
                                                          cc
2.      The Select Variables screen (Chapter 3) will appear.

           •   Select one or more variable(s) from the Select Variables screen.

           •   If graphs are  to  be produced by using a Group variable, select a group variable by
               clicking the arrow below the Select Group Column (Optional) button. This will result
               in a drop-down list of available variables. The user should select an appropriate variable
               representing a group variable as shown below.
           The default option for Graph by Groups is Group Graphs. This option produces side-by-
           side box plots for all groups included in the selected Group ID Column (e.g., Zone here). The
           Group Graphs option is used when multiple graphs categorized by a group variable need to
           be produced on the same graph. The Individual Graphs option generates individual graphs
           for each selected variable or one box plot for each group for the variable categorized by a
           Group ID column (variable).
102

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                                     Graphs by Groups

                                     O Individual Graphs
                                                     * Group Graphs
                                                 Label
                                                            Value
                                      1  0   Screening Level=


                                      2  D   C

                                      3  D
                                     4  D
              While generating box plots, one can display horizontal lines at specified screening levels
              or a BTV estimate (e.g., UTL95-95) computed using a background data set. For data sets
              with NDs, a horizontal  line is also displayed at the largest reported DL associated with a
              ND value. The use of this option may provide information about the analytical methods
              used to analyze field samples.

              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 (Group Graph)
                    Selected options:  Label (Screening Level), Value (12)
                                           Box Plot for Cu
          Meซ Non-Deled Value 20
Making Changes in Graphs using Toolbar

One can use the toolbar to make changes in a graph generated by ProUCL. The toolbar can be activated
by right clicking the mouse on your graph. The context menu on the box plot shown below appears. By
using the context menu, one can change color, title, font size, legend box and label points. For example,
one can edit the title by clicking title in the context menu. These are typical windows operations which
                                                                                          103

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can also be used in ProUCL. However, it should be pointed out that options which affect the computation
of statistics displayed on a graph are  not  wired and can yield incorrect results.  For example,  changing
scales along the  x-axis  or y-axis (e.g.,  to log scale) will not automatically displayed statistics  in the
changed log- scale. The resulting graph using options that affect computed statistics will be incorrect and
should be avoided. These operations are illustrated by several screen captures displayed as follows.
 f R.e Edit ฃ:Ete/ฃ;noU SEES  B!s SimUater  S-iph: Statist tal Tefc  Jpper Limils/B~Vj UOs/EPCs Wrdow;  Help
                                                   Box Plot for Aluminum
Activating the toolbar shown above.
  Fie Edit S:it,'ฃซnolซ Sizat  EliSimUirttr G-iphs Statical lets  JppaLrrvk'rV; UZU'EPCs Wrdow:  Help
                                                   Box Plot for Aluminum
Changing color of the graph shown in the above graph.
104

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5 ProUCL5.1-[BoxFb::JU3;tl

•y Fit  Ecft  S-^te'EsrvoItSizs  BISSiiriLlatcr  G-ipht  Statistcal TKtE  JppsrlimiteTVs  UCU'EPCs  Wrdow:   Help
                             •#- '
                                                        *
a   x
  _ s >
Changing title of the graph.
r1 ProUCLS,!-[BQxFh;:j|.?a]

-} fit  Edit  STst.'Scnal* 5cEt  BtiSimi,|jtcr  G-aphE  SUtirt cal TK*E  JpperUmitr:..'rV;  UCU'EPC;  Wrdow;  Help
               ^ •'  -
                             >-  "    •     ^ li?lKjr
                                                                              Box Plot for Aluminum
            Edit the title of the  graph.
                                                                                                                                                       105

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6.2     Histogram
1.
Click Graphs ^- Histogram
                                                               )UCL 5.0 - [Zn-Cu-ND-data.xls]
Graphs | Statistical Tests Upper Limits/BTVs UCLs/EPCs Windows Help
Box Plot >
Multiple Box Plots t
Histogram ^
Q-Q Plots ^
Multiple Q-Q Plots ^

56789



Full (w/o NDs)
WithNDs
1







10



11



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 Select Group Column (Optional) button. This will result
               in a drop-down list of available variables. The user should select an appropriate variable
               representing a group variable as shown below.

           •   When the option button is clicked for data sets with NDs, the following window will be
               shown.  By default, histograms are generating using the RLs for NDs.
                                       Graphs by Groups

                                        O Individual Graphs
                                                 * Group Graphs
                                       Select How to Handle Nondetect Values

                                        • Use Reported Detection Unit

                                        O Use Detection Unit Divided by 2.0

                                        O Do not Display Nondetects
               The default selection for histograms (and for all other graphs) by a group variable is
               Group  Graphs. This option  produces multiple histograms  on the same graph.  If
               histograms are needed to be  displayed individually, the user should check the radio
               button next to Individual Graphs.

               Click on the OK button to continue or on the Cancel button to cancel the histogram (or
               other selected graphical) option.
106

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                                     Histogram Output Screen
                                 Selected options: Group Graphs
                                    Histogram for Cu
                              Reported values used for nondetects
                                                                            alluvial fan
                                                                             Number of D
                                                                             N umbei of
                                                                             Number of
                                                                             Mean of D
                                                                                       f Detects
                                                                                       f Nondetects
                                                                                    d Less Bins
                                                                                    D More Bins
Notes: ProUCL does not perform any GOF tests when generating histograms. Histograms are generated
using the development software ChartFx and not many options are available to alter the histograms. The
labeling along the x-axis is done by the development software and it is less than perfect. However, if one
hovers the mouse on a bar, relevant statistics (e.g., begin point, midpint and end point) about the bar will
appear on  the screen.  The  Histogram option  automatically generates a normal probability  density
function (pdf) curve irrespective of the data distribution.  At this time, ProUCL 5.1 does not display a pdf
curve for any other distribution  (e.g., gamma) on  a historgram. The  user  can  increase or decrease the
number of bins to be used in a histogram.
6.3     Q-Q Plots
1.
Click Graphs
be shown.
                         Q-Q Plots. When that option button is clicked, the following window will
                             Graphs  Statistical Tests   Upper Limits/BTVs   UCLs/E
                                Box Plot
                                Multiple Box Plots
                                Histogram
                                Q-Q Plots
                                Multiple Q-Q Plots
                                                        Full (w/o NDsJ
                                                With NDs
        Q-Q Plots can be generated for data sets With NDs and without NDs [Full (w/o NDs)].

        •    Select either Full (w/o NDs) or With NDs option.

        •    The Select Variables screen (Chapter 3) will appear.
                                                                                               107

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            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 Select Group Column (Optional) 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 as shown below.

            Click on the OK button to continue or on the Cancel button to cancel the selected Q-Q plots
            option. The  following options screen  appears providing choices on how to treat NDs.  The
            default option  is to use the reported values for all NDs.
                                                          .QQ_Plot_w
                                            Graphs by Groups

                                             ;  Individual Graphs
                                                             • Group Graphs
                                            Select How to Handle Nondetect Values

                                             • Use Reported Detection Limit

                                             O Use Detection Unit Divided by 2.0

                                             O Do not Display Nondetects
            Click on the OK button to continue or on the Cancel button to cancel the selected Q-Q plots
            option. The following  Q-Q plot appears when used on  the  copper concentrations of two
            zones: Alluvial Fan and Basin Trough.

                                 Output Screen for Q-Q plots (With NDs)
                             Selected options: Group Graph, No Best Fit Line
                                          Q-Q Plot for Cu
                                   Reported values used for nondetects
                                      Theoretical Quantiles (Standard Normal)
                                        NDs Displayed in smaller font
             alluvial far, • tajn ll.augh
alluvial fen
 Total Numbs ol Dal
 Mumbet of Ncn-Oets
                                                                                      Slope [displayed data] = 4.045
                                                                                      Intercept (displayed dataM 815
                                                                                      Co-relation, H = 1359
                                                                                      Detected Sd = 5.214
                                                                                      Slopaldisp^ed data] -4.534
                                                                                      Intelcapt Idiplayed dataN 5 49
                                                                                      Cwtdation. Fl =0.903
        Note: The font size of ND values is smaller than that of the detected values.
108

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6.4    Multiple Q-Q Plots

6.4.1  Multiple Q-Q plots (Uncensored data sets)

1.     Click Graphs >• Multiple Q-Q Plots
2.
Multiple Q-Q Plots can be generated for data sets With NDs and without NDs [Full (w/o
NDs)].

•   When that Option button is clicked, the following window will be shown.
Graphs Statistical Tests Upper Limits/BTVs UCLs/EPCs
Box Plot t
Multiple Box Plots t
Histogram t
Q-Q Plots >
Multiple Q-Q Plots >
5 3.6 1.4
567
i d-spjengt d_sp*idth djjHe
.2 1 1
.2 1 1
Full (w/o NDs)
With NDs

\
\
       •   Select either Full (w/o NDs) or With NDs.

       •   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 Select Group Column (Optional) 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 as  shown below.

       •   Click OK to continue or Cancel button to cancel the selected Multiple Q-Q Plots option.

Example 6-1: The following graph is generated by using Fisher's (1936) data set for 3 Iris species.

                        Output Screen for Multiple Q-Q Plots (Full w/o NDs)
                           Selected Options:  Group Graph, Best Fit Line
                                         Normal Q-Q Plot
                                      Theoretical Quantises [Standard Normal]
                                                                                           109

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If the user does not want the regression lines shown above, click on the Best Fit Line and all regression
lines will disappear as shown below.
                                          Normal Q-Q Plot
                                                                                    N-EO
                                                                                    Mean * E 006
                                                                                    Sd-0952
                                                                                    Elope - 0 356
                                                                                    Intercept * 5.008
                                                                                    Carelation. R - 0 93!
                                                                                    N.50
                                                                                    Mean - 5 336
                                                                                    Sd-0.518
                                                                                    Stow-0522
                                                                                    Intercept - 5.J36
                                                                                    Correlation, R - 0.33

                                                                                   .p-lengthfj)
                                                                                    Correlation. R = 0.9E

                                                                                   sp-widthti)
                                                                                    N = EO

                                                                                    Sd = 0 379
                                                                                    Slope - 0.379
                                                                                    Intercept = 3.428
                                      .
                                      Theoretical Quantiles (Standard Normal)
                lhri) • sp-lenslh[2] • tp-lengt)i(3j 9 sp™dtli(1) O sp-wUlh[2] Q sp-widthฎ
Notes: For Q-Q plots and Multiple Q-Q plots 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=l). Quantiles for other distributions (e.g., Gamma
distribution) are used when using the Goodness-of-Fit (GOF, G.O.F.) test option.
6.5    Multiple Box Plots

6.5.1  Multiple Box plots (Uncensored data sets)

^.      Click Graphs >  Multiple Box Plots

2.      Multiple Q-Q Plots can be generated for data sets With NDs and without NDs [Full (w/o
        NDs)].
        •   When the option button is clicked, the following window will be shown.
Graphs
Box
Statistical Tests Upper Limits/BTVs UCLs/EPCs
Plot >
Multiple Box Plots >
Histogram t
Q-Q Plots >
Multiple Q-Q Plots >

5
6
7
Full (w/o NDs) krt-te
With NDs
.2
1

1

            Select either Full (w/o NDs) or With NDs.

            The Select Variables screen (Chapter 3) will appear.

            Select one or more variable(s) from the Select Variables screen.
110

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    •   If graphs have to be produced by using a Group variable, then select a group variable by
       clicking the arrow below the Select Group Column (Optional) 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 as shown below.

    •   Click on the OK button to continue or on the Cancel button to cancel the selected Multiple
       Box Plots options. The following graph is generated by using the above options.

Example 6-1 (continued): The following graph is generated by using the above options on Fisher's
(1936) Iris data set collected from 3 species of Iris flower.

                     Output Screen for Multiple Box Plots (Full w/o NDs)
                               Selected options: Group Graph
                                       Multiple Box Plots
                                                                                        111

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                                         Chapter 7


                                Classical Outlier Tests


Outliers are inevitable in data sets originating from environmental applications. In addition to informal
graphical displays (e.g., Q-Q plots and box plots) and classical outlier tests (Dixon test, Rosner test), there
exist several robust outlier identification methods (e.g., Biweight, Huber, PROP, MCD) for identifying
any number of multiple  outliers present in data  sets of various sizes (Scout 2008; EPA 2009). It is well
known that the classical outlier tests:  Dixon test and Rosner test suffer from masking (e.g.,  extreme
outliers may mask intermediate outliers)  effects. The use  of robust outlier  identification procedures is
recommended for identifying multiple outliers,  especially when dealing with  multivariate  (having
multiple constituents) data sets. However, those  preferred and more efficient robust outlier identification
methods are beyond the scope of ProUCL 5.1. Several robust outlier identification methods (e.g., based
upon Biweight, Huber, and PROP influence functions, Singh and Nocerino 1995) are  available in the
Scout 2008 vl.O software package (EPA, 2009).

The two classical outlier tests: Dixon and Rosner tests (EPA 2006a; Gilbert 1987) are  available in
ProUCL 4.0 and higher versions of the ProUCL software. These tests can be used on data sets with and
without ND observations. These tests require the assumption of normality  of the data set without the
outliers; as data sets consisting of outliers seldom follow a normal distribution. 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
requiring further investigations. Therefore, for data sets with NDs, two options are available in ProUCL to
deal with data sets with outliers. These  options are: 1) exclude NDs 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 processes. To compute the various statistics of interest, ProUCL uses rigorous statistical
methods suited for left-censored data sets with multiple DLs.

It is suggested that the outlier identification procedures be supplemented with graphical displays such as
normal Q-Q plots  and box plots.  On a normal  Q-Q plot, observations that are well-separated from the
bulk of the data typically represent potential outliers needing further investigation. Also, significant and
obvious jumps and breaks  in a normal Q-Q plot can be indications of the  presence of more than one
population and/or data gaps due to lack of enough data points (data sets of smaller  sizes).  Data sets of
large sizes (e.g.,  >100) exhibiting such behavior on Q-Q plots may need to be partitioned out into
component sub-populations before estimating EPCs or BTVs.
112

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Outlier tests in ProUCL 5.1 are available under the Statistical Tests module.
                         Statistical Tests   Upper Limits/BTVs   UCLs/EPCs   Windows
Outlier Tests >
Goodness-of-Fit Tests >
Single Sample Hypothesis ^
Two Sample Hypothesis >
Oneway AN OVA ^
OLS Regression
Trend Analysis ^
Full (w/o NDs) C
With NDs >
1 ~T
1 ~T
0 ~~6~
i " ~T
1 i






Dixon's Outlier Test (Extreme Value Test): Dixon's test is used to identify statistical outliers when the
sample size is < 25. This test identifies outliers or extreme values in the left tail (Case 2) and also in the
right tail  (Case 1) of a data distribution. In environmental  data sets,  outliers found in the right tail,
potentially representing impacted locations, are of interest. The Dixon test assumes that the data without
the suspected outlier (s) are normally distributed. If the user wants to perform a normality test on the data
set, he should first remove the outliers before performing the normality test.  This test tends to suffer from
masking in the presence of multiple outliers.  This means that if more than one outlier (in either tail) is
suspected, this test may fail to identify all of the outliers.

Rosner Outlier Test: This test can be used to identify up to 10 outliers in data sets of sizes 25 and higher.
This test also  assumes that the data set without the suspected outliers  is normally distributed. Like the
Dixon test, if the user wants to perform a normality test on the data set, he should first remove the outliers
(which are not known in advance) before performing the normality test.  The detailed discussion of these
two tests is given in the associated ProUCL Technical Guide. A couple of examples illustrating the
identification of outliers in data sets with NDs are described in the following sections.

7.1     Outlier Test for Full Data Set

1.       Click Outlier Tests ^ Full (w/o NDs) ^ Compute
                         Statistical Tests  Upper Limits/BTVs   UCLs/EPCs  Windows  Help
Outlier Tests
Goodness-of-Fit Tests
Single Sample Hypothesis
Two Sample Hypothesis
Oneway ANOVA
OLS Regression
Trend Analysis ^
Full (w/o NDs) >
With NDs
1 '
1
1
D
1
1

1
1
1
D
1
1
ป

Compute
f
I
1
1
0
1
1











2.      The Select Variables screen (Chapter 3) will appear.

        •    Select one or more variable(s) from the Select Variables screen.

        •    If outlier test needs to be performed by using a Group variable, then select a group variable
            by clicking the arrow below the Select Group Column (Optional) 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.
                                                                                               113

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           If at least one of the selected variables (or group) has 25 or more observations, then click the
           option button for the Rosner Test. ProUCL automatically performs the Dixon test for data
           sets of sizes <25.
                               Number of Outliers for Rosner Test 2
                               Applicable to Rosner's Test (N >= 25) Only
                               Dixon Test is for Ns25 and tests for 1 outlier
       •   The default option for the number of suspected outliers is 1. To use the Rosner test, the user
           has to obtain an initial guess about the number of suspected outliers that may be present in the
           data set. This can be done by using graphical displays such as a Q-Q plot. On a 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 Test.

7.2    Outlier Test for Data Sets with NDs

Two options: exclude NDs; or replace NDs by their respective DL/2 are available in ProUCL to perform
outlier tests on data sets with NDs.
1.
Click Outlier Tests ^ With NDs ^ Exclude NDs
                                         UCLs/EPCs   Windows   Help
                G o o d n ess- of - Fit Tests

                Single Sample Hypothesis

                Two Sample Hypothesis

                Oneway AN OVA

                OLS Regression

                Trend Analysis
                                                        DL/2 Estimates
Note:  The above screen  shot was generated using ProUCL 5.0; the exactly similar screen is
generated using ProUCL  5.1 with the exception that in the title of the screen shot, ProUCL 5.0
will be replaced by ProUCL 5.1. Therefore, to save time,  many intermediate screen shots used in
the ProUCL 5.0 User Guide have been used in this ProUCL 5.1  User Guide.
114

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                                  Output Screen for Dixon's Outlier Test
               Dixon's Outlier Test for TCE-1
     Total N = 12
     Number NDs= 4
     Number Detects = 8
     10% critical value: 0.479
     5% critical value: 0.554
     1'"-. critical value: 0.683
     Note: NDs excluded from Outlier Test
                                                    2. Data Value 0.75 is a Potential Outlier (Lower Tail)?
     1. Data Value 9.29 is a Potential Outlier (Upper Tail)?
                                                    Test Statistic: 0.011
     Test Statistic: 0.392

     For 10% significance level. 9.29 is not an outlier.
     For 5% significance level. 9.29 is not an outlier.
     For 1% significance level, 9.29 is not an outlier.
For 10%. significance level, D.75 is not an outlier.
For 5% significance level, 0.75 is not an outlier.
For 1% significance level, 0.75 is not an outlier.
                 Q-Q plot without Four Nondetect Observations are Shown as Follows
Q-Q Plot for TCE-1 TCE-I
Nondetects not displayed ^^"^'^




8


6
4

2









OK
'• Number of Detects - 8
Detected Mean - 2 941
Detected Sd = 3 12
Slope (displayed daia] = 2 934
Intercept (displayed data(= 2.941
Cwwlatiotx R - 1875
QBeslFitLra
5.37

298
V9






•1.5 -1.0 -05 0.0 0.5 1 0 1.5
Theoretical Quantiles (Standard Normal)
NDs Displayed in smaller font
Example: Rosner's Outlier Test by a Group Variable, Zone
         •    Selected Options: Number of Suspected Outliers  = 4

         •    NDs excluded from the Rosner Test
         •    Outlier test performed using the Select Group Column (Optional)
                                                                                                           115

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                     Output Screen for Rosner's Outlier Test for Zinc in Zone: Alluvial Fan
Rosnt
IT'S Outlier
Total N
Number NDs
Number Detects
Mean of Detects
SD of Detects
Number of data
Number of suspected outliers
s not included in the following:
U Mean sd
1 27.88
2 16.04
3 15.35
4 14.83
84.13
8.776
7.356
6.485
Test for 4 Outliers in Zn (alluvia
67
16
51
27.88
85.02
51
4






Potential Obs. Test
outlier Number value
520 26 7.034
50 28 3.87
40 27 3.352
33 29
2.801
Ifan)
I





Critical
value (5%)
3.137
3.127
3.11B
3.108




Critical
value fTA)
3.4S8
3.478
3.469
3.468

	













                         For 5% significance level, there are 3 Potential Outliers
                         620, 50.40
                         For 1% Significance Level, there are 2 Potential Outliers
                     Q-Q plot for Zinc Based upon Detected Data (Alluvial Fan)
Q-Q Plot for Zn (alluvial fan) z- ซNป™IW
Nondetects not displayed OซNUI.I,ซ,>ID.I..B
umbel ol N on-D eleclr -16







400
= 300
s
200
too

HjCU











? n ซ 5'ฐ


Detected Sd- 85.02
Slope (displayed data) = 35.4
lntelCept(diSplayeddalaH27.aB


n Best fit Line





-1.8 -1.2 -0.6 0.0 0.6 1.2 1.8
Theoretical Quantiles (Standard Normal)
NDs Displayed in smaller font
116

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Total N 50
Number NDs 4
Number Detects 4G
Mean of Detects 23.13
SD of Detects 19.03
Number of data 46




















 Number of suspected outliers
s not included in the following:
    Output Screen for Rosner's Outlier Test for Zinc in Zone: Basin Trough

           Rosner's Oilier Test for 4 Outliers in Zn (basin trough)
Potential
# I Mean sd outlier
1 23.13 18.82 90
2 21.64 16.32 70
3 20.55 14.73 60
4 19.63 13.57 60
Obs.
Number
45
21
3
22
Test
value
3.553
2.963
2.679
2.975
Critical
value (5%)
3.09
3.09
3.08
2.07
Critical
value (1%)
3.45
3.44
3.43
3.41
For 5% significance level, there are 4 Potential Outliers
90. 70. 60. 60




For Insignificance Level, there is 1 Potential Outlier
                                                                                     117

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118

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                                      Chapter 8

 Goodness-of-Fit (GOF) Tests for Uncensored and Left-Censored
                                      Data Sets

GOF tests are available under the Statistical Test module of ProUCL 5.0/ProUCL 5.1. Throughout this
User Guide and in ProUCL software, "Full" represents uncensored data sets without ND observations.
The details and usage of the various GOF tests are described in the associated ProUCL Technical Guide.
In ProUCL 5.1, critical values associated with Lilliefors normality test  are computed using a more
efficient algorithm as described in the associated ProUCL 5.1 Technical  Guide. Therefore, tables and
graphs involving Lilliefors test have been re-generated using ProUCL 5.1.

8.1    Goodness-of-Fit Test in ProUCL

Several GOF tests for uncensored full  (Full (w/o NDs)) and left-censored (With NDs) data sets are
available in the ProUCL software.

       •   Full (w/o NDs)
Statistical Tests Upper Limits/BTVs UCLs/EPCs Windows
Outlier Tests ^
G o o d n ess- of - Fit Tests >
Single Sample Hypothesis ป
Two Sample Hypothesis ^
Oneway AN OVA ^
OLS Regression
Trend Analy:i: ^
678
Normal
Gamma
Lognormal
G.O.F, Statistics
0.071 22 0.043E
0.427 32 0.001 3f
              o   This option is used on uncensored full data sets without any ND observations. This
                 option can be used to determine GOF for normal, gamma, or lognormal distribution
                 of the variable(s) selected using the Select Variables option.

              o   Like all other methods in ProUCL, GOF tests can  also be performed on variables
                 categorized by a Group ID variable.

              o   Based upon the hypothesized distribution (normal, gamma, lognormal), a Q-Q plot
                 displaying all statistics of interest including the derived conclusion is also generated.

              o   The G.O.F. Statistics option generates a detailed output log (Excel type spreadsheet)
                 showing all GOF test statistics (with derived conclusions) available in ProUCL. This
                 option helps a user to determine  the distribution of a data set before generating a
                 GOF Q-Q plot for the hypothesized distribution. This option was included  at the
                 request of some users in earlier versions of ProUCL.
                                                                                      119

-------
       •   With NDs
               o   This option performs GOF tests on data sets  consisting of both nondetected and
                  detected data values.

               o   Several sub-menu items shown below are available for this option.
Statistical Tests
Outlier Test
Upper Limits/BTVs UCLs/EPCs Windows Help
> >
Goodness-of-Fit Tests >
Single Sample Hypothesis I
Two Sample Hypothesis ^
Oneway ANOVA >
OLS Regression
Trend Analysis >
6789
Full (w/o NDs) >
With NDs >








10

11

Exclude NDs >
Normal-ROS Estimates >
Gamma-ROS Estimates *
Log-ROS Estimates K
01^2 Estimates K
G.O.F. Statistics
12


13

14

Normal
Gamma
Lognormal






               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 detected values and imputed nondetects.

                  o   Three ROS methods for normal, lognormal (Log), and gamma distributions are
                      available.  This option imputes the NDs based upon the specified distribution and
                      performs the specified GOF test on the data set consisting of detects and imputed
                      nondetects.

               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.
                  ProUCL does  not make any recommendations based upon this option.

               4.  G.O.F. Statistics: Like full uncensored data sets, this option generates an output log
                  of all  GOF test statistics available in ProUCL for data sets with nondetects. The
                  conclusions about the data distributions for all selected variables are also  displayed
                  on the generated output file (Excel-type spreadsheet).

       Multiple variables: When multiple variables are selected from the Select Variables screen, one
       can use one of the following two options:

           o   Use the Group Graphs option to produce multiple  GOF Q-Q plots  for all selected
               variables in a single graph. This option may be used when a selected variable has data
               coming  from  two or more  groups or populations. The  relevant statistics (e.g.,  slope,
               intercept,  correlation,  test  statistic  and  critical value) associated with  the  selected
               variables are shown on the right panel of the GOF Q-Q plot.  To capture all the graphs
               and results shown on the window screen, it is preferable to print the graph  using the
120

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               Landscape option.  The user may also want to turn off the Navigation Panel and Log
               Panel.

           o   The Individual Graphs option is used to generate individual GOF Q-Q plots for each of
               the  selected variables,  one variable at a time (or for  each  group individually of the
               selected variable categorized by a Group ID). This is the most commonly used option to
               perform GOF tests for the selected variables.

    •  GOF Q-Q plots for hypothesized distributions: ProUCL computes the  relevant test statistic
       and the associated critical value, and prints them on the  associated Q-Q plot (called GOF Q-Q
       plot). On a GOF Q-Q plot, the program informs the user if the data are gamma, normally, or
       lognormally distributed.

       o   For  all options  described above,  ProUCL  generates  GOF Q-Q  plots  based upon the
           hypothesized distribution (normal, gamma, lognormal). All  GOF Q-Q  plots display several
           statistics of interest including the derived conclusion.

       o   The linear pattern displayed by a GOF Q-Q plot suggests an approximate GOF for the
           selected  distribution. 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) may be an indication of a good fit for that distribution; however, the
           high correlation should  exhibit a definite linear pattern in the Q-Q plot without breaks and
           discontinuities.

       o   On a GOF  Q-Q plot,  observations that are well  separated from the  majority of the data
           typically represent potential outliers needing further investigation.

       o   Significant and obvious jumps and breaks and curves in a  Q-Q plot are indications of the
           presence of more than one population. Data sets exhibiting such behavior of Q-Q plots may
           require partitioning of the  data set into component  subsets  (representing  sub-populations
           present in a mixture data set) before computing upper limits to estimate EPCs or BTVs. It is
           recommended that both graphical and formal goodness-of-fit tests 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 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). However, the
           Lilliefors test (generalized  Kolmogorov Smirnov  [KS] 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. S-W)  Test:  a test used for samples  of size  smaller than or equal to
           2000 (<= 2000). In ProUCL 5.0, the SW test uses the  exact SW critical values for samples of
           size 50 or less. For samples of size, greater than 50, the SW test statistic is displayed along
           with the/"-value of the test (Royston 1982, 1982a).

Notes: As with other statistical tests, sometimes these two GOF tests might lead to different conclusions.
The user is advised to exercise caution when interpreting these test results.  When  one the  GOF tests
passes the  hypothesized distribution, ProUCL 5.0/ProUCL 5.1 determines that the data set follows an
                                                                                            121

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approximate hypothesized distribution. It should be pointed out that for data sets of smaller sizes (e.g.,
<50), when Lilliefors tests determines that the data set follows a normal (lognormal) distribution and
Shapiro-Wilk's test determines that the data set does not  follow a normal (lognormal) distribution, the
user may not use the approximate normality (lognormality) conclusion derived using the Lilliefors test.
Under these situations, the user may determine the  data distribution based upon the highest p-value
associated with GOF test statistics for other distributions or may determine that the  data set does not
follow a discernible distribution.

    •  GOF test for Gamma Distribution: In addition to the graphical gamma Q-Q plot, two formal
       empirical distribution function (EDF) procedures are also available to test the gamma distribution
       of a data set. These tests are the AD test and the KS 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 (known or estimated) of the  shape parameter,  k (k hat)  should lie in the
           interval  [0.01,  100.0]. Consult the  associated ProUCL  Technical Guide for a  detailed
           description of the gamma distribution and its parameters, including k. Extrapolation of critical
           values beyond these sample sizes and values of k is not recommended.

       Notes: Even though, the GOF 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.

 8.2   Goodness-of-Fit Tests for Uncensored Full Data Sets
1.
2.
Click Goodness-of-Fit Tests ^ Full (w/o NDs)

                                                     roUCL 5.0 - [pyrene.xls]
 Statistical Tests I  Upper Limits/BTVs   UCLs/EPCs  Windows  Help
Outlier Tests >
Goodness-of-Fit Tests >
Single Sample Hypothesis t
Two Sample Hypothesis t
Oneway AN OVA >
OLS Regression
Trend Analysis >•
6
7
8 3
Full (w/o NDs) >
With NDs t




1C
11
Normal
Gamma
Lognormal
G.O.F. Statistics








Select the distribution to be tested: Normal, Lognormal, or Gamma

    •   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.
122

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8.2.1  GOF Tests for Normal and Lognormal Distribution
1.     Click Goodness-of-Fit Tests ^- Full (w/o NDs) ^- Normal or Lognormal
                                                            •UCL 5.0 - [pyrene.:
      Statistical Tests I  Upper Limits/BTVs    UCLs/EPCs   Windows   Help
2.
          Outlier Tests
          Goodness-of-Fit Tests
          Single Sample Hypothesis    >
          Two Sample Hypothesis     >
          Oneway AN OVA           >
          OLS Regression
          Trend Analysis            >
                                                       8
                                                                  10
                 11
                                         Full (w/o NDs)
                                  With NDs
Normal
Gamma
                                                         Lognormal
                                                         G.O.F. Statistics
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 Select Group Column (Optional) 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.
                               Select Goodness-of-Fit Opti
                       Select Confidence Coefficient
                         O 99*            ฎ 95*          O 90%
                       Select GOF Method
                         *  Shapiro-Wilk         O Lilliefors
                       Graphs by Groups
                         O  Individual Graphs     (ง) Group Graphs
                                                 Cancel
                  The default option for the Confidence Level is 95%.
                                                                                         123

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                   The default GOF Method is Shapiro-Wilk.
                   The default option for Graphs by Group is Group Graphs. If you want to see the
                   plots  for all  selected variables  individually, and then check the  button next to
                   Individual Graphs.
                   Click OK button to continue or Cancel button to cancel the GOF tests.
Notes: This option for Graphs by Group is specifically provided for 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 of COPC-
Arsenic) collected from two or more groups (e.g., upgradient wells, monitoring wells, residential wells).
Example  8-la Consider the chromium concentrations data set used in Example 1-1  of Chapter 1. The
lognormal and normal  GOF test results on chromium concentrations are shown in the following figures.
                    Output Screen for Lognormal Distribution (Full (w/o NDs))
                                  Selected Options: Shapiro-Wilk
                                  Lognormal Q-Q Plot for Chromium
                                                                                      Slope - 0.579
                                                                                      nlercepl - 2.334
                                                                                      Conelalion. R * 0 987
                                                                                      Shapilo-WilkTesI
                                                                                      ExaclTestStalijnc-0.979
                                                                                      CrilicalValue[0.05]-0916
                                                                                      Dala^pear Lognormal
                                                                                      Appro*. TestValue-0.378
                                                                                      p-Value - 0.849
                                   Theoretical Quantiles (Standard Normal)
124

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                     Output Screen for Normal Distribution (Full (w/o NDs))
                   Selected Options: Shapiro-Wilk, Best Fit Line not Displayed
                                   Normal Q-Q Plot for Chromium
                                   Theoretical Quantises (Standard Normal}
                                                                                  Mean-11.97
                                                                                  Sd = 6 892
                                                                                  Stops-S.58


                                                                                  Coitelafoi). R - D 324
                                                                                  Shspkc-Wik Test
                                                                                  Appro*. Test Vd>.ie = 0 ฃ70


                                                                                  D Best Fit Line
8.2.2  GOF Tests for Gamma Distribution

1.      Click Goodness-of-Fit Tests ^ Full (w/o NDs) ^ Gamma
Outlier Tests ^
Goodness-of-Fit Tests >
Single Sample Hypothesis >
Two Sample
Oneway AN
OLS Regress
Trend Analj
; Hypothesis t
OVA >
ion
sis >
6
7
S 9 1C 11
Full (w/o NDs) c
With NDs >•






Normal
Gamma
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 Select Group Column (Optional) 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.
                                                                                              125

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                                   Select Goodness-of-Fit Options

                             Select Confidence Coefficient

                              O 99*          ฎ 95X        O
                             Select GOF Method

                               *  Aiderson Dariing     ' Kolmogorov Smiinov


                             Graphs by Groups

                              O Individual Graphs   • Group Graphs
               o   The default option for the Confidence Coefficient is 95%.

               o   The default GOF method is Anderson Darling.

               o   The default option for  Graph by Groups  is Group  Graphs. If you want to see
                  individual graphs, then check the radio button next to Individual Graphs.

               o   Click the OK button to continue or the Cancel button to cancel the option.

               o   Click OK button to continue or Cancel button to cancel the GOF tests.

   Example 8-lb. Consider arsenic  concentrations  data set used in Example 1-1  of Chapter  1. The
   Gamma GOF test results for arsenic concentrations, are shown in the following G.O.F. Q-Q plot.

                     Output Screen  for Gamma Distribution (Full (w/o NDs))
                      Selected Options: Anderson Darling with Best Line Fit
                                   Gamma Q-Q Plot for Arsenic
                                  Theoretical Quantiles of Gamma Distribution
126

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8.3    Goodness-of-Fit Tests Excluding NDs

This option is the most important option for a GOF test applied to data sets with ND observations. Based
upon the skewness and distribution of detected data, ProUCL computes the appropriate decision statistics
(UCLs, UPLs, UTLs, and USLs) which accommodate data skewness. Specifically, depending upon the
distribution of detected data,  ProUCL uses KM estimates in parametric or nonparametric upper limits
computation formulae (UCLs, UTLs) to estimate EPC and BTV estimates.
1.
Click Goodness-of-Fit Tests ^ With NDs ^ Exclude NDs
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.

8.3.1  Normal and Lognormal Options

1.     Click Goodness-of-Fit Tests ^  With NDs ^ Excluded NDs ^ Normal or Lognormal
r Graphs

senic
1
1
1
1
1
1
1
0
0
2




Statistical Tests
Upper Limits/BTVs UCLs/EPCs Windows Help
Outlier Tests >
Goodness-of-Fit Tests >
Single Sample Hypothesis t
Two Sample Hypothesis ฅ
Oneway ANOVA >
OLS Regression
Trend Analysis ป









6
7
8 9
Full (w/o NDs) >
With NDs




>



10

11

Exclude NDs >
Normal-ROS Estimates >
Gamma-RQS Estimates >
Log-ROS Estimates ฅ
DL/2 Estimates >
Using Censored Plot >
G.O.F. Statistics
12 13 14

Normal
Gamma
Lognormal



Note: In ProUCL 5.1, the censored probability plot option has been added as shown in the above drop-
down menu  as "Using  Censored Plot." This option is very much the same  as the Q-Q  plot option
generated by excluding NDs except that the hypothesized quantiles displayed along the x-axis adjust for
quantiles associated with NDs. There is not much  difference (except for the correlation, slope and
intercept of the optional line displayed on the Q-Q plot) between these two graphs from the decision
making point of view. Censored probability (Q-Q) plots do not provide additional information than tools
and graphs already available in ProUCL 5.0. This was the  reason that censored Q-Q plots  were not
included in ProUCL 5.0  and its earlier versions.

2.     The Select Variables screen (Chapter 3) will appear.

           •   Select one or more variable(s) from the Select Variables screen.
                                                                                        127

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              If graphs have to be produced by using a group variable, then select a group variable by
              clicking the arrow below the Select Group Column (Optional) 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, following window appears.
                                     ?ct Goodness-of-Fit Options

                            Select Confidence Coefficient

                             O MX          ฎ 35X         O 90%
                            Select GOF Method

                              * Shapiro-Wilk          Lilliefors
                            Graphs by Groups

                             (_J Individual Graphs
>:•:' Group Graphs
                                        OK
                                                  Cancel
              o   The default option for the Confidence Coefficient is 95%.

              o   The default GOF Method is Shapiro-Wilk.

              o   The default option for Graphs by Group is Group Graphs. If you want to see the
                  plots for all selected variables individually, then check the button next to Individual
                  Graphs.

           •  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 GOF tests.

       Example  8-2a. Consider the  arsenic Oahu data set with  NDs discussed in the literature  (e.g.,
       Helsel 2012; NADA in R [Helsel 2013]). The normal and lognormal GOF test results based upon
       the detected data are shown in the following two figures. Censored Q-Q plots are also displayed
       along with Q-Q plots based upon detected data.
128

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  GOF Q-Q Plot for Normal Distribution (Exclude NDs)
                 Selected Options: Shapiro-Wilk
     Normal Q-Q Plot (Statistics using Detected Data) for Arsenic
                                                                                   -.,	     Total Nui
                                                                                           NumbeiofNDs = 13
                                                                                           MaxDL=2
                                                                                           Percent NDs = 54ฃ
                                                                                           Mean = 1.236
                                                                                           Srfปtt9G5
                                                                                           Slope = 0.906
                                                                                           Intercept 1.23G
                                                                                           Collection, R - Q 887
                                                                                           ShapiioWikTest
                                                                                           Enact Test Value-0.777
                                                                                           Critical Val{0.05) = 0.850
                                                                                           p-Value = O.OOG73
                                                                                           Distribution Test Suspect
                                                                                           D Best Fit Line
                         0.7      0.7
                   Theoretical Quantiles (Standard Normal)
Censored Probability (Q-Q) Plot for Normal Distribution
     Selected Options: Shapiro-Wilk and Best Line Fit
                  Normal Q-Q Plot for Arsenic
Statistics Displayed using Censored Quantiles and Detected Data
                                                                                      Numbeiol Detecli -11
                                                                                      M inimum Censored Quamtile - -1 347
                                                                                      Percent MDx = 54ฃ
                                                                                      Mean of Detected fete = 1.236
                                                                                      Sd of Detected Data = 1965
                                                                                      Slope = 0.776
                                                                                      Intercept = 1.53
                                                                                      rniiflnhon.n-ll'ir*
                                                                                      Stmpiio WifcTesl
                                                                                      Exacl TeslValue - IJ /.'/
                                                                                      C.ilicalV3l|0.05) = a350
                                                                                      DataNol Normal
                                                                                      Approx. Test Value = II ?i51
                                                                                      p-Value - 0 OOB73
                                                                                      Oistiibution Test Suspect
                Theoretical Quantiles (Standard Normal)
                  Only Quantiles for Detects Displayed
                                                                                                       129

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                           GOF Q-Q Plot for Lognormal Distribution (Exclude NDs)
                                             Selected Options:  Lilliefors Test
                              Lognormal Q-Q Plot (Statistics using Detected Data) for Arsenic
                                                   0.557    -0.557
                                                                                                                      NumbeiofNDs = 13
                                                                                                                      MaxDL=2
                                                                                                                      N =11
                                                                                                                      Percent NO s = 54ฃ
                                                                                                                      Mean— 0.0255
                                                                                                                      LillieforsTest
                                                                                                                      Test Statistic = 0.223
                                                                                                                      CriticalValuefG 05) - 0.251
                                                                                                                      D ata Appeal Lognoimal
                                                                                                                      D isli ibulion T est Suspect
                                                                                                                      Q Best Fit Line
                                              Theoretical Quantiles (Standard Normal)
                          Censored Probability (Q-Q) Plot for Lognormal Distribution
                                 Selected options: Lilliefors Test with Best Fit Line
                                           Lognormal Q-Q Plot for Arsenic
                           Statistics Displayed using Censored Quantiles and Detected Data
Total Number nl Data = 24
Number of Detects = 11
M inimum Censoted U uam tile = 1 347
Percent NDs = 54ฃ
MaKDL=2
H eanx of Delects = -0.0255
StdvxofDetects = aG94
Slope = 0.559
                                                                                                                  Coirel'-rfion, !"! - 0 97C,
                                                                                                                  LilliefoisTest
                                                                                                                  Teil Statistic = 0.229
                                                                                                                  CiiticalValuetO. 05) =0.251

                                                                                                                  I > isu ii H ji urn T est Suspect
                                                                                                                  • Best Fit Line
                                           Theoretical Quantiles (Standard Normal)
                                             Only Quantiles for Detects Displayed
130

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8.3.2  Gamma Distribution Option

1.     Click Goodness-of-Fit Tests ^ With NDs *• Excluded NDs ^ Gamma
2.
Statistical Tests ] Upper Limits/BTVs UCLs/EPCs Windows Help
Outlier Tests >
Goodness-of-Fit Tests It
Single Sample Hypothesis
Two Sample Hypothesis
Oneway AN OVA
OLS Regression
Trend Analysis
3 0.344
515657826
*
6789
Full (w/o NDs) t
With NDs >

1
1
1
1
1











10

11

Exclude NDs >
Normal-ROS Estimates ^
Gamma-ROS Estimates >
Log-ROS Estimates >
DL/2 Estimates >
G.O.F. Statistics
12

13

14

Normal
Gamma
Lognormal













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 Select  Group Column (Optional) 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.
                                    Select Goodness-of-Fit Options
                       Select Confidence Coefficient

                        O 99*         ฎ 9SX
                                                          O
                              Select GOF Method

                               O Anderaon Dating   * Kblmogorov Smimov


                              Graphs by Groups

                               O Individual Graphs   ป Group Graphs
                  o   The default option for the Confidence Coefficient is 95%.

                  o   The default GOF test method is the Anderson Darling test.
                  o
               The default option for Graph by Groups is Groups Graphs. If you want to
               display all  selected variables on separate graphs, check  the  button next to
               Individual Graphs.
                                                                                           131

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             •    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 GOF tests.

         Example  8-2b (continued).  Consider the  arsenic Oahu  data  set with NDs as  discussed  in
         Example 8-2a above. The gamma GOF test results based upon the detected data are shown in the
         following GOF Q-Q plot.

                          Output Screen for Gamma Distribution (Exclude NDs)
                       Selected Options: Anderson-Darling Test with Best Fit Line
                               Gamma Q-Q Plot (Statistics using Detected Data) for Arsenic
                                                                                                    Mean = 1 2364
                                                                                                    khal = 2256G
                                                                                                    Ihela hat = 0.5479
                                                                                                    Slope -1.0345
                                                                                                    Intercept = -0.0177

                                                                                                    nders on-Darling Ted
                                                                                                    est Statistic = 1787
                                                                                                    riticalValue|0.05) = 0,738
                                                                                                       nt 6 amma Distributed
                                                                                                    Distribution T est Suspect
                                                                                                    • Best Fit Line
                                         Theoretical Quantiles of Gamma Distribution
                        Censored Probability (Q-Q) Plot for Gamma Distribution
                        Selected options: Anderson-Darling Test with Best Fit Line
                                         Gamma Q-Q Plot for Arsenic
                           Statistics Displayed using Censored Quantiles and Detected Data
Anraifc
 TotalNumbeiofOata = 24
 Number of Delects -11
 Number or NonDetects - 13
 Minimum Censored Quamtile - 0.2
 Percent NDi- 54%
 Mean of Detects = 1.2364
 khalolDetects-22566
 ! 11. t.i hat ui Deiecti - 0.5479
 Slope = 0.9205
 Intercept = OVf05B
 Correlation. R = 0.3882
 Anderson Darling Test
 lestStahstb^ttTB?
 Critical Value[0.05) - 0738
 Data not Gamma Distributed
 Distribution Test Suspect
                                       Theoretical Quantiles of Gamma Distribution
                                         Only Quantiles for Detects Displayed
132

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8.4    Goodness-of-Fit Tests with ROS Methods

1.      Click Goodness-of-Fit Tests ^ With NDs ^ Gamma-ROS Estimates or Log-ROS Estimates
Statistical Tests | UpperLimits/BTVs UCLs/EPCs Windows Help
Outlier Tests ป•
Goodness-of-Fit Tests >
Single Sample Hypothesis k
Two Sa
Onewa
OLSRe
Trend L
.9 0 344
mple Hypothesis k
• ANOVA >
jression
analysis >
515657826
6 7
S 9
Full (w/o NDs) >
With NDs >
1
1
1
1
1






1C

11 12 13 14

Exclude NDs >
Normal-ROS Estimates >
Gamma-ROS Estimates >
Log-ROS Estimates >
DL/2 Estimates >
G.O.F. Statistics











Normal
Gamma


Lognormal







2.      Select the distribution to be tested: Normal, Lognormal, or Gamma

          •   To test for normal distribution, click on Normal from the drop-down menu list.

          •   To test for gamma distribution, click on Gamma from the drop-down menu list.

          •   To test for lognormal distribution, click on Lognormal from the drop-down menu.


8.4.1  Normal or Lognormal Distribution (Log-ROS Estimates)

1.      Click Goodness-of-Fit Tests ^ With NDs ^  Log-ROS Estimates ^ Normal, Lognormal

                                      ProUCL 5.0 - [Oahu-chapters 4 -5.xls]
^^^^^H
Statistical Tests
Outlier Test
Upper Limits/BTVs UCLs/EPCs Windows Help
:

Goodness-of-Fit Tests K
Single Sample Hypothesis

Two Sample Hypothesis >
Oneway AN OVA >
OLS Regression
Trend Analysis >
.9 0.344
15 0.737
515657826
515657826

6
789
Full (w/o NDs) >
With NDs >

0

10

11

Exclude NDs >
Normal-ROS Estimates >
Gamma-ROS Estimates k
Log-ROS Estimates >
DL/2 Estimates
G.O.F, Statistics
K
12



13



14




Normal
Gamma
Lognormal



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 Select Group Column (Optional) button. This will result
                                                                                     133

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              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 appears.
                             Select Goodness-of-Fit Options
                      Select Confidence Coefficient

                       O 99%             ฎ 95%           O 90%


                      Select GOF Method

                       *  Shapiro Wilk         O LiHiefors


                      Graphs by Groups

                       O hdviduaJ Graphs     (•) Group Graphs
                                    OK
              o   The default option for the Confidence Coefficient is 95%.

              o   The default GOF test Method is Shapiro-Wilk.

              o   The default option for Graphs by Group is Group Graphs. If you want to display
                 graphs for all selected variables individually, check  the button next to  Individual
                 Graphs.

          •   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 GOF tests.

       Example 8-2c (continued). Consider the arsenic Oahu data set with NDs considered earlier in
       this chapter. The lognormal GOF test results on LROS data (detected and imputed LROS NDs) is
       shown in the following GOF Q-Q plot.
134

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                 Output Screen for Lognormal Distribution (Log-ROS Estimates)
                       Selected Options: Shapiro Wilk test with Best Line Fit
                                    Lognormal Q-O Plot for Arsenic
                              Statistics using ROS Lognormal Imputed Estimates
                                    Theoretical Quantiles (Standard Normal)
                                     Imputed NDs Displayed in smaller font
                                                                                   Mean ' -0.209
                                                                                   Sd - 0 571
                                                                                   Slope = 0.568
                                                                                   Intercept * -0.20S
                                                                                   Correlation. R = 0.363
                                                                                   Shapiro-Will Test
                                                                                   AppFox.TestValue-OS24
                                                                                   p-Value - 0 0742
Note: The font size of ND values is smaller than that of the detected values.

8.4.2  Gamma Distribution (Gamma-ROS Estimates)

1.      Click Goodness-of-Fit Tests ^ With NDs ^  Gamma-ROS Estimates ^ Gamma
Statistical Tests | Upper Limits/BTVs UCLs/EPCs Windows Help
Outlier Tests >
Goodness-of-Fit Tests >
Single Sample Hypothesis t
Two Sample Hypothesis >
Oneway ANOVA >
OLS Regression
Trend Analysis >
6 7
8 9
Full (w/o NDs) >
With NDs ^
1
1
1
1
9^ 0.344 515657826 1




10


11

Exclude NDs >
Normal-ROS Estimates k
Gamma-ROS Estimates >
Log-ROS Estimates >
DL/2 Estimates ^
G.O.F. Statistics
12


13

14


Normal
Gamma
Lognormal



	

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 generated by using a Group variable, then select a group variable by
               clicking the arrow below the Select Group Column (Optional) 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 will be shown.
                                                                                               135

-------
                                    Select Goodness-of-Fit Options

                              Select Confidence Coefficient

                               O 99*           • 95%         O 90*


                              Select GOF Method

                                  Anderson Darling   • KolmogorDV Smimov


                              Graphs by Groups

                               O Individual Graphs   * Group Graphs
               o   The default option for the Confidence Coefficient is 95%.

               o   The default GOF test Method is Anderson Darling.

               o   The default option for Graph by Groups is Group Graphs. If you want to generate
                   separate graphs for all selected variables, the check the button next to Individual
                   Graphs.

           •   Click the OK button to continue or the Cancel button to cancel the GOF tests.

       Example 8-2d (continued). Consider the arsenic Oahu data set with NDs considered earlier. The
       gamma GOF test results on GROS data (detected and imputed GROS NDs) are shown in the
       following GOF Q-Q plot.

                 Output Screen for Gamma Distribution (Gamma-ROS Estimates)
                               Selected Options: Anderson Darling
                                    Gamma Q-Q Plot for Arsenic
                              Statistics using ROS Gamma Imputed Estimates
                                   Theoretical Ouantiles of Garnma Distribution
                                    Imputed NDs Displayed in smaller font
136

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Note: The font size of ND values in the above graph (and in all GOF graphs) is smaller than that of
detected values.

8.5    Goodness-of-Fit Tests with DL/2 Estimates

1.     Click Goodness-of-Fit Tests ^ With NDs ^ DL/2 Estimates

2.     Select the 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 a gamma distribution, click on Gamma from the drop-down menu list.


8.5.1  Normal or Lognormal Distribution (DL/2 Estimates)

1.     Click Goodness-of-Fit Tests ^ With NDs ^ DL/2 Estimates ^ Normal or Lognormal
                                        ProUCL 5.0 - [Oahu-chapters 4 -5.xls]
     Statistical Tests I Upper Limits/BTVs  UCLs/EPCs  Windows  Help
Outlier Tests >
Goodness-of-Fit Tests t
Single Sample Hypothesis
Two Sample Hypothesis
Oneway ANOVA
OLS Regression
Trend Analysis
3 ฃ TV .'
PT~ C.737
515657826
515657826
>

6
7
2
Full (w/o NDs) >
With NDs K
1
1
1
1
1
0



9

Exclud
Norm;
Gamrr
Log-R
10

eNDs
I-ROS Estin
a-ROSEstir
3S Estimate
11

K
nates >
nates K
S K
DL/2 Estimates >
G.O.F.

Statistics



12


13



14


	
Normal
Gamma
Lognormal



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 generated by using a group variable, then select a group variable by
              clicking the arrow below the Select Group Column (Optional) 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.

       The rest of the process to  determine the distribution (normal, lognormal, and gamma) of the data
       set thus obtained is the same as described in earlier sections.

8.6    Goodness-of-Fit Test Statistics

The G.O.F. option displays all GOF test statistics available in ProUCL. This option is used when the user
does not know which GOF test to use to  determine the data distribution.  Based upon the  information
                                                                                         137

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provided by the GOF test results, the user can perform an appropriate GOF test to generate GOF Q-Q plot
based upon the hypothesized distribution. This option is available for uncensored as well as left censored
data sets. Input and output screens associated with the G.O.F statistics option for data sets with NDs are
summarized as follows.

    1.  Click Goodness-of-Fit ^ With NDs ^ G.O.F. Statistics
ซ3 File Edit Stats/Sample Sizes Graphs
Navigation Panel
Name
Worksheet ;ds
Well 10jds
WMW-rath NDsxIs
ASHALL7groupsxls
Box Plot Full.gst
Box Plot Full_a.gst
Box Plot Full_b.gst
1
2
3
4
5
6
7
0

backgroun
1
4
5
7
12
15
18


Statistical Tests | Upper Limits/BTVs UCLs/EPCs
Outlier Tests
^
Goodness-of-Fit Tests *
Single Sample Hypothesis ^
Two Sample Hypothesis
Oneway ANOVA
OLS Regression
Trend Analysis
34 1 1
"

5
Windows Help
S 7
Full (w/o NDs) t
With NDs t








I


B

Exclude NDs
N
G
L
D
ormal-RQS
amma-RO?
Dg-ROS Est
b'2 Estimat
9 10

N
Estimates ^
Estimates ^
mates ^
es ^
G.O.F. Statistics




11





    2.  The Select Variables screen (Chapter 3) will appear.

          •   Select one or more variable(s) from the Select Variables screen.

          •   When the option button is clicked, the following window will be shown.
                                      GOF_Conf Level F
                           Select Confidence Coefficient

                             O 397.        ฉ 95%       O 90%
                                            OK
Caned
           •   The default confidence level is 95%.

           •   Click the OK button to continue or the Cancel button to cancel the option.
138

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Example 8-2e (continued). Consider the arsenic Oahu data set with NDs discussed earlier. Partial GOF
test results, obtained using the G.O.F. Statistics option, are summarized in the following table.

    Sample Output Screen for G.O.F. Test Statistics on Data Sets with Nondetect Observations
Asenic


Raw Statistics


Statistics (Non-Detects Only)
Statistics (Detects Only)
Statistics (All: NDs treated as D Lvalue)
Statistics {All: NDs treated as DL/2 value)
Statistics (Normal ROS Imputed Data)
Statistics (Gamma ROS Imputed Data)
Statistics (LDgnormal ROS Imputed Data}
Statistics (Detects Only)
Statistics (NDs = DL)
Statistics (NDs = DL/2)
Statistics (Gamma ROS Estimates)
Statistics (Lognormal ROS Estimates)
Num Obs
24
Num Miss
0
Num Valid
24
Number Minimum Maximum
13 0.9 2
11 0.5 3.2
24 0.5 3.2
24 0.45 3.2
24 -0.09S5 3.2
24 0.119 3.2
24 0.349 3.2
K hat K Star Theta hat
2.257 1.702 0.54B
3.538 3.124 C.4C6
3.233 2.857 0.31
2.071 1.84 0.461
Detects
11
Mean
1.ECS
1.236
1.438
1.002
0.997
0.956
0.972
Log Mean
-C.C2EE
0.215
-0.16
-0.209

NDs
13
Median
2
0.7
1.25
0.95
0.737
0.7
0.7
Log Stdv
0.694
0.574
0.542
0.571

*ANDs
54.17%
SD
0.517
0.965
0.761
0.699
0.776
0.75B
0.718
LogCV
-27.26
2.669
-3.381
-2.727
Normal GOF Test Results
No NDs NDs=DL NDs = DL/2NormalROS
Correlation Coefficient R
Shapiro-Wilk (Detects Only)
Shapiro-Wilk (NDs = DL)
Shapiro-Wilk (NDs = DL/2)
Shapiro-Wilk (Normal ROS Estimates)
Lilliefors (Detects Only)
Lilliefors (NDs = DL)
Lilliefors (NDs = DL/2)
Lilliefors (Normal ROS Estimates)
0.887 0.948 0.833
Test value
0.777
0.89
0.701
O.S6S
0.273
0.217
0.335
0.17
Crit. (0.05)
O.S5
0.916
0.916
0.916
0.251
0.177
0.177
0.177
0.828


Conclusion with Alpha(0.05)
Data Not Normal
Data Not Normal
Data Not Normal
Data Not Normal
Data Not Normal
Data Not Normal
Data Not Normal
Data Appear Normal
                                                                                           139

-------
      Sample Output Screen for G.O.F. Test Statistics on Data Sets with Nondetect Observations
                                                         (continued)
                                                 Gamma GOF Test Results
                                                    NoNDs   NDs = DL  NDs = DL/2Gamma ROJ
                                Correlation Coefficient R    0.964     0.956      0.924     0.975
                                                   Test value  Crit. (0.05)
                          Anderson-Darling (Detects Only)    0.787     0.738
                                                          Conclusion with Alpha(O.OS)
                        Kolmogorov-Smirnov (Detects Only)    0.254     0.258
                             Anderson-Darling (NDs = DL)    0.98      0.75
                          Kolmogorov-Smirnov (NDs = DL)    0.214     0.179
                           Anderson-Darling (NDs = DL/2)    1.492     0.751
                         Kolmogorov-Smirnov (NDs = DU2)    0.261     0.179
                  Anderson-Darling (Gamma ROS  Estimates)    0.48      0.755
                    Kolmogorov-Smirnov (Gamma  ROS Est.)    0.126     0.18
                                                  Detected Data appear Approximate Gamma Disti
                                                  Data Not Gamma Distributed
                                                  Data Not Gamma Distributed
                                                  Data Appear Gamma Distributed
                                                   Lognormal GOF Test Results
                                  Correlation Coefficient R
                                                        NoNDs
                                                          0.939
NDs = DL
   0.959
                                                          s = DIJ2  Log ROS
                                                           0.933      0.963
                                                       Test value
                                S hapiro-Wilk (D elects 0 nly)    0.86
                                  Shapiro-Wilk (NDs = DL)    0.906
                                                          0.365
                                                          0.924
                                                          0.229
                                                          0.214
                                                          0.217
                                                          0.143
            Shapiro-Wilk (NDs = DL/2]
Shapiro-Wilk (Lognormal ROS  Estimates)
                Lilliefors (Detects Only)
                  Lilliefors(NDs = DL)
                Lilliefors (NDs = DL/2)
Crit. (0.05)           Conclusion with Alpha(0.05)
   0.85    Data Appear Lognormal
   0.916   Data Not Lognormal
   0.916   Data Not Lognormal
                        Lilliefors (Lognormal ROS  Estimates)
   0.916   D ata Appear Lognormal
   0.251   D ata Appear Lognormal
   0.177   Data Not Lognormal
   0.177   Data Not Lognormal
   0.177   Data Appear Lognormal
              Note: Substitution methods such as DL 01 DL/2 are not recommended.
140

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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 the  ProUCL software.  All hypothesis tests are available under the
Statistical Tests module of ProUCL 5.0/ProUCL 5.1. ProUCL software can perform these hypotheses
tests on data sets with and without ND observations. It should be pointed out that, when one wants to use
two-sample hypotheses tests on data sets with NDs,  ProUCL  assumes that samples from both  of the
samples/groups have ND observations. All this means is that, a ND column (with 0 or 1 entries only)
needs to be provided for the variable in each of the two samples. This has to be done even if one of the
samples (e.g., Site) has all detected  entries; in this case the associated ND column will have all entries
equal to '!.' 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 with
pre-specified  Cs  or CLs.  The single-sample hypotheses tests are useful when the  environmental
parameters such as the Cs, action  level, or  CLs are known, and the objective  is to compare  site
concentrations with those known pre-established 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 an AOC (or a MW) exceeds a certain specified proportion
(e.g., 1%, 5%, 10%).

ProUCL  5.1  can  perform  these  hypotheses tests on data sets with  and without ND  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 Csor a CL (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 Wilcoxon
Signed Rank (WSR) test is 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 CL exceeds a specified proportion, P0 (e.g., 5%, 10%).  ProUCL
has graphical methods that may be used  to visually compare the concentrations of a site AOC with an
action level. This can be done using a box plot of site data with horizontal lines displayed at action levels
on the same graph. The details of the various single-sample hypotheses testing approaches are provided in
the associated ProUCL Technical Guide.
                                                                                         141

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9.1.1  Single-Sample Hypothesis Testing for Full Data without Nondetects

1.      Click Single Sample Hypothesis  ^- Full (w/o NDs)

                                                 ProUCL 5.0 - [Zn-Cu-ND-data.xls
       Statistical Tests  I Upper Limits/BTVs   UCLs/EPCs  Window:  Help
2.
Outlier Tests ^
Goodness-of-Fit Tests ^
Single Sample Hypothesis ^
Two Sample Hypothesis >
Oneway AN OVA >
OLS Regression
Trend Analysis ป
an 1 1
e

7

8

9

Full (w/o NDs) t
With NDs ^




10

11

12

tTest
Proportion
Sign Test
Wilcoxon Signed Rank






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 (WSR) test, click on Wilcoxon Signed Rank from the
       drop-down menu.

All single-sample hypothesis tests for uncensored and left-censored data sets can be performed by
a group variable. The user selects a group variable by clicking the arrow below the Select Group
Column (Optional) 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.
                                             Select Variable
	
Available Variables Selected Variable
Name ID
Y3 1
ป Name ID
X3 0
ซ < ^H >
^m
Select Group Column (Optional)
1 -1
Options <— Select an action level
OK Cancel


142

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9.1.1.1        Single-Sample t-Test

1.     Click Single Sample Hypothesis ^ Full (w/o NDs) > t-Test

                                                            . EPA (2006)-cl
        Statistical Tests I Upper Limits/BTVs   UCLs/EPCs  Windows   Help
Outlier Tests ^
Goodness- of -Fit Tests >
Single Sample Hypothesis >
Two Sample Hypothesis >
Oneway AN OVA ^

Trend Analysis >
6

Fi
W



7

jll (w/o ND.
ith NDs



8

0






t
tt



9 10

[ tTest
Proportion
Sign Test
Wilcoxon >

11




gned Rank

12













2.      The Select Variables screen will appear.

           •   Select variable(s) from the Select Variables screen.

           •   When the Options button is clicked, the following window will be shown.
                                  Select Uncensored t Test Options


                              Select Null Hypothesis Form

                              ป  Sample Mean <= Action Level (Form 1)

                             O Sample Mean >= Action Level (Form 2)

                             O Sample Mean => Action Level - S (Form 2)

                             O Sample Mean = Action Level (Two Sided)

                                           Confidence Level

                               Substantial Difference. S (Form 2)

                                               Action Level

                                                  OK
Cancel
                   Specify the Confidence Level; default is 0.95.
               o   Specify meaningful values for Substantial Difference, S and the Action Level. The
                   default choice for S is "0."

               o   Select form of Null Hypothesis; default is Sample Mean <= Action Level (Form 1).

               o   Click on OK button to continue or on Cancel button to cancel the test.
                                                                                              143

-------
         Example 9-la.  Consider the WSR data set described in EPA (2006a). One Sample t-test results
         are summarized as follows.
                           Output for Single-Sample t-Test (Full Data w/o NDs)
                                              From File  WSR EPA (2006H:hapter S-USerxIs
                                            Full Precision  OFF
                                     Confidence Coefficient  35%
                                      Substantial Difference  0.000
                                            Action Levd  800.000
                                    Selected Null Hypothesis  Mean <= Action Level (Form 1)
                                      Alternative hypothesis  Mean > the Action Level
                              WSR1
                                              One Sample t-Test

                                                Raw Statistics
                                          Number of Valid Observations   10
                                         Number of Distinct Observations   10
                                                       Minimum  750
                                                       Maximum  1161
                                                         Mean  925.7
                                                        Median  388
                                                           SD  136.7
                                                     SE of Mean   43.24
                              HO: Sample Mean <= 800  (Form 1)
                                                      Test Value   2.907
                                                Degrees of Freedom   9
                                                 Critical Value (0.05)   1.833
                                                        P-Value   0.00869
                              Conclusion with Alpha = 0.05
                                Reject HO. Conclude Mean > 800
                                P-Value < Alpha (0.05)
9.1.1.2         Single-Sample Proportion  Test
1.       Click Single Sample Hypothesis ^ Full (w/o NDs) ^ Proportion
        Statistical Tests   Upper Limits/BTVs   UCLs/EPCs   Windows    Help
Outlier Tests >
Goodness-of-Fit Tests >
Single Sample Hypothesis >
Two Sample Hypothesis >
Oneway AN OVA >
OLS Regression
Trend Analysis >
6

7

8

9

Full (w/o NDs) > 1
With NDs >












10

11

12

tTest
Proportion
Sign Test
Wilcoxon Signed Rank







144

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2.      The Select Variables screen will appear.

          •   Select variable(s) from the Select Variables screen.

          •   When the Options button is clicked, the following window will be shown.
                       Select Uncensored Proportion Opti
                       Select Null Hypothesis Form

                       ฎ Sample 1 Proportion. P <= PO (Fonn 1)

                       O Sample 1 Proportion. P >= PO (Form 2)

                       O Sample 1 Proportion. P = PO (Two Sided)
                                    Confidence Level

                                      Proportion. PO

                      Action Level (For % Exceedances)
 0.95
  0.2
 800
                                            OK
Cancel
             o   Specify the Confidence level; default is 0.95.

             o   Specify the Proportion level and a meaningful Action Level.

             o   Select the  form of Null Hypothesis; default is  Sample  1  Proportion  <= PO
                 (Form 1).

             o   Click on OK button to continue or on Cancel button to cancel the test.
                                                                                    145

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         Example 9-lb (continued). Consider the WSR data set described in EPA  (2006a). One  Sample
         proportion test results are summarized as follows.
                    Output for Single-Sample Proportion Test (Full Data without NDs)
         User Selected Options
     Date/Time of Computation  3/17/201310:29:33 P M
                From File  WSR EPA (2006}chapter 9-USerjds
                        OFF
                        95X
                              Full Precision
                        Confidence Coefficient
                      User Specified Proportion  0.200 (PO of Exceedances of Action Level)
                       Action/compliance Limit  800.000
                        Select Null Hypothesis  Sample Proportion. P of Exceedances of Action Level >= User Specified Proportion (Form 2}
                        Alternative Hypothesis  Sample Proportion. P of Exceedances of Action Level cthe User Specified Proportion

WSR1

One Sample Proportion Test




















                  Raw Statistics
            Number of Valid Observations
           Number of Distinct Observations
                          Minimum
                                                   10
                                                   10
                                                  750
                                          Maximum
                                            Mean
                                           Median
                                             SD
1161
925.7












                       SE of Mean
               Number of Exceedances
         Sample Proportion of Exceedances

HO: Sanple Proportion >= 0.2  [Form 2)
                                                  136.7
                                                  43.24
                                                   B
                                                   0.8
                Conclusion with Alpha = 0.05
                  Do Not Reject HO. Conclude
                  P-Value > Alpha (0.05)



) 1

'roportion >= 02




























9.1.1.3          Single-Sample Sign Test
1.       Click Single Sample Hypothesis ^  Full (w/o NDs) >  Sign test
       Statistical Tests    Upper Limits/BTVs
                              UCLs/EPCs   Windows    Help
Outlier Tests ^
Goodness-of-Fit Tests ^
Single Sample Hypothesis ^
Two Sa m p 1 e Hy p oth esi s >
Oneway AN OVA >
OLS Regression
Trend Analysis >
6 7


8

9 10

Full (w/o NDs) >
With NDs >













11

12

tTest
Proportion
Sign Test
Wilcoxon Signed Rank






146

-------
2.      The Select Variables screen will appear.

          •   Select variable(s) from the Select Variables screen.

          •   When the Options button is clicked, the following window will be shown.
                         Select Uncensored Sign Test Options
                     Select Null Hypothesis Form

                    <.._.'!  Sample Median <= Action Level (Form 1)

                    (j  Sample Median >= Action Level (Form 2)

                    O  Sample Median >= Action Level + S (Form 2)

                     •  Sample Median = Action Level (Two Sided)
                                     Confidence Level

                      Substantial Difference. S (Form 2)


                                         Action Level
                                                             Cancel
              o   Specify the Confidence Level; default choice is 0.95.

              o   Specify meaningful values for Substantial Difference, S and Action Level.

              o   Select the form of Null Hypothesis; default is Sample Median <= Action Level
                 (Form 1).

              o   Click on OK button to continue or on Cancel button to cancel the test.
                                                                                      147

-------
         Example 9-lc (continued). Consider the WSR data set described in EPA (2006a). The Sign test
         results are summarized as follows.
                      Output for Single-Sample Sign Test (Full Data without NDs)
                                  From Rle
                              Full Precision
                       Confidence Coefficient
                        Substantial Difference
                               Action Level
                      Selected Null Hypothesis
                        Alternative Hypothesis
WSR EPA(200ffH;hapterS-USer.xls
OFF
95%
0.000
5.000
Median = Action/compliance Limit (2 Sided Alternative)
Median o Action/compliance Limit
              WSR2
                                               One Sample Sign Test
                                                  Raw Satisfies
                                       Number of Valid Observations    49
                                     Number of Distinct Observations    44
                                                      Minimum    1.09
                                                      Maximum    7.5
                                                         Mean    5.048
                                                       Median    5.55
                                                          SD    1.775
                                                    SE of Mean    0.254
                                        Number Above .Action Level    29
                                        Number Equal Action Level    0
                                        Number Below .Action Level    20
              HO: Sample Median = 5
                                       Large Sample Z Test Statistic    1.236
                                                       P-Value    0.199
              Conclusion with Alpha = 0.05
                Do Not Reject HO at the specified level of significance (0.05). Conclude Median = 5
                P-Value > Alpha (0.05)
148

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9.1.1.4       Single-Sample Wilcoxon Signed Rank (WSR) Test
1.      Click Single Sample Hypothesis ^- Full (w/o NDs) ^- Wilcoxon Signed Rank
                                              CL 5.0 - [WSR EPA (2006)-chapter 9-U
Statistical Tests | Upper Limits/BTVs UCLs/EPCs Windows Help
Outlier Tests *•
Goodness-of-Fit Tests t
Single Sample Hypothesis *
Two Sample Hypothesis ^
Oneway AN OVA >
OLS Regression
Trend Analysis t
678


5 10

Full [w/o NDs) >
With NDs >






11

12

tTest
Proportion
Sign Test
Wilcoxon Signed Rank




2.      The Select Variables screen will appear.
          •   Select variable(s) from the Select Variables screen.
          •   When the Options button is clicked, the following window will be shown.
                           Select Null Hypothesis Form
                           •  Mean/Median <= Action Level (Form 1)
                           O Mean/Median >= Action Level (Form 2)
                           O Mean/Median >= Action Level + S (Form 2)
                           O Mean/Median = Action Level (Two Sided]

                                       Confidence Level

                                           Action Level
                                               OK
Cancel
              o   Specify the Confidence Level; default is 0.95.
              o   Specify meaningful values for Substantial Difference, S, and Action Level.
              o   Select form of Null Hypothesis; default is Mean/Median <= Action Level (Form 1).
                                                                                        149

-------
                o   Click on OK button to continue or on Cancel button to cancel the test.

        Example 9-ld (continued). Consider the WSR data set described in EPA (2006a). One Sample
        WSRtest results are summarized as follows.

           Output for Single-Sample Wilcoxon Signed Rank Test (Full Data without NDs)

                            Confidence Coefficient   95%
                            Substantial Difference   0.000
                                    Action Level   800.000
                          Selected Null Hypothesis   Mean/Median <= .Action Level (Form 1}
                            Alternative Hypothesis   Mean/Median > the Action Level

                   WSR1

                             One Sample Wilcoxon Spied Rank Test

                                         Raw Statistics
                                  Number of Valid Observations    10
                                Number of Distinct Observations    10
                                                  Minimum   750
                                                 Maximum  1161
                                                    Mean   925.7
                                                   Median   883
                                                      SD   136.7
                                               SE of Mean    43.24
                                   Number Above .Action Level    8
                                    Number Equal .Action Level    G
                                    Number Below Action Level    2
                                                   T-plus    50
                                                  T-minus    5

                   HO: Sample Mean/Median <= 800  (Form 1)

                                          Exact Test Statistic    50
                                         Critical Value {0.05}    45
                                                  P-Value   0.0098

                   Conclusion with Alpha = 0.05
                     Reject HO. Conclude Mean/Median > 800
                     P-Value < Alpha {0.05)
150

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9.1.2  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 ND values
assume that all ND observations lie below the specified action level, Ao. These single-sample tests are not
performed if ND observations exceed the action levels. Single-sample hypothesis tests for data sets with
NDs are shown in the following screen shot.
1.
2.
Click on Single Sample Hypothesis ^- With NDs
       Statistical Tests   Upper Limits/BTVs
                              UCLs/EPCs   Windows   Help
Outlier Tests >
Goodness-of- Fit Tests ^
Single Sample Hypothesis >
Two Sam pie Hypothesis >
Oneway AN OVA >

Trend Analysis >
6

Fi
W



7

ill [w/o ND
ith NDs



Q

0






>
>



9 10 11 12


Proportion
Sign Test
Wilcoxon Signed Rank

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 Single Sample Hypothesis ^- With NDs ^- Proportion

                                                  ProUCL 5.0 -
Statistical Tests | Upper Limits/BTVs
Outlier Tests >
G o o d n ess- of - Fit Tests >
Single Sample Hypothesis ป
Two Sample Hypothesis ป
Oneway AN OVA ^
OLS Regression
Trend Analysis ^
UCLs/EPCs Windows Help
673

Full (w/o NDs) f
Wrth NDs >

3 1C 11 12

Proportion
Sign Test
Wilcoxon Signed Rank

                                                                                         151

-------
2.       The Select Variables screen will appear.

           •   Select variable(s) from the Select Variables screen.

           •   If hypothesis test has to be performed by using a Group variable, then select a group
               variable by clicking the arrow below the Select Group Column (Optional) 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. This option has been used in the
               following screen shot for the single-sample proportion test.
                                  Available Variables
                   Selected Variable
                                  Name
                                  Cu
ID
D
Name
Zn
ID
1
                                                              Select Group Column (Optional)
                                                      Options   <- Sdect an action level

                                                              [  OK        Caned
               When the Options button is clicked, the following window will be shown.
                                     Select Censored Proportion Options
                                   Select Null Hypothesis Form

                                   @ Sample 1 Proportion. P <= PO (Form 1)

                                   O Sanple 1 Proportion. P >= PO (Form 2)

                                   O Sai^le 1 Proportion. P = PO (Two Sided)

                                              Confidence Level

                                                Proportion. PO

                                  Action Level (For % Bcceedances)
                    tions
                                                     OK
                                                                Cancel
               o   Specify the Confidence Level; default is 0.95.

               o   Specify meaningful values for Proportion and the Action Level (=15 here).

               o   Select form of Null Hypothesis; default is Sample 1 Proportion, P <= PO (Form 1).

               o   Click on OK button to continue or on Cancel button to cancel the test.
152

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Example 9-2a. Consider the copper and zinc  data set collected from two zones: Alluvial Fan and Basin
Trough discussed in the literature (Helsel 2012, NADA in R [Helsel 2013]).  This data set is used here to
illustrate the  one sample proportion test on a data set with NDs. The output sheet generated by ProUCL
5.1 is presented below.
 Output for  Single-Sample Proportion Test  (with NDs) by Groups:  Alluvial Fan and Basin Trough
            User Selected Options
        Date/Time of Computation  3/1 a/2013 3:55:58 AM
                  From File  Zn-Cu-ND-dataxis
               Full Precision  OFF
          Confidence Coefficient  35%
         User Specified Proportion
               Action Level
             " 5K :?D of Exceedances of Action Level)
             15.000
Select Null Hypothesis  Sample Proportion, P of Btceedances of Action Level c= User Specified Proportion (Form 1)
Alternative Hypothesis
                        Sample Proportion, P uf F_-ceedancer of Action Level > User Specified Proportion
    Zn (alluvial fan)
               One Sample Proportion Ted
    Note: All nondetects are treated as detects at values (eg.. DLs) included in Data Rle

Raw Statistics
Number of Valid Data 67
Number of Missing Observations 1
Number of Distinct Data 1 5
Numberof Non-Detects 16
Number of Detects 51
Percent Non-Detects 23.85%
Minimum Non-detect 3
Maximum Non-detect 10
Minimum Detect 5
Maximum Detect 62D
Mean of Detects 27.88
Median of Detects 11
SD of Detects 35.02
Number of Exceedances 24
Sample Proportion of Exceedances 0.358



































































(




                                                                            HO: Sample Proportion <= 09  (Form 1)

                                                                                        Large Sample i-Test Statistic   -14.58
                                                                                             Critical Value (0.05)   1.645
                                                                                                    P-Value   1

                                                                            Conclusion with Alpha = 0 05
                                                                              Do Not Reject HO. Conclude Sample Proportion <= 0.9

                                                                              P-Value > Alpha (0 OS)
Zn (basin trough)




One Sample Proportion Test
Note: All nondetects are treated as detects at values (eg. DLs) included in Data H
Raw Statistics
Number of Valid Data 50
Number of Distinct Data 20
Numberof Non-Detects 4
Number of Detects 46
Percent Non-Detects S.OOH
Minimum Non-detect 3

Maximum Non-detect 10
Minimum Detect 3
Maximum Detect 90
Mean of Detects 23.13

SD of Detects 19.03

Number of Exceedances 27
Sample Proportion of Exceedances 0.54
















e










HO: Sample Proportion <= 0.9 (Form 1)
Exact P-Value 1

Conclusion with Alpha = 0.05
Do Not Reject HO. Conclude Sample Proportion <= 0.9
P-Value > Alpha (0.05)
                                                                                                                       153

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9.1.2.2        Single-Sample Sign Test with NDs

1.     Click Single Sample Hypothesis ^ With NDs ^ Sign test
2.
      Statistical Tests   Upper Limits/BTVs   UCLs/EPCs   Windows   Help
Outlier Tests ป
G o o d n ess- of - Fit Tests ป
Single Sample Hypothesis ป
Two Sample Hypothesis ป
Oneway AN OVA >
OLS Regression
Trend Analysis ป
6

7

8

9 10

Full (w/o NDs) (•
With NDs >










11

12


Proportion
Sign Test
Wilcoxon Signed Rank
E


The Select Variables screen will appear.

    •   Select variable(s) from the Select Variables screen.

    •   When the Options button is clicked, the following window will be shown.
                                Select Censored Sign Test Options
                          Select Null Hypothesis Form

                          O Sample Median <= Action Level (Form 1)

                          O Sample Median >= Action Level (Form 2)

                          •  Sample Median = Action Level (Two Sided)
                                            Confidence Level

                                                Action Level
                                                    OK
                                                        Cancel
               o   Specify the Confidence Level; default is 0.95.

               o   Select an Action Level.
               o   Select the form of Null Hypothesis; default is Sample Median <= Action Level
                  (Form 1).

               o   Click on OK button to continue or on Cancel button to cancel the test.
154

-------
Example 9-2b (continued). Consider the copper and zinc data set collected from two zones: Alluvial Fan
and Basin Trough discussed above.  This data set is used here to illustrate the Single-Sample Sign test on
a data set with NDs. The output sheet generated by ProUCL 5.0 follows.
                       Output for Single-Sample Sign Test (Data with Nondetects)
                     Zn (alluvial fan)
                           Selected Null Hypothesis
                            .Alternative Hypothesis
Median = Action/compliance Limrt (Two Sided Alternative}
Median o Action/compliance Limit
                                                  One Sample Sign Test
                            Note All nondetects are treated as detects at values (e.g.. DU) included in Data Rle
                                                     Raw Statistics
                                             Number of Valid Data   67
                                       Number of Missing Observations   1
                                           Number of Distinct Data   19
                                           Number of Non-Detects   16
                                              Number of Detects   51
                                             Percent Non-Detects 23.SS%
                                             Minimum Non-detect   3
                                             Maximum Non-detect
                                                Minimum Detect
                                                Maximum Detect
                                                Mean of Detects
                                              Median of Detects
                                                 SD of Detects
                                         Number Above Action Level
                                         Number Equal Action Level
                                         Number Below Action Level
                 10
                 5
                620
                 27.88
                 11
                 85.02
                 24
                 0
                 43
                     HO: Sample Median = 15
                               Standardized Test Value using Normal Appx.   -2.321
                                                     P-Value   0.0203
                     Conclusion with Alpha = 0.05
                       Reject HO at the specified level of significance (0.05). Conclude Median o 15
                       P-Vdue < Alpha (0.05)
9.1.2.3         Single-Sample Wilcoxon Signed Rank Test with NDs
1.       Click Single Sample Hypothesis ^- With NDs ^- Wilcoxon Signed Rank
                                                             ProUCL 5.0 - [Zn-Cu-ND-
        Statistkal Tests  I  Upper Limits/BTVs    UCLs/EPCs   Windows    Help
Outlier Tests >
Goodness-of-Fit Tests >
Single Sample Hypothesis ^
Two Sample Hypothesis ป
Oneway AN OVA >
OLS Regression
Trend Analysis >
6

7

&

5

Full (w/o NDs) >
With NDs >










10


Proportion
Sign Test
11



12



Wilcoxon Signed Rank



                                                                                                               155

-------
2.
The Select Variables screen will appear.

    •   Select variable(s) from the Select Variables screen.
    •   When the Options button is clicked, the following window will be shown.
                             <
                            Select Censored WSR Test Options

                     Select Null Hypothesis Form
                        Sample Mean/Median <= Action Level (Form 1)
                     •  Sample Mean/Median >= Action Level (Form 2)
                     O Sample Mean/Median = Action Level (Two Sided)
                                      Confidence Level

                                          Action Level
                                                OK
                                                                 Cancel
                o   Specify the Confidence Level; default is 0.95.
                o   Specify an Action Level.
                o   Select form of Null Hypothesis; default is Sample Mean/Median <= Action Level
                    (Form 1).
                o   Click on OK button to continue or on Cancel button to cancel the test.
Example 9-2c (continued). Consider the copper and zinc data set collected from two zones: Alluvial Fan
and Basin Trough discussed earlier in this chapter. This data set is used here to  illustrate one sample
Wilcoxon Signed Rank  test on  a data set with NDs. The output sheet generated by ProUCL  5.0  is
provided as follows.
           Output  for Single-Sample Wilcoxon Signed Rank Test (Data with Nondetects)
                         One Sample Wilcoxon Signed Rank Test for Data Sets with Non-Detects
               User Selected Options
           Date/Time of Computation  3/1S/2013 1:43:46 PM
                      From File  Zn-Cu-ND-datajds
                   Full Precision  OFF
              Confidence Coefficient  95%
                   Action Level  15.000
            Selected Null Hypothesis  Mean/Median >= Action Level (Form 2)
              Alternative Hypothesis  Mean/Median =: the Action Level
156

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     Output for Single-Sample Wilcoxon Signed Rank Test (Data with Nondetects)- continued
          Zn (basin trough)
                One Sample Wilcoxon Signed Rank Test
Raw Statistics
Number of Valid Data
Number of Distinct Data
Number of Non-Detects
Number of Detects
Percent Non-Detects
Minimum Non-detect
Maximum Non-detect
Minimum Detect
Maximum Detect
Mean of Detects
Median of Detects
SD of Detects
Median of Processed Data used in WSR
Number Above Action Level
Number Equal Action Level
Number Below Action Level
T-plus
T-minus
50
20
4
46
8.00X
3
10
3
90
23.13
20
1903
1S.5
27
1
22
764
461






HO: Sample Median >=
15 (Form 2)


Large Sample z-Test Statistic
Critical Value (0.05)


P-Value
Conclusion with Alpha = 0.05
Do Not Reject HO. Conclude Mean/Meet;
F -Value > Alpha (0.05)

1.269
-1.645
0898

n>= 15


Dataset contains multiple Non Detect values!
All NDs are replaced by their respective DL/2
9.2    Two-Sample Hypotheses Testing Approaches

The two-sample hypotheses testing approaches available in ProUCL are described in this section. Like
Single-Sample  Hypothesis,  the  Two-Sample  Hypothesis options are available under the Statistical
Tests module of ProUCL 5.0/ProUCL 5.1. These approaches are used to compare the parameters and
distributions of two populations (e.g., Background vs. AOC) based upon data sets collected from those
populations. Several forms (Form 1 and Form 2, and Form 2 with Substantial Difference, S) of the two-
sample hypothesis testing approaches are available in ProUCL. The methods are available for full-
uncensored data sets as well as for data sets with ND observations with multiple detection limits.

       •   Full (w/o NDs) - performs parametric and nonparametric hypothesis tests on uncensored data
           sets consisting of all detected values. The following 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.

       •   With  NDs - performs hypothesis tests on left-censored data sets consisting of detected and
           ND 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.
                                                                                           157

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               o   Gehan's test is useful when multiple detection limits may be present.

               o   Tarone-Ware test is useful when multiple detection limits may be present.

The details of these methods can be found in the ProUCL Technical Guides (2013, 2015) and are also
available in EPA (2002b, 2006a, 2009a, 2009b).  It is emphasized 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 data sets may consist of
NDs with multiple detection limits and observations from multiple populations  (e.g., mixture samples
collected from various onsite locations) and outliers.

Notes: As mentioned before, when one wants to use two-sample hypotheses tests  on data sets with NDs,
ProUCL assumes that samples from both of the groups have ND 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; the
user will have to provide a ND column (with 0 or 1 entries only) for the selected  variable of each of the
two samples/groups. Thus when one of the samples (e.g., site arsenic) has no ND value, the  user supplies
an associated ND column with all entries equal to  T. 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.


9.2.1 Two-Sample Hypothesis Tests for Full Data

Full (w/o NDs): This option is used to analyze data sets consisting of all detected values. The following
two-sample tests are available in ProUCL 5.1.

           •   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.

    •   Student's!-Test

           o   Based upon collected data sets, this test is used to compare the mean concentrations of
               two populations/groups provided the populations are normally distributed. The data sets
               are represented by independent random observations, XI, X2, . . ., Xn collected from one
               population (e.g., site), and independent random observations, Yl, Y2, .  . . , Ym collected
               from another (e.g., background) population. The same terminology is used for all other
               two-sample tests discussed in the following sub-sections of this section.

           o   Student's t-test also assumes  that the spreads  (variances)  of the  two populations are
               approximately equal.

           o   The F-test can be used to the check the equality of dispersions of two populations.  A
               couple of other tests (e.g., Levene 1960) are also available in the literature to compare the
               variances of two populations. Since the F-test performs fairly well, other  tests are not
               included in the ProUCL software. For more details refer to ProUCL Technical Guides.
158

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    •  Satterthwaite t-Test
               This test is used to compare the means of two populations when the variances 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 the two populations. However, this test alone
               is more powerful test to compare the means 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
               sensitive to departures from normality.

    •  Two-Sample Nonparametric WMW Test

           o   This test is used to determine the comparability 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 (mean, median) of the two populations are
               significantly different.

           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 WMW test
               statistic can be approximated by a normal distribution.

Notes: 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, cleanup and remediation decisions have
to be made in the 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.
1.
Click on Two Sample Hypothesis ^- Full (w/o NDs)
                    Upper Limits/BTVs   UCLs/EPCs
                                         ProUCLS.O-
                                       Windows   Help
Outlier Tests ^
G o o d n ess- of - Fit Tests ^
Single Sample Hypothesis ป
Two Sample Hypothesis >
Oneway AN OVA >
OLS Regression
Trend Analysis ^
6 7
ln-89
4600
8 9
MW5 MN9
9 22DD
Full (w/o NDs) >
With NDs ^
1790
1730
10


11
MN-99
2200
12
D.MN-99
1
tTest
Wilcoxon-Mann-Whitney
9T 2150
9 222^

21 ED
222 D
1
0
                                                                                            159

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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.


9.2.1.1        Two-Sample t- Test without NDs

1.      Click on Two Sample  Hypothesis ^ Full (w/o NDs) ^ t Test

2.              The Select Variables screen will appear.

           •   Select variable(s) from the Select Variables screen.
                      Available Variables
                                               tVarial

                                                 Without Group Variable
                      Name    ID
                      Well ID   0
                      Mn     1
                      MW-ID   2
                      Manganese 3
                      MW-89   5
                      MWS     8
                      MN9    9
Count
48
32
32
                      MN99
                      index
              Sample 1
              Sample 2
                                                          Name
                                               • Using Group Variable
                                               Variable



                                               Group Variable


                                               Sample 1


                                               Sample 2
Name
Mn-89
ID
6
Count
32


MW-89
( Count = 32 )

V
               Without Group Variable: This option is used when the  sampled data of the variable
               (e.g.,  lead) for the  two populations (e.g.,  site vs. background) are given in separate
               columns.

               With  Group Variable: This option is used when sampled data of the variable (e.g., lead)
               for the two populations (e.g., site vs. background) are given  in the same column.

               The values  are separated into  different populations  (groups)  by  the  values  of an
               associated Group ID Variable. The group  variable may represent several populations
               (e.g.,  background, surface, subsurface,  silt,  clay, sand,  several AOCs, MWs). The user
               can compare two groups at a time by using this option.

               When the Group option is used, the user then selects  a  variable by using the Group
               Variable Option. The user should  select an appropriate variable representing a group
               variable. The user can use letters, numbers, or alphanumeric labels for the group names.
160

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                  When the Options button is clicked, the following window will be shown.
                                     Select t Test Options
                              Select Null Hypothesis Form

                              • Sample 1 <= Sample 2 (Form 1)

                             O Sample 1 >= Sample 2 (Form 2)

                             O Sample 1 >= Sample 2 + S (Ftxm 2)

                             O Sample 1 = Sample 2 (Two Sided)

                              Select Confidence Coefficient

                             O 99 9*    O 99.5%    O 99*

                             O 97.5%    (ง) 95%      O 90%
                                         OK
              o   Specify a useful Substantial Difference, S value. The default choice is 0.

              o   Select the Confidence Coefficient. The default choice is 95%.

              o   Select the form of Null Hypothesis. The default is Sample 1 <= Sample 2 (Form 1).

              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  Sample 1 versus
              Sample 2 Comparison.

Example 9-3. Consider the manganese concentrations data set collected from three wells: MW1, an
upgradient well, and MW8 and MW9, two downgradient wells. The two-sample t-test results, comparing
Mn concentrations in MW8 vs. MW9, are described as follows.
                                                                                         161

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                         Output for Two-Sample t-Test (Full Data without NDs)
Confidence Coefficient
Substantial Difference (S)
Selected Null Hypothesis
Alternative Hypothesis
95*i
0.000
Sample 1 Mean = Sample 2 Mean (Two Sided Alternative}
Sample 1 Mean o Sample 2 Mean


                 Sample 1 Data: Mn-89(8)
Sample 2 Data: Mn-KS'3)



Number of Valid




Raw Statistics
Sample 1
Observations 16
Number of Distinct Observations






Sample I vs

Minimum
Maximum
Mean
Median
SD
SE of Mean
16
1270
4600
1998
1750
S3S.B
2C3.7


Sample 2
16
15
1050
3080
1968
2C55
500.2
125










Sample 2 Two-Sample t-Test







                HO: Mean of Sample 1 = Mean of Sample 2
                                                          t-Test
                                                          Value
                                                          0.123
                                                          0.123
Method                           DF
Pooled (Equal Variance)              30
Welch-Satterthvvalte (Unequal Variam    24.5
Pooled SD: 690.548
Conclusion with Alpha = O.D50
 Student! (Pooled): Do Not Reject HD, Conclude Sample 1 = Sample 2
 Welch-Satterthwaite: Do Not Reject HD. Conclude Sample 1 = Sample 2
Lower C.Val Upper C.Val
 t (0.025)   t (0.975)    P-Value
  -2.042     2.W2      C.SC3
  -2.0S4     2.064      C<.BC3
                                     Test of Equality of Variances
                                     Variance of Sample 1
                                     Variance of Sample 2
                                        703523
                                        250190
                    Numerator DF        Denominator DF
                         15                  15
                Conclusion with Alpha = 0.05
                 Two variances appear to be equal
                                             F-Test Value
                                               2.812
            P-Value
            0.054
162

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9.2.12        Two-Sample Wilcoxon-Mann-Whitney (WMW) Test without NDs

1.     Click on Two Sample Hypothesis Testing ^ Full (w/o NDs) ^ Wilcoxon-Mann-Whitney
          Statistical Tests   Upper Limits/BTVs   UCLs/EPCs   Windows   Help
Outlier Tests K
Goodness-of- Fit Tests >
Single Sample Hypothesis k
Two Sample Hypothesis ฅ•
Oneway AN OVA >
OLS Regression
Trend Analysis K
6
1n-89
7

S 5
MW9 MN9
10

4600 9 2200
Full (w/o NDs) >
With NDs >
11
MN-99
12
D_MN-99
2200 1
tTest
Wilcoxon-Mann-Whitney
1790 9 2150
173.

9 2220
2150 1
2220
D
2.      The Select Variables screen will appear.
                   Available Variables
                                               • Without Group Variable
                                               Sample 1
                                               Sample 2
                                                           Name     ID    Count
                                                           X3       0     24
Name     ID
Y3       1
Count
25
                                                 Using Group Variable
                                               Variable



                                               Group Variable


                                               Sample 1


                                               Sample 2
                                                           Name     ID    Count
               Select variable(s) from the Select Variables screen.

               Without Group Variable: This option is used when the  data values of the variable
               (arsenic) are given in separate columns.

               With Group Variable: This option is used when data of the variable (arsenic) are given
               in the same column. The values are  separated into  different samples (groups) by the
               values of an associated Group Variable.

               When the  Group option is used, the user then selects a group variable/ID by using the
               Group Variable Option. The user should select an appropriate variable representing a
                                                                                             163

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              group variable. The user can use letters, numbers, or alphanumeric labels for the group
              names.

       Notes:  ProUCL documents  have been  written  using environmental  terminology  such  as
       performing background versus site comparisons. However, all tests and procedures incorporated
       in ProUCL can be used on data sets from any other application. For other applications such as
       comparing a new treatment drug versus older treatment drug, the group variable may represent
       the two groups:  Control Drug and New Drug.

          •   When the Options button is clicked, the following window is shown.
                       a~? Select Wilcoxon- Mann- Whitney

                           Select Null Hypothesis Form
                           O Sample 1  <= Sample 2 (Form 1)

                           O Sample 1  >= Sample 2 (Form 2)

                           O Sample 1  >= Sample 2 + S (Form 2)

                           *  Sample 1 = Sample 2 (Two Sided)

                           Select Confidence Coefficient

                           O 99 9*    O 99.5%     O 99%

                           O 37.5%    •:•:•  95%      O 90%


              o   Specify a Substantial Difference, S value. The default choice is 0.

              o   Choose the Confidence Coefficient. The default choice is 95%.

              o   Select the form of Null Hypothesis. The default is Sample 1<= Sample 2 (Form 1).

              o   Click on OK button to continue or on Cancel button to cancel the selected options.

          •   Click on OK to continue or on Cancel to cancel Sample 1 vs. Sample 2 comparison.
164

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Example 9- 4. The two-sample Wilcoxon Mann Whitney (WMW) test results on a data set with ties are
summarized as follows.

           Output for Two-Sample Wilcoxon-Mann-Whitney Test (Full Data with ties)
Selected Null Hypothesis Sample 1 Mean/Median
Alternative Hypothesis Sample 1 Mean/Median
= Sample 2 Mean/Median (Two Sided Memative)
o Sample 2 Mean/Median

Sample 1 Data: X3
Sample 2 Data: Y3
Raw Statistics
Sample 1






Number erf Valid Observations
Number of Distinct Observations
Minimum
Maximum
Mean
Median
SD
SE of Mean
24
i
5.687
31.2
17.38
17.56
7.421
1.515
Sample 2
25
19
1,85
79.06
39.8
44.63
19.39
3.878

HO: Mean/Median of Sample 1 = Mean/Median of Sample 2

Sample 1 Rank Sum W-Stat 396
WMW U-Stat %
Standardized WMW U-Stat -4.0B3
Mean (U) 300
SD(U)-Mjties 49.97
Lower Approximate U-Stat Critical Value (0.025) -1 .96
Upper Approximate U-Stat Critical Value {0.975} 1 .96
P-Value (Adjusted for Ties) 4.4474E-5

Conclusion with Alpha = 0.05
Reject HO. Conclude Sample 1 o Sample 2

P Value < alpha (0.05)
9.2.2  Two-Sample Hypothesis Testing for Data Sets with Nondetects

1.      Click Two Sample Hypothesis ^- With NDs
                                         •oUCL 5.0 - [Zn-Cu-ND-data-chapter 9-u
     Statistical Tests I  Upper Limits/BTVs   UCLs/EPCs   Windows   Help
Outlier Tests ^
Goodness-of-Fit Tests K
Single Sample Hypothesis ป
Two Sample Hypothesis t-
Oneway AN OVA ^
OLS Regression
Trend Analysis ป
6

7

a,

9


Full (w/o NDs) >
With NDs *









10



11



12







Gehan
Tarone-Ware
Wilcoxon-Mann-Whitney
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 Tarone-Ware test, click on Tarone-Ware from the drop-down menu.
                                                                                       165

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9.2.2.1       Two-Sample Wilcoxon-Mann-Whitney Test with Nondetects

1.      Click Two Sample Hypothesis ^- With NDs ^- Wilcoxon-Mann-Whitney
2.
Statistical Tests | Upper Limits/BTVs UCLs/EPCs Windows Help
Outlier Tests >
Goodness-of-Fit Tests t
Single Sample Hypothesis t
Two Sample Hypothesis >
Oneway AN OVA >
OLS Regression
Trend Analysis ^
6 7

8

9 10


Full (w/o NDs) >
With NDs >




11


12





Gehan
Ta rone- Ware
Wilcoxon-Mann-Whitney
The Select Variables screen shown below will appear.

Available Variables • Without Group Variable
Name ID Count
ป
Sample 1
ซ
" Sample 2
ซ
O Using Group Vs
ป
Variable
ซ
Croup Variable
Sample 1
Sample 2
Options
Name ID
Site 1
Count
10

Name ID
Background 0

liable
Name ID

OK

Count
10


Count
v
Caned

              Select variable(s) from the Select Variables screen.

              Without Group Variable: This option is used when the data values of the variable (e.g.,
              TCDD 2,3,7,8) for the site and the background are given in separate columns.

              With Group Variable: This option is used when data values of the variable (TCDD 2, 3,
              7,  8) are given in the same column. The  values are  separated  into different samples
              (groups) by the values of an associated Group Variable. When using this  option, the
              user should select an appropriate variable representing groups such as AOC1, AOC2.
              AOC3 etc.

              When the Options button is clicked, the following window will be shown.
166

-------
                         a  Select Wilcoxon-Mann-Whitne...
                             Select Null Hypothesis Form

                            '.._) Sample 1 <= Sample 2 (Form 1)

                             ป Sample 1 >= Sample 2 (Form 2)

                            O Sample 1 = Sample 2 (Two Sided)


                             Select Confidence Coefficient

                            O 99.9%    O 99-5*    O 99*

                            O 97.5%    * 95%      O 90%
                                                 Cancel
              o  Choose the Confidence Coefficient. The default choice is 95%.

              o  Select the form of Null Hypothesis. The default is Sample 1 <= Sample 2 (Form 1).

              o  Click on OK button to continue or on Cancel button to cancel the selected options.

           •   Click on OK to continue or on Cancel to cancel the Sample 1 vs. Sample 2 comparison.

Example 9-5. Consider a two sample data set with nondetects and multiple detection limits. Since the
data sets have more than one detection limit, the  WMW test is not recommended for this data set.
However, sometimes, the users tend to use the WMW test on data sets with multiple detection limits. The
WMW test results are summarized as follows.
                                                                                       167

-------
            Output for Two-Sample Wilcoxon-Mann-Whitney Test (with Nondetects)
Date/Time of Computation 3/1 8/201 3 6:43:04 PM
From File WMW-NDs-Chapter&user_ajds
Full Precision OFF
Confidence Coefficient 95";






Selected Null Hypothesis Sample 1 Mean/Median s= Sample 2 Mean/Median (Form 2)
Alternative Hypothesis Sample 1 Mean/Median : Sample 2 Mean/Median
Sample 1 Data: Site
Sample 2 Data: Background


Raw Statistics
Sample 1
Number of Valid Data 11
Number of Non-Detects
Number of Detect Data
Minimum Non-Detect
Maximum Non-Detect
Percent Non-detects
Minimum Detect
Maximum Detect
Mean of Detects
Median of Detects
SD of Detects
3
8
4
11
27.27%
2
43
27
29.5
13.71


Sample 2
11
3
4
9
27.27%
1
27
15.5
16.5
9.196

















WMW test is meant fora Single Detection Limit Case
of Gehan or T-W test is suggested when multiple detection limits are pres
All observations <- 1 1 (Max DL) are ranked the same
WMW test is meant for a Single Detection Limit Case
Use of Gehan or T-W test is suggested when multiple detection limits are present
All observations <= 1 1 (Max DL) are ranked the same

Wilco.on-Mar.n-Whitney (WMW) Test

HO Mean/Median of Sample 1 >= Mean/Median of Sample 2

Sample 1 Rank Sum W-Stat 144.5
WMW U-Stat 78.5
Mean (U) 60.5
SD(U)-Aijties 15.22
WMW U-Stat Critical Value (0.05) 35
Standardized WMW U-Stat 1191
Approximate P-Value O.SS3

Conclusion with Alpha - 0.05
Do Not Reject HO. Conclude Sample 1 >= Sample 2
Notes: 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
tends to reduce the associated power of the WMW test considerably. This in turn may lead to an incorrect
conclusion.

9.2.2.2       Two-Sample Gehan Test for Data Sets with Nondetects

1.       Click Two Sample Hypothesis ^- With NDs ^- Gehan
     Statistical Tests   Upper LimitE/BTVs
UCLs/EPCs  Windows   Help
Outlier Tests >
G o o d n ess- of - Fit Tests ^
Single Sample Hypothesis t
Two Sample Hypothesis ป
Oneway AN OVA ^
OLS Regression
Trend Analysis ^
6

7

8


9


Full (w/o NDs) t
With NDs >




10


11


12






Gehan
Ta rone- Ware
Wi 1 c oxo n - M a n n - Wh itn ey
168

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2.
The Select Variables screen will appear.
                        Available Variables
                                                  : Without Group Variable
                                   Count
                                   118
                                                 Sample 1
                                                 Sample 2
                                                            Name    ID    Counl
                                                 • Using Group Variable


                                                 Variable
                                                            Zn
                                                           ID    Counl
                                                 Group Variable   Zone (Counl-118)


                                                 Sample 1      alluvial fan
                                                 Sample 2
                Select variable(s) from the Select Variables screen.

                Without Group Variable: This option  is used  when the data values of the variable
                (Zinc) for the two data sets are given in separate columns.

                With Group Variable: This option is used when data values of the variable (Zinc) for
                the two data sets are given in the same column. The values are separated into different
                samples (groups) by  the values of an associated  Group  Variable. When using this
                option, the user should select a group variable representing groups/populations such as
                Zone 1, Zone2, ZoneS, etc.

                When the Options button is clicked, the following window will be shown.
                                             Select Gehan Options


                                        Select Null Hypothesis Form

                                       O Sample 1 <= Sample 2 (Form 1)

                                        •  Sample 1 >= Sample 2 (Form 2)

                                       O Sample 1 = Sample 2 (Two Sided)

                                        Select Confidence Coefficient

                                       O 99-9*   O 99 5%       99%

                                       O 97-5X   ฎ 95%      O 90%


                                              OK
                                                                                                 169

-------
                o   Choose the Confidence Coefficient. The default choice is 95%.
                o   Select the form of Null Hypothesis. The default is  Sample 1 <= Sample 2 (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  Sample 1 vs.
                Sample 2 Comparison.
Example 9-6a. Consider the copper and zinc data set collected from two zones: Alluvial Fan and Basin
Trough discussed in the literature (Helsel 2012).  This data set is used here to  illustrate the Gehan two-
sample test. The  output sheet generated by ProUCL 5.0  (similar sheet is generated using  ProUCL 5.1,
therefore the following output is not replaced) follows.
                        Output for Two-Sample Gehan Test (with Nondetects)
                           Confidence Coefficient   95%
                         Selected Null hypothesis   Sample 1 Mean/Median >= Sample 2 Mean/Median (Form 2}
                           Alternative Hypothesis   Sample 1 Mean/Median : Sample 2 Mean/Median
                   Sample 1 Data: Zn(alluvial fan)
                   Sample 2 Data: Zn(basin trough)
                                           Raw Statistics
                                                     Sample 1   Sample 2
                                                       67
                                                        1
                                                       16
                                                       51
                                                        3
                                                       10
               Number of Valid Data
        Number of Missing Observations
              Number of Non-Detects
              Number of Detect Data
               Minimum Non-Detect
               Maximum Non-Detect
               Percent Non-detects   23.88%
                   Minimum Detect     5
                   Maximum Detect
                   Mean of Detects
                  Median of Detects
                     SD of Detects
          EC
          0
          4
          46
          3
          10
       8.00%
          3
620       3D
 27.8B     23.13
 11        2D
 S5.D2     19.03
                                  Sample 1 vs Sample 2 Gehan Test
                   HO: Mean of Sample 1 >= Mean of background
                                       Gehan 2 Test Value   -3.037
                                          Critical z (0.05)   -1.645
                                               P-Value   0.0012
                   Conclusion with Alpha = 0.05
                     Reject HO. Conclude Sample 1 < Sample 2
170

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9.2.2.3        Two-Sample Tarone-Ware Test for Data Sets with Nondetects

The two-sample Tarone-Ware (T-W) test (1978) for data sets with NDs is new in ProUCL 5.0.

1.      Click Two Sample Hypothesis Testing ^ Two Sample ^ With NDs ^ Tarone-Ware
Statistical Tests
Outlier Test
Goodness-c
Single Samf
Upper Limits/BTVs UCLs/EPCs Windows Help
: K
f-Fit Tests >
le Hypothesis ป
Two Sample Hypothesis ^
Oneway AN OVA >
OLS Regression
Trend Analysis >
6 7 8 9 10 11 12








Full (w/o NDs] >
With NDs ป>















Gehan
Tarone-Ware
Wikoxon- Mann-Whitney
       The Select Variables screen will appear.
                  Available Variables
                                                Without Group Variable
                  Name
                  Cu
ID
0
Counl
118
                                              Sample 1
                                              Sample 2
                                              ฎ Using Group Variable

                                              Variable
                                 Name     ID    Counl
                                 Zn       1     118
                                              Group Variable   Zone (Count-118)


                                              Sample 1       alluvial fan
                                              Sample 2
                                                          basin trough
               Select variable(s) from the Select Variables screen.

               Without Group Variable: This option is used when the data values of the variable (Cu)
               for the two data sets are given in separate columns.

               With Group Variable: This option is used when data values of the variable (Cu) for the
               two data  sets are given  in the same column. The values are separated into different
               samples (groups)  by the values  of an associated Group Variable. When using this
                                                                                            171

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              option, the user should select a group variable/ID by clicking the arrow next to the
              Group Variable option for a drop-down list of available variables. The user selects an
              appropriate group variable representing the groups to be tested.

              When the Options button is clicked, the following window will be shown.
                                 Select Tarone-Ware Options


                              Select Null Hypothesis Form

                              *  Sample 1 <= Sample 2 (Form 1}

                              O Sample 1 >= Sample 2 (Form 2)

                              i.J1 Sample 1 = Sample 2 [Two Sided)

                              Select Confidence Coefficient

                              O 99.9%    O 99.5%    O 99%

                              O 97-5*    '* 95%      O 90%
              o   Choose the Confidence Coefficient. The default choice is 95%.

              o   Select the form of Null Hypothesis. The default is Sample 1 <= Sample 2 (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 Sample 1 vs.
              Sample 2 Comparison.
172

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Example 9-6b (continued). Consider the copper and zinc data set used earlier. The data set is used here
to illustrate the T-W two-sample test. The output sheet generated by ProUCL 5.0/ProUCL 5.1 is
described as follows.

                     Output for Two-Sample Tarone-Ware Test (with Nondetects)
                          Confidence Coefficient   95%
                        Selected Null Hypothesis   Sample 1 Mean/Median <= Sample 2 Mean/Median (Form 1}
                          Memative Hypothesis   Sample 1 Mean/Median > Sample 2 Mean/Median
                  Sample 1 Data: Zn(alluvial fan)
                  Sample 2 Data: Zn(basin trough)

                                           Raw Statistics
                                                       Sample 1   Sample 2
                                    Number of Valid Data      67       50
                            Number of Missing Observations      1        0
                                  Number of Non-Detects      16       4
                                      Number of Detects      51       46
                                    Minimum Non-Detect      3        3
                                    Maximum Non-Detect      10       1fl
                                    Percent Non-detects    23.88%    8.00%
                                        Minimum Detect      5        3
                                        Maximum Detect     620       M
                                        Mean of Detects      27,88     23.13
                                      Median of Detects      11       20
                                         SD of Detects      85.02     19.03

                               Sample 1 vs Sample 2 Tarone-Ware Test

                  HO: Mean/Median of Sample 1 <= Mean/Median of Sample 2

                                            TW Statistic   -2.113
                                    TW Critical Value (0.05)    1.645
                                               P-Value    0.383

                  Conclusion with Alpha  = 0.05
                    Do Not Reject HO, Conclude Sample 1  <= Sample 2
                    P-Value >= alpha (0.05)
                                                                                                        173

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174

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                                     Chapter 10


    Computing  Upper Limits to Estimate Background Threshold
      Values Based Upon  Full Uncensored  Data Sets and Left-
                    Censored  Data Sets with Nondetects


This chapter illustrates the computations of parametric and nonparametric statistics and upper limits that
can be used as estimates of BTVs and other not-to-exceed  values. The BTV estimation methods are
available for data sets with and without ND observations. Technical details about the computation of the
various limits can be found  in the associated ProUCL 5.1 Technical Guide. For each selected variable,
this option computes various upper limits such as UPLs, UTLs, USLs and upper percentiles to estimate
the BTVs that are used in site versus background evaluations.

Two choices are available to  compute background statistics for data sets:

          •   Full (w/o NDs)  - computes background statistics for uncensored full data sets without
              any ND observation.

          •   With NDs -  computes background statistics for data sets consisting of detected as well as
              nondetected  observations with multiple detection limits.

The user specifies the confidence coefficient (probability) associated with each interval estimate. ProUCL
accepts a CC value in the interval [0.5,  1), 0.5 inclusive. The default choice is 0.95. For data sets with and
without NDs, ProUCL  5.0/ProUCL 5.1  can compute the following upper limits to estimate BTVs:

          •   Parametric and nonparametric upper percentiles.

          •   Parametric and  nonparametric UPLs for a single observation,  future or next  k (>  1)
              observations, mean of next k observations. Here  future k,  or next k observations may
              represent k observations from another population  (e.g., site) different from the sampled
              (background) population.

          •   Parametric and nonparametric UTLs.

          •   Parametric and nonparametric USLs.

Note on Computing  Lower Limits: In  many environmental applications (e.g., groundwater monitoring),
one needs  to compute lower limits including: lower prediction  limits (LPLs), lower tolerance  limits
(LTLs), or lower simultaneous limit (LSLs). At present, ProUCL does not directly compute a LPL, LTL,
or a LSL.  It should be noted that for data sets with and without nondetects, ProUCL outputs several
intermediate results  and critical values (e.g.,  khat,  nuhat, K, d2max) needed to compute the interval
estimates and lower limits. For  data sets with and without NDs,  except for the bootstrap methods, the
same critical value (e.g., normal z value, Chebyshev critical value, or t-critical value) can be used to
compute a parametric LPL, LSL, or  a LTL (for  samples  of size >30 to be able to  use Natrella's
approximation in LTL) as used in the computation of a UPL, USL, or a UTL (for samples of size >30).
Specifically, to compute a LPL, LSL, and LTL (n>30) the '+'  sign  used in the  computation of the


                                                                                     175

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corresponding UPL, USL, and UTL (n>30) needs to be replaced by the '-' sign in the equations used to
compute UPL, USL, and UTL (n>30). For specific details, the user may want to consult a statistician. For
data sets without ND observations, the user may want to  use the Scout 2008 software package  (EPA
2009c) to compute the various parametric and nonparametric LPLs, LTLs (all sample sizes), and LSLs.

10.1   Background Statistics for Full  Data Sets without Nondetects

1.      Click Upper Limits/BTVs ^ Full (w/o NDs)
Statistical Tests
3
Upper Limits/BTVs UCLs/EPCs Windows Help
Full (w/o NDs) >
With NDs >

Normal
Gamma
Lognormal
Non-Parametric
All
10 11 12 13
2.
Select Full (w/o NDs)
              To compute background statistics assuming the normal distribution, click on Normal
              from the drop-down menu list.

              To compute background statistics assuming the gamma distribution, click on Gamma
              from the drop-down menu list.

              To  compute  background  statistics  assuming the lognormal  distribution, click on
              Lognormal from the drop-down menu list.

              To compute 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, click on the All option
              from the drop-down menu list. ProUCL will display data distribution, all parametric and
              nonparametric background statistics in an Excel type spreadsheet. The user may use this
              output sheet to select the most appropriate statistic to estimate a BTV.
176

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10.1.1 Normal or Lognormal Distribution

1.     Click Upper Limits/BTVs ^- Full (w/o NDs) ^- Normal or Lognormal

                             ProUCL 5.0 - FCE-NDs-Blanks-data-BTVs-UCL-chaps10.xls]
           Upper Limits/BTVs I  UCLs/EPCs  Windows  Help
2.
Full (w/o NDs) >
With NDs >










Normal
Gamma
Lognormal
Non-Parametric
All




1D



11



12



13



The Select Variables screen (Chapter 3) will appear.

   •   Select a variable(s) from the Select Variables screen.

   •   To compute BTV estimates by a group variable, select a group variable by clicking the
       arrow below the Select Group Column (Optional) to obtain a drop-down list of
       available variables and select an appropriate group variable.
              When the Option button is clicked, the following window will be shown.
                                             Confidence Level

                                                  Coverage

                                 Different or Future K Observations    1

                                                   I  OK       I Caned
                                                                                           177

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                 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  0.95.
                     Remember, a UTL is an upper confidence limit (e.g.,  with confidence level =  0.95)
                     for a 95% (e.g., with coverage = 0.95) percentile.

                 o   Specify the Different or Future K Observations. The default choice is 1. It is noted
                     that when K = 1, the  resulting interval will be a UPL for a single future observation.
                     In the example shown above, a value of K =  1 has been used.

                 o   Click on OK button to continue or on Cancel button to cancel this option.

             •   Click on OK to continue or on Cancel button to  cancel the  Upper Limits/BTVs options.

Example 10-la. Consider the real  data  set described in Example  1-1  of Chapter  1  collected from  a
Superfund site. Aluminum  concentrations follow  a normal  distribution and manganese concentrations
follow a lognormal distribution.  The normal and lognormal distribution  based estimates of BTVs are
summarized in the following two tables.

          Aluminum - Output Screen for BTV Estimates Based upon a Normal Distribution
                                       (Full -  Uncensored Data Set)
                     User Selected Options
                  Date/Time of Computation   3/18/2013 3:2S:21 PM
                            From File   SuperFundjds
                         Full Precision   OFF
                    Confidence Coefficient   95';
                           Coverage   95%
                New or Future K Observations   1
              General Satisfies
                               Total Number of Observations   24
                                           Minimum  171(3
                                       Second Largest  15400
                                          Maximum  1620-3
                                            Mean  7789
                                   Coefficient of Variation   0.547
                                    Mean of logged Data   8.798
                      Number of Distinct Observations   24
                                 First Quartie  405S
                                   Median  7010
                                Third Quartie  10475
                                      SO  4264
                                  Skewness   0.542
                             SD of logged Data   0.61
                                    Critical Values for Background Threshold Values (BTVs)
                               Tolerance Factor K (For UTL)   2.309
                                                                                d2max for USL)   2.644
                                 Shapiro Wilk Test Statistic
                               5'-, Shapiro Wilk Critical Value
                                    Ulliefors Test Statistic
                                  5% Ulliefors Critical Value
Normal GOF Test
 0.939
 0.916
 0.109
 0.1S1
     Shapiro Wilk GOF Test
Data appear Norma! at 5% Significance Level
      Lillief ors GOF Test
Data appear Normal at 5% Significance Level
                                        Data appear Normal at 5% Significance Level
                                      Background Statistics Assuming Normal Distribution
                               95% UTL with 95% Coverage  176.3.5
                                         95% UPL})  15248
                                          95% USL  19063
                              90% Percentile (i)  13254
                              95% Percentile h]  14803
                              99% Percentile fz)  17708
178

-------
          Manganese -Output Screen for BTV Estimates Based upon a Lognormal Distribution
                                                (Full-Uncensored Data Set)
                                           Lognormal Background Statistics for Uncensored Full Data Sets
                         User Selected Options
                                   From Rle  SuperFundjds
                                 Full Precision
                           Confidence Coefficient
                                 OFF
                                 95%
                                   Coverage  95?.
                      New or Future K Observations  1
                     Number of Bootstrap Operations  2000
                   Manganese
                   General Statistics
                                       Total Number of Observations   24
                                                     Minimum   3.6
                                                Second Largest  440
                                                    Maximum  530
                                                      Mean  113.8
                                            Coefficient of Variation   1181
                                             Mean of logged Data   4.192
                                                                           Number of Distinct Observations   23
                                                                                        First Quartile   29 3
                                                                                          Median   71.25
                                                                                       Thiid Quartile
                                                                                             SD
                                                                                         Skewness
                                                                                              SD of logged Data
122.5
134.5
 2.17
 1.084
                                             Critical Values for Background Threshold Values (BTVs)
                                       Tolerance Factor K {For UTL)   2.309
                                                                                               d2max for USD    2.644
                                                           Lognormal GOF Test
                                         Shapiro Wilk Test Statistic   0.972            Shapiro Wilk Lognormal GOF Test
                                       5% Shapiro Wilk Critical Value   0.916          Data appear Lognomial at 5'., Significance Level
                                             Ulliefors Test Statistic   012               Ljllietors Loyr.orm.al GOF Test
                                          5% Ljlliefors Critical Value   0.131          Data appear Lognormal at 5% Significance Leve!
                                                Data appear Lognormal at 5X Significance Level
                                    Background Statistics assuming Lognormal Distribution
                             95%UTLwth 95%Coverage   808.1
                                        95%UPLซ)   440.6
                                          95'iUSL   1162
                                                                                              90% Percentile (z)   265.4
                                                                                              95% Percentile (z)   393.5
                                                                                              99% Percentile (z)   823.5
10.1.2  Gamma Distribution
1.         Click Upper Limits/BTVs ^ Full  (w/o NDs) ^ Gamma
                  Upper Limits/BTVs    UCLs/EPCs   Windows   Help
Full (w/o NDs) o
With NDs i>










Normal
Gamma
Lognormal
Non-Parametric
All





10



11



12



13





2.
The Select Variables screen (Chapter 3) will appear.
     •    Select a variable(s) from the  Select Variables screen.
                                                                                                                                179

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Available Variables Selected Variables













Name ID
Aluminum 0
Areenic 1
Chromium 2
Iran 3
Lead 4
Thallium 6
Vanadium 7
Benzo[a)pyrene S
Naphthalene 9
Benzo{a)pyrene ... 10

<^^^H >

I ป I

ซ








Name ID
Manganese 5








<^^^^^H >











Select Group Column (Optional)
I I
| Option. OK Cancel
              If needed, select a group variable by clicking the arrow below the Select Group Column
              (Optional) to obtain a drop-down list of variables, and select a proper group variable.

              When the Option button is clicked, the following window will be shown.
                                         Enter BTV level Options

                                                Confidence Level

                                                      Coverage
0.95
095
                                   Different or Future K Observations    1
                                     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 Coverage level; a number in interval (0.0, 1). Default choice is 0.95.

              o   Specify the Future 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 to continue or on Cancel button to cancel the Upper Limits/BTVs options.
180

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Example 10-lb  (continued). Manganese  concentrations also  follow a gamma distribution. The gamma
distribution based BTV estimates are summarized  in the following table  generated by ProUCL 5.0. The
Gamma GOF test is shown in the following figure.
                                               Gamma Q-Q Plot for Manganese
                                              Theoretical Quantiles of Gamma Distribution
                                                                                                                 Mn.-in- MJ it,*HI
                                                                                                                 k slai - 0.9531
                                                                                                                 Ihelaitat -119.4215
                                                                                                                 Slope = 1.1632
                                                                                                                 Intercept -1&5912
                                                                                                                 CnirHc)linn,R-0%l')
                                                                                                                 ."nder-ton IjailimT'Sf
                                                                                                                 Test Statistic = 0.575
                                                                                                                 CiihcalValue(0.05) = 0.771
                                                                                                                 Data appear Gamma Dปtribut
      Gamma GOF Test for Manganese Data Set
           Manganese - Output Screen for BTV Estimates Based Upon a Gamma Distribution
                                                 (Full-Uncensored Data Set)
                     Manganese
                     General Statistics
                                         Total Number of Observations
                                                      Minimum
                                                              24
                                                              3.6
                                                 Second Largest  440
                                                     Maximum  530
                                                        Mean
                         Number of Distinct Observations
                                     first Quartile
                                        Median
                                                                                                            23
                                             Coefficient of Variation
                                              Mean of logged Data
113.3
  1.181
  4192
                      29.3
                      71.25
           Third Quartile   122.5
                 SD   134.5
             Skewness
        SD of logged Data
                                                                                                             2.17
                                               Critical Values for Background Threshold Values (BTVs)
                                         Tolerance Factor K (For UTL)   2.309
                                                                                                d2max for USD    2.644
                                                             Gamma GOF Test
                                                A-D Test Statistic   0.575            Anderson-Darling Gamma GOF Test
                                              5% A-D Critical Value   D.771    Detected data appear Gamma Distributed at 5S Signrficance Level
                                                K-S Test Statistic   0168           Kclmogrov-Smirnoff Gamma GOF Test
                                              5% K-S Critical Value   0.183    Detected data appear Gamma Distributed at 5% Signrficance Level
                                           Detected data appear Gamma Distributed at 5*4 Significance Level
                                                    khat(MLE)
                                                 Theta hat (MILE)
                                                   nuhat(MLE)
                                          MLE Mean tolas corrected)
Gamma Statistics
  1.05B
107.6
 50.76
113.8
   k star bias corrected MLE)    0.953
Theta star (bias corrected MLE)   1194
     nu star (bias corrected)   45.75
     M LE Sd (bias corrected)   116.6
                                                Background Statistics Assuming
                              95% Wilson Hiferty (WH) Approx. Gamma UPL  353.6
                             957. Hawkins Wrxley (HW) Approx. Gamma UPL  364.2
                           95% WH Approx. Gamma UTL with 95% Coverage  503.8
                           95% HW Approx. Gamma UTL with 95% Coverage  540.3
                                                   95XWHUSL  611.3
                                    90% Percentile
                                    95% Percentile
                                    99% Percentile
                     265.2
                     346.8
                     537
                                                                                                 95% HW USL   6723
                                                                                                                                  181

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The mean manganese concentration is 113.8 with sd = 134.5, and the maximum value = 530. The UTL
based upon a lognormal distribution is 808.1 which is significantly higher than the largest value of 530. It
is noted that the sd of the log-transformed data is 1.084. By comparing BTV estimates computed using
lognormal and gamma distributions, it is noted that the lognormal distribution based upper limits, UTL
and UPL, are  significantly higher than those based upon a gamma distribution confirming the statements
made earlier that the use of a lognormal distribution tends to yield inflated values of the upper limits used
to estimate environmental parameters (e.g., BTVs, EPCs). These upper limits are summarized as follows.
          UTL95-95
          UPL95
Lognormal
   808.1
   440.6
Gamma (WH)
     504
    353.6
Gamma (HW)
    540.3
    364.2
Mean = 113.8, Max value = 530.
10.1.3 Nonparametric Methods

1.     Click Upper Limits/BTVs ^ Full (w/o NDs) ^ Non-Parametric
Upper Limits/BTVs UCLs/EPCs Windows Help
Full (w/o NDs) >
With NDs >








7 8 9 10 11 12 13
Normal
Gamma
Lognormal


All




2.     The Select Variables screen (Chapter 3) will appear.

           •   Select a variable(s) from the Select Variables screen.

           •   If needed, select a group variable by clicking the arrow below the Select Group Column
              (Optional) to obtain a drop-down list of variables, and select a proper group variable.

           •   When the Option button is clicked, the following window will be shown.
                                     Enter BTV Level Options
                                           Confidence Level    0.95

                                                 Coverage    0.95

                                Number of Bootstrap Operations  2000
                                                    OK
                                                            Cancel
182

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                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 0.95.
                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 OK button to continue or Cancel button to cancel the Upper Limits/BTVs options.
Example  10-2.  Lead  concentrations  data  set used  in  Example  1-1 does not  follow  a discernible
distribution. Nonparametric BTV estimates are summarized as follows.  ProUCL 5.1 also outputs the
sample size needed to compute a nonparametric UTL needed to achieve the specified CC.

                      Lead - Output Screen for Nonparametric BTVs Estimates
                                      (Full-Uncensored Data Set)
    General Statistics
                         Total Number of Observations   24
                                      Minimum   4.9
                                  Second Largest   98.5
                                      Maximum  109
                                        Mean   22.49
                             Coefficient of Variation   1.193
                              Mean of logged Data   2.743
                                               Number of Distinct Observations   18
                                                            First Quartile   10.43
                                                               Median   14
                                                           Third Quartile   19.25
                                                                  SD   26.83
                                                             Skewness   2.665
                                                        SD of logged Data   0.771
                                                                                  d2max (for USL)   2.644
       Critical Values for Background Threshold Values (BTVs)
T olerance Factor K (For U T L]    2.309

        Nonparametric Distribution Free Background Statistics
          Data do not follow a Discernible Distribution [005}
                              Nonparametric Upper Limits for Background Threshold Values
                                Order of Statistic, r   24                         35% UTL with 95% Coverage  109
                                   Approximate f    1.263  Approximate Actual Confidence Coefficient achieved by UTL   0.708
                                                      Approximate Sample Size needed to achieve specified CC   59
          95% Percentile Bootstrap UTL with 35% Coverage   109               95% BCA Bootstrap UTL with 35% Coverage  109
                                      95% UPL   106.4                                  30% Percentile   44.81
                              30% Chebyshev UPL   104.6                                  35% Percentile   91.72
                              35% Chebjishev UPL   141.8                                  33% Percentile  106.6

To compute nonparametric upper limits providing the specified coverage (e.g., 0.95), sizes of the data sets
should be fairly large  (e.g., > 59). For details, consult the associated ProUCL Technical Guide.  In this
example the sample size is only 24,  and the  confidence coefficient (CC) achieved by the nonparametric,
UTL is only 0.71 which is significantly lower than the desired CC of 0.95.
                                                                                                     183

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10.1.4 All Statistics Option
1.      Click Upper Limits/BTVs ^ Full (w/o NDs) ^ All
                             ICL5.0 - [TCE-NDs-Blanks-data-BTVs-UCL-chaps1u.xl
     Upper Limits/BTVs  I  UCLs/EPCs   Windows   Help
         Full [w/o NDs)
         With NDs
2.
                         Normal
                         Gamma
                         Lognormal
                         Non-Parametric
                                All
10
11
12
13
The Select Variables screen (Chapter 3) will appear.
   •   Select a variable(s) from the Select Variables screen.
   •   If needed, select a group variable by clicking the arrow below the Select Group Column
       (Optional) to obtain a drop-down list of variables, and select a proper group variable.
   •   When the Option button is clicked, the following window will be shown.
                                    Enter BTV Level Options

                                           Confidence Level
                                                 Coverage
                               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 Coverage level; a number in the interval (0.0, 1). Default is 0.9.
              o   Specify the Future 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 to continue or on Cancel button to cancel the Upper Limits/BTVs options.
184

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Example  10-lc  (continued). The various BTV  estimates  based upon the manganese  concentrations
computed  using the All option of ProUCL are  summarized as follows.  The  All  option computes and
displays  all available  parametric  and nonparametric BTV estimates. This  option also informs the user
about the distribution(s)  of the  data set. This option is specifically useful when one has to process  many
analytes  (variables) without any knowledge about their probability distributions.
            Manganese
                              Manganese - Output Screen for All BTVs Estimates
                                              (Full-Uncensored Data Set)
            General Statistics
                                 Total Number of Observations    24
                                                Minimum    8.6
                                           Second Largest   440
                                               Maximum   53S
                                                  Mean   113.8
                                      Coefficient of Variation    1.181
                                       Mean of logged Data    4.192
                           Number of Distinct Observations   23
                                         First Quartile   29.3
                                            Median   71.25
                                        Third Quartile  122.5
                                               SD  134.5
                                          Skewness   2.17
                                    SD of logged Data   1.084
                                        Critical Values for Background Threshold Values (BTVs)
                                  Tolerance Factor K {For UTL)    2.309
                                                                                             d2max for USL)    2.644
                                    Shapiro Wilk Test Statistic
                                   ; Shapiro Wilk Crtical Value
                                       Ljlliefors Test Statistic
                                     5'i Ulliefors Crtical Value
Normal GOF Test
 0.637
 0.916
 0.233
 0.181
                                              Shapiro Wilk GOF Test
                                         Data Not Normal at 5% Significance Level
                                               Ulliefors GOF Test
                                         Data Not Normal at 5% Significance Level
                                              Data Not Normal at 5% Significance Level
         Background Statistics Assuming Normal Distribution
95% UTL with  35% Coverage   424.3
                                                                                            30% Percentile {z}
                                                                                            35% Percentile (z)
                                                                                            33% Percentile (z)
                                                   286.1
                                                   335
                                                   426.6
            95%UPLJ)   349
              95% USL   469.3
                      Gamma GOF Test
         A-D Test Statistic    0.575             Anderson-Darting Gamma GOF Test
      5% A-D Critical Value    0.771     Detected data appear Gamma Distributed at 5% Significance Level
         K-S Test Statistic    0168            Kolmogrov Smirnoff Gamma GOF Test
      5% K-S Critical Value    0.183     Detected data appear Gamma Distributed at 5% Significance Level
   Detected data appear Gamma Distributed at 5% Significance Level
                                                       Gamma Statistics
                                             khat(MLE)    1.058
                                          Theta hat {MLEj   107.6
                                            nu hat (MLE)   50.76
                                   P>1LE Mean ibias corrected)   113.8
                               k star (bias corrected MLE)   0.953
                           Theta star (bias corrected MLE}  119.4
                                 nu star (bias corrected)   45.75
                                 MLE Sd (bias corrected)  116.6
                                          Background Statistics Assuming Gamma Distribution
                      35% Wilson HiKerty (W H) Approx. Gamma LI P L   353.ซ
                     95% Hawkins Wixley (HW) .Approx. Gamma UPL   364.2
                   95% WH Approx. Gamma UTL with  95% Coverage   503.8
                   35% HW Approx. Gamma UTL with  95% Coverage   54D.3
                                            35%WHUSL   611.3
                                       90% Percentile   2S5.2
                                       95% Percentile   346.8
                                       93% Percentile   537

                                       35% HW USL   672.3
                                                                                                                         185

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                     Manganese - Output Screen for All BTVs Estimates - Continued
                                          (Full-Uncensored Data Set)
                                                  Lognormal GOF Test
                                  Shapiro Wilk Test Statistic    0 972            Shapiro Wilk Lognormal GOF Test
                               5% Shapiro Wilk Critical Value    0.916          Data appear Lognomial at 5', Significance Level
                                     Llefors Test Statistic    D 12               Lilliefors Lognoimal GOF Test
                                  5% Ulliefors Cntical Value    0.181          Data appear Lognormal at 5% Significance Level
                                        Data appear Lognormal at 5% Significance Level
                                      Background Statistics assuming Lognormal Distribution
                               95% UTL with  95% Coverage   80B.1
                                          95*4 UPLJ)   440.6
                                            95%USL   1162
                                      Nonparametric Distribution Free Background Statistics
                                     Data appear Gamma Distributed at 5% Significance Level
                 90% Percentile (z)  265.4
                 95% Percentile (z)  393.5
                 99% Percentile (z)  823.5

                                   Nonparametric Upper Limits for Background Threshold Values
                                      Order of Statistic, r   24
                                         Approximate f    1.263
                  95% Percentile Bootstrap UTL with  95% Coverage   530
                                            95%UPL   507.5
                                     mChebyshevUPL   525.5
                                     95%ChebyshevlJPL   712
                                            95% USL   530
         95% UTL with 95% Coverage
Confidence Coefficient (CC) achieved by UTL
95% BCA Bootstrap UTL with 95% Coverage
                   90% Percentile
                   95% Percentile
                   99% Percentile
530
 0708
530
280
425
5C5.3
10.2   Background Statistics with NDs
1.       Click Upper Limits/BTVs  ^  With NDs
         Upper Limits/BTVs    UCLs/EPCs    Windows   Help
Full (w/o NDs) t
With NDs t









7
8 9
Normal
Gamma
Lognormal
Non-Parametric
All




10




11




12




13




2.       Select the With NDs option.
                  To compute  background  statistics assuming  the normal distribution,  click on Normal
                  from the drop-down menu list.

                  To compute  background  statistics assuming the  gamma distribution, click on  Gamma
                  from the drop-down menu list.
186

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           •   To  compute  background  statistics  assuming the  lognormal  distribution, click  on
              Lognormal from the drop-down menu list.

           •   To  compute  background  statistics using  distribution-free  methods, click on  Non-
              Parametric from the drop-down menu list.

           •   To compute all available background statistics in ProUCL, click on the All option from
              the drop-down menu list.

10.2.1 Normal or Lognormal Distribution

1.     Click Upper Limits/BTVs ^ With NDs ^ Normal or Lognormal
                    ProUCL 5.0 - [TCE-NDs-Blanks-data-BTVs-UCL-chaps10.x
Upper Limits/BTVs  |  UCLs/EPCs   Windows   Help
    Full (w/o NDs)     *
         With NDs
                                       8
                                Normal
                                Gamma
                                Lognormal
                                Non-Parametric

                                All
                                                 10
11
12
13
2.
The Select Variables screen (Chapter 3) will appear.

   •   Select a variable(s) from the Select Variables screen.

   •   If needed, select a group variable by clicking the arrow below the Select Group Column
       (Optional) to obtain a drop-down list of variables, and select a proper group variable.

   •   When the option button is clicked, the following window will be shown.
                     "=!
                              Enter BTV level Options
                                            Confidence Level
                                                  Coverage
                               Different or Future K Observations
                                Number of Bootstrap Operations  2000
                                                     OK
                                                               Caned
                  Specify the Confidence Level; a number in the interval (0.5, 1), 0.5 inclusive. The
                  default choice is 0.95.
                                                                                         187

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                o   Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.95.

                o   Specify the Future 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 to continue or on  Cancel button to cancel the Upper limits/BTVs options.

    Example 10-3a. Consider a small real TCE data set of size ซ=12 consisting of 4 ND  observations.
    The detected data  set  of  size  8 follows a normal as well  as a lognormal distribution. The  BTV
    estimates using the  LROS method, normal  and  lognormal distribution on KM  estimates, and
    nonparametric Chebyshev inequality and bootstrap methods on KM estimates are summarized in the
    following two tables. It is noted that upper limits including UTL95-95  and UPL95 based  upon the
    robust LROS method yield much higher values than the other methods including KM estimates in
    normal and lognormal  equations to  compute the upper limits.  It is noted that the  detected  data also
    follows a gamma distribution.  The  gamma distribution (of detected data) based BTV  estimates are
    described in the next section.

 TCE - Output Screen for BTV Estimates Computed Using Normal Distribution of Detected Data
                                  (Left-Censored Data Set with NDs)
       I      User Selected Options
                       From File  TCE-NDs-Hanks-data-BTVs-UCL-chapsjds
                    Full Precision  OFF
              Confidence Coefficient  95%
                      Coverage  95%
        Different or Future K Observations  2
        TCE
                           Total Number of Observations
                         Number of Missing Observations
                                 Number of Detects
                             Number of Distinct Detects
                                   Minimum Detect
                                   Maximum Detect
                                 Variance Detected
                                   Mean Detected
                         Mean of Detected Logged Data
                                              General Satisfies
                                                U
0.75
9.29
9.732
2.941
0.634
Number of Distinct Observations   9

     Number of Non-Detects   4
Number of Distinct Non-Defects   1
       Minimum Non-Detect   G.68
      Maximum Non-Detect   G.S8
       Percent Non-Detects   33,33%
           SD Detected   3.12
  SD of Detected Logged Data   0.978
                                 Critical Values for Background Threshold Values (BTVs)
                           Tolerance Factor K (For UTLj    2.736
                                 d2maxforUSL)    2.285
                                         Normal GOF Test on Detects Only
                             Shapiro Wilk Test Statistic    0.765                Shapiro Wilk GOF Test
                           5% Shapiro Wilk Critical Value    0.818            Data Not Normal at 5% Significance Level
                                Ulliefors Test Statistic    0.256                  LHIiefors GOF Test
                              5% Uliefors Critical Value    0.313        Detected Data appear Normal at 5% Significance Level
                            Detected Data appear Approximate Normal at 5% Significance Level
188

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 TCE (continued) - Output Screen for BTV Estimates Computed Using Normal Distribution of
                             Detected Data (Left-Censored Data Set with NDs)
                           Kaplan Meier (KM) Background Satisfies Assuring Normal Distribution
                                             Mean     2.183                                                 SD     2.61
                                                      9.323                                       95%KMUPLft}     7.CK7
                                                      S.167               95% KM UPLfor Mean of Next 2 Observations     5.768
                                                      14.03                                  90% KM Percentile (z)     5.533
                                                      S.481                                  99% KM Percentile {2}     8.26
           95% UTL95% Coverage
95% KM UPLfor Next 2 Observations
          95% KM Chebyshev UPL
            95% KM Percentile (z)
Output Screen for BTV Estimates Computed Using a Lognormal Distribution of Detected Data
                                      (Left-Censored Data Set with NDs)
                                  Lognormal Background Statistics for Data Sets with Non Detects
              User Selected Options
                         From File
                      Full Precision
               Confidence Coefficient
                        Coverage
        Different or Future K Observations
         Number of Bootstrap Operations
              TCE-NDs-Hanks-data-BTVs-UCL-chapslOxIs
              OFF
              95%
              95*',
              2
              2000
       TCE
                             Total Number of Observations
                            Number of Missing Observations
                                     Number of Detects
                                Number of Distinct Detects
                                       Minimum Detect
                                       Maximum Detect
                                     Variance Detected
                                       Mean Detected
                            Mean of Detected Logged Data
                                General Statistics
                                  12
                                  2
                                  0.75
                                  9.29
                                  9.732
                                  2.941
                                  0.634
Number of Distinct Observations    9

      Number of Non-Defects    4
Number of Distinct Non-Detects    1
        Minimum Non-Detect    Q.68
       Maximum Non-Detect    G.GS
        Percent Non-Detects    33.33ฐ;
             SD Detected    3.12
  SD of Detected Logged Data    0.97S
                                    Critical Values for Background Threshold Values (BTVs)
                              Tolerance Factor K (For UTL)    2.736
                                                                         d2max for USL)    2.285
                                      Lognormal GOF Test on Detected Observations Only
                                Shapiro Wilk Test Statistic    O.S65                   Shapiro Wilk GOF Test
                             5% Shapiro Wilk Critical Value    0.81 S        Detected Data appear Lognormal at 5'i Significance Level
                                   Lilliefors Test Statistic    0.258                     Lilliefors GOF Test
                                 5*% Uliefors Critical Value    0.313        Detected Data appear Lognormal at 5% Significance Level
                                   Detected Data appear Lognormal at 5% Significance Level
                                                                                                                       189

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    Continued: Output Screen for BTV Estimates Computed Using a Lognormal Distribution of
                         Detected Data (Left-Censored Data Set with NDs)

                         Kaplan Meier (KM) Background Statistics Assuming Normal Distribution

Mean 2.18S
95% UTL95% Coverage 9.329
95% KM UPLfor Next 2 Observations B.167
95% KM Chebyshev UPL 14.03

Background

95% KM Percentile (z) 6.4S1
95% KM USL 8.152
Lognormal ROS Statistics Assuming
Mean in Original Scale 2.01 S
SD in Original Scale 2.S3S
95% UTL.95% Coverage 50.54
95% Bootstrap (%) UTL95% Coverage 9.29
95
% UPLfor Next 2 Observations 25.78
90% Percentile (z) 5.606
99% Percentile (z) 27.2


Statistics using KM estimates on Logged
SD
35% KM UPL 
-------
                                           Set BTV level Options
                                            ^^^^^^^^^^^^^^^^^^^^^H
                                                   Confidence  Level

                                                           Coverage
                                  Different or Future K Observations
                                                               OK
                 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 0.95.

                 o    Click on the OK button to continue or on the Cancel button to cancel option.

            •    Click on OK to continue or on Cancel button to cancel the Upper Limits/BTVs options.

Example  10-3b (continued).  It is noted that the  detected TCE data considered in Example 10-3 also
follows a gamma distribution. The gamma distribution based upper limits are summarized as follows.

TCE - Output Screen for BTV Estimates Computed Using Gamma Distribution of Detected Data
                                    (Left-Censored Data Set with NDs)
     TCE
                                                General Statistics
                          Total Number of Observations   12
                         Number of Missing Observations    2
                                 Number of Detects    8
                            Number of Distinct Detects    8
                                   Minimum Detect    0.75
                                   Maximum Detect    9.29
                                 Variance Detected    9.732
                                   Mean Detected    2.841
                         Mean of Detected Logged Data    0.634
Number of Distinct Observations   9

     Number of Non-Delects   4
Number of Distinct Non-Detects   1
       Minimum Non-Detect   0.68
       Maximum Non-D elect   0.68
       Percent N on-D etects  33.33%
            SD Detected   3.12
  S D of D elected Logged D ata   0.978
                                  Critical Values for B ackground T hreshold Values (BTVs)
                           Tolerance Factor K (For UTL]   2.736
          d2max (for USL)   2.285
                                    G amma GOFTestsonD elected 0 bservations Or Jy
                                  A-D Test Statistic    0.624                Anderson-Darling GOF Test
                                5%A-D Critical Value    0.732    Detected data appear Gamma Distributed at 5% 5ignificance Level
                                  K-S Test Statistic    0.274                 Kolmogomv SmirnovGOF
                                5XK-S Critical Value    0.3     Detected data appear Gamma Distributed at 5% Significance Level
                              Detected data appear Gamma Distributed at 5% Significance Level
                                                                                                        191

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   TCE (continued) - Output Screen for BTV Estimates Computed Using Gamma Distribution of
                            Detected Data (Left-Censored Data Set with NDs)
                                         Gamma Statistics on Detected Data Only
                                       khat(MLE)   1.265
                                    Theta hat (MLE)   2.326
                                      nu hat (MLE)   20.23
                                                  2.941
                                                  3.147
 MLE Mean (bias corrected)
   MLE Sd (bias corrected)
    k star (bias corrected MLE)    0.874
 Theta star (bias corrected MLE)    3.366
       nu star (bias corrected)   13.98

95? Percentile of Chisquare (2k)    5.492
                                     G amma ROSS tatistics using I mputed N on-Detects
                                         Minimum   0.01
                                         Maximum   9.29
                                             SD   2.377
                                       k hat (MLE)   0.372
                                    Theta hat (MLE)   5.274
                                      nu hat (MLE)   8.938
                             MLE Mean (bias corrected)   1.964
                         95% Percentile of Chisquare (2k)   2.956
                                     '35% Percentile   8.668
                                                                   Mean
                                                                  Median
                                                                     CV
                                                     k star (bias corrected MLE)
                                                  Theta star (bias corrected MLE)
                                                        nu star (bias corrected)
                                                       MLE Sd (bias corrected)
                                                             90% Percentile
                                                             98% Percentile
                        The following statistics are computed using Gamma ROS Statistics on Imputed Data
                           Upper Limits using Wilson Hilferty (WH] and Hawkins Wixley (HW) Methods
                                         WH      HW                                         WH
      95% Appro*. Gamma UTL with 95% Coverage   19.62     27.19                   95% Appro*. Gamma UPL    9.793
                          35% Gamma USL   13.95     17.89
                                   Estimates of Gamma Parameters using KM Estimates
                                       Mean (KM)   2.188
                                                  6.813
                                                  0.702
                                                  13.99
                                                  3.115
                                                  3.606
         Variance (KM)
            k hat (KM)
           nu hat (KM)
         theta hat (KM)
80% gamma percentile (KM)
35% gamma percentile (KM)
                 SD (KM)
           SE of Mean (KM)
               k star (KM)
              nu star (KM)
             theta star (KM)
   30% gamma percentile (KM)
                                                  7.957
                                                                                99% gamma percentile (KM)
                          1.964
                          0.845
                          1.465
                          0.335
                          5.865
                          8.037
                          3.394
                          5.709
                         16.26
                                                                          HW
                                                                          11.66
2.61
0.806
0.582
13.98
3.757
5.728
13.36
                         The following statistics are computed using gamma distribution and KM esbnates
                           U pper Limits using Wilson H ilf eity (WH) and H awkins Wixley (HW) Methods
                                         WH     HW                                         WH      HW
      95% Appro*. Gamrna UTL with 95% Coverage   11.34     11.95                   95%Approx. Gamma UPL    6.88      6.896
                    35% KM Garnma Percentile    5.955      5.902                        85% Gamma USL    8.836     9.063

The detected data set does not follow a normal distribution based upon the  S-W test, but follows a normal
distribution based upon the  Lilliefors test.  Since the  detected data set is of small size (=8), the normal
GOF  conclusion is suspect. The detected  data follow a gamma  distribution.  There are  several NDs
reported with  a  low  detection limit  of 0.68,  therefore, GROS method may yield  infeasible negative
imputed values. Therefore, the  use of a gamma distribution  on KM estimates is  preferred for computing
the BTV estimates. The gamma KM UTL95-95 (HW) =11.34, and gamma KM UTL95-95 (WH) =  11.95.
Any one of these two limits can be used to estimate the BTV.
192

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1'0.2.3 Nonparamethc Methods (with NDs)

1.      Click Upper Limits/BTVs^ With NDs ^ Non-Parametric
2.
                                  >.0 - [TCE-NDs-Blanks-data-BTVs-UCL-chaps10.xls]
Upper Li mits/BTVs UCLs/EPCs Windows Help
Full (w/o NDs) >
With NDs >










7 8 S 10 11 12 13
Normal
Gamma
Lognormal
Non-Parametric
All





The Select Variables screen (Chapter 3) will appear.

   •   Select a variable(s) from the Select Variables screen.

   •   If needed, select a group variable by clicking the arrow below the Select Group Column
       (Optional) to obtain a drop-down list of variables, and select a proper group variable.

   •   When the Option button is clicked, the following window will be shown.
                                        : BTV level Optic

                                            Confidence Level

                                                  Coverage

                               Different or Future K Observations
                                                     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 0.95.

              o   Click on the OK button to continue or on the Cancel button to cancel the option.

              Click on OK to continue or on Cancel button to cancel the Upper Limit/BTVs option.
                                                                                         193

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Example 10-3c (continued). The nonparametric upper limits based the TCE data considered in Example
10-3 are summarized in the following table.

                          TCE - Output Screen for Nonparametric BTV Estimates
                                       (Left-Censored Data Set with NDs)
            TCE
                                                      General Statistics
                                 Total Number of Observations   12
                               Number of Missing Observations   2
                                        Number of Detects   8
                                   Number of Distinct Detects   8
                                          Minimum Detect   0.75
                                          Maximum Detect   9.29
                                        Variance Detected   9.732
                                          Mean Detected   2.941
                               Mean of Detected Logged Data   0.634
Number of Distinct Observations
     Number of Non-Detects   4
Number of Distinct Non-Detects   1
       Minimum Non-Delect   0.68
       Maximum Non-Detect   0.68
       Percent Non-Detects   33.33%
             SD Detected   3.12
  SD of Detected Logged Data   0.978
                                         Critical Values for Background Threshold Values (BTVs)
                                 Tolerance Factor K (For UTL]   2.736

                                         Nonparametric Distribution Free Background Statistics
          d2max(forUSL)   2.285
                                   Data appear to follow a Discernible Distribution at 5% Signficance Level

                                   Kaplan Meier [KM] Background Statistics Assuming Normal Disbrfajtjon
                                                 Mean   2.188                                        SD   2.61
                                     35% UTL95S Coverage   9.329                                95% KM UPL (t)   7.067
                                    35% KM Chebjishev UPL   14.03                            90% KM Percentile (2)   5.533
                                      95S KM Percentile (z)   6.481                            99% KM Percentile (:)   8.26
                                            95%KMU8L   8.152
                            N onparametric Upper Limits for B TVs(no distinction made between detects and nondetects)
                                        Order of Statistic, r   12                           95X UTL with95% Coverage   9.29
                                           Approximate f   0.632  Approximate Actual Confidence Coefficient achieved by UTL   0.46
              Approximate Sample Size needed to achieve specified CC   59                                       95% UPL   9.29
                                              95KUSL   9.29                           95% KM Chebyshev UPL   14.03
10.2.4 All Statistics Option

1.       Click Upper Limits/BTVs ^  With NDs ^ All

                                     ProUCL 5.0 - [TCE-NDs-Blanks-data-BTVs-UCL-chaps1u.xls]
                Upper Limits/BTVs  |_UCLs/EPCs   Windows   Help
                    Full (w/o NDs)
                    With NDs
                                                     B
                                              Normal
                                              Gamma
                                              Lognormal
                                              Non-Parametric
                                              All
                                                                        1C
                                                                                 11
                                                                                           12
                                                                                                    13
2.       The Select Variables screen (Chapter 3) will appear.
              •    Select a variable(s) from the Select Variables screen.
194

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                If needed, select a group variable by clicking the arrow below the Select Group Column
                (Optional) to obtain a drop-down list of variables, and select a proper group variable.

                When the Option button is clicked, the following window will be shown.
                                           Enter BTV level Options


                                                 Confidence Level    0.95
                                                               I	

                                                      Coverage     0-95
                                      Different or Future K Observations   1
                                       Number of Bootstrap Operations  2000

                                                      I  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 0.95.

                o    Specify the Future 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 to continue or on Cancel button to cancel the Upper Limits/BTVs option.

Example 10-3d (continued).  BTV estimates using the  All option for the TCE data are summarized as
follows. The detected data set is of small size (ซ=8) and follows a gamma distribution. The gamma GOF
Q-Q plot based upon detected data is shown in the following figure.  The relevant statistics have been
high-lighted in the output table provided after the gamma GOF Q-Q plot.
                            Gamma Q-Q Plot (Statistics using Detected Data) for TCE
                                                                                    lotalNumbclofDjta 14
                                                                                    Slope = 1.0'
                                                                                    Intercept = -0.1201
                                                                                    Conelation.R = 0.3M3
                                                                                    Anderson-Daring Test


                                                                                    Critical Value(Q.05] = 11732
                                                                                    Data appeal Gamma D iiliibuted
                                                                                    • Best Fil Line
                                   Theoretical Quantiles of Gamma Distribution
                                                                                                  195

-------
              TCE -  Output Screen for All BTV Estimates (Left-Censored Data  Set with NDs)
            TCE
                                                               General Statistics
                                     Total Number of Observations T  12
                                   Number of Distinct Observations r  9
                                              Number of Defects'  8
                                        Number of Distinct Detects r  8
                                                Minimum Detect'  0.75
                                                Maximum Detectr  9.29
                                              Variance  Detected r  9.732
Number of Missing Observations f  2
                                                Mean Detected '  2.941
                                    Mean of Detected Logged Data '  0.634
       Number of Non-Detects
 Number of Distinct Non-Detects r  1
         Minimum Non-Detect'  0.6S
         Maximum Non-Detect r  0.68
         Percent Non-Detects r  33.33ฐi
               SD Detected '  3.12
  SD of Detected Logged Data '  0.978
                                             Critical Values lor Background Threshold Values (BTVs)
                                      Tolerance Factor K (For UTL)r  2.736
             d2rnaxfforUSL)''  2.285
                                                       Normal GOF Test on Detects Only
                                        Shapiro Wilk Test Statistic "  0.765                     Shapiro Wilk GOF Test
                                     5% Shapiro Wilk Critical Value r  0.818                Data Not Normal at 5ฐ. Significance Level
                                            Lilliefors Test Statistic r  0.256                       Ljlliefors GOF Test
                                         5% Ljlliefors Critical Value r  0.313          Detected Data appear Normal at 55; Significance Level
                                       Detected Data appear Approximate Normal at 5% Significance Level
                                      Kaplan Meier (KM) Background Statistics Assuming Normal Distribution
                                                        Meanr  2.188                                                 SD'   2.61
                                          95% UTL35% CoverageT  9.329                                       955; KM UPLซr   7.067
                                         95%KMCnebysnevUPLr  14.03                                   90% KM Percentile (z)r   5.533
                                            955; KM Percentile (z)r  6.481                                   99% KM Percentile (z)r   8.26
                                                  95%KMUSLr  S.I 52  '                                                 '
                                               Gamma GOF Tests on Detected Observations Only
                                               A-D Test Statistic '  0.624                   Anderson-Darling GOF Test
                                            5% A-D Critical Value *  0.732      Detected data appear Gamma Distributed at 55; Significance Level
                                               K-S Test Statistic r  0.274                     Kolmogrov Smirnoff GOF
                                            5*; K-S Critical Value '  0.3        Detected data appear Gamma Distributed at 55; Significance Level
                                        Detected data appear Gamma Distributed at 5% Significance Level

                                                    Gamma Statistics on Detected Data Only
                                                   khat{MLE)r  1.265
                                                Thetahat{MLE)r  2.326
                                                  nuhat(MLE)r  20.23
                                       MLE Mean (bias corrected)'  2.941   r
    k star (bias corrected MLE)'  0.874
 Theta star Ibias corrected MLE)r  3.366
       nu star (bias corrected)r  13.98
                                          MLE Sd (bias corrected)r  3.147
95% Percentile of Chisquare (2k)T  5.492
196

-------
     TCE (continued) - Output Screen for All BTV Estimates (Left-Censored Data Set with NDs)
                       The following statistics are computed using Gamma ROS Statistics on Imputed Data
                           Upper Limits using Wilson Hilferty (WH) and Hawkins Wixley (HW) Methods
                                          WH     HW    T
         95% Appro*.Gamma UTLwith 95%Coverage'' 19.62  r 27.19                   95% Approx. Gamma UPL r
                            95% Gamma USLr 13.95  T 17.89   r                                r
                         WH
                          9.793

HW
11.66
                         The following statistics are computed using gamma distribution and KM estimates
                           Upper Limits using Wilson Hrferty (WH) and Hawkins Wbdey (HW) Methods
                                         k hat (KM)'  0.702                                    nu hat (KM)'  16.86
                                          WH     HW    r                                    WH     HW
         95%Approx. Gamma UTL with 95% Coverage' 11.34  *  11.95                   957. Approx. Gamma UPLr  6.SS   '  6.896
                            95% Gamma USL r  S.S36  r  9.063  r                                  r        r
                                    Lognormal GOF Test on Detected Observations Only
                               Shapiro Wilk Test Statistic '  O.S65                  Shapiro Wilk GOF Test
                            5% Shapiro Wilk Critical Value r  0,818        Detected Data appear Lognoimal at 5% Significance Level
                                  Ulliefors Test Statistic '  0.258                    Lilliefors GOF Test
                               5% Lilliefors Critical Value r  0.313        Detected Data appear Lognormal at 5% Significance Level
                                  Detected Data appear Lognormal at 5% Significance Level
                 Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Defects
                                 Mean in Original Scale r  2.018
                                   SD in Original Scale '  2.S38
                                95% UTL95% Coverage r  50.54
                       95% Bootstrap {%) UTL95% Coverage r  9.29
                                    90%Percentile(z)r  5.606
                                    99%Percentile(z)r  27.2
                  Mean in Log Scale r -0.214
                    SD in Log Scale T  1.512
           95% BCAUTL95% Coverage r  9.29
                       95%UPL95% Coverage '  15.25
                                 KM SD of Logged Data r  0.888                           95% KM UPL (Lognormal)'  7.06
                          95*/. KM Percentile Lognormal (z)'  5.7S4                           95% KM USL (Lognormal)T  10.21
                    Nonparametric Llppper Limits for BTVs(no distinction made between detects and nondetects)
                                   Order of Statistic. r    12                          95% UTL with 95% Coverage    9.29
                                       .Approximate f    D.632
                                         95% UPL    9.29
Confidence Coefficient ICC) achieved by UTL
                        95% USL
 0.46
 9.29
Note: Even though the data set failed the Shapiro-Wilk test of normality, based upon Lilliefors test it was
concluded that the data set follows  a normal distribution. Therefore, instead of saying that the data  set
does not follow a normal distribution, ProUCL outputs that the data set follows an approximate normal
distribution. In practice the two tests can lead to different conclusions,  especially when the data set is of
small size. In such instances, the user may want to select a distribution (if any) passing both of the GOF
tests.  It is also  suggested that the user supplements test results with graphical displays to derive the final
conclusion.

As noted, detected data follow a gamma as well as a lognormal distribution.  The various upper  limits
using Gamma  ROS and  Lognormal  ROS methods and Gamma and Lognormal  distribution on KM
estimates are summarized as follows.
                                                                                                             197

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 Summary of Upper Limits Computed using Gamma and Lognormal Distribution of Detected Data
               Sample Size = 12, No. of NDs = 4, % NDs = 33.33, Max Detect = 9.29
Upper Limits
Mean (KM)
Mean (ROS)
UPL95 (ROS)
UTL95-95 (ROS)
UPL95 (KM)
UTL95-95 (KM)
Gamma Distribution
Result
2.188
1.964
9.79
19.62
6.88
11.34
Reference/
Method of Calculation
-
-
WH- ProUCL(ROS)
WH- ProUCL(ROS)
WH - ProUCL (KM-
Gamma)
WH - ProUCL (KM-
Gamma)
Lognormal Distribution
Result
0.29
2.018
13.63
50.54
7.06
15.25
Reference/
Method of Calculation
Logged
-
Helsel(2012),EPA(2009)-
LROS
Helsel(2012),EPA(2009)-
LROS
KM-Lognormal EPA (2009)
KM- Lognormal EPA(2009)
Note: All computations have been performed using the ProUCL software. In the above table, methods
proposed/described in the literature have been cited in the Reference Method of Calculation column. The
statistics summarized above demonstrate the merits of using the gamma distribution based upper limits to
estimate decision parameters (BTVs) of interest.  These results summarized in the above tables suggest
that the use of a gamma distribution cannot be dismissed just because it is easier to use a lognormal
distribution to model skewed data sets as stated by some practitioners.
198

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199

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                                     Chapter 11


Computing Upper Confidence Limits (UCLs) of Mean  Based Upon
   Full-Uncensored Data Sets and Left-Censored  Data Sets with
                                     Nondetects

Several parametric and nonparametric UCL methods for full-uncensored and  left-censored data sets
consisting of ND observations with multiple DLs are available in ProUCL 5.1. Methods such as the
Kaplan-Meier (KM) and regression on order statistics (ROS) methods incorporated in ProUCL can handle
multiple detection limits. For details regarding the goodness-of-fit tests and UCL computation methods
available in ProUCL, consult the ProUCL Technical Guides, Singh, Singh, and Engelhardt (1997); Singh,
Singh, and laci (2002); and Singh, Maichle, and Lee (2006).

In ProUCL 5.0/ProUCL 5.1, two choices are available for computing UCL statistics:

       •   Full (w/oNDs): Computes UCLs for full-uncensored data sets without any nondetects.

       •   With NDs: Computes UCLs for data sets consisting of ND observations with multiple DLs or
          reporting limits (RLs).

       •   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 a 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 CLT, adjusted  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 output sheet also informs the user if data are normal, gamma, lognormal, or a non-
             discernible 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.
200

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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, ProUCL outputs most
              of the relevant UCLs available in ProUCL. The program  informs the user about the
              distribution of the underlying  data  set,  and offers  advice  regarding the use  of an
              appropriate UCL.

              For lognormal data  sets, ProUCL can compute  90%,  95%, 97.5%, and 99% Land's
              statistic-based H-UCL of the mean. For all other methods, ProUCL can compute a UCL
              for any confidence coefficient (CC) in the interval (0.5,  1.0), 0.5 inclusive. If you have
              selected a distribution, then ProUCL will provide a recommended UCL method for 0.95,
              confidence  level. Even though  ProUCL  can compute  UCLs for any  confidence
              coefficient level in the  interval (0.5,  1.0), the recommendations are provided only for
              95% UCL; as EPC term  is estimated by a 95% UCL of the mean.

Notes: Like all other methods, it is recommended that the user identify a few low probability (coming
from extreme tails) outlying observations that may be present in the  data set. Outliers distort statistics of
interest including  summary statistics, data distributions,  test statistics, UCLs and BTVs. Decisions based
upon distorted statistics may be misleading and incorrect. The  objective is to compute decision statistics
based upon the majority of the data set representing the  main dominant population. The project team
should decide about the disposition (to include or not to include) of outliers before computing estimates
of EPCs and BTVs. To determine the influence of outliers on UCLs and background statistics, the project
team may want to compute statistics twice: once using the data set with outliers, and once  using the data
set without outliers.

Note  on  Computing Lower Confidence  Limits  (LCLs) of the  Mean:  In  several  environmental
applications, one needs to compute a LCL of the population mean. At present, ProUCL does not directly
compute LCLs of mean. It should be pointed out that for data  sets with and without NDs,  except for the
bootstrap methods, gamma distribution (e.g., samples  of sizes  <50),  and H-statistic based LCL of mean,
the  same critical value (e.g., normal z value, Chebyshev critical value, or t-critical value) are used to
compute a LCL of mean as used in the computation of the UCL of mean.  Specifically, to compute a LCL,
the  '+' sign used in the computation of the corresponding UCL needs to be replaced by the '-' sign in the
equation used to compute that UCL  (excluding gamma, lognormal  H-statistic, and bootstrap methods).
For specific details, the user may  want to consult a statistician.  For data  sets without nondetect
observations, the user may want to use the Scout 2008  software package (EPA 2009c) to directly compute
the various parametric and nonparametric LCLs of mean.
                                                                                          201

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11.1    UCLs for Full (w/o NDs) Data Sets



11.1.1 Normal Distribution (Full Data Sets without NDs)



1.       Click UCLs/EPCs ^ Full (w/o NDs) ^ Normal



           ^^^^^^^^^^^^^H
           ;s  ISM Simulator Graphs  Statistical Tests  Upper Limrts/BTVs UCLs/EPCs I  Windows Help
2.
D
TCE
1
D_TCE
2

3

4

5

Full (w/o NDs) >
With NDs t
0.81 1
0.68 0
0.68 0

0.95 1
0.68 51
51
61
0.68 61
9.29 |
1.9
0.88
2.98
0.75
5.97






































Normal
Gamma
Lognormal
Non-Parametric

All
































12

13






















The Select Variables screen (Chapter 3) will appear.



    •   Select a variable(s) from the Select Variables screen.



    •   If needed, select a group variable by clicking the arrow below the Select Group Column

       (Optional) to obtain a drop-down list of available variables to select a group variable.



                                     :lect Variables


                    Available Variables            Selected Variables
                            Name
                                     ID
                                                      Name


                                                      TCE
                                                        ID


                                                        0
                                                     Select Group Column (Optional)
               When the Option button is clicked, the following window will be shown.
202

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                                            Select Confidence Lev
                                        Confidence Level
                  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 to continue or on Cancel to cancel the UCL computation option.
Example  11-1.  Consider  the  data used in  Example 1-1  collected from  a  Superfund site; vanadium
concentrations follow a normal  distribution. The normal  distribution based  95%  UCLs  of mean  are
summarized in the following table.
                Vanadium - Output Screen for Normal Distribution (Full Data w/o NDs)
                                        Normal UCL Statistics for Uncensored Full Data Sets
                      User Selected Options
                  Date/Time of Computation  3/25/2013 3:53:05 PM
                             From File  SuperFundjds
                          Full Precision  OFF
                    Confidence Coefficient  95V.
             Vanadium
                                                    General Statistics
                                Total Number of Observations   20
                                             Minimum   7.2
                                             Maximum   32
                                                SD   3.075
       Number of Distinct Observations   17
       Number of Missing Observations   0
                       Mean   17.34
                      Median   16.5
               SD of logged Data   0.492
                                     Coefficient of Variation   0.4S8
                                                                                         Skewness    0.429
                                                    Normal GOF lest
                                   Shapiro Wilk Test Statistic   0.925
                                                                         Shapiro Wilk GOF Test
                                5!i Shapiro Wilk Critical Value   0.905          Data appear Normal at 5% Significance Level
                                     ulliefore Test Statistic   0.146                 LJIIiefors GOF Test
                                   5% Lilliefore Critical Value   0.198          Data appear Normal at 5ซ Significance Level
                                          Data appear Normal at 5% Significance Level
                                                Assuming Normal Distribution
                              95% Normal UCL
                                      SE:= Students4 UCL   20.4S
95% UCLs (Adjusted for Skewness)
     95% Adjusted-CLT UCL (Chen-1995)   20.49
     95'i Modified4 UCL (Johnson-1978)   20.49
                                                  Suggested UCL to Use
                                      95'i Student's-t UCL   20.46
                                                                                                                 203

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11.1.2 Gamma, Log normal, Nonparamethc, All Statistics Option (Full Data without NDs)

1.      Click UCLs/EPCs ^ Full (w/o NDs) ^ Gamma, Lognormal, Non-Parametric, or All
2.
Upper Limits/BTVs
1



5





UCLs/EPCs
Windows Help
Full (w/o NDs] >
With NDs >



Normal
Gamma
Lognormal
Non-Parametric
All




12


13



The Select Variables screen (Chapter 3) will appear.

   •   Select a variable(s) from the Select Variables screen.

   •   If needed, select a group variable by clicking the arrow below the Select Group Column
       (Optional) 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.
                                       sleet UCL Option
                                         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.

              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 to continue or on Cancel to cancel the selected UCL computation option.

Example 11-2: This skewed data set of size n=25  with mean=44.09 was used in Chapter 2  of the
Technical Guide. The data follows a lognormal and a gamma distribution. The data are: 0.3489, 0.8526,
2.5445, 2.5602, 3.3706, 4.8911, 5.0930, 5.6408, 7.0407, 14.1715, 15.2608, 17.6214, 18.7690, 23.6804,
25.0461, 31.7720, 60.7066, 67.0926, 72.6243, 78.8357, 80.0867, 113.0230, 117.0360, 164.3302, and
169.8303. UCLs based upon Gamma, Lognormal, Non-parametric, and All options are summarized in
the following tables.
204

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  Output Screen for Gamma Distribution Based UCLs (Full [w/o NDs])
                                               General Statistics
                       Total Number of Observations    25
                                      Minimum
                                      Maximum
                                          SD
                            Coefficient of Variation
             0.349
            163.8
             51.34
             1.164
Number of Distinct Observations
Number of Missing Observations
                   Mean
                  Median
          SD of logged Data
                                                                                  25
                                                                                  0
                                                                                  4409
                                                                                  18.77
                                                                                  1.SB
                                                                                  1.294
                                               Gamma GOF Test
                                A-D Test Statistic     0.374              Anderson-Darting Gamma GOF Test
                             5% A-D Critical Value     0.794         Data appear Gamma Distributed at 5% Significance Level
                                K-S Test Statistic     0.113             Kblmogrov-Smimoff Gamma GOF Test
                             5% K-S Critical Value     0.183         Data appear Gamma Distributed at 5'i Significance Level
                             Data appear Gamma Distributed at 5% Significance Level
                                               Gamma Statistics
                                    k hat (MLE)     0.643
                                 Tneta hat (MLE)    68.58
                                   nu hat (MLE)    32.15
                         MLE Mean (bias corrected)    44.09

                      Adjusted Level of Significance    0.0395
                                             k star (bias corrected MLE)    0.592
                                          Theta star (bias corrected MLE)   74.42
                                                nu star Ibias corrected)   29.62
                                               M LE Sd (bias corrected)   57.28
                                      Approximate Chi Square Value (D.05)   1 8.2
                                             Adjusted Chi Square Value   17.59
                                          Assuming Gamma Distribution
         95% Approximate Gamma UCl (use when n> =50)    71.77                95V. Adjusted Gamma UCL (use when nc50)    74.27
                                            Suggested UCL to Use
                         95% Adjusted Gamma UCL    74.27
Output Screen for Lognormal Distribution  Based UCLs (Full  [w/o NDs])
Total Number of Observations

                Minimum
               Maximum
                    SO]
     Coefficient of Variation
                                               General Statistics
                                                25
                                                0.349
                                               169.8
                                                51.34
                                                1.164
                                          Number of Distinct Observations
                                          Number of Missing Observations
                                                             Mean
                                                            Median
                                                    Std. Error of Mean
                                                          Skewness
                                             Lognomal GOF Test
                         Shapiro Wilk Test Statistic    0.943               Shapiro Wilk Lognomal GOF Test
                      5% Shapiro Wilk Critical Value    0.918            Data appear Lognormal at 5'i Significance Level
                             Lilliefors Test Statistic    0.135                 Lilliefors Lognormal GOF Test
                          5% Lilliefors Critical Value    D.177            Data appear Lognormal at 55 Significance Level
                                 Data appear Lognormal at 5% Significance Level
                                              Lognormal Statistics
                          Minimum of Logged Data    -1.053
                         Maximum of Logged Data     5.135
                                                  Mean of logged Data
                                                    SD of logged Data
                      95% Chebyshev (MVUE) UCL
                      35/i Chebyshev (MVUE) UCL
    Assuming Lognormal Distribution
951'. H-UCL  229.2
           176.3
           323
                            25
                            0
                            4409
                            18.77
                            10.27
                            1.234
                             2.835
                             1.68
                                                                               Ml Chebyshev (MVU E) UCL   140.6
                                                                              97.5% Chebyshev (MVUE) UCL   225.8
                                            Suggested UCL to Use
                             Data appear Gamma. May want to try Gamma Distribution
                                                                                                                            205

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                          Output Screen for Nonparametric UCLs (Full [w/o NDs])

Nonparametric Distribution Free UCLs
95%CLTUCL 60.9!
95% Standard Bootstrap UCL 60.*
95% Halls Bootstrap UCL G2.1-
95%JackknifeUCL
* 95% Bootstrap-t UCL
) 95% Percentile Bootstrap UCL
61.66
65
61.42
95% BCA Bootstrap UCL 63.63
90% ChebyshevfMean, Sd) UCL 74.89 95% ChebyshevfMean
97.5% ChebyshevfMean , Sd) UCL 1 08.2

99%Chebyshev(Mean

Sd} UCL
Sd) UCL
3S.35
146.3

Suggested UCL to Use
Data appear Gamma. May want to try Gamma_Distribution
                           Output Screen for All Statistics Option (Full [w/o NDs])
                         Total Number of Observations
                                                  General Statistics
                                                   25
                                         Minimum
                                        Maximum
                                             SD
                              Coefficient of Variation
                            Shapiro Wilk Test Statistic
                         5?; Shapiro Wilk Critical Value
                               Lilliefons Test Statistic
                            5% Lilliefons Critical Value
Normal GOF Test
  0.799
  0.918
  0.245
  0.177
     Shapiro Wilk GOF Test
Data Not Normal at 55i Significance Level
       Lilliefors GOF Test
Data Not Normal at 5% Significance Level
C.34S
169.8
51.34
1.164
Number of Missing Observations
Mean
Median
Std. Error of Mean
Skewness
0
44.09
18.77
10.27
1.294
                                       Data Not Normal at 5% Significance Level
                                            Assuming Normal Distribution
                      95% Normal UCL
                                957. Students4 UCL
                                                   61.66
                      95% UCLs (Adjusted for Skewness)
                            95% Adjusted-CLT UCL (Chen-1995)    63.82
                            95% Modified-t UCL (Johnson-197S)    62.1
                                  A-D Test Statistic
                                5% A-D Critical Value
                                  K-S Test Statistic
                                5% K-S Critical Value
Gamma GOF Test
  0.374               Anderson-Darling Gamma GOF Test
  0.794      Detected data appear Gamma Distributed at 5% Significance Level
  0.113              Kblmogrov-Smirnoff Gamma GOF Test
                                                    0.183
                                                             Detected data appear Gamma Distributed at 5% Significance Level
                            Detected data appear Gamma Distributed at 5% Significance Level
206

-------
                   Continued:  Output Screen for All Statistics Option (Full [w/o NDs])
                                        khat(HLE)
                                     Thetahat(MLE)
                                       nu hat (MLE)
                             MLE Mean Ibias corrected)
Gamma Statistics
  G.W3
 68.58
 32.15
 fijoa
                           Adjusted Level of Significance    0.0395
       k star bias corrected MLE)
    Theta star (bias corrected MLE)
          nu star (bias corrected)
         MLE Sd (bias corrected)
Approximate Chi Square Value (0.05)
       Adjusted Chi Square Value
                                             Assuring Gamma Distribution
              95% Approximate Gamma UCL (use when n>=50)    71."
                                                 Lognormal GOF Test
                              Shapiro Wilk Test Statistic    0.948              Shapiro Wilk Lognoimal GOF Test
                           5% Shapiro Wilk Critical Value    0.918           Data appear Lognormal at 5% Significance Level
                                 LJIIiefors Test Statistic    0.135                Lilliefors Lognonnal GOF Test
                               5% Lilliefors Critical Value    0.177           Data appear Lognormal at 5% Significance Level
                                     Data appear Lognonnal at 5% Significance Level
                                                 Lognormal Statistics
                               Minimum of Logged Data   -1.053
                              Maximum of Logged Data    5.135
                                     Mean of logged Data
                                      SD of logged Data
                                            Assuming Lognormal Distribution
                                        95% H-UCL   229.2
                           95% Chebyshev (MVU E) UCL   176.3
                           99% Chebyshev (MVU E) UCL   323
                                      Nonparametric Distribution Free UCL Statistics
                           Data appear to follow a Discernible Distribution at 5% Significance Level
                                          Nonparametric Distribution Free UCLs
                                      95%CLTUCL   60.98
                            95% Standard Bootstrap UCL   60.45
                               35% Halls Bootstrap UCL   63.51
                               95% BCA Bootstrap UCL   64.96
                          90% Chebyshev (Mean, Sd) UCL   74.89
                        97.5% Chebyshev{Mean, Sd) UCL   108.2
0.592
74.42
29.62
57.28
18.2
17.59
                    55% Adjusted Gamma UCL [use when rK5D)    74.27
                              2.835
                              1.68
                              90% Chebyshev fMVUE) UCL   140.6
                             97.5% Chebyshev (MVUE) UCL   225.S
                                     95%JackknifeUCL   61.66
                                    95% Bootstrap^ UCL   65.83
                              95% Percentile Bootstrap UCL   61,84
                             95% ChebyshevfMean, Sd) UCL   88.85
                             99% Chebyshev(Mean, Sd) UCL   146.3

                                                Suggested UCL to Use
                             95% Adjusted Gamma UCL   74.27
Notes:  Once  again,  the  statistics  summarized  above  demonstrate  the  merits of using  the  gamma
distribution based UCL of mean  to estimate EPCs.  The use  of a lognormal  distribution tends to yield
unrealistic UCLs without practical merit (e.g., Lognormal UCL = 229.2 and the maximum = 169.8 in the
above example). The results summarized in the above tables suggest that the use of a gamma distribution
(when a data set follows a gamma distribution) cannot be dismissed just because it is easier (Helsel and
Gilroy 2012) to use a lognormal distribution to model skewed data sets.
                                                                                                               207

-------
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 values is useful when using bootstrap methods. Specifically, it is
not desirable to use bootstrap methods on data sets with only a few distinct values.

11.2  UCL for Left-Censored Data Sets with NDs
1.
Click UCLs/EPCs ^ With NDs
                  .tatistical Tests  Upper Limits/BTVs  UCLs/EPCs  Windows  Help
3

4

5

Full (w/o NDs) f


Wrth NDs t

3 10 11 12 13
Norma
Gamma
Lognormal
Non-Parametric
All


       Choose the Normal, Gamma, Lognormal, Non-Parametric, or All option.

       The Select Variables screen (Chapter 3) will appear.

           •  Select a variable(s) from the Select Variables screen.

           •  If needed, select a group variable by clicking the arrow below the Select Group Column
              (Optional) 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    2000

                                                  I  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 to continue or on Cancel to cancel the selected UCL computation option.
208

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Example 11-3. This real data set of size n=55 with 18.8% NDs (=10) is also used in Chapters 4 and 5 of
the ProUCL Technical Guide. The minimum detected value is 5.2 and the largest detected value is 79000,
sd of detected logged data is 2.79 suggesting that the data set is highly skewed. The detected data follow a
gamma as well as a lognormal distribution.  It is noted that GROS data set with imputed values follows a
gamma distribution and LROS data set with imputed values follows a lognormal distribution (results not
included). The lognormal  Q-Q  plot based  upon  detected data is shown  in the following  figure. The
various UCL  output  sheets:  normal, nonparametric, gamma, and  lognormal generated by ProUCL are
summarized in tables following the lognormal  Q-Q plot on detected data. The main results have been
high-lighted in the output screen provided after the lognormal GOF Q-Q plot.
                               Lognormal Q-Q Plot (Statistics using Detected Data) for A-DL
                                                                                     FWwtNDi-H*
                                                                                            n
                                                                                     ]ซISL*ihr.=aiW
                                                                                     Cmc4VikปIDQ5|-am
                                                                                     D*siซ>ซa Loarnyiwl
                                        Theoretical QuanWes (Standard Normal)
  Output Screen for UCLs based upon Normal, Lognormal, and Gamma Distributions (of Detects)
            A-DL



General Statistics
Total Number of Obseivations ' 55
Number of Detects' 45
Number of Distinct Detects ' 45
Minimum Detect ' 5.2
Maximum Detect '79000
Variance Detects I5.954EปB
Mean Detects '1 D556
Median Detects '1940
Skevmess Detects ' 2.632
Mean of Logged Detects ' 7.031

Number of Distinct Observations '
Number of Non-Detects '
Number of Distinct Non-Detects '
Minimum Non-Detect '
Maximum Non-Detect '
Percent Non-Detects '
SD Detects '
CV Detects '
Kurtosis Detects '
SD of Logged Detects '
53
10
8
3.3
124
18.18%
19S86
1.884
6.496
2.788
Normal GOF Test on Detects Only
Shapiro Wilk Test Statistic ' 0.575
St. Shapiro Wilk Critical Value ' 0.945
Shapiro Wilk GOF Test
Detected Data Not Normal at 5% Significance Level


                                  Ulliefors Test Statistic '  0.298                LJIIiefore GOF Test
                                5% Ulliefors Critical Value '  0.132        Detected Data Not Normal at 5% Significance Level
                                    Detected Data Not Noraal at 55 Significance Level


                        Kaplan-Meier (KM) Statistics using Normal Critical Values and other Nonparametric UCLs
                                          Mean'8638                        Standard Error of Mean ' 2488
                                            S D '18246                          55% KM (BCA) UCL '13562
                                     95% KM () LICL '12802                  95% KM (Percerrtile Bootstrap) UCL '13040
                                     95% KM (z) UCL'12731                       95% KM Bootstrap! UCL'15221
                                90% KM Chebyshev UCL '16102                       95% KM Chebyshev UCL '19483
                               97.5% KM Chebyshev UCL '24176                       99% KM Chebyshev UCL '33394
                                                                                                     209

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  Continued:  Output Screen for UCLs based upon Normal, Lognormal, and Gamma Distributions
                                                             (of Detects)
                                             Gamma GOF Tests on Detected Observations Only
                                             A-D Test Statistic T   0.591                   Anderson-Darling GOF Test
                                          5% A-D Critical Value '   0.86       Detected data appear Gamma Distributed at 5% Significance Level
                                             K-S Test Statistic '   0.115                     Kdmogrov-Smimoff GOF
                                          5% K-S Critical Value '   0.143      Detected data appear Gamma Distributed at 5% Significance Level
                                      Detected data appear Gamma Distributed at 5% Significance Level
                                                 Gamma Statistics on Detected Data Only
                                                 khat{MLE)r   0.307
                                              Thetahat(MLE)r34333
                                                nuhat{MLE)r  27.67
                                     MLE Mean (bias corrected) "10556
             k star (bias corseted MLE)r   0.302
          Theta star (bias corrected MLE) r34380
                nu star (bias corrected)   27.16
               MLE Sd (bias corrected) "1 5216
                                                    Gamma Kaplan-Meier (KM) Statistics
                                                  khat(KM)r   0.224                                          nu hat (KM)
                           Approximate Chi Square Value (24.66, a)'   14.35                       Adjusted Chi Square Value (24.66, p)
                 95% Gamma Approximate KM-UCL (use when n>=5D) "14844                95*/; Gamma Adjusted KM-UCL (use when n<50)
                                                GROS Statistics using imputed NDs
                                                    MinimumT  0.01
                                                   Maximum "7SOOQ
                                                        SD "18415
                                                 k hat (MLE)r  0.18
                                              Theta hat (MLE) "47915
                                                nuhat(MLE)r  19,83
                                     MLE Mean (bias corrected)T8637
                           Approximate Chi Square Value (20.08, a)'  10.91
                    95% Gamma Approximate UCL (use when n>=50) "15896
                              Mean
                             Median
                                CV
             k star (bias corrected MLE)
          Theta star (bias corrected MLE)
                 nu star (bias corrected)
                MLE Sd Ibias corrected)
        Adjusted Level of Significance (3)
      Adjusted Chi Square Value (20.08. p)
95% Gamma Adjusted UCL (use when n<50)
   24.66
   14.14
r15066

r8637
T 588
    2.132
    0.183
r47314
   20.08
"20215
   0.0456
   10.73
'16167
.
)
                                            Lognormal GOF Test on Detected Observations Only
                                      Shapiro Wilk Test Statistic r   0.939                      Shapiro Wilk GOF Test
                                   5% Shapiro Wilk Critical Value '   0.945           Detected Data Not Lognormal at 5% Significance Level
                                          Lilliefors Test Statistic T   0.104                        LJIIiefors GOF Test
                                       5% LJIIiefons Critical Value '   0.132         Detected Data appear Lognormal at 5% Significance Level
                                   Detected Data appear Approximate Lognormal at 5% Significance Level
                                            Logrormal ROS Statistics Using Imputed Non Detects
                                         Mean in Original Scale r 8638
                                           SD in Original Scale r1S414
                        95% t UCL (assumes normality of ROS data) "12793
                                       95% BCA Bootstrap UCL "14069
                                        95% H-UCL (Log ROS) "1855231
                    Mean in Log Scale T  5.983
                      SD in Log Scale r  3.391
           95% Percentile Bootstrap UCL'13090
                  95%Bootstrap! UCL"15524
                     UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognormally Distributed
                                            KM Mean flogged)"  6.03                                   95% H-UCL {KM -Log) 1173988
                                              KM SD (logged)'  3.286                            95% Critical H Value (KM-Log)'  5.7
                              KM Standard Error of Mean flogged)"  0.449  "                                                 "
                                                          Suggested UCL to Use
                                      95% KM (Chebyshev) UCL "19483                        95% GROS Approximate Gamma UCL "15896
                                95% Approximate Gamma KM-UCL"!4844    "                                                "
210

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Detected data follow a gamma as well as a lognormal distribution. The various upper limits using Gamma
ROS and  Lognormal ROS methods and Gamma and Lognormal distribution on KM estimates are
summarized in the following table.

 Upper Confidence Limits Computed using Gamma and Lognormal Distributions of Detected Data
                      Sample Size = 55, No. of NDs=10, % NDs = 18.18%
Upper Limits
Min (detects)
Max (detects)
Mean (KM)
Mean (ROS)
UCL95 (ROS)
UCL (KM)
Gamma Distribution
Result
5.2
79000
8638
8637
15896
14844
Reference/
Method of Calculation
-
-
-
-
ProUCL5.0-GROS
ProUCL 5.0 - KM-Gamma
Lognormal Distribution
Result
1.65
11.277
6.3
8638
14863
12918
1173988
Reference/
Method of Calculation
Logged
Logged
Logged
-
bootstrap-t on LROS,
ProUCL 5.0
percentile bootstrap on
LROS,Helsel(2012)
H-UCL, KM mean and
sd on logged data, EPA
(2009)
   All computations have been performed using the ProUCL software. In the above table, methods
   proposed/described in the literature have been cited in the Reference Method of Calculation column.
   The results summarized in the above table re-iterate that the use of a gamma distribution cannot be
   dismissed just because it is easier to use a lognormal distribution to model skewed data sets.  These
   results also demonstrate that for skewed data sets, one  should use bootstrap methods which adjust for
   data skewness (e.g., bootstrap-1 method) rather than using percentile bootstrap method.
                                                                                        211

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                                      Chapter 12


Sample Sizes Based  Upon User Specified Data Quality Objectives
                        (DQOs) and Power Assessment

One of the most  frequent problems in  the application of statistical theory to practical  applications,
including environmental projects, is to determine the minimum number of samples needed for sampling
of reference/background areas and survey units (e.g., potentially impacted site areas,  areas of concern,
decision units) to  make cost-effective and defensible decisions about the population  parameters based
upon the sampled discrete data. The sample size determination formulae for estimation of the population
mean (or some other parameters) depend upon certain decision parameters including  the confidence
coefficient, (1-a) and the  specified error margin (difference), A from the unknown true population mean,
fj,.  Similarly, for hypotheses testing approaches, sample size determination formulae depend upon pre-
specified values of the decision parameters selected while describing the data quality objectives (DQOs)
associated with an environmental project. The  decision parameters associated with hypotheses testing
approaches include Type I (false positive  error rate, a) and Type II (false negative error rate, /?=!-power)
error rates; and the allowable width, A of the gray region. For values of the parameter of interest (e.g.,
mean, proportion) lying in the gray  region, the consequences of committing  the two types of errors
described above are not significant from both human health and cost-effectiveness point of view.

Note: Initially, the Sample Sizes module was incorporated  in ProUCL 4.0/ProUCL 4.1. Not many
changes have been made  in ProUCL 5.0/ProUCL 5.1 except those described below. Therefore, many
screenshots generated using an earlier 2010 version of ProUCL have been used in the examples described
in this chapter.

Both parametric (assuming normality) and nonparametric (distribution free) sample size determination
formulae as  described in  guidance documents  (MARS SIM 2000, EPA 2002c and 2006a) have been
incorporated in the ProUCL software. Specifically, the DQOs Based Sample Sizes module of ProUCL
can be used to determine sample sizes  to estimate the mean, perform parametric and nonparametric
single-sample and two-sample hypothesis tests, and apply acceptance sampling approaches to address
project needs of the various CERCLA and RCRA site projects. The details can be found in Chapter 8 of
the ProUCL Technical Guide and in EPA  guidance documents (EPA 2006a, 2006b).

New in ProUCL 5.0/ProUCL 5.1: The Sample size module in ProUCL 5.0/ProUCL 5.1  can be used at
two different stages of a project.   Most of the sample  size formulae require some estimate of the
population standard deviation (variability). Depending upon the project stage,  a standard  deviation: 1)
represents a preliminary estimate of the population (e.g., study area) variability needed to compute the
minimum sample  size during the planning and design stage;  or  2)  represents the sample standard
deviation computed using the data collected  without considering DQOs process which is used to  assess
the power of the test based upon the collected data. During the power assessment stage, if the computed
sample size is larger than the size of already collected data set, it can be inferred that  the size  of the
collected data set is not large enough to achieve the desired power. The formulae to compute the sample
sizes during the planning stage and after performing a statistical test are the same except that the estimates
of standard deviations are computed/estimated differently.

Planning stage before collecting data: Sample size formulae are commonly used during the planning stage
of a project to determine  the minimum sample sizes needed to address project objectives (estimation,
hypothesis testing) with specified values  of the decision parameters  (e.g.,  Type I and II errors, width of
212

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gray region). During the planning stage, since the data are not collected a priori, a preliminary rough
estimate of the population standard  deviation (to  be  expected in sampled data) is obtained from other
similar sites, pilot studies, or expert opinions. An estimate of the expected standard deviation along with
the specified values of the other decision  parameters are used to compute the minimum sample sizes
needed to address the project objectives during the sampling planning stage; the project team is expected
to collect the number of samples thus obtained. The detailed discussion of the sample size determination
approaches during the planning stage can be found in MARSSIM 2000 and U.S. EPA 2006a.

Power  assessment  stage after performing  a statistical method:  Often,  in practice,  environmental
samples/data sets are collected without taking the DQOs process into consideration. Under this scenario,
the project team  performs statistical tests on the available already collected data  set. However, once a
statistical test (e.g., WMW test) has been performed, the project team can assess the power associated
with the test in retrospect. That is for specified DQOs and decision errors (Type I error and power of the
test [=l-Type  II error]), using the sample standard deviation computed based upon the already collected
data, the minimum sample size needed to perform the  test for specified values of the decision parameters
is computed.

    •   If the computed sample size obtained using the sample variance is less than  the size of the already
        collected  data set used to perform  the test, it may be determined that the  power of the test has
        been achieved. However, if the sample size of the collected data is less than the minimum sample
        size computed in retrospect, the user may want to collect additional  samples to assure that the test
        achieves the desired power.

    •   It should be pointed out  that there could be  differences in  the  sample sizes computed in two
        different  stages due to the differences  in the values of the estimated variability.  Specifically, the
        preliminary estimate of the  variance  computed using information  from similar sites could  be
        significantly different from the variance computed using the available data already collected from
        the study area under investigation which will yield different values of the sample  size.

Sample size determination methods in ProUCL can be used for both stages. The only difference will be in
the input value of the standard deviation/variance.  It is user's responsibility to input  a correct value for
the standard deviation during  the two stages.
      File  Edit  Stats/Sample Sizes  Graphs  Statistical Tests  Upper Limits/BTVs  UCLs/EPCs Windows  Help
Navigation F
Name
Worksheet it>
General Statistics ^
mputed NDs using ROS Methods >
DQOs Based Sample Sizes t
WMW^ith NDsxIs
ASHALL7groupsxls
Box Plot Full.gst
Box Plot Full_a.gst
Box Plot Full_b.gst

3
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Fฐ\ D-Ste
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Estimate Mean
Hypothesis Tests
^
Acceptance Sampling
1 1
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1 1


6

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8 9 1


Single Sample Tests *
Two Sample Tests ^






0 11


tTest
Proportion
Sign Test
Wilcoxon Sign

sd Rank

                                                                                              213

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12.1   Estimation of Mean
1.      Click Stats/Sample Sizes^ DQOs Based Sample Sizes ^ Estimate Mean
•a File Edit
Navigation F
Name
Worksheet jds
Well 10 Jds
WMW-with NDsxl
ASHALL7groupsj<
Box Plot Full.gst
Stats/Sample Sizes | Graphs Statistical Tests Upper Limits/BTVs UCLs/EPCs Windows Help
General Statistics
>
Imputed NDs using ROS Methods ^
DQOs Based Sample Sizes >
s
s
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Estimate Mean
Hypothes s Tests ^
Acceptance Sampling

2d| 1 1





6



7




2.      The following options window is shown.
                                         Select Sample Size Options
                                                   Confidence Level
                                 Allowable Error Margin in Mean Estimate
                                        Estimate of Standard Deviation

                                                         I   OK
            •   Specify the Confidence Coefficient. Default is 0.95.
            •   Specify the Estimate of standard deviation. Default is 3.
            •   Specify the Allowable Error Margin in Mean Estimate. Default is 10.
            •   Click on OK button to continue or on Cancel button to cancel the options.
  Output Screen for Sample sizes for Estimation of Mean (CC = 95%, sd = 25, Error Margin = 10)
                              S ample 5 ize f 01C si imation of M ean
                              Based on Specified Values of Decision Par ameters/DQ Os [Data Quafty Objectives)
            Date/Time of Computation  2/26/201012:12:37 PM
               User Selected Options
               Confidence Coefficient  35%
               Allowable Error Margin  10
          Estimate of Standard Deviation  25
                   95% Confidence Coefficient:
proximate Minimum Sample Size
26


















214

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12.2   Sample Sizes for Single-Sample Hypothesis Tests
12.2.1 Sample Size for Single-Sample t-Test
1.      Click DQOs Based Sample Sizes ^- Hypothesis Tests ^- Single Sample Tests ^-1 Test
ab1 File Edit Stats/Sample Sizes Graphs Statistical Tests Upper Limits/BTVs UCLs/EPCs Windows Help
Navigation FJ General Statistics ^
Name r
nputed NDs using ROS Methods >
Wo*Sheetjls 1 DQOs Based Sample Sizes t
WMW-with NDsxIs
ASHALL7groupsxls
Box Plot Full.gst
Box Plot Full_a gst
Box Plot Full_b.gst
2
4
S
6
7
8
5 8
7 17
12 20
15 25
18 34
21 35
3-56789 10 11
J^" OS* |
Estimate Mean
Hypothesis Tests > Single Sample Tests ฅ
Acceptance Sampling Two Sample Tests t
1 1
1 T[
tTest
Proportion
Sign Test
Wilcoxon Signed Rank

1 1
       The following options window is shown.
               False Rejection Rate [Alpha]
                     False Acceptance Rate [Beta]
                        O 0.005
                        O 0.010
                        O O.025
                        (I> 0.050
                        O O.100
                        O 0.150
                        O 0.200
                        O 0.250
[0.5%]
[1.0%]
[25%]
[5.0%]
[10.%]
[15%]
[20.%]
[25.%]
O 0.005
O 0.010
O 0.025
O 0.050
ฎ 0.100
O 0.150
O 0.200
O 0.250
[0.5%]
[10%]
[2.5%]
[5.0%]
[10.%]
[15%]
[20.%]
[25%]
               Estimate of Population SD

                   Preliminary Estimate (planning stage)
                   Sample SD using collected data
                   (to assess power)
                     Width of Gray Region [Delta]
                                                             OK
                                        Cancel
           •   Specify the False Rejection Rate (Alpha, a). Default is 0.05.

           •   Specify the False Acceptance Rate (Beta, P). Default is 0.1.

           •   Specify the Estimate of standard deviation. Default is 3.

           •   Specify the Width of the Gray Region (Delta, A). Default is 2.

           •   Click on OK button to continue or on Cancel button to cancel the options.
                                                                                          215

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    Output Screen for Sample Sizes for Single-Sample t-Test (ซ = 0.05, /? = 0.2, sd = 10.41, A = 10)
                                    Example from EPA 2006a (page 49)
                                         ! Sample Sizes for Single Sample (Test
                                          Based on Specified Values of Decision Parameteis/DQOs (Data Quafty Objectives)
                       D ate/T ime of Computation   2/26/2010 12:41:58 PM
                          User Selected Options
                      False Rejection Rate [Alpha]   0.05
                     False Acceptance R ate [B eta]   0.2
                      Width of Q ray R egion [D elta]   10
                    EstimateofStandardDeviation   10.41
                                          Approximate Minimum Sample Size
                  Single Sided Alternative Hypothesis:
                   Two Sided Alternative Hypothesis:
11
12.2.2  Sample Size for Single-Sample Proportion Test
1.       Click DQOs Based Sample Sizes ^- Hypothesis Tests ^-  Single Sample Tests ^-  Proportion
     ซJ File  Edit
                 Stats/Sample Sizes  Graphs  Statistical Tests  Upper Limits/BTVs   UCLs/EPCs   Windows  Help
Navigation F
Name
Worksheet xls
Well 10xls
WMW-wth NDsxl
ASHALL7groupsjt
Box Plot Full .gst
Box Plot Full_a.gst
Box Plot Full_b. gst
General Statistics ^
Imputed NDs using ROS Methods >
DQOs Based Sample Sizes
3
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T 17
12 20
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18 34
>




*sro
3
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4

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Estimate Mean
HypothessTests t
Acceptance Sampling
1
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1





6

7

8 9 10 11


Single Sample Tests ^
Two Sample Tests ^





tTest


Sign Test
VYilcoxon Signed Rank

2.       The following options window is shown.
                                 Single Sample Proportion Test Sample Size Options
                      False Rejection Rate [Alpha]               False Acceptance Rate [Beta]
                               O 0.005 [0.5%]
                               O 0.010 [i.o%]
                               O 0.025 [2.5%]
                               ฎ 0.050 [5.0%]
                               O 0.100 [10.%]
                               O 0.150 [15.%]
                               O 0.200 [20.%]
                               O 0.250 [25.%]

                      Desirable Proportion [PO]

                          Preliminary Estimate (planning stage)
                          Sample Proportion using collected data
                          Qo assess power)

                                     0.3
                 O 0.005 [0.5%]
                 O 0.010 [1.0%]
                 O 0.025 [2.5%]
                 O 0.050 [5.0%]
                 ฎ 0.100 [10.%]
                 O 0.150 [15.%]
                 O 0.200 [20.%]
                 O 0.250 [25.%]

           Width of Gray Region [Delta]
                       0.15
216

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            •   Specify the False Rejection Rate (Alpha, a). Default is 0.05.
            •   Specify the False Acceptance Rate (Beta, /?). Default is 0.1.
            •   Specify the Desirable Proportion (PO). Default is 0.3.
            •   Specify the Width of the Gray Region (Delta, A). Default is 0.15.
            •   Click on OK button to continue or on Cancel button to cancel the options.
  Output Screen for Sample Size for Single-Sample Proportion Test (ซ = 0.05, /? = 0.2, PO = 0.2, A
                               0.05) Example from EPA 2006a (page 59)
                                      i S ample S izes for Single S ample Proportion Test
                                       Based on Specified Values of Decision Parameters/DQ Os (Data Quafty Objectives)
                      DateATime of Computation  2/26^2010 12:50:52 PM
                        User Selected Options
                    False Rejection Rate [Alpha]  0.05
                    False Acceptance Rate [Beta]  0.2
                    Width of Gray Region [Delta]  0.05
                    Proportion/Action Level [PO]  0.2
                                      Approximate Minimum Sample Size
                 R ight S ided Alternative Hypothesis:           413
                  Left S ided Alternative Hypothesis:           363
                 Two Sided Alternative Hypothesis:
man[471,528)
72.2.3 Sample Size for Single-Sample Sign Test

1.      Click DQOs Based Sample Sizes ^ Hypothesis Tests^- Single Sample Tests^- Sign Test
                                                                                      - [WMW-i
a^ File Edit
Navigation F
Name
Worksheet*
Well 10*
WMW-wth NDsxl
ASHALL7graupsx
Box Plot Full.gst
Box Plot Full_a.gst

Stats/Sample Sizes
General Statist!
mputed NDs u
Graphs Statistical Tests Upper Limits/BTVs UCLs/EPCs Windows Help
:: t
sing ROS Methods ^
DQOs Based Sample Sizes t
s
s
2
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5
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15
18

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25
3?
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^=r:i C_Site
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Estimate Mean
Hypothes
s Tests
>
Acceptance Sampling
1 1
1 1

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a 9 1

0 11


Single Sample Tests >
Two Sample Tests >



tTest
Proportion

Sign Test
Wilcoxon Signed Rank


2.      The following options window is shown.
                                                                                                     217

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                                  Single Sample Sign Test Sample Size Options
                    False Rejection Rate [Alpha]
                       False Acceptance Rate [Beta]
                              O 0.005
                              O 0.010
                              O 0.025
                              ฎ 0.050
                              O 0.100
                              O 0.150
                              O 0.200
                              O 0.250
[0.5%]
[10%]
[2.5%]
[5.0%]
[10.%]
[15.%]
[20.%]
[25.%]
O 0.005
O 0.010
O 0.025
O 0.050
'ft 0.100
O 0.150
O 0.200
O 0.250
[0.5%]
[10%]
[2.5%]
[5.0%]
[10.%]
[15.%]
[20.%]
[25.%]
                    Estimate of Population SD

                        Preliminary Estimate (planning stage)
                        Sample SD using collected data
                        (to assess power)
                       Width of Gray Region [Delta]

                                                                       OK
             •   Specify the False Rejection Rate (Alpha, a). Default is 0.05.

             •   Specify the False Acceptance Rate (Beta, P). Default is 0.1.

             •   Specify the Estimate of standard deviation. Default is 3

             •   Specify the Width of the Gray Region (Delta, A). Default is 2.

             •   Click on OK button to continue or on Cancel button to cancel the options.

Output Screen for Sample Sizes for Single-Sample Sign Test (Default Options)
                                       I Sample Sizes for Single Sample Sign Test
                                        Based on Specified Values of Decision Paiameteis/DQOs [Data Qua% Objectives)
                    Date.'Tirne of Computation   2/26/2010 12:15:27 PM
                       User Selected Options
                   False Rejection Rate [Alpha]   0.05
                  False Acceptance Rate [Beta]   0.1
                   Width of Bray Region [Delta]   2
                  Estimate of Standard Deviation   3
                                        Approximate Minimum Sample Size
               Single Sided Alternative Hypothesis:            35
                      	
                Two Sided Alternative Hypothesis:            43
218

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72.2.4 Sample Size for Single-Sample Wilcoxon Signed Rank Test

1.       Click DQOs Based Sample Sizes ^- Hypothesis Tests ^- Single Sample Tests ^- Wilcoxon
         Signed Rank
                                                                                      . 5.0 - [WMW-with NDs.xls]
     File  Edit
      Navigation F
              Stats/Sample Sizes  Graphs   Statistical Tests  Upper Limte/BTVs  UCLs/EPCs  Windows  Help
   Name
   Worksheet jds
   Well 10:ds
   WMW^ith NDsxIs
   ASHALL7groupsjds
   Box Plot Full.flst
   Box Plot Full_a.gst
   Box Plot Full_b.gst
                 General Statistics
                 Imputed NDs using ROS Methods
DQOs Based Sample Sizes
                    17
             12
             15
             18
                                               D Site
                             Estimate Mean
                             Hypothesis Tests
                             Acceptance Sampling
                                                                                10
                                                   Single Sample Tests
                                                   Two Sample Tests
20
25
34
                                                    tTest
                                                    Proportion
                                                    Sign Test
Wilcoxon Signed Rank
2.      The following options window is shown.
                            Single Sample Wilcoxon Signed Rank Test Sample Size Options
       False Rejection Rate [Alpha]

               O 0.005 [0.5%]
               O 0.010(1.0%]
               O 0.025 [2.5%]
               (?) 0.050 [50%]
               O 0.100 [10%]
               O 0.150(15.%]
               O 0.200 [20 %]
               O 0.250 [25.%]

       Estimate of Population SD

          Preliminary Estimate (planning stage)
          Sample SD using collected data
          (to assess power)
                                                            False Acceptance Rate [Beta]

                                                                  O 0.005 [0.5%]
                                                                  O 0.010(1 0%]
                                                                  O 0.025 [2.5%]
                                                                    0.050 [5.0%]
                                                                  ฎ 0.100(10%]
                                                                  O 0.150(15.%]
                                                                  O 0.200 [20.%]
                                                                  O 0.250 [25 %]

                                                            Width of Gray Region [Delta]
                 Specify the False Rejection Rate (Alpha, a). Default is 0.05.

                 Specify the False Acceptance Rate (Beta, P). Default is 0.1.

                 Specify the Estimate of standard deviation of WSR Test Statistic. Default is 3

                 Specify the Width of the Gray Region (Delta, A). Default is 2.

                 Click on OK button to continue or on Cancel button to  cancel the options.
                                                                                                         219

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  Output Screen for Sample Sizes for Single-Sample WSR Test (ซ = 0.1, /? = 0.2, sd = 130, A = 100)
                                    Example from EPA 2006a (page 65)
                       Date/Time of Computation
                         User Selected Options
                     False Rejection Rate [Alpha]
                    False Acceptance Rate [Beta]
                     Width of Gray Region [Delta]
                    Estimate of Standard Deviation
                                         Sample Sizes toi Single Sample WilcoKon Signed Rank Test
                                         Based on Specified Values of Decision Parameters/DQOs (Data Quafty Objectives)
                                         2/26/20101:13:58PM
                 Single Sided Alternative Hypothesis:
                  Two Sided Alternative Hypothesis:
 0.1
 0.2
 100
 130
Approximate Minimum Sample Size
          10
          14
12.3    Sample Sizes for Two-Sample Hypothesis Tests
12.3.1 Sample Size for Two-Sample t-Test
1.       Click  DQOs Based Sample Sizes ^ Hypothesis Tests ^ Two Sample Tests ^ t Test
                                                                                  CL5,0-[WMW-withNC
    File  Edit  I Stats/Sample Sizes   Graphs  Statistical Tests  Upper Limits/BTVs   UCLs/EPCs  Windows  Help
Navigation F
Name
WofcSheeWs
Well Hbds
WMW-wBi NDsjt
ASHALL7graupsj
Box Plot Full .git
Box Plot Full_a.gsl
Rnx Pint Fi ill h nsl
General Statistics ^
Imputed NDs using ROS Methods >
DQOs Based Sample Sizes ^
5
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2
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5
6
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7
12
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17
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25



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Estimate Mean
Hypothesis Tests ^
Acceptance Sampling

7

8 9 10 11 12


Single Sample Tests t
Two Sample Tests >
1 1
1 1

ties, |
Wilcoxon-Mann-Whitney

The following options window is shown.
                                       Two Sample t Test Sample Size Options
                         False Rejection Rate [Alpha]
                                 O 0.005 [0.5%]
                                 O 0.010 [i.o%]
                                 O O.025 [2.5%]
                                 ฎ 0.050 [5.0%]
                                 O o.ioo [io.%]
                                 O O.150[15.%]
                                 O 0.200 [20.%]
                                 O O.250 [25.%]
                         Pooled Estimate of Population SD
                            Preliminary Estimate (planning stage)
                            Sample SD using collected data
                            (to assess power)

                                       3
                    False Acceptance Rate [Beta]

                          O 0.005 [0.5%]
                          O 0.010(1.0%]
                          O 0.025 [2.5%]
                          O O.050 [5.0%]
                          ฎ 0.100 [10.%]
                          O O.150[15.%]
                          O 0-200 [20.%]
                          O 0.250 [25.%]

                    Width of Gray Region [Delta]
220

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            •   Specify the False Rejection Rate (Alpha, a). Default is 0.05.
            •   Specify the False Acceptance Rate (Beta, P). Default is 0.1.
            •   Specify the Estimate of standard deviation. Default is 3
            •   Specify the Width of the Gray Region (Delta, A). Default is 2.
            •   Click on OK button to continue or on Cancel button to cancel the options.
    Output Screen for Sample Sizes for Two-Sample t-Test (ซ = 0.05, /? = 0.2, sp = 1.467, A = 2.5)
                                 Example from EPA 2006a (page 68)
                                    i Sample Sizes for Two Sample (Test
                                     B ased on Specified Values of Decision Parameters/DQ0s [Data Qunity Objectives)
                  Date/Time of Computation   2/26/20101:17:57 PM
                     User Selected Options
                 False R ejection R ate [Alpha]   0.05
                False Acceptance Rate [Beta]   0.2
                 Width of Gray Region [Delta]   2.5
                     E stirnate of Pooled SD   1.467
                                     Approximate Minimum Sample Size
             Single Sided Alternative Hypothesis:            5
              Two Sided Alternative Hypothesis:            7
72.3.2 Sample Size for Two-Sample Wilcoxon Mann-Whitney Test
1.
        Click DQOs Based Sample Sizes ^ Hypothesis Tests ^ Two Sample Tests ^-
        Wilcoxon-Mann-Whitney
                                                                         DUCL 5.0 - [WMW-with NDs.>
    File  Edit   Stats/Sample Sizes   Graphs  Statistical Tests  Upper Limits/BTVs  UCLs/EPCs  Windows  Help
Navigation F
Name
Work Sheet jds
Well 10.xb
WMW-wlh NDsxl
ASHALL7groupsi
Box Plot Full.gst
Box Plot FulLa.gst
Rnv Pint Fl ill h nit
General Stat sties *
Imputed NDs using ROS Methods ^
DQOs Based Sample Sizes >
s
s
2
3

S
6
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7 17
12 20
15 25
34567
f ro DSite |

Estimate Mean
Hypothesis Tests *
Acceptance Sampling


S


Single SampleTests >
Two SampleTests >
1 1
1 1




9


10



11




tTest
Wilcoxon-Mann-Whitney
12




2.      The following options window is shown.
                                                                                                  221

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                           Two Sample Wilcoxon Mann-Whitney Test Sample Size Options
                      False Rejection Rate [Alpha]

                               O 0.005 [0.5%]
                               O 0.010 [1.0%]
                               O 0.025 [2.5%]
                               ฎ 0.050 [5.0%]
                               O 0.100 [10.%]
                               O 0.150 [15.%]
                               O 0.200 [20.%]
                               O 0.250 [25.%]

                      Pooled Estimate of Population SD

                          Preliminary Estimate (planning stage)
                          Sample SD using collected data
                          (to assess power)
False Acceptance Rate [Beta]

      O 0.005 [0.5%]
      O 0.010 [1.0%]
      O 0.025 [25%]
      O 0.050 [5.0%]
      ฎ 0.100 [10.%]
      O 0.150 [15.%]
      O 0.200 [20.%]
      O 0.250 [25.%]

Width of Gray Region [Delta]
             •   Specify the False Rejection Rate (Alpha, a). Default is 0.05.

             •   Specify the False Acceptance Rate (Beta, P). Default is 0.1.

             •   Specify the Estimate of standard deviation of WMW Test Statistic. Default is 3

             •   Specify the Width of the Gray Region (Delta, A). Default is 2.

             •   Click on OK button to continue or on Cancel button to cancel the options.

           Output Screen for Sample Sizes for Single-Sample WMW Test (Default Options)
                                        i  S ample S izes for T wo S ample Witc oxon- M ann -Whitney Test
                                          Based on Specified Values of Decision Paiamelets/DOOs [Data Quaify Objectives)
                      Date/Time of Computation   2/26V2010 12:18:47 PM
                         User Selected Options
                     False Rejection Rate [Alpha]   0.05
                    False Acceptance Rate [Beta]   0.1
                     Width of G ray R egion [D elta]   2
                    Estimate of Standard Deviation   3
                                         Approximate Minimum Sample Size
                 Single Sided Alternative Hypothesis:            46
                  Two Sided Alternative Hypothesis:            56
222

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12.4   Sample Sizes for Acceptance Sampling
1.      Click DQOs Based Sample Sizes ^- Acceptance Sampling
r^ File Edit
Navigation F
Name
Worksheet jds
Well Iflxls
WMW-with NDsjd
ASHALL7groupsa
Stats/Sample Sizes Graphs Statistical Tests
General Statistics ป I
Imputed NDs using ROS Methods > rk
DQOs Based Sample Sizes > |
s
s
2
3
4
5 a
71 17

Upper Limits/BTVs
3
9ro D_Site
4

UCLs/EPCs Windows Help
567

Estimate Mean
Hypothesis Tests >
Acceptance Sampling






2.      The following options window is shown.
                                         OptionsSampteSizeAcceptance
                               Confidence Coefficient [C
                               Pre-specffied Proportion [P] of non-comforming items/drums
                                                   0.05

                               Number of Allowable non-conforming items/drums
                                                I    o
        •   Specify the Confidence Coefficient. Default is 0.95.
        •   Specify the Proportion [P] of non-conforming items/drums. Default is 0.05.

        •   Specify the Number of Allowable non-conforming items/drums. Default is 0.
        •   Click on OK button to continue or on Cancel button to cancel the options.

             Output Screen for Sample Sizes for Acceptance Sampling (Default Options)
                                            ! Acceptance Sampling for Pie-specified Proportion of Non-confonwig Items
                                             Based on Specified Values of Decision Parameters/DHOs
                         Date/Time of Computation   2/26^2010 12:20:34 PM
                            User Selected Options
                            Confidence Coefficient   0.95
    Pre-specified proportion of non-conforming items in the lot   0.05
        Number of allowable non-conforming items in the lot   0
                                             Approximate Minimum Sample Size
                       Exact Binomial/Beta Distribution           59
         Approximate Chisquare Distribution (Tukey-Scheffe)           59
                                                                                                    223

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                                     Chapter 13


                              Analysis of Variance

Oneway Analysis of Variance (ANOVA) is a statistical technique that is used to compare the measures of
central tendencies:  means or medians of more than two populations/groups. Oneway ANOVA is often
used to perform inter-well comparisons in groundwater monitoring projects. Classical Oneway ANOVA
is a generalization of the two-sample t-test (Hogg and Craig 1995); and nonparametric ANOVA, Kruskal-
Wallis test (Hollander and Wolfe 1999) is a generalization of the two- sample Wilcoxon Mann Whitney
test. Theoretical details  of Oneway  ANOVA are given in the ProUCL Technical Guide. Oneway
ANOVA is available  under the Statistical Tests module of ProUCL 5.0/ProUCL 5.1. It is advised to use
these  tests  on raw  data in  the original  scale without transforming  the  data  (e.g., using  a log-
transformation) .
B^ File Edit Stats/Sample Sizes Graphs
Navigation Panel
Name
Work Sheet jds
Well Kbds
WMW-with NDsjds
ASHALL7graups;
-------
2.      The Select Variables screen will appear.
            •   Select the variables for testing.
            •   Select a Group variable by using the arrow under the Group Column option.
            •   Click OK to continue or Cancel to cancel the test.
Example 13-la. Consider Fisher's (1936) 3  species (groups) Iris flower data set. Fisher collected data on
sepal length, sepal width, petal length and petal width for each of the 3 species.  Oneway ANOVA results
with conclusions for the variable sepal-width (sp-width) are shown as follows:
                               Output for a Classical Oneway ANOVA
                 Date/Time of Computation
                          From Rle
                        Full Precision
Classical Oneway ANOVA
3/26/2013 10:45:03 AM
FULLIRIS-rdsjds
OFF
sp -width
Group Obs Mean
1 50 3.428
2 50 2.77
3 50 2.974
Grand Statistics (All data) 150 3.057
SD Variance
0379 0144
0.314 0.0985
0322 0.104
0.436 0.19

Classical One-Way Analysis of Variance Table
Source SS DOF MS
Between Groups 11.34 2 5.872
Within Groups 16.96 147 0.115
Total 28.31 149
Pooled Standard Deviation 0.34
R-Sq 0.401
V.R.(FStat) P-Value
49.16 0



Note: A p-value •:- 0.05 (or some other selected level) suggests that there are significant deferences in
mean/median characteristics of the various groups at 0.05 or other selected level of significance
             A p-value > 0.05 (or other selected level) suggests that mean/median characteristics of the various groups are comparable
                                                                                                   225

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13.2   Nonparametric ANOVA

Nonparametric  Oneway ANOVA or the Kruskal-Wallis (K-W) test is a generalization of the Mann-
Whitney two-sample test. This is a nonparametric test and can be used when data from the various groups
are not normally distributed.

1.       Click Oneway ANOVA ^ Nonparametric
a^ File Edit Stats/Sample Sizes Graphs
Navigation Panel
Name
Worksheet :ds
Wdl 10jds
WHW-with NDsxIs
ASHALL7groupsjds
Box Plot Full.gst
Box Plot Full_a.gst

1
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Statistical Tests | Upper Limits/BTVs UCLs/EPCs Windows Help
Outlier Tests I
Goodness-of-Fit Tests >
Single Sample Hypothesis >
Two Sample Hypothesis ^
Oneway ANOVA ^
OLS Regression
Trend Analysis >
5

6

7





Classical
Nonparametric

Like classical Oneway ANOVA, nonparametric ANOVA also requires that the data file  used should
follow the data format as shown above; the  data file should consist of a group variable  defining the
various groups to be evaluated using the Oneway ANOVA module.

2.     The Select Variables screen will appear.


           •   Select the variables for testing.


           •   Select the Group variable.


           •   Click OK to continue or Cancel to cancel the test.


Example  13-lb (continued). Nonparametric Oneway ANOVA results with conclusion for sepal-length
(sp-length) are shown as follows.
                              Output for a Nonparametric ANOVA

                               Nonparametric Oneway ANOVA (Kruskal-Wallis Test)
                 Date/Time of Computation  3/26/2013 11:11:32 AM
                         From Be  FULLI
                       Full Precision  OFF
                      sp4ength



                    Group
1
2
3
Overall

K-W (H-Stat)
96.76
96.34

50
50
50
150
5
5.9
6.5
5.8

DOF
2
2
P-Value
0
0

29.64
82.65
114.2
-9142
1.425
7.716
75.5
(Approx. Chisquare)
(Adjusted for Ties)























Hole: A p-value <= 0.05 (or some other selected level) suggests that there are significant differences in
mean/median characteristics of the various groups at 0.05 or other selected level of significance
             A p-value > 0.05 (or other selected level) suggests that mean/median characteristics of the various groups are comparable-
226

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227

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                                       Chapter 14


     Ordinary  Least Squares of Regression and Trend Analysis


The OLS of regression and trend tests are often used to determine trends potentially present in constituent
concentrations at polluted sites, especially in GW monitoring applications. The OLS regression and two
nonparametric trend tests: Mann-Kendall test and Theil-Sen test are available under the Statistical Tests
module  of ProUCL  5.0/ProUCL 5.1. The  details of these tests can be found in Hollander and Wolfe
(1999) and  Draper and Smith  (1998). Some time series plots, which are useful in comparing trends in
analyte concentrations of multiple groups (e.g., monitoring wells), are also available in ProUCL.

The two nonparametric trend tests: M-K test and Theil-Sen test are meant to identify trends in time series
data (data collected  over a certain period of time such as daily,  monthly, quarterly,  etc.) with distinct
values of the time variable (time of sampling events). If multiple observations are collected/reported at a
sampling event (time),  one or more pairwise slopes used in the computation of the Theil-Sen test may not
be computed (become infinite). Therefore, it is suggested that the Theil-Sen test only be used on data sets
with one measurement collected at each sampling event. If multiple measurements  are collected  at a
sampling event, the user may want to use the average (or median, mode, minimum or maximum) of those
measurements resulting in a time series with one measurement per sampling time event. Theil-Sen test in
ProUCL has  an option which can be used to average multiple  observations reported for the various
sampling events. The use of this option also computes M-K test statistic and OLS statistics based upon
averages of multiple observations collected at the various sampling events.

New in ProUCL 5.1:  In  addition to slope and intercept of the nonparametric Theil-Sen (T-S) trend line,
ProUCL 5.1 computes residuals based upon the T-S trend line.

The trend tests in ProUCL software  also assume that the user has entered data in chronological order. If
the data are not entered properly in chronological order, the graphical trend displays may be meaningless.
Trend Analysis and OLS Regression  modules  handle missing values  in both response variable (e.g.,
analyte concentrations) as well as the sampling event variable (called independent variable in OLS).
14.1   Simple Linear Regression

1.     Click Statistical Tests ^ OLS Regression.
                                                                       CL 5.0 - [WMW-with N[
of File Edit Stats/Sample Sizes Graphs
Navigation Panel
Name
Worksheet.^
WeJIDjds
WMW™th NDs:xis
ASHALL7groupsxls
Box Plot Full.gst
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Rnv Pint Fi ill h net
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Statistical Tests | Upper Limits/BTVs
Outlier Tests ^
Goodness-of-Fit Tests ^
Single Sample Hypothesis ^
Two Sample Hypothesis ^
Oneway ANOVA ^
OLS Regression
Trend Analysis >





UCLs/EPCs Windows Help
5



6 7






8



9



10



11







228

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2.      The Select Regression Variables screen will appear.

       •    Select the Dependent Variable and the Independent Variable for the regression analysis.

Available Variables
Name ID
Time (days)-! 0
BTEXConc. @... 1
Time (days>2 2
BTEXConc. @... 3
Time [days)-3 4
BTEXConc. @... 5
Time (daysH 6
BTEXConc. @... 7
Time (days}-5 8
BTEXConc. @... 9
< >

ป
ซ
Dependent Variable
Name ID
MW-28 1 1
< >

Independent Variable
ซ
Option
Name ID
Time(days)-6 10
< >

Select Group Column (Optional)
V

OK Cancel
               Select a group variable (if any) by using the arrow below the Select Group Column
               (Optional). The analysis will be performed separately for each group.

               When the  Options button is clicked, the following options window will appear.
                                             Regression Options
                                    EH Display Intervals

                                         Confidence Level

                                              095

                                    @ Display Regression Table

                                    I  I Display Diagnostics
                                  Graphics Options

                                    0 Display XY Plot

                                          XY Plot Title
                                      Classical Regression
                                    0 Display Confidence Interval

                                    0 Display Prediction Interval
                                       OK
                   Select Display Intervals for the confidence limits and the prediction limits of each
                   observation to be displayed at the  specified Confidence Coefficient. The interval
                   estimates will be displayed in the output sheet.
                                                                                              229

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               o   Select Display Regression Table to display Y-hat, residuals and the standardized
                   residuals in the output sheet.

               o   Select "XY Plot" to generate a scatter plot display showing the regression line.

               o   Select Confidence Interval and Prediction Interval to display the confidence and
                   the prediction bands around the regression line.

               o   Click on OK button to continue or on Cancel button to cancel the option.

           •   Click OK to continue or Cancel to cancel the OLS Regression.

               o   The use of the above options will display  the following graph on  your computer
                   screen which can be copied using the Copy Chart (To Clipboard) in a Microsoft
                   documents (e.g., word document) using the File  ^-Paste combination.

               o   The above options will also generate an Excel-Type  output  sheet. A partial output
                   sheet is  shown below following the OLS Regression Graph.

Example 14-la. Consider analyte concentrations, X collected from a groundwater (GW) monitoring well,
MW-28  over  a certain period of time. The objective  is to determine  if there is any trend in  GW
concentrations, X of the MW-28.  The OLS regression line with inference about  slope and intercept are
shown in the  following figure. The slope  and its  associated p-value suggest  that there is a significant
downward trend in GW concentrations of MW-28.
               OLS Regression Graph without Regression and Prediction Intervals
                                     Classical Regression
                                                                                Slope
                                                                                Intercept
                                                                                  Je(Reg)
                                                                                  je (Slope]
2.163.7246
 0.8432
 •09183
 372.6302
 00000
 00000
              2117
                                                                                SUndaldizedS    5.0756
                                                                                Approximate p-vatje  O.OOfH
                                          4246       '"--....
                                        Time (days)-6
230

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OLS Regression Graph with Regression and Prediction Intervals
                         Classical Regression
                                                                                          •1.6372
                                                                                        3,1637245
                                                                                          OB432
                                                                              P-valtie (Reg)
                                                                              P-value (Slope)

                                                                              Mann-Kendall
                                                                              S
                                                                              SDoJS



                                                                              Confidence Coefficient

                                                                              Red - Prediction Inierva
-135.0000
 26.4009
 •S075S
160    260    360    460    560   660   760   860
                             Time (days)-6
                                       960    1060   116C
             Partial Output of OLS Regression Analysis
Ordinary Least Squares Linear Regression Output Sheet

User Selected Options
Date/Time of Computation



From File
Full Precision


Paramater
intercept
Tme (daysJ-6

3/27/2013 11:51:45 AM
Trend-MW-data-use jds
OFF







Number Reported W-values) 1 8
Dependendant Variable MW-28
Independent Variable Time (dsys}f
Regression Estimates and Inference Table
Estimates Std. Error T^alues p^/alues
2164
-1.637
165.3 13.09 5.793E-10
0.176 -9.276 7.7292E-8

OLS ANOVA Table
Source of Variation SS DOF MS
Regression 11952431 1 11952431


Error 2222368 16 138898
Total 14174799 17
R Square 0.843
.Adjusted R Square 0.833

Obs
1
2
3
4
5
6
7
8
9
Y Vector
2880
2117
1633
1845
1706
1719
1065
831.8
920.6
Sqrt(HSE) = Scale 372.7
Regression Table
Yhat Residuals Res/Scale
2164 716.3 1.922
2028 B9.17 0.239
1900 -267.6 -0.718
1748 97.13 0.261
1587 118.2 0.317
1307 411.1 1.103
1154 -SB. 55 -0.23B
1009 -177.7 -0.477
1009 -B8.87 -0.238







F-Value
8605







P-Value
0.0000




























                                                                                               231

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Verifying Normality of Residuals: As shown in the above  partial  output, ProUCL displays residuals
including standardized residuals on the OLS output sheet. Those residuals can be imported (copying and
pasting) in an excel file to assess the normality of those OLS  residuals. The parametric trend evaluations
based  upon the OLS slope (significant slope,  confidence interval and  prediction interval) are valid
provided the OLS residuals  are normally distributed.  Therefore, it is suggested that the user assesses the
normality of OLS  residuals before drawing trend conclusions using a parametric test based upon the OLS
slope estimate. When the assumptions are  not  met,  one can use graphical  displays and  nonparametric
trend tests (e.g., T-S test) to  determine potential trends present in a time series data set.
14.2   Mann-Kendall Test

1.      Click Statistical Tests ^ Trend Analysis ^ Mann-Kendall.
                                                     ^L b.U - llrend-MW-data-uiapi4.xlsJ
           Statistical Tests  I Upper Limits/ETVs   UCLs/EPCs  Windows   Help
       Outlier Tests

       G o o d n ess- of - Fit Tests

       Single Sample Hypothesis

       Two Sample Hypothesis

       Oneway ANOVA

       OLS Regression
               Trend Analysis
                                                      8
                                                                     1C
                                                                             11
                                                                                    12
                                  Mann-Kendall
                                          Theil-Sen

                                          Time Series Plot   >
2.
The Select Trend Event Variables screen will appear.
                         Available Variables
                         Name
                                    ID
                                                Selected Variable
                                                Values/Measured Data
                                                       Name
                                                       MW-2S
                                                           ID
                                                           1
                                                       Optional EventfTime
                                                    Not Required - Index data will
                                                       be generated for graphics
                                                      Select Group Column (Optional)
232

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    Select the Event/Time variable. This variable is optional to perform the Mann-Kendall
    (M-K) Test; however, for graphical display it is suggested to provide a valid Event/Time
    variable (numerical values only). If the user wants to generate a graphical display without
    providing an Event/Time variable, ProUCL generates an index variable to represent
    sampling events.

    Select the Values/Measured Data variable to perform the trend test.

    Select a group  variable (if  any) by using the  arrow below the Select Group Column
    (Optional). When a group variable is chosen, the analysis is performed separately for
    each group represented by the group variable.

    When the Options button is clicked, the following window will be shown.
                      Select Mann Kendall Options
                    Confidence Level
                                 0.95
                    Graphics Options

                    0 Display Graphics

                    0 Display Theil-Sen Trend Line

                    0 Display OLS Regression Line

                           Title for Graph
                       Mann-Kendall Trend Test
    o   Specify the Confidence Level; a number in the interval (0.5, 1), 0.5 inclusive. The
       default choice is 0.95.

    o   Select  the trend lines to be displayed: OLS Regression Line and/or Theil-Sen
       Trend Line. If only Display Graphics is chosen, a time series plot will be generated.

    o   Click on OK button to continue or on Cancel button to cancel the option.

•   Click OK to continue or Cancel to cancel the Mann-Kendall test.
                                                                               233

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14- Ib (Continued). The M-K test results are shown in the following figure and in the following M-K
test output sheet. Based upon the M-K test, it is concluded that there is a statistically significant
downward trend in GW concentrations of the MW-28.
                     Mann Kendall Test Trend Graph displaying all Selected Options
                                            Mann-Kendall Trend Test
                                                                                                       O.OMMI
                                                                                             SiwMtad Deviate rt S  26 4MB
                                                                                                 dVaijfrtS  -SOTS
                                   Mann-Kendall Trend Test Output Sheet
                                                       Mann-Kendall Trend Test Analysis
                                      User Selected Options
                                    Date/Time of Computation  3/27/7013 12:19:26 PM
                                               From File  Tnend-MW-data-Chap14jds
                                             Full Precision  OFF
                                      Confidence Coefficient  0.95
                                       Level of Significance  0.05

                                                MW-28
                                            General Statistics
                                                   Number of Events
                                            Number Values Reported In}
                                                         Minimum
                                                        Maximum
                                                           Mean
                                                    Geometric Mean
                                                          Median
                                                  Standard Deviation
                                           Mam-Kendall Test
                                                     Test Value 
-------
14.3   Theil - Sen Test
To perform the Theil-Sen test, the user is required to provide numerical values for a sampling event
variable (numerical values only) as well as values of a characteristic (e.g., analyte concentrations) of
interest observed at those sampling events.
1.
Click Statistical Tests ^ Trend Analysis ^ Theil-Sen.
aJ File Edit Stats/Sample Sizes Graphs
Navigation Panel
Name
Worksheet xis
Well 10xls
WMW^ith NDsjds
ASHALL7groups*is
Box Plot Full.gst
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Box Plot Full_b.gst
1
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Statistical Tests
Upper Limto/BTVs UCLs/EPCs Windows Help
Outlier Tests t
Goodness-of-Fit Tests t
Single Sample Hypothesis >
Two Sample Hypothesis >
Oneway ANOVA t
OLS Regression
Trend Analysis ^
34 1
21 35 1
1
1
5 6

7





Mann-Kendall
Theil-Sen
Time Series Plot




ป
2.
The Select Variables screen will appear.
                              Available Variables
                              Name
                                       ID
                                                     Selected Event/Time
                                                        Event/Time Data
Name
Time (daysX
ID
D
                                                      Selected Variable
                                                      Values/Measured Data
Name
MW-28
ID
1
                                                     Select Group Column (Optional)
               Select an Event/Time Data variable.

               Select the Values/Measured Data variable to perform the test.

               Select a group variable (if any) by using the arrow below the Select Group Column
               (Optional). When a group variable is chosen, the analysis is  performed separately for
               each group represented by the group variable.

               When the Options button is clicked, the following window will be shown.
                                                                                              235

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                                         Confidence Level
                                         Graphics Options

                                          0 Display OLS Regression Line
                                          0 Display Theil-Sen Trend Line

                                                Title for Graph
                                          Theil-Sen Trend Line and Ol
                 o   Specify the Confidence Level; a number in the interval (0.5, 1), 0.5 inclusive. The
                     default choice is 0.95.

                 o   Select the trend lines to be displayed: OLS Regression Line and/or Theil-Sen
                     Trend Line.

                 o   Click on OK button to continue or on Cancel button to cancel the option.

             •   Click OK to continue or Cancel to cancel the Theil-Sen Test.

14-lc (continued). The Theil-Sen test results are shown in the following figure and  in  the following
Theil-Sen test Output Sheet. It is concluded that there is a statistically significant downward trend in GW
concentrations of MW-28. Theil-Sen test  results  and residuals are summarized in tables  following the
trend graph shown below.

                      Theil-Sen Test Trend Graph displaying all Selected Options
                            Theil-Sen Trend Line and OLS Regression Line
                                                                                         Theil Sen T
-------
                                   Theil-Sen Trend Test Output Sheet
Date/Time of Computation 3/27/20132:19:55 PM
From Rle Trend-MW-data-Chap14jds
Full Precision OFF
Confidence Coefficient 0.95
Level of Significance 0.05
MW-28
General Statistics
Number of Events
Number Values Reported In}
Minimum
Maximum
Mean
Geometric Mean
Median
Standard Deviation
18
18
1.7
2880
8-64.6
174.8
628.2
913.1
                                                        Approximate inference for Theil-Sen Trend Test
                                                                      Mann-Kendall Statistic (S)   -137
                                                                       Standard Deviation of S    26.4
                                                                       Standardized Value of S   -5.151
                                                                         Approximate p-value  1.2930E-7
                                                                           Number of Slopes   153
                                                                            Theil-Sen Slope   -1.705
                                                                          Theil-Sen Intercept  1917
                                                                                    M2    98.21
                                                                One-sided 95% upper limit of Slope   -1.365
                                                                      95% LCL of Slope (0.025)   -2.222
                                                                      95% UCL of Slope (0.975)   -1.268
                                                      Statistically significant evidence of a decreasing
                                                      trend at the specified level of significance.
                                  Then-Sen Trend Test Estimates and Residuals
Notes: As with other statistical test statistics, trend test statistics: M-K test statistic, OLS regression and
Theil-Sen slopes may lead to different trend conclusions,  hi such instances it is suggested that the user
supplements statistical conclusions with graphical displays.
Averaging of Multiple  Measurements at Sampling Events:  hi practice, when multiple observations  are
collected/reported at  one or more sampling events  (times),  one or more pairwise slopes  may become
infinite, resulting in a failure to compute the Theil-Sen test statistic. In such cases, the user may want to
                                                                                                        237

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pre-process the data before using the Theil-Sen test. Specifically, to assure that only one measurement is
available at each sampling event, the user pre-processes the time  series  data by computing average,
median, mode, minimum, or maximum of the multiple observations collected at those sampling events.
The Theil-Sen test in ProUCL 5.0/ProUCL 5.1 provides the option of averaging multiple measurements
collected at the various sampling  events. This option also computes M-K test and  OLS regression
statistics using the averages of multiple measurements collected at the various sampling event. The OLS
regression and M-K test can be performed on data sets with multiple measurements taken at the various
sampling time events. However, often it is desirable to use the averages (or median) of measurements
taken at the various sampling events to determine potential trends present in a time-series data set.

14-lc  (continued). The data set used in Example 14-lc has some  sampling events where multiple
observations were taken. Theil-Sen test results based upon averages of multiple observations is shown as
follows. The data set is included in the ProUCL Data directory which comes  with ProUCL 5.1.

                    Theil-Sen Test Trend Graph displaying all Selected Options
           Multiple Observations Taken at Some Sampling Events Have Been Averaged
                         Theil-Sen Trend Line and OLS Regression Line
T heil-S en T lend Analysis


 Level ol Significance


OLS Regression Line (Blue]
 OLSRegiessionSlope
 OLS Regression Intercept


T heil-S en T lend Line (Red)
 Theil Sen Slope
 Theil-Sen Intercept
                                                                                             -1.8539
                                                                                            2,183.2196
                                                                                Ml            32.6972
                                                                                M2            72.3028
                                                                                LCLol9lope         2.3073
                                                                                UCLofSlope        -1.3503
                                                                                Statistically significant evidence oi
                                                                                trend at the specified level ol sigrafcance.
                                     Time (days)-6
14.4   Time Series Plots

This option of the Trend Analysis module can be used to determine and compare trends in multiple
groups over the same period of time.

This option is specifically useful when the user wants to compare the concentrations of multiple groups
(wells) and the exact sampling event dates are not be available (data only option). The user may just want
to graphically compare the time-series data collected from multiple groups/wells during several quarters
(every year, every 5 year, etc.). When the user wants to use this module using the data/event option,
each group  (e.g., well)  defined by a group variable must have the same number of observations and
should share the same sampling event values. That is the number of sampling events and values (e.g.,
quarter ID, year ID, etc.) for each group (well) must be the same for this option to work. However, the
238

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exact sampling dates (not needed to use this option) in the various quarters (years) do not have to be the
same as long as the values of the sampling quarters/years (1,3,5,6,7,9,..) used in generating time-series
plots for the various groups (wells) match. Using the geological and hydrological information, this kind of
comparison may help the project team in identifying non-compliance wells (e.g., with upward trends in
constituent concentrations) and associated reasons.
1.
Click Statistical Tests ^ Trend Analysis ^ Time Series Plots
                                                                  /IW89-C
jraphs
2
MW-ID






Statistical Tests | Upper Limrts/BTVs UCLs/EPCs Windows Help
Outlier Tests >
Goodness-of-Fit Tests >
Single Sample Hypothesis >
Two Sample Hypothesis t
Oneway ANOVA >
OLS Regression
Trend Analysis >
1 460 8
1 547 &
1 605 8
6
f-Mn-89
4600
2760
1270
1860
OgD
7



Mann-Kenda
Thai-Sen
8
MW9
9
9
9
9
i


Time Series Plot >
1610 <
H jnn i
9
MN9
2200
2340
2340
2420
2150
10

11
MN-99
2200
2340
2340
' 1 2420

215C
2220 2220
2050 2050
Data Only
2060
Event/Data 1770
2.
When the Data Only option is clicked, the following window is shown:

                                   : Trend Data Variable

                                                Selected Variable
                                                Values/Measured Data
                       Mn-GW
                       MW-ID
                       Manganese
                       MW-E9
                       GW-Mn-B9
                       MW9
                       MN9
                       MN-99
                                                       Name
                                                                  ID
                                              Select Group Column (Optional)
       This option is used on the measured data only. The user selects a variable with measured values
       which are used in generating a time  series plot. The time series plot option is specifically useful
       when data come from multiple groups (monitoring wells during the same period of time).

       Select a group variable (is any) by using the arrow shown below the Group Column (Optional).
                                                                                             239

-------
*
Select Trend Data Variable - n
Available Variables
Name ID
Well ID 0
MW-ID 2
Manganese 3
MW-B9 5
GW-Mn-89 6
MW9 8
MN9 9
MN-99 11
index 14

Selected Variable
Values/Measured Data
ป
ซ
<
Options
Name ID
Mn-GW 1

felect Group Column (Optional)


OK Cancel


           When the Options button is clicked, the following window will be shown.
                      Confidence Coefficient

                                  0.95


                      Set Event/Index Label
                                                    Set Initial Start Value
                                 Event
                      Plot Graphs Together
                      0 Group Graphs

                       Must select a Group Column
                       All Groups the Same Size!
                     0 Display Thei I -Sen Trend Line
                       Minimal Theil-Sen Stats Provided

                     D Display OLS Regression Line
             1
         Event/Index




Set Event/Index Increments

             1
     Greater Than Zero [0]


       Title for Graph
                                                     Time-Series Trend Analysis
        The user can opt to display graphs for each group individually or for all groups together on the
        same graph by selecting the Group Graphs option. The user can also display the OLS line and/or
        the Theil-Sen line for all  groups displayed on the same graph. The user may  pick an initial
        starting value and an increment value to display the measured data. All statistics will be computed
        using the data displayed on the graphs (e.g., selected Event values).

        o   Input a starting value for the index of the plot using the Set Initial Start Value.

        o   Input the increment steps for the index of the plot using the Set Index/Event Increments.

        o   Specify the lines (Regression and/or Theil-Sen) to be displayed on the time series plot.

        o    Select Plot Graphs Together option for comparing the time  series trends for more than one
           group on the same graph.
240

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            If this option is not selected but a Group Variable is selected, different graphs will be plotted
            for each group.

            Click on OK button to continue or on Cancel button to cancel the Time Series Plot.
3.
When the Event/Data option is clicked, the following window is shown:
                            Available Variables
Name
Well ID
MVi'-ID
Manganese
MW-S9
GW-Mn-39
MW9
HNS
MN-99
ID
0
2
3
5
6
S
5
11
                                                Opt
                                                Selected Event/Time
                                                   Event/Time Data
                                                         Name
                                                         index
                                                           ID
                                                           14
                                                         Selected Variable
                                                         Values/Measured Data
Name
Mn-GW
ID
1
                                                       Select Group Column (Optional)
        Select a group variable (is any) by using the arrow shown below the Group Column (Optional).

        This option  uses both the  Measured Data and the Event/Time Data.  The  user selects  two
        variables; one representing the Event/Time variable and the other representing the Measured Data
        values which will be used in generating a time series plot.

        When the Options button is clicked, the following window will be shown.
                                          elect Time Series

                                          fidence Coefficient

                                                 0.95

                                        Display OLS Regression Line

                                        Display Thei l-Sen Trend Line
                                      Plot Graphs Together
                                      EH Group Graphs

                                       Must select a Group Column
                                       All Groups the Same Size!
                                              Title for Graph
                                        Time-Series Trend Analysis
                                                                                                 241

-------
        The user can select to display graphs individually or together for all groups on the same graph by
        selecting the Plot  Graphs Together option. The  user can also display the OLS line and/or the
        Theil-Sen line for all groups displayed on the same graph.

        o   Specify the lines (Regression and/or Theil-Sen) to be displayed on the time series plot.

        o   Select Plot Graphs Together  option for  comparing time series trends  for more  than one
            group on the same graph.

        o   If this option is not selected but a Group Variable is selected, different graphs will be plotted
            for each group.

        o   Click on OK button to continue or on  Cancel button to cancel the options.

    •   Click OK to continue or Cancel to cancel  the Time Series Plot.
Notes:  To use this option, each group (e.g., well) defined by a group variable must have the same number
of observations and should share the same sampling event values (if available).  That is the  sampling
events (e.g., quarter ID, year ID, etc.) for each group (well) must be the same for this option to work.
Specifically, the exact sampling  dates within the various quarters (years) do  not have to be the same as
long as the sampling quarters (years) for the various wells match.

Example 14-2. The following graph has three (3) time series plots comparing manganese concentrations
of the three GW monitoring wells (1 upgradient well  [MW1]  and 2 downgradient wells  [MW8 and
MW9]) over the period of 4 years (data collected  quarterly).  Some trend statistics are displayed in the
side panel.

        Output for a Time Series Plot - Event/Data Option by a Group Variable (1, 8, and 9)
                                   Time-Series Trend Analysis
                                                   496    478   508   469   475
     0.0   1.0    2.0    3.0   4.0    5.0   6.0   7.0    8.0   9.0   10.0    11.0   12.0   13.0   14.0   15.0   16.0   17.0
                                           index
Tiroe-SeriesTiend Analysis
 ttEvents/Time Periods
OLS Regression Line [Bkjel
Theil-SenTlend Line [Red)
                                                                                       1
                                                                                         LSRegiession Slope   -6.9618
                                                                                         LSReglessionlntercept 561.5500
                                                                                         heil-SenSlope     -7.8333
                                                                                         heilSenlntetcept    568.5333
                                                                                         LS Regression Slope   -42.8371
                                                                                         LSRegiessionlnlocept 2.362.7500
                                                                                         heil-SenSlope     -4.7619
                                                                                         heil-Senlnt<*cept   1.790.4762
                                                                                         LS Regression Slope   40.8382
                                                                                         LSRegiessionlnleicept 2.315.2500
                                                                                         heil SenSlope     -72.5000
                                                                                         heil-SenlnleicepI   2.671.2500
242

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243

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                                      Chapter 15


           Background Incremental Sample Simulator (BISS)
   Simulating BISS Data from a Large Discrete Background Data


The Background Incremental Sample Simulator (BISS) module was incorporated in ProUCLS.O at the
request of the Office of Superfund Remediation and Technology Innovation (OSRTI).  However, this
module is currently under further investigation and research, and therefore it is not available for general
public use. This module has been retained in ProUCL 5.1. This module may be released in a future
version of the ProUCL software, along with strict conditions and guidance for how it is applied. The main
text for this chapter is not included in this document for release to general public. Only a brief placeholder
write-up is provided here.

The following scenario describes the Site or project conditions under which the BISS module could be
useful:  Suppose there is a long history of soil sample collection  at a Site. In addition to having a large
amount of Site data, a robust background data set (at least 30 samples from verified background
locations) has also been collected. Comparison of background data to on-Site data has been,  and will
continue to be, an important part of this project's decision-making strategy. All historical  data is from
discrete samples, including the background data. There is now a desire to switch to incremental sampling
for the Site. However,  guidance for incremental sampling makes  it clear that it is inappropriate to
compare  discrete  sample  results  to  incremental sample results.  That includes comparing  a Site's
incremental results directly to discrete background results.

One option is to recollect all background data in the form of incremental samples from background DUs
that are designed to match Site  DUs  in geology, area, depth, target soil particle size, number of
increments, increment sample support, etc. If project decision-making uses a BTV strategy to  compare
Site DU results one at a time against background, then an appropriate number (the default is no less than
10) of background DU incremental samples would need to  be collected to determine the BTV for the
population of background DUs. However, if the existing discrete background data show background
concentrations to  be low (in comparison to Site concentrations) and fairly consistent (relative  standard
deviation, RSD <1), there is a second option described as follows.

When a robust discrete  background data set that meets the above conditions already exists, the following
is an alternative to automatically recollecting ALL background data as incremental samples.

Step 1.  Identify 3  background DUs and collect at least 1 incremental sample from each for a minimum of
3 background incremental samples.

Step 2. Enter the discrete background data set (n > 30) and the >3 background incremental  samples into
the BISS module (the BISS module will not run unless both data sets are entered).

•      The BISS module will generate a specified (default is 7) simulated incremental samples from the
       discrete data set.

•      The module will then run a t-test to compare the simulated background incremental data set (e.g.,
       with n = 7) to the actual background incremental data set (n > 3).
244

-------
           o   If the t-test finds no difference between the 2 data sets, the BISS module will combine
               the  2 data sets  and determine the statistical distribution, mean,  standard  deviation,
               potential UCLs and potential BTVs for the combined data set. Only this information will
               be supplied to the general user.  The individual values of the simulated incremental
               samples will not be provided.

           o   If the t-test finds a difference between the actual and  simulated data sets, the BISS
               module will not combine the data sets nor provide a BTV.

           o   In  both cases, the  BISS  module will report summary statistics  for  the  actual and
               simulated data sets.

Step 3. If the BISS module reported out statistical analyses from the combined data set, select the BTV to
use with Site  DU incremental sample  results. Document the procedure used to generate the  BTV in
project reports. If the BISS module reported that  the simulated and actual data sets were different, the
historical discrete data set  cannot be used to simulate incremental results.  Additional background DU
incremental samples will need to be collected to obtain a background DU incremental data set with the
number of results appropriate for the intended use of the background data set.

The  objective  of the BISS module is  to take advantage of the information provided  by the  existing
background discrete samples.  The availability of a large discrete data set collected from the background
areas with geological formations and conditions comparable to the Site DU(s) of interest is a requirement
for successful  application of this module.  There are fundamental differences between incremental and
discrete samples. For  example,  the sample  supports of discrete  and  incremental  samples are very
different. Sample support has a profound effect  on sample results so samples with different  sample
supports should not be compared directly, or compared with great caution.

Since incremental sampling is  a relatively new approach,  the performance of the BISS module  requires
further investigation. If you would like to try this strategy for your project, or if you  have  questions,
contact Deana Crumbling, crumbling.deana(g),epa.gov.
                                                                                            245

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                                        Chapter 16

                                         Windows

The Windows Menu performs typical Windows program options.
 File  Edit  State/Sample Sizes   BIS Simulator   Graphs  Statistical Tests  Upper Limits/BTVs  UCLs/EPCs
                                                                                       3UCL 5.0
    Navigation Panel
 Name
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  many  windows (e.g., >40) open; the
               navigation panel only holds the first 40 windows.
246

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                                      Chapter 17
                 Handling the Output Screens and  Graphs

17.1   Copying and Saving Graphs
Graphs can be copied into Word, Excel, or PowerPoint files in two ways.
1.      Click the Copy Chart (To Clipboard) shown below; a graph must be present to be copied to the
       clipboard.
       File ^ Copy Chart (To Clipboard)
                            File  Edit  Stats/Sample Sizes  BIS Simulator  Graphs  Statistical Tests
                               New
                               Open Single File Sheet
                               Open Excel File with Multiple Sheets
                               Close
                               Save
                               Save As...
                               Save Graph
                               Copy Chart (To Clipboard)
                               Page Setup
                               Print
                               Print Preview
                               Exit
       Once  the user has clicked Copy Chart  (To Clipboard), the graph is ready to be imported
       (pasted) into most Microsoft office applications (e.g., Word, Excel, and PowerPoint) by clicking
       the Edit ^- Paste option in those Microsoft applications as shown below.
                                 TheJI-Sen Trend Line and OLS Regression Line
                                                                                        247

-------
2.      Graphs can be saved using the  Save Graph Option in the Navigation Panel as a Bitmap file
        with .bmp extension. The user can import the saved bitmap file into a desired document such as a
        word document or a PowerPoint presentation by using the Copy and Paste options available in
        the selected Microsoft application.
        File ^-  Save Graph
•ff
N
<:>,
w
w
A
Br
Bf
Bi
'.S
Tl
Tr
File Edit Stats/Sample Sizes BIS Simulator Graphs Statistical Tests Upf
New
Open Single File Sheet
Open Excel File with Multiple Sheets
Close
Save
Save As..,
Save Graph
Copy Chart (To Clipboard)
Page Setup
Print
Print Preview
Exit



17.2   Printing Graphs
1.
Click the graph you want to print in the Navigation Panel.
 Zn-Cu-ND-data-chapter 9-
 Box Plot wNDs gsi
 Box Plot wNDs_agst
                                                       Q-Q Plot for Cu
                                                  Theoretical Quantiles (Standard Normal)
                                                                                                      Mean-4815
                                                                                                      Sd-4.E3S
                                                                                                      Elope -4.045
                                                                                                      Intercept -4.815


                                                                                                     basin tiou^i
                                                                                                      tt-sa
                                                                                                      Mean-5.49
                                                                                                      Sd-4.92S
                                                                                                      Elope - 4.534
                                                                                                      Ititeicept = 5.49
                                                                                                      Correlation fi - 0 90
248

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Click File ^-  Page Setup.
                              New
                              Open Single File Sheet
                              Open Excel File with Multiple Sheets
                              Close
                              Save
                              Save As.,.
                              Save Graph
                              Copy Chart (To Clipboard)
                              Page Setup
                              Print
                              Print Preview
Check the button next to Portrait or Landscape (shown below), and click OK. In some cases,
with larger headings and captions, it may be desirable to use the Landscape printing option.
                                Paper
                                        Letter
                                        Auto
                                Orientation
                                '•' Portrait
                                O Landscape
                                           Margins indies)
                                           Left:   D.5     Right: |0.5
                                           Top:  J0.5     Bottom: |o.5
Click File ^  Print to print the graph, and File
before printing.
                                                     Print Preview to preview (optional) the graph
                             File  Edit   Stats/Sample Sizes  BIS Simulator  Graphs

M
'!',
VI
Vv
A
Bi
a
B<
Vv
Tl
A
New
Open Single File Sheet
Open Excel File with Multiple Sheets
Close
Save
Save As...
Save Graph
Copy Chart (To Clipboard)
Page Setup
Print
Print Preview
Exit





i
                                                                                                   249

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17.3  Making Changes in Graphs using Tools and Properties

ProUCL uses a couple of development tools such as FarPoint spread (for Excel type input and output
operations) and ChartFx (for graphical displays). ProUCL generates box plots using the built-in box plot
feature in  ChartFx. The programmer has no  control  over computing various statistics (e.g., Ql, Q2, Q3.
IQR) using ChartFx.  So box plots generated by ProUCL can differ slightly from box plots generated by
other programs (e.g.,  Excel). Box plots generated using ChartFx round values to the nearest integer. For
increased precision of graphical displays (all graphical displays generated by ProUCL), the user can use
the process described as follows.

Position your mouse  cursor on the graph and right-click, a popup menu will appear. Position the mouse
on Properties and right-click; a windows form labeled Properties will appear.  There are three choices at
the top; General, Series  and Y-Axis.  Position the mouse  cursor over the Y-Axis choice and left-click.
You can change the number of decimals to increase the precision, change the step to increase or decrease
the number Y-Axis values displayed and/or change the direction of the label. To show values on the plot
itself, position your mouse cursor on the graph and right-click; a popup menu will appear. Position the
mouse on  Point  Labels and right-click. There  are other options available in this popup  menu including
changing font sizes.
17.4  Printing Non-graphical Outputs

1.       Click/Highlight the output you want to save or print in the Navigation Panel.
   File  Edit  BIS Simulator  Graphs  Statistical Tests  Upper Limits/BTVs  UCLs/EPCs  Windows  Help
Navigation Panel
Name
Worksheet xis
Well 10xls
WMW^th NDsxis
ASHALL7groLjpsxis
Box Plot Full.gst
Box Plot Full_a.gst
Box Plot Full_b.gst
WorkSheet_axls
Theil-Senxls
Trend Test.gst
WorkSheet_bxis
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
A B C D
E F G H
Thei I Sen Trend Test Analysis
User Selected Options
D atsTi me of Computation 8C7Q01 3 9:23:00 PM
From File Well 10.xls
Full Precision OFF


Average Replicates Replicates at sampling events will be averaged!
Confidence Coefficient 0.95
Level ofSignificance 0.05
U(mg/L)




General Statistics
NumberofEvents 8
Number Values Observations
Number Values Missing
Number of Values Reported In)
Number of Values After Averaging
Number of Replicates
Minimum
Maximum
Mean
Geometric Mean
Median
Standard Deviation

Mann-Kendall Statistics
TestValue(S)
Tabulated p- value
Standard Deviation of S
Standa rdized Value of S
15
7
8
8
0
0.584
1.78
0.853
0.799
0.724
O39
12
0.089
8.083
1.361
250

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2.
Click File ^- Print or File ^- Print Preview if you wish to see the preview before printing.
                                File   Edit   BIS Simulator   Graphs   Statisti
                                    New
                                    Open Single File Sheet
                                    Open Excel File with Multiple Sheets
                                    Close
                                    Save
                                    Save As...
                                    Print
                                    Print Preview
                                    Exit
17.5   Saving Output Screens as Excel Files
ProUCL 5.0/ProUCL 5.1 saves output files and data files as Excel files with .xls or .xlsx extensions.
1.       Click on the output you want to save in the Navigation Panel List.
2.       Click File  ^- Save or File ^- Save As
                                     Edit   BIS Simulator   Graphs   Statist!
                            V',
                                   New
                                   Open Single File Sheet
                                   Open Excel File with Multiple Sheets
                                   Close
                                   Save
                                   Save As..,
                                   Print
                                   Print Preview
                                   Exit
        Enter the desired file name you want to use, and click Save, and save the file in the desired folder
        using your browser as shown below.
                                                                                              251

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'
ฉ - t M
Save As
^ Anita Singh ^ Deslctop ^ sprfnd2012 > Danang-AFB-33 v C Search Danang-AFB-33 p
Organize1* Newfolder |EE •* ''
-i Favorites
E Desktop
4!- Downloads
..'_ Recent places
W. Desktop
^ Libraries
*^ Homegroup
f-^ Anita Singh
ฃ Anita Singh
3^' Computer
Qfi Network
Filename:
Save as type:
* Hide Folders
A Name Date modified Type
@3 Trend Results-by-source-treat-Averaged Data 6/19/2013 5:32 PM Microsoft Office E...
S] Trend Results- by-source- Averaged Data 6/19/2013 5:30 PM Microsoft Office E,,,
ป13 Trend Results-Averaged Data 6/19/2013 3:56 PM Microsoft Office E...







Gamma-UCLs v
Excel Files (.xlsx) v

Save Cancel

252

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                                     Chapter 18

 Summary and Recommendations to Compute a 95% UCL for Full
          Uncensored and  Left-Censored Data Sets with NDs

This chapter briefly summarizes recommendations and the process to compute upper confidence limits of
the population mean based upon data sets with and without ND observations. The recommendations are
made based upon the simulation studies summarized in Singh,  Singh, and Engelhardt (1997, 1999);
Singh, Singh, and laci (2002); Singh and Singh (2003); and Singh, Maichle, and Lee (2006). Some details
can be found in Chapters 2 and 4 of the associated ProUCL 5.1 Technical Guide. Depending upon the
data size, data distribution  (e.g., normal, gamma,  lognormal, and nonparametric), and data skewness,
ProUCL suggests using one  or more 95% UCL to estimate the population mean. The project team
collectively should determine which of the suggested UCLs  will be most appropriate for their project. If
needed, the user may want to consult a statistician for additional insight.

18.1   Computing UCL95s of the Mean  Based Upon Uncensored Full Data Sets

    •   Formal GOF tests and GOF Q-Q plots are used first to determine the data distribution so that
       appropriate parametric or nonparametric UCL95s can be computed.

    •   For a normally or approximately normally distributed data set, the user is advised to use Student's
       t-distribution-based  UCL of the mean. Student's t UCL or modified-t-statistic based UCL can be
       used to estimate the EPC when the data set is symmetric (e.g., skewness =
is smaller than 0.2-
       0.3) or mildly skewed; that is, when a or a is less than 0.5. In practice, for mildly skewed data
       sets (with sd of logged  data <0.5), all parametric UCLs computation methods available in
       ProUCL tend to yield comparable results.

       For gamma  or approximately gamma distributed data sets, the  user is advised to:  1) use the
       approximate gamma UCL when k>l and n >50; 2) use the adjusted gamma UCL when k>\ and
       ซ<50; 3) use the bootstrap-t method or Hall's bootstrap method when k < 1 and the sample size, n
       < 15-20; 4)  use approximate gamma UCL for k < 1 and sample size, n >  50; and 5) use the
       adjusted gamma UCL (if available) for k < 1 and sample size, 50> n > 15. If the adjusted gamma
       UCL is not  available  (e.g., when an unusual  CC level such as 0.935 is selected), then use the
       approximate gamma UCL as an estimate of the EPC. When the bootstrap-t method or Hall's
       bootstrap method yields an erratic inflated UCL (e.g., when outliers are present) result, the UCL
       may be computed using the adjusted gamma UCL (if available) or the approximate gamma UCL.

       For lognormally distributed data sets, ProUCL recommends a UCL computation method based
       upon the  sample size,  n,  and standard deviation of the  log-transformed  data,  a. These
       suggestions are summarized in Table 2-10 of the ProUCL 5.1 Technical Guide.

       For nonparametric  data sets, which  are not  normally, lognormally, or gamma distributed, a
       nonparametric UCL is used to estimate the EPC. Methods used to estimate EPC terms based
       upon nonparametric data sets are summarized in Table 2-11 of the ProUCL 5.1 Technical Guide.
       For example for mildly skewed nonparametric data sets of smaller sizes (e.g., <30), one may use
       a modified-t UCL or BCA bootstrap UCL; and  for larger samples one may use a CLT-UCL,
                                                                                    253

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       adjusted-CLT UCL, or a BCA bootstrap UCL. These nonparametric UCLs computation methods
       do not provide desired coverage to the mean for moderately skewed to highly skewed data sets.

    •   For moderately skewed to highly skewed nonparametric data sets, the use of a Chebyshev (Mean,
       Sd) UCL is suggested. It is noted that for extremely skewed data sets (e.g., with a  exceeding
       3.0), even  a Chebyshev inequality-based 99% UCL of the mean  fails to provide the desired
       coverage (e.g., 0.95) of the population mean.

    •   For highly skewed data sets with  a exceeding 3.0, 3.5, pre-processing the data is suggested. It is
       very likely that the  data  consist of outliers  and/or  come  from  multiple  populations. The
       population  partitioning methods may be used to identify mixture populations present in the data
       set. For defensible conclusions,  the decision statistics such as EPC terms may  be  computed
       separately for each of the identified sub-population present in the mixture data set.

18.2   Computing UCLs Based  Upon Left-Censored Data Sets with Nondetects

The parametric maximum likelihood estimation (MLE) methods (e.g., Cohen 1991)  and expectation
maximization (EM) method (Gleit 1985) assume normality or lognormality of data sets and tend to work
only when the data set has NDs with only one detection limit. These days, due to modern analytical tools
and equipment, an environmental data sets consists of NDs with multiple detection limits.  Since it is not
easy to verify (perform goodness-of-fit) the distribution of a left-censored data set consisting of detects
and NDs  with  multiple  detection  limits, some  poor performing estimation methods  including  the
parametric MLE and EM methods and the winsorization method are not retained in ProUCL 5.0/ProUCL
5.1. In ProUCL,  emphasis is given to the use of nonparametric UCL computation methods and hybrid
parametric methods based upon  KM estimates which account for data skewness in the computation of
UCL95. Avoid the use of transformations (to achieve symmetry) while computing upper limits  based
upon left-censored data sets. It is not easy to correctly interpret the statistics computed in the transformed
scale.  Moreover, the results and statistics computed in the original scale do not suffer from transformation
bias.  Like full uncensored data  sets, when the standard deviation of the log-transformed  data becomes
>1.0,  avoid the use of a lognormal model even when the data appear to be lognormally distributed.   Its
use often results in unrealistic statistics of no practical merit (Singh, Singh, and Engelhard 1997; Singh,
Singh, and laci 2002). It is also recommended to identify  potential outliers representing observations
coming from population(s) different from the main dominant population and investigate them separately.
Decisions about the disposition of outliers should be made by all interested members of the project team.

       •   It is recommended to avoid the use of the DL/2  (t) UCL method, as the DL/2 UCL does not
           provide the desired coverage  (for any distribution and sample size) for the population mean,
           even for censoring levels as low as 10%,  15%. This is contrary to the conjecture and assertion
           (e.g., EPA 2006a) made that the DL/2 method can be used for lower (e.g., < 20%) censoring
           levels.  The coverage provided by the  DL/2 (t) method deteriorates fast as  the censoring
           intensity increases. The  DL/2  (t) method is not recommended by the authors  or developers of
           this text and ProUCL software.

       •   The use of the KM estimation method is a preferred method as it can  handle multiple
           detection limits. Therefore, the use  of KM estimates is suggested to compute the decision
           statistics based upon methods which adjust for data skewness. Depending upon the data set
           size,  distribution of the detected data, and data skewness, the  various nonparametric and
           hybrid  KM UCL95 methods  including KM (BCA), bootstrap-t  KM  UCL,  Chebyshev KM
254

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UCL, Gamma-KM  UCL based upon the  KM estimates provide good  coverages for the
population mean. All of these methods are available in ProUCL 5.1.
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