4>EPA
United States
Environmental Protection
Agency
   ProUCL Version 4.00.02
          User Guide
      RESEARCH AND DEVELOPMENT

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                                                                EPA/600/R-07/038
                                                                     April 2007
                                                                   www.epa.gov
    ProUCL  Version  4.00.02
                    User Guide
                              Prepared for:

                            Brian Schumacher

                     U.S. Environmental Protection Agency
                      Office of Research and Development
                     National Exposure Research Laboratory
                       Environmental Sciences Division
                         Technology Support Center
                     Characterization and Monitoring Branch
                           944 E. Harmon Ave.
                           Las Vegas, NV89119

                              Prepared by:

                            Anita Singh, Ph.D.1
                             Robert Maichle1
                          Ashok K. Singh, Ph.D.2
                            Sanghee E. Lee1
                             Narain Armbya1

                     1 Lockheed Martin Environmental Services
                     1050 E. Flamingo Road, Suite N240
                     Las Vegas, NV 89119

                     Department of Hotel Management
                     University of Nevada, Las Vegas
                     Las Vegas, NV 89154

                       Editor and Technical Direction by:

                             John Nocerino
                     U.S. Environmental Protection Agency
                      Office of Research and Development
                     National Exposure Research Laboratory
                       Environmental Sciences Division
                     Characterization and Monitoring Branch
                           944 E. Harmon Ave.
                           Las Vegas, NV89119
Notice: Although this work was reviewed by EPA and approved for publication, it may not necessarily reflect official
     Agency policy. Mention of trade names and commercial products does not constitute endorsement or
     recommendation for use.


                   U.S. Environmental Protection Agency
                   Office of Research and Development
                         Washington, DC 20460                          129cmb07

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                                          Notice

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

ProUCL software was developed by Lockheed Martin under a contract with the EPA and is made
available through the EPA Technical Support Center in Las Vegas, Nevada. Use of any portion of
ProUCL that does not comply with the ProUCL User Guide is not recommended.

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

ProUCL software provided by the EPA was scanned with McAfee VirusScan and is certified free of
viruses.

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

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  Changes from ProUCL 4.0 (version 4.00.00) to ProUCL 4.00.002

Although extensive changes were made in the code for ProUCL 4.0 (version 4.00.00) to produce ProUCL
4.00.02, those changes are transparent to the users. Most of those changes were made so that
ProUCL 4.00.02 is compatible with our developing statistical software, Scout (e.g., both programs share
the same statistical libraries).  ProUCL will also reside as a separate module in Scout as a research tool.

There is a very minor correction of a displayed value in one of the hypothesis tests, the two sample t-test.
The p-value associated with the t-test was computed in two different ways: one way is correct and the
other way, although it produced  subtle differences, is incorrect. The incorrect method has been removed
from ProUCL 4.00.02.

Several extra warning messages  have been added to ProUCL 4.00.02, mainly in regard to attempting tests
when a data set is very small (n < 5), when the number of detected values is small (e.g., only zero, one, or
two), or when all of the values are non-detected values. For an example, some screens depicting those
warning messages are included in the newly added Section 2.11 (page 40) of this ProUCL 4.00.02 User
Guide.

The only software files that were changed from ProUCL version 4.0 (4.00.00) to version 4.0.02 were
updates in the ProUCL.exe file, and updates to the StatsLib.dll file to produce a more advanced
ScoutLib.dll file. Very minor changes were made to this  ProUCL 4.00.02 User Guide, including: changes
to avoid inappropriate user inputs (warnings), changes to  the title page, the inclusion of an
acknowledgement page, and the  inclusion of a contact information page.

No changes were made to the ProUCL 4.0 Technical Guide; that is, the ProUCL 4.0 Technical Guide is
still applicable to the ProUCL 4.00.02 software and User  Guide.
                 Contact Information for ProUCL 4.00.002

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

Felicia Barnett, (HSTL)
US EPA, Region 4
61 Forsyth Street, S.W.
Atlanta, GA  30303-8960
barnett.felicia@epa.gov
(404) 562-8659
Fax: (404) 562-8439
IV

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Executive Summary


Statistical inference, including both estimation and hypotheses testing approaches, is routinely used to:

           1.   Estimate environmental parameters of interest, such as exposure point concentration
               (EPC) terms, not-to-exceed values, and background level threshold values (BTVs) for
               contaminants of potential concern (COPC),
           2.   Identify areas of concern (AOC) at a contaminated site,
           3.   Compare contaminant concentrations found at two or more AOCs of a contaminated site,
           4.   Compare contaminant concentrations found at an AOC with background or reference
               area contaminant concentrations, and
           5.   Compare site concentrations with a cleanup standard to verify the attainment of cleanup
               standards.

Several exposure and risk management and cleanup decisions in support  of United States Environmental
Protection Agency (EPA) projects are often made based upon the mean concentrations of the COPCs. A
95% upper confidence limit (UCL95) of the unknown population (e.g., an AOC) arithmetic mean (AM),
[j.\, can be used to:

           •   Estimate the EPC term of the AOC under investigation,
           •   Determine the attainment of cleanup standards,
           •   Compare site mean concentrations with reference area mean concentrations, and
           •   Estimate background level mean contaminant concentrations. The background mean
               contaminant concentration level may be used to compare the mean of an area of concern.
               It should be noted that it is not appropriate to compare individual point-by-point site
               observations with the background mean concentration level.

It is important to compute a reliable and stable UCL95 of the population mean using the available data.
The UCL95 should approximately provide the 95% coverage for the unknown population mean, ^. Based
upon the  available background data, it is equally important to compute reliable and stable upper
percentiles, upper prediction limits (UPLs), or upper tolerance limits (UTLs). These upper limits based
upon background (or reference) data are used as estimates of BTVs, compliance limits (CL), or not-to-
exceed values.  These upper limits are often used in site (point-by-point) versus background comparison
evaluations.

Environmental scientists  often encounter trace level concentrations of COPCs when evaluating sample
analytical results. Those low level analytical results cannot be measured accurately and, therefore, are
typically  reported as less  than one or more detection limit (DL) values (also called nondetects). However,
practitioners need to obtain reliable estimates of the population mean, //;, and the population standard
deviation, GI, and upper limits including the UCL of the population mass  or mean, the UPL, and the UTL
based upon data sets with nondetect (ND) observations.  Additionally, they may have to use hypotheses
testing approaches to verify the  attainment of cleanup standards, and compare site and background
concentrations  of COPCs as mentioned above.

Background evaluation studies,  BTVs, and not-to-exceed values should be estimated based upon
defensible background data sets. The estimated  BTVs or not-to-exceed values are then used to identify the
COPCs, to identify the site AOCs or hot spots, and to compare the contaminant concentrations at a site

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with background concentrations. The use of appropriate statistical methods and limits for site versus
background comparisons is based upon the following factors:

           o   Objective of the study,
           o   Environmental medium (e.g., soil, groundwater, sediment, air) of concern,
           o   Quantity and quality of the available data,
           o   Estimation of a not-to-exceed value or of a mean contaminant concentration,
           o   Pre-established or unknown cleanup standards and BTVs, and
           6.   Sampling distributions (parametric or nonparametric) of the concentration data sets
               collected from the site and background areas under investigation.

In background versus site comparison evaluations, the environmental population parameters of interest
may include:

           •   Preliminary remediation goals (PRGs),
           •   Soil screening levels (SSLs),
           •   RBC standards,
           •   BTVs, not-to-exceed values, and
           •   Compliance limit, maximum concentration limit (MCL), or alternative concentration
               limit (ACL), frequently used in groundwater applications.

When the environmental parameters listed above are not known or pre-established, appropriate upper
statistical limits are used to estimate those parameters. The UPL, UTL, and upper percentiles are used to
estimate the BTVs and not-to-exceed values. Depending upon the site data availability, point-by-point site
observations are compared with the estimated (or pre-established) BTVs and not-to-exceed values. If
enough site and background data are available, two-sample hypotheses testing approaches are used to
compare  site concentrations with background concentrations levels. These statistical methods can also be
used to compare contaminant concentrations of two site AOCs, surface and subsurface contaminant
concentrations,  or upgradient versus monitoring  well contaminant concentrations.

The ProUCL Version 4.0 (ProUCL 4.0) is an upgrade of ProUCL Version 3.0 (EPA, 2004). ProUCL 4.0
contains statistical methods to address various environmental issues for both full data sets without
nondetects and for data sets with NDs (also known as left-censored data sets).

ProUCL 4.0 contains:

           o   Rigorous parametric and nonparametric (including bootstrap methods) statistical methods
               (instead of simple ad hoc or substitution methods) that can be used on full data sets
               without nondetects and on data sets with below detection limit (BDL) or ND
               observations.

           2.   State-of-the-art parametric and nonparametric UCL, UPL, and UTL computation
               methods. These methods can be used on full-uncensored data sets without nondetects and
               also on data sets with BDL observations. Some of the methods (e.g., Kaplan-Meier
               method, ROS methods) are applicable on left-censored data sets having multiple
               detection limits. The UCL and other upper limit computation methods cover a wide range
               of skewed data sets with and without the BDLs.
VI

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           3.  Single sample (e.g., Student's t-test, sign test, Proportion test, Wilcoxon Singed Rank
               test) and two-sample (Student's t-test, Wilcoxon-Mann-Whitney test, Gehan test, quantile
               test) parametric and nonparametric hypotheses testing approaches for data sets with and
               without ND observations. These hypothesis testing approaches can be used to: verify the
               attainment of cleanup standards, perform site versus background comparisons, and
               compare two or more AOCs, monitoring wells (MWs).

           4.  The single sample hypotheses testing approaches are used to compare site mean, site
               median, site proportion, or a site percentile (e.g., 95th) to a compliance limit (action level,
               regularity limit). The hypotheses testing approaches can handle both full-uncensored data
               sets without nondetects, and left-censored data sets with nondetects.  Simple two-sample
               hypotheses testing methods to compare two populations are available in ProUCL 4.0,
               such as two-sample t-tests, Wilcoxon-Mann-Whitney (WMW) Rank Sum test, quantile
               test, Gehan's test, and dispersion test. Variations of hypothesis testing methods (e.g.,
               Levene's method to compare dispersions, generalized WRS test) are easily available in
               most commercial and freely available software packages (e.g., MINITAB, R).

           5.  ProUCL 4.0 also includes graphical methods (e.g., box plots, multiple Q-Q plots,
               histogram) to compare two  or more populations. ProUCL 4.0 can also be used to display
               a box plot of one population (e.g., site data) with compliance limits or upper limits (e.g.,
               UPL) of other population (background area) superimposed on the same graph. This kind
               of graph provides a useful visual comparison of site data with a compliance limit or
               BTVs. Graphical displays of a data set (e.g., Q-Q plot) should be used to gain insight
               knowledge contained in a data set that may not otherwise be clear by looking at simple
               test statistics such as t-test,  Dixon test statistic, or Shapiro-Wilk (S-W) test statistic.

           6.  ProUCL 4.0 can process multiple contaminants (variables) simultaneously and has the
               capability of processing data by groups. A valid group column should be included in the
               data file.

           7.  ProUCL 4.0 provides GOF  test for data sets with nondetects. The user can create
               additional columns to store  extrapolated (estimated) values for nondetects based upon
               normal ROS, gamma ROS, and lognormal ROS (robust ROS) methods.

ProUCL 4.0 retains all of the capabilities of ProUCL 3.0, including goodness-of-fit (GOF) tests for a
normal, lognormal, and a gamma distribution and computation of UCLs based upon full data sets without
nondetects. Graphical displays and GOF tests for data sets with BDL observations have also been
included in ProUCL 4.0. It is re-emphasized that the computation of appropriate UCLs, UPLs, and other
limits is based upon the assumption that the data set under study represents a single a single population.
This means that the data set used to compute the limits should represent a single statistical population. For
example, a background data set should represent a defensible background data set free of outlying
observations.  ProUCL 4.0 includes simple and commonly used classical outlier identification procedures,
such as the Dixon test and the Rosner test. These procedures are included as an aid to identify outliers.
These simple  classical outlier tests often suffer from masking effects in the presence  of multiple outliers.
Description and use of robust and resistant outlier procedures is beyond the scope of ProUCL 4.0.

It is suggested that the classical outlier procedures should always be accompanied by graphical displays
including box plots and Q-Q plots. The use of a Q-Q plot is useful to identify multiple or mixture samples
that might be  present in a data set. However, the decision regarding the proper disposition of outliers (e.g.,
                                                                                             vn

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to include or not to include outliers in statistical analyses; or to collect additional verification samples)
should be made by members of the project team and experts familiar with site and background conditions.
Guidance on the disposition of outliers and their accommodation in a data set by using a transformation
(e.g., lognormal distribution) is discussed in Chapter 1 of this User Guide.

ProUCL 4.0 has improved graphical methods, which may be used to compare the concentrations of two or
more populations such as:

           o   Site versus background populations,
           o   Surface versus subsurface concentrations,
           o   Concentrations of two or more AOCs, and
           o   Identification of mixture samples and/or potential outliers

These graphical methods include multiple quantile-quantile (Q-Q) plots, side-by-side box plots, and
histograms. Whenever possible, it is desirable to supplement statistical results with useful visual displays
of data sets. There is no substitute for graphical displays of a data set. For example, in addition to
providing information about the data distribution, a normal Q-Q plot can also help identify outliers and
multiple populations that may be present in a data set. On a Q-Q plot, observations well separated from
the majority of the data may represent potential outliers, and jumps and breaks of significant magnitude
may suggest the presence of observations from multiple populations in the data set. It is suggested that
analytical outlier tests (e.g., Rosner test) and goodness-of-fit (G.O.F.) tests (e.g., SW test) should always
be supplemented with the graphical displays such as Q-Q plot and box plot.

ProUCL 4.0 serves as a companion software package for Calculating Upper Confidence Limits for
Exposure Point Concentrations at Hazardous Waste Sites (EPA, 2002a) and Guidance for Comparing
Background and Chemical Concentrations in Soil for CERCLA Sites (EPA, 2002b). ProUCL 4.0 is also
useful to verify the attainment of cleanup standards (EPA, 1989). ProUCL 4.0 can also be used
to perform two-sample hypotheses tests and to compute various upper limits often needed in
groundwater monitoring applications (EPA, 1992 and EPA, 2004).
Vlll

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                  Acronyms and Abbreviations
%NDs

ACL
A-D, AD
AM
AOC

BC
BCA
BDL
BTV
BW

CERCLA

CL
CLT
CMLE
COPC
CV

DL
DL/2 (t)

DL/2 Estimates

DQO

EA
EOF
EM
EPA
EPC
Percentage of Nondetect observations

alternative concentration limit
Anderson-Darling test
arithmetic mean
area(s) of concern

Box-Cox-type transformation
bias-corrected accelerated bootstrap method
below detection limit
background threshold value
Black and White (for printing)

Comprehensive Environmental Response, Compensation, and
Liability Act
compliance limit
central limit theorem
Cohen's maximum likelihood estimate
contaminant(s) of potential concern
Coefficient of Variation
detection limit
UCL based upon DL/2 method using Student's t-distribution
cutoff value
estimates based upon data set with nondetects replaced by half
of the respective detection limits
data quality objective
exposure area
empirical distribution function
expectation maximization
Environmental Protection Agency
exposure point concentration
                                                                                 IX

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      FP-ROS (Land)
UCL based upon fully parametric ROS method using Land's H-
statistic
      Gamma ROS (Approx.)   UCL based upon Gamma ROS method using the gamma
      Gamma ROS (BCA)
      GOF, G.O.F.
approximate-UCL method
UCL based upon Gamma ROS method using the bias-corrected
accelerated bootstrap method
goodness-of-fit
      H-UCL
                             UCL based upon Land's H-statistic
      ID
      IQR
identification code
interquartile range
      K
      KM (%)

      KM (Chebyshev)

      KM(t)

      KM(z)

      K-M, KM
      K-S, KS
Next K, Other K, Future K
UCL based upon Kaplan-Meier estimates using the percentile
bootstrap method
UCL based upon Kaplan-Meier estimates using the Chebyshev
inequality
UCL based upon Kaplan-Meier estimates using the Student's t-
distribution cutoff value
UCL based upon Kaplan-Meier estimates using standard normal
distribution cutoff value
Kaplan-Meier
Kolmogorov-Smirnov
      LN
      Log-ROS Estimates
lognormal distribution
estimates based upon data set with extrapolated nondetect values
obtained using robust ROS method
      MAD
      Maximum
      MCL
      Mean
      Median
      Minimum
      MLE
      MLE (t)
Median Absolute Deviation
Maximum value
maximum concentration limit
classical average value
Median value
Minimum value
maximum likelihood estimate
UCL based upon maximum likelihood estimates using Student's
t-distribution cutoff value
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MLE (Tiku)

Multi Q-Q
MVUE
ND
NERL
NumNDs
NumObs

ORD

PRO

Q-Q

RBC
RCRA
ROS
RU
SD, Sd, sd
SSL
S-W, SW

UCL
UCL95, 95% UCL
UPL
UPL95, 95% UPL
USEPA
UTL

Variance

WMW
WRS
WSR
UCL based upon maximum likelihood estimates using the
Tiku's method
multiple quantile-quantile plot
minimum variance unbiased estimate
nondetect or nondetects
National Exposure Research Laboratory
Number of Nondetects
Number of Observations

Office of Research and Development

preliminary remediation goals

quantile-quantile

risk-based cleanup
Resource Conservation and Recovery Act
regression on order statistics
remediation unit

substantial difference
standard deviation
soil screening levels
Shapiro-Wilk

upper confidence limit
95% upper confidence limit
upper prediction limit
95% upper prediction limit
United States Environmental Protection Agency
upper tolerance limit

classical variance

Wilcoxon-Mann-Whitney
Wilcoxon Rank Sum
Wilcoxon Signed Rank
                                                                                  XI

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                             Acknowledgements

We wish to express our gratitude and thanks to all of the many people that reviewed, tested, and
gave some very helpful suggestions for the development of ProUCL 4.0. We wish to especially
acknowledge: Nadine Adkins, Ray Bienert, Cheri L. Butler, Lucia Casabo, John P. Christopher,
Marion Edison, Evan Englund, Tim Ehli, Kuen Huang-Farmer, Dolores Gardner, Philip E.
Goodrum,  Scott A. Grubisich, Dennis R. Helsel, Jennifer Hubbard, Richard O. Gilbert, Sarah
Levinson, Jayne Michaud, Jeff Myers, Yassine Nachti,Gareth Pearson, Marian Olsen, Kristina
Rayner, Nancy Rios-Jafolla, Shahrokh Rouhani, Peter Rousseeuw, Dan Sahagun, Terry W.
Schulz, Mike Schum, Tom Siard, Christopher Sibert, Bob Stewart, Rick Sugatt, J. Keith Tolson,
Robert Tucker, John Warren, and Edward P. Wosika.
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                                   Table of Contents

Notice	iii

Changes from ProUCL 4.0 (4.00.00) to ProUCL 4.00.02	iv

Contact Information for ProUCL 4.00.02	iv

Executive Summary	v

Acronyms and Abbreviations	ix

Acknowledgements	xii

Introduction	1
      The Need for ProUCL Software	1
      ProUCL 4.0 Capabilities	2
      ProUCL Applications	4
      ProUCL Methods	5
      Background versus Site Comparison Evaluations	7
      Graphical Capabilities	11
      Technical Guide	12
      Minimum Hardware Requirements	12
      Software Requirements	12
      Installation Instructions	13
      Getting Started	13

Chapter 1    Guidance on the Use of Statistical Methods and Associated Minimum
             Sample Size Requirements	15
       1.1    Background Data Sets	16
       1.2    Site Data Sets	17
       1.3    Discrete Samples or Composite Samples?	18
       1.4    Upper Limits and Their Use	18
       1.5    Point-by-Point Comparison of Site Observations with BTVs, Compliance Limits,
             and Other Threshold Values	21
       1.6    Hypothesis Testing Approaches and Their Use	23
             1.6.1   Single Sample Hypotheses - BTVs and Not-to-Exceed Values are Known
                    (Pre-established)	23
             1.6.2   Two-sample Hypotheses - When BTVs and Not-to-Exceed
                    Values are Unknown	24
       1.7    Minimum Sample Size Requirements	25
             1.7.1   Minimum Sample Size for Estimation and Point-by-Point Site
                    Observation Comparisons	26
             1.7.2   Minimum Sample Size Requirements for Hypothesis Testing Approaches	26
       1.8    Sample Sizes for Bootstrap Methods	27
       1.9    Statistical Analyses by a Group ID	27
       1.10   Use of Maximum Detected Values as Estimates of Upper Limits	27
             1.10.1  Use of Maximum Detected Values to Estimate BTVs and
                    Not-to-Exceed Values	28
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              1.10.2  Use of Maximum Detected Values to Estimate EPC Terms	28
              1.10.3  Samples with Nondetect Observations	29
              1.10.4  Avoid the Use of DL/2 Method to Compute UCL95	29
              1.10.5  Samples with Low Frequency of Detection	30
       1.11    Other Applications of Methods in ProUCL 4.0	30
              1.11.1  Identification of COPCs	31
              1.11.2  Identification ofNon-Compliance Monitoring Wells	31
              1.11.3  Verification of the Attainment of Cleanup Standards, Cs	31
              1.11.4  Using BTVs (Upper Limits) to identify Hot Spots	32

Chapter 2    Entering and Manipulating Data	33
       2.1     Creating a New Data Set	33
       2.2     Opening an Existing Data Set	33
       2.3     Input File Format	34
       2.4     Number Precision	34
       2.5     Entering and Changing a Header Name	35
       2.6     Saving Files	36
       2.7     Editing	36
       2.8     Handling Nondetect Observations	37
       2.9     Caution	38
       2.10    Summary Statistics for Data Sets with Nondetect Observations	39
       2.11    Warning Messages and Recommendations for Datasets with Insufficient
              Amount of Data	40
       2.12    Handling Missing Values	42
       2.13    User Graphic Display Modification	44
              2.13.1  Graphics Tool Bar	44
              2.13.2  Drop-Down Menu Graphics Tools	44

Chapter 3    Select Variables Screen	46
       3.1     Select Variables Screen	46
              3.1.1   Graphs by Groups	48

Chapter 4    Summary Statistics	50
       4.1     Summary Statistics with Full Data Sets	50
       4.2     Summary Statistics withNDs	52

Chapters    Estimating Nondetects  Using ROS Methods	56

Chapter 6    Graphical Methods (Graph)	58
       6.1     Box Plot	59
       6.2     Histogram	61
       6.3     Multi-QQ	63
              6.3.1   Multi-QQ (Full)	63
              6.3.2   Multi-QQ (withNDs)	64

Chapter 7    Simple Classical Outlier Tests 	68
       7.1     Outlier Test for Full Data Set	70
       7.2     Outlier Test for Data Set with NDs	70
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Chapter 8    Goodness-of-Fit (G.O.F.) Tests	74
       8.1     ROS Estimated (Est.) NDs - Saving Extrapolated NDs	77
       8.2     Goodness-of-Fit Tests with Full Data Sets	78
              8.2.1   GOF Tests for Normal and Lognormal Distribution	78
              8.2.2   GOF Tests for Gamma Distribution	81
       8.3     Goodness-of-Fit Tests Excluding NDs	83
              8.3.1   Normal and Lognormal Options	84
              8.3.2   Gamma Distribution Option	87
       8.4     Goodness-of-Fit Tests with Log-ROS Estimates	89
              8.4.1   Normal or Lognormal Distribution (Log-ROS Estimates)	90
              8.4.2   Gamma Distribution (Log-ROS Estimates)	93
       8.5     Goodness-of-Fit Tests with DL/2 Estimates	95
              8.5.1   Normal or Lognormal Distribution (DL/2 Estimates)	96
              8.5.2   Gamma Distribution (DL/2 Estimates)	99
       8.6     Goodness-of-Fit Tests Statistics	101

Chapter 9    Single Sample and Two-Sample Hypotheses Testing Approaches	104
       9.1     Single Sample Hypotheses Tests	104
              9.1.1   Single Sample Hypothesis Testing for Full Data without Nondetects	105
                     9.1.1.1  Single Sample t-Test	105
                     9.1.1.2  Single Sample Proportion Test	107
                     9.1.1.3  Single Sample Sign Test	109
                     9.1.1.4  Single Sample Wilcoxon Signed Rank (WSR) Test	Ill
              9.1.2   Single Sample Hypothesis Testing for Data Sets with Nondetects	114
                     9.1.2.1  Single Proportion Test on Data Sets with NDs	114
                     9.1.2.2  Single Sample Sign Test with NDs	116
                     9.1.2.3  Single Sample Wilcoxon Signed Rank Test with NDs	118
       9.2     Two-Sample Hypotheses Testing Approaches	121
              9.2.1   Two-Sample Hypothesis Tests for Full Data	122
                     9.2.1.1  Two-Sample t-Test without NDs	124
                     9.2.1.2  Two-Sample Wilcoxon-Mann-Whitney (WMW) Test without NDs 126
                     9.2.1.3  Two-Sample Quantile Test for Full Data without NDs	129
              9.2.2   Two-Sample Hypothesis Testing for Data Sets without Nondetects	132
                     9.2.2.1  Two-Sample Wilcoxon-Mann-Whitney Test with Nondetects	132
                     9.2.2.2  Two-Sample Gehan Test for Data Sets with Nondetects	136
                     9.2.2.3  Two-Sample Quantile Test for Data Sets with Nondetects	138

Chapter 10   Background Statistics	142
       10.1    Background Statistics  for Full Data Sets without Nondetects	143
              10.1.1  Normal or Lognormal Distribution	143
              10.1.2  Gamma Distribution	147
              10.1.3  Nonparametric Methods	149
              10.1.4  All Statistics Option	151
       10.2    Background Statistics  with NDs	153
              10.2.1  Normal or Lognormal Distribution	153
              10.2.2  Gamma Distribution	158
              10.2.3  Nonparametric Methods (with NDs)	160
              10.2.4  All Statistics Option	162
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Chapter 11   Computing Upper Confidence Limits (UCLs) of Mean	166
       11.1   UCLs for Full Data Sets	167
             11.1.1 Normal Distribution (Full Data Sets without NDs)	167
             11.1.2 Gamma, Lognormal, Nonparametric, All Statistics Option
                   (Full Data without NDs)	170
       11.2   UCL for Data Sets with NDs	175

Chapter 12   Windows	186

Chapter 13   Help	188

Chapter 14   Handling the Output Screens and Graphs	190
             Copying Graphs	190
             Printing Graphs	191
             Printing Non-graphical Outputs	192
             Saving Output Screens as Excel Files	193

Chapter 15   Recommendations to Compute a 95% UCL (Estimate of EPC Term)
             of the Population Mean, /Ji, Using Symmetric and Positively Skewed
             Full Data Set without any Nondetects	196
       15.1   Normally or Approximately Normally Distributed Data Sets	196
       15.2   Gamma Distributed Skewed Data Sets	197
       15.3   Lognormally Distributed Skewed Data Sets	198
       15.4   Data Sets without a Discernable Skewed Distribution - Nonparametric Methods for
             Skewed Data Sets	200
       15.5   Should the Maximum Observed Concentration be Used as an Estimate of
             the EPC Term?	203

Chapter 16   Recommendations to Compute a 95% UCL of the Population Mean,
             Pi, Using Data Sets with Nondetects with Multiple Detection Limits	206
       16.1   General Recommendations and Suggestions	206
       16.2   Recommended UCL95 Methods for Normal (Approximate Normal) Distribution	208
       16.3   Recommended UCL95 Methods for Gamma Distribution	209
       16.4   Recommended UCL95 Methods for Lognormal Distribution	209
       16.5   Recommended Nonparametric UCL Methods	210

Glossary	214

References	216
xvi

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                                      Introduction

The Need for ProUCL Software

Statistical inferences about the sampled populations and their parameters are made based upon defensible
and representative data sets of appropriate sizes collected from the populations under investigation.
Statistical inference, including both estimation and hypotheses testing approaches, is routinely used to:

           1.   Estimate environmental parameters of interest such as exposure point concentration
               (EPC) terms, not-to-exceed values, and background level threshold values (BTVs) for
               contaminants of potential concern (COPC),
           2.   Identify areas of concern (AOC) at a contaminated site,
           3.   Compare contaminant concentrations found at two or more AOCs of a contaminated site,
           4.   Compare contaminant concentrations found at an AOC with background or reference
               area contaminant concentrations,
           5.   Compare site concentrations with a cleanup standard to verify the attainment of cleanup
               standards.

Statistical inference about the sampled populations and their parameters are made based upon defensible
and representative data sets of appropriate sizes collected from the populations under investigation.
Environmental data sets originated from the Superfund and RCRA sites often consist of observations
below one or more detection limits (DLs). In order to address the statistical issues arising in: exposure and
risk assessment applications; background versus site comparison and evaluation studies; and various other
environmental applications, several graphical, parametric, and nonparametric statistical methods for data
sets with nondetects and without nondetects have been incorporated in ProUCL 4.0.

Exposure and risk management and cleanup decisions in support of United States Environmental
Protection Agency (EPA) projects are often made based upon the mean concentrations of the COPCs. A
95% upper confidence limit (UCL95) of the unknown population (e.g., an AOC) arithmetic mean (AM),
Hi, can be used to:

    •  Estimate the EPC term of the AOC under investigation,
    •  Determine the attainment of cleanup standards,
    •  Compare site mean concentrations with reference  area mean concentrations, and
    •  Estimate background level mean contaminant concentrations. The background mean contaminant
       concentration level may be used to compare the mean of an AOC. It should be noted that it is not
       appropriate to compare individual point-by-point site observations with the background mean
       concentration level.

It is important to compute a reliable and stable UCL95 of the population mean using the available data.
The UCL95 should approximately provide the 95% coverage for the unknown population mean, H\- Based
upon the available background data, it is equally important to compute reliable and stable upper
percentiles, upper prediction limits (UPLs), or upper tolerance limits (UTLs). These upper limits based
upon background (or reference) data are used as estimates of BTVs, compliance limits (CL), or not-to-
exceed values. These upper limits are often used in site (point-by-point) versus background comparison
evaluations.

Environmental scientists often encounter trace level concentrations of COPCs when evaluating sample

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analytical results. Those low level analytical results cannot be measured accurately, and therefore are
typically reported as less than one or more detection limit (DL) values (also called nondetects). However,
practitioners often need to obtain reliable estimates of the population mean, //;, the population standard
deviation, ah and upper limits, including the upper confidence limit (UCL) of the population mass or
mean, the UPL, and the UTL based upon data sets with nondetect (ND) observations. Hypotheses testing
approaches are often used to verify the attainment of cleanup standards, and compare site and background
concentrations of COPCs.

Background evaluation studies, BTVs, and not-to-exceed values should be estimated based upon
defensible background data sets. The estimated BTVs or not-to-exceed values are then used to identify the
COPCs, to identify the site AOCs or hot spots, and to compare the contaminant concentrations at a site
with background concentrations. The use of appropriate statistical methods and limits for site versus
background comparisons is based upon the following factors:

    1.  Objective of the study,
    2.  Environmental medium (e.g., soil, groundwater, sediment, air) of concern,
    3.  Quantity and quality of the available data,
    4.  Estimation of a not-to-exceed value or of a mean contaminant concentration,
    5.  Pre-established or unknown cleanup standards and BTVs, and
    6.  Sampling distributions  (parametric or nonparametric) of the concentration data sets collected
       from the site and background areas under investigation.

In background versus site comparison evaluations, the environmental population parameters of interest
may include:

    •  Preliminary remediation goals (PRGs),
    •  Soil screening levels (SSLs),
    •  Risk-based cleanup (RBC) standards,
    •  BTVs, not-to-exceed values, and
    •  Compliance limit, maximum concentration limit (MCL), or alternative concentration limit  (ACL),
       frequently used in groundwater applications.

When the environmental parameters listed above are not known or have not been pre-established,
appropriate upper statistical limits are used to estimate the parameters. The UPL, UTL, and upper
percentiles are used to estimate the BTVs and not-to-exceed values. Depending upon the site data
availability, point-by-point site  observations are compared with the estimated (or pre-established) BTVs
and not-to-exceed values. If enough site and background data are available, two-sample hypotheses
testing approaches are used to compare site  concentrations with background concentrations levels. These
statistical methods can also be used to compare contaminant concentrations of two site AOCs, surface and
subsurface contaminant concentrations, or upgradient versus monitoring well contaminant concentrations.

ProUCL 4.0 Capabilities

ProUCL Version 4.0 (ProUCL 4.0) is an upgrade of ProUCL Version 3.0 (EPA, 2004). ProUCL 4.0
contains statistical methods to address various environmental issues for both full data sets without
nondetects and for data sets with NDs (also  known as left-censored data sets).

ProUCL 4.0 contains:

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o   Rigorous parametric and nonparametric (including bootstrap methods) statistical methods
    (instead of simple ad hoc or substitution methods) that can be used on full data sets
    without nondetects and on data sets with below detection limit (BDL) or nondetect (ND)
    observations.

o   State-of-the-art parametric and nonparametric UCL, UPL, and UTL computation
    methods. These methods can be used on full-uncensored data sets without nondetects and
    also on data sets with BDL observations. Some of the methods (e.g., Kaplan-Meier
    method, ROS methods) are applicable on left-censored data sets having multiple
    detection limits. The UCL and other upper limit computation methods cover a wide range
    of skewed data sets with and without the BDLs.

o   Single sample (e.g., Student's t-test, sign test, proportion test, Wilcoxon Singed Rank
    test) and two-sample (Student's t-test, Wilcoxon-Mann-Whitney test, Gehan test, quantile
    test) parametric and nonparametric hypotheses testing approaches for data sets with and
    without ND observations. These hypothesis testing approaches can be used to: verify the
    attainment of cleanup standards, perform site versus background comparisons, and
    compare two or more AOCs, monitoring wells (MWs).

o   The single  sample hypotheses testing approaches are used to compare site mean, site
    median, site proportion, or a site percentile (e.g., 95th) to a compliance limit (action level,
    regularity limit). The hypotheses testing approaches  can handle both full-uncensored data
    sets without nondetects, and left-censored data sets with nondetects. Simple two-sample
    hypotheses testing methods to compare two populations are available in ProUCL 4.0,
    such as two-sample t-tests, Wilcoxon-Mann-Whitney (WMW) Rank  Sum test, quantile
    test, Gehan's test, and dispersion test. Variations of hypothesis testing methods (e.g.,
    Levene's method to compare dispersions, generalized WRS test) are easily available in
    most commercial and freely available software packages (e.g., MINITAB, R).

o   ProUCL 4.0 includes graphical methods (e.g., box plots, multiple  Q-Q plots, histogram)
    to compare two or more populations. Additionally, ProUCL 4.0 can also be used to
    display a box plot of one population (e.g., site data) with compliance  limits or upper
    limits (e.g., UPL) of other population (background area) superimposed on the  same
    graph. This kind of graph provides a useful visual comparison of site  data with a
    compliance limit or BTVs. Graphical displays of a data set (e.g., Q-Q plot) should be
    used to gain insight knowledge contained in  a data set that may not otherwise be clear by
    looking at simple test statistics such as t-test, Dixon test statistic, or Shapiro-Wilk (S-W)
    test statistic.

6.   ProUCL 4.0 can process multiple contaminants (variables) simultaneously and has the
    capability of processing data by groups. A valid group column should be included in the
    data file.

7.   ProUCL 4.0 provides a GOF test for data sets with nondetects. The user can create
    additional columns to store extrapolated (estimated)  values for nondetects based upon
    normal ROS, gamma ROS, and lognormal ROS (robust ROS) methods.

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ProUCL Applications

The methods incorporated in ProUCL 4.0 can be used on data sets with and without BDL and ND
observations. Methods and recommendations as incorporated in ProUCL 4.0 are based upon the results
and findings of the extensive simulation studies as summarized in Singh and Singh (2003), and Singh,
Maichle, and Lee (EPA, 2006). It is anticipated that ProUCL 4.0 will serve as a companion software
package for the following EPA documents:

           •   Calculating Upper Confidence Limits for Exposure Point Concentrations at Hazardous
              Waste Sites (EPA, 2002a), and
           •   The revised Guidance for Comparing Background and Chemical Concentrations in Soil
              for CERCLA Sites (EPA, 2002b).

Methods included in ProUCL 4.0 can be used in various other environmental applications including the
verification of cleanup standards (EPA, 1989), and computation of upper limits needed in groundwater
monitoring applications (EPA,  1992 and EPA, 2004).

In 2002, EPA issued guidance for calculating the UCLs of the unknown population means for
contaminant concentrations at hazardous waste sites. The ProUCL 3.0 software package (EPA, 2004) has
served as a companion software package for the EPA (2002a) guidance document for calculating UCLs of
mean contaminant concentrations at hazardous waste sites. ProUCL 3.0 has several parametric and
nonparametric statistical methods that can be used to compute appropriate UCLs based upon full-
uncensored data sets without any ND  observations. ProUCL 4.0 retains the capabilities of ProUCL 3.0,
including goodness-of-fit (GOF) and the UCL computation methods for data sets without any BDL
observations. However,  ProUCL 4.0 has the additional capability to perform GOF tests and computing
UCLs and other upper limits based upon data sets with BDL observations.

ProUCL 4.0 defines log-transform (log) as the natural logarithm (In) to the base e. ProUCL 4.0 also
computes the maximum likelihood estimates (MLEsj and the minimum variance unbiased estimates
(MVUEs) of unknown population parameters of normal, lognormal, and gamma distributions. This, of
course, depends upon the underlying data distribution. ProUCL 4.0 computes the (7 - a)100% UCLs of
the unknown population mean, ^i, using 5 parametric and 10 nonparametric methods. It should be pointed
out that ProUCL 4.0 computes the simple summary statistics for detected raw and log-transformed data
for full data sets without NDs, as well as for data sets with BDL observations. It is noted that estimates of
mean and sd for data sets with NDs based upon rigorous statistical methods (e.g., MLE, ROS, K-M
methods) are note provided in the summary statistics. Those estimates and the associated upper limits for
data sets with NDs are provided under the  menu options: Background and UCL.

It is emphasized that throughout this User Guide, and in the ProUCL 4.0 software, it is assumed that one
is dealing with a single population. If multiple populations (e.g., background and site data mixed together)
are present, it is recommended to first separate them out (e.g., using appropriate statistical population
partitioning techniques), and then compute appropriate respective 95% UCLs separately for each of the
identified populations. Outliers, if any, should be identified and thoroughly investigated. ProUCL 4.0
provides two commonly used simple classical outlier identification procedures: 1) the Dixon test and 2)
the Rosner test. Outliers distort most parametric statistics (e.g., mean, UCLs, upper prediction limits
(UPLs), test statistics) of interest. Moreover, it should be noted that even though outliers might have
minimal influence on  hypotheses testing statistics based upon ranks (e.g., WMW test), outliers do distort
those nonparametric statistics (including bootstrap methods), which are based upon higher order statistics
such as UPLs and UTLs. Decisions about the disposition (exclusion or inclusion) of outliers in a data set

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used to estimate the EPC terms or BTVs should be made by all parties involved (e.g., project team, EPA,
local agency, potentially responsible party, etc.) in the decision making process.

The presence of outlying observations also distorts statistics based upon bootstrap re-samples. The use of
higher order values (quantiles) of the distorted statistics for the computation of the UCLs or UPLs based
upon bootstrap t and Hall's bootstrap methods may yield unstable and erratic UCL values. This is
especially true  for the upper limits providing higher confidence coefficients such as 95%, 97.5%, or 99%.
Similar behavior of the bootstrap t UCL is observed for data sets having BDL observations. Therefore, the
bootstrap t and Hall's bootstrap methods should be used with caution. It is suggested that the user should
examine various other UCL results and determine if the UCLs based upon the bootstrap t and Hall's
bootstrap methods represent reasonable and reliable UCL values of practical merit. If the results based
upon these two bootstrap methods are much higher than the rest of methods, then this could be an
indication of erratic behavior of those bootstrap UCL values, perhaps distorted by outlying observations.
In case these two bootstrap methods yield erratic and inflated UCLs, the UCL of the mean should be
computed using the adjusted or the approximate gamma UCL computation method for highly skewed
gamma distributed data sets of small sizes. Alternatively,  one may use a 97.5% or 99% Chebyshev UCL
to estimate the  mean of a highly skewed population. It should be noted that typically, a Chebyshev UCL
may yield conservative and higher values of the UCLs than other methods available in ProUCL 4.0 This
is especially true when data are moderately skewed and sample size is large. In such cases, when the
sample size is large, one may want to use a 95% Chebyshev UCL or a Chebyshev UCL with lower
confidence coefficient such as 92.5% or 90% as estimate of the population mean.

ProUCL  Methods

ProUCL 4.0 provides 15 UCL computation methods for full data sets without any BDL observations; 5
are parametric and 10 are nonparametric methods. The nonparametric methods do not depend upon any
assumptions about the data distributions. The five parametric UCL computation methods are:

           o  Student's t-UCL,
           o  Approximate gamma UCL using chi-square approximation,
           o  Adjusted gamma UCL (adjusted for level significance),
           o  Land's H-UCL, and
           o  Chebyshev inequality-based UCL (using  MVUEs of parameters of a lognormal
              distribution).

The 10 nonparametric methods are:

           1.  The central limit theorem (CLT)-based UCL,
           2.  Modified-t statistic (adjusted for skewness)-based UCL,
           3.  Adjusted-CLT (adjusted for skewness)-based UCL,
           4.  Chebyshev inequality-based UCL (using  sample mean and sample standard deviation),
           5.  Jackknife method-based UCL,
           6.  UCL based upon standard bootstrap,
           7.  UCL based upon percentile bootstrap,
           8.  UCL based upon bias-corrected accelerated (BCA) bootstrap,
           9.  UCL based upon bootstrap t, and
           10. UCL based upon Hall's bootstrap.

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Environmental scientists often encounter trace level concentrations of COPCs when evaluating sample
analytical results. Those low level analytical results cannot be measured accurately, and therefore are
typically reported as less than one or more DL values. However, the practitioners need to obtain reliable
estimates of the population mean, //;, and the population standard deviation, ah and upper limits including
the UCL of the population mass (measure of central tendency) or mean, UPL, and UTL. Several methods
are available and cited in the environmental literature (Helsel (2005), Singh and Nocerino (2002), Millard
and Neerchal (2001)) that can be used to estimate the population mean and variance. However, till to date,
no specific recommendations are available for the use of appropriate methods that can be used to compute
upper limits (e.g., UCLs, UPLs) based upon data sets with BDL observations. Singh, Maichle, and Lee
(EPA, 2006) extensively studied the performance of several parametric and nonparametric UCL
computation methods for data sets with BDL observations. Based upon their results and findings, several
methods to compute upper limits (UCLs, UPLs, and UTLs) needed to estimate the EPC terms and BTVs
have been incorporated in ProUCL 4.0.

In 2002, EPA issued another Guidance for Comparing Background and Chemical Concentrations in Soil
for CERCLA Sites (EPA, 2002b).  This EPA (2002b) background guidance document is currently being
revised to include statistical methods that can be used to estimate the BTVs and not-to-exceed values
based upon data sets with and without the BDL observations. In background evaluation studies, BTVs,
compliance limits, or not-to-exceed values often need to be estimated based upon defensible background
data sets. The estimated BTVs or  not-to-exceed values are then used for screening the COPCs, to identify
the site AOCs or hot spots, and also to determine if the site concentrations (perhaps after a remediation
activity) are comparable to background concentrations, or are approaching the background level
concentrations. Individual point-by-point site observations (composite  samples preferred)  are sometimes
compared with those not-to-exceed values or BTVs. It should be pointed out that in practice, it is
preferred to use hypotheses testing approaches to compare site versus background concentrations
provided enough (e.g., at least 8-10 detected observations from each of the two populations) site  and
background data are available. Chapter 1 provides practical guidance on the minimum sample size
requirements to estimate and use the BTVs, single and two-sample hypotheses testing approaches to
perform background evaluations and background versus site  comparisons. Chapter 1 also briefly
discusses the differences in the definitions and uses of the various upper limits as incorporated in ProUCL
4.0. Detailed discussion of the various methods to estimate the BTVs and other not-to-exceed values for
full-uncensored data sets (Chapter 5) without any nondetect values and for left-censored data sets
(Chapter 6) with nondetect values are given in the revised background  guidance document.

ProUCL 4.0 includes statistical methods to compute UCLs of the mean, upper limits to estimate the
BTVs, other not-to-exceed values, and compliance limits based upon data sets with one or more detection
limits. The use of appropriate statistical methods and limits for exposure and risk assessment, and site
versus background comparisons, is based upon several factors:

           1.  Objective of the study;
           2.  Environmental medium (e.g., soil, groundwater, sediment, air) of concern;
           3.  Quantity and quality of the available data;
           4.  Estimation of a not-to-exceed value or of a mean contaminant concentration;
           5.  Pre-established or unknown cleanup standards and BTVs; and
           6.  Sampling distributions (parametric or nonparametric) of the concentration data sets
               collected from the site and background areas under investigation.

In background versus site comparison studies, the population parameters of interest are typically
represented by upper threshold limits (e.g., upper percentiles, upper confidence limits of an upper

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percentile, upper prediction limit) of the background data distribution. It should be noted that the upper
threshold values are estimated and represented by upper percentiles and other values from the upper tail
of the background data distribution. These background upper threshold values do not represent measures
of central tendency such as the mean, the median, or their upper confidence limits. These environmental
parameters may include:

           •   Preliminary remediation goals (PRGs), Compliance Limits,
           •   Soil screening levels (SSLs),
           •   Risk-based cleanup (RBC) standards,
           •   BTVs, compliance limits, or not-to-exceed values, and
           •   Maximum concentration limit (MCL) or alternative concentration limit (ACL) used in
               Groundwater applications.

When the environmental parameters listed above are not known or pre-established, appropriate upper
statistical limits are used to estimate those parameters. The UPL, UTL, and upper percentiles are typically
used to estimate the BTVs, not-to-exceed values, and other parameters listed above. Depending upon the
availability of site data, point-by-point site observations are compared with the estimated (or pre-
established) BTVs and not-to-exceed values. If enough site and background data are available, two-
sample hypotheses testing approaches (preferred method to compare two populations) are used to
compare site concentrations with background concentrations levels. The hypotheses testing methods can
also be used to compare contaminant concentrations of two site AOCs, surface and subsurface
contaminant concentrations, or upgradient versus monitoring well contaminant concentrations.

Background versus Site Comparison  Evaluations

The following statistical limits have been incorporated in ProUCL 4.0 to assist in background versus site
comparison evaluations:

Parametric Limits for Full-Uncensored Data  Sets without Nondetect Observations

           •   UPL for a single observation (Normal, Lognormal) not belonging to the original data set
           •   UPL for next k (k is user specified) or k future observations (Normal, Lognormal)
           •   UTL, an upper confidence limit of a percentile (Normal, Lognormal)
           •   Upper percentiles (Normal, Lognormal, and Gamma)

Nonparametric Limits for Full-Uncensored Data Sets without Nondetect  Observations

Nonparametric limits are typically based upon order statistics of a data set such as a background or a
reference data set. Depending upon the size of the data set, higher order statistics (maximum, second
largest, third largest, and so on) are used as these upper limits (e.g., UPLs, UTLs). The details of these
methods with sample size requirements can be found in Chapter 5 of the revised Guidance for Comparing
Background and Chemical Concentrations in Soil for CERCLA Sites (EPA, 2002b). It should be, noted
that the following statistics might get distorted by the presence of outliers (if any) in the data set under
study.

           •   UPL for a single observation not belonging to the original data set
           •   UTL, an upper confidence limit of a percentile
           •   Upper percentiles

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           •   Upper limit based upon interquartile range (IQR)
           •   Upper limits based upon bootstrap methods

For data sets with BDL observations, the following parametric and nonparametric methods to compute
the upper limits were studied and evaluated by Singh, Maichle, and Lee (EPA, 2006) via Monte Carlo
Simulation Experiments. Depending upon the performances of those methods, only some of the methods
have been incorporated in ProUCL 4.0. Methods (e.g., Delta method, DL method, uniform (0, DL)
generation method) not included in ProUCL 4.0 do not perform well in comparison with other methods.

Note: When the percentage ofNDs in a data set is high (e.g., > 40%-50%), especially when multiple
detection limits might be present, it is hard to reliably perform GOF tests (to determine data distribution)
on those data sets with many NDs. The  uncertainty associated with those  GOF tests will be high,
especially when the data sets are of small sizes  (< 10-20). It should also be noted that the parametric
MLE methods (e.g., for normal and lognormal distributions) often yield unstable estimates of mean and
sd. This is especially true when the number ofnondetects exceeds 40%-50%. In such situations, it is
preferable to use nonparametric (e.g., KM method) methods to compute statistics of interest such as
UCLs, UPLs,  and UTLs. Nonparametric methods do not require any distributional assumptions about the
data sets under investigation. Singh, Maichle, and Lee (EPA, 2006) also concluded that the performance
of the KM estimation method is better (in terms of coverage probabilities) than various other parametric
estimation (e.g., MLE, EM, ROS) methods.

Parametric Methods to Compute Upper Limits for Data Sets with Nondetect Observations

           •   Simple substitution (proxy) methods (0, DL/2, DL)
           •   MLE method, often known as Cohen's MLE method - single detection limit
           •   Restricted MLE method - single detection limit - not in ProUCL 4.0
           •   Expectation maximization (EM) method - single detection limit - not in ProUCL 4.0
           •   EPA Delta log method - single detection limit - not in ProUCL 4.0
           •   Regression method on  detected data and using slope and  intercept of the OLS regression
               line as estimates of standard deviation, sd,  and mean (not a recommended method)
           •   Robust ROS (regression on order statistics) on log-transformed  data - nondetects
               extrapolated (estimated) using robust ROS; mean, sd, UCLs, and other statistics
               computed using  the detected and extrapolated data in original scale - multiple detection
               limits
           •   Normal ROS - nondetects extrapolated (estimated) using normal distribution, mean, sd,
               UCLs, and other statistics computed using the detected and extrapolated data - multiple
               detection limits.
           •   It is noted that the estimated NDs often become negative  and even larger than the
               detection limits (not a recommended method)
           •   Gamma ROS - nondetects extrapolated (estimated) using gamma distribution, mean, sd,
               UCLs, and other statistics computed using the detected and extrapolated data - multiple
               detection limits

Nonparametric Methods to Compute  Upper Limits for Data Sets with Nondetect Observations

           •   Bootstrap Methods
               o   Percentile Bootstrap on robust ROS
               o   Percentile Bootstrap

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               o  BCA Bootstrap
               o  Bootstrap t

           •   Jackknife Method
               o  Jackknife on robust ROS

           •   Kaplan-Meier (KM) Method
               o  Bootstrap (percentile, BCA) using KM estimates
               o  Jackknife using KM estimates
               o  Chebyshev Method using KM estimates

           •   Winsorization Method

For uncensored full data sets without any NDs, the performance (in terms of coverage for the mean) of
the various UCL computation methods was evaluated by Singh and Singh (2003). The performance of the
parametric and nonparametric UCL methods based upon data sets with nondetect observations was
studied by Singh, Maichle, and Lee (EPA, 2006). Several of the methods listed above have been
incorporated in ProUCL 4.0 to compute the estimates of EPC terms (95% UCL), and of BTVs (UPLs,
UTLs, upper percentiles). Methods that did not perform well (e.g., poor coverage or unrealistically large
values, infeasible and biased estimates) are not included in ProUCL 4.0. Methods not incorporated in
ProUCL 4.0 are: EPA Delta Log method, Restricted MLE method, and EM method, substitution method
(0, and DL), and Regression method.

Note: It should be noted that for data sets with NDs,  the DL/2 substitution method has been incorporated
in ProUCL 4.0 only for historical reasons and also for its current default use. It is well known that the
DL/2 method (with NDs replaced by DL/2) does not perform well (e.g., Singh, Maichle, and Lee (EPA,
2006)) even when the percentage of NDs is only 5%-10%. It is strongly suggested to avoid the use of
DL/2 method for estimation and hypothesis testing approaches used in various  environmental
applications. Also, when the % of NDs becomes high (e.g., > 40%-50%), it is suggested to avoid the use
of parametric MLE methods. For data sets with high percentage of NDs (e.g., > 40%),  the distributional
assumptions needed to use parametric methods are hard to verify; and those parametric MLE methods
may yield unstable results.

It should also be noted that even though the lognormal distribution and some statistics based upon
lognormal assumption (e.g., Robust ROS, DL/2 method) are available in ProUCL 4.0, ProUCL 4.0 does
not compute MLEs of mean and sd based upon a lognormal distribution. The main reason is that the
estimates need to be computed in the original scale via back-transformation (Shaarawi, 1989, and Singh,
Maichle, and Lee (EPA, 2006)). Those back-transformed estimates often suffer from an unknown amount
of significant bias. Hence, it is also suggested to avoid the use of a lognormal distribution to compute
MLEs of mean and sd, and associated upper limits, especially UCLs based upon those MLEs obtained
using a lognormal distribution.

ProUCL 4.0 recommends the use of an appropriate UCL to estimate the  EPC terms. It is desirable that the
user consults with the project team and experts familiar with the site before using those recommendations.
Furthermore, there does not seem to be a general agreement about the  use of an upper limit (e.g., UPL,
percentile, or UTL) to estimate not-to-exceed values  or BTVs to be used for screening of the COPCs and
in site versus background comparison studies. ProUCL 4.0 can compute both parametric and
nonparametric upper percentiles, UPLs, and UTLs for uncensored and censored data sets. However, no
specific recommendations have been made regarding the use of UPLs, UTLs, or upper percentiles to

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estimate the BTVs, compliance limits, and other related background or reference parameters. However,
the developers of ProUCL 4.0 prefer the use of UPLs or upper percentiles to estimate the background
population parameters (e.g., BTVs, not-to-exceed values) that may be needed to perform point-by-point
site versus background comparisons.

The standard bootstrap and the percentile bootstrap UCL computation methods do not perform well (do
not provide adequate coverage to population mean) for skewed data sets. For skewed distributions, the
bootstrap t and Hall's bootstrap (meant to adjust for skewness) methods do perform better (in terms of
coverage for the population mean) than the other bootstrap methods. However, it has been noted (e.g.,
Efron and Tibshirani (1993), and Singh,  Singh, and laci (2002b)) that these two bootstrap methods
sometimes yield erratic and inflated UCL values (orders of magnitude higher than the other UCLs). This
may occur when outliers are present in a data set. Similar behavior of the bootstrap t UCL is observed
based upon data sets with NDs. Therefore, whenever applicable, ProUCL 4.0 provides cautionary
statements regarding the use of bootstrap methods.

ProUCL 4.0 provides several state-of-the-art parametric and nonparametric UCL, UPL, and UTL
computation methods that can be used on uncensored data sets (full data sets) and on data sets with BDL
observations. Some of the methods (e.g., Kaplan-Meier method, ROS methods) incorporated in ProUCL
4.0 are applicable on left-censored data sets having multiple detection limits. The UCLs and other upper
limits computation methods in ProUCL 4.0 cover a wide range of skewed data distributions with and
without the BDLs arising from the environmental applications.

ProUCL 4.0 also has parametric and nonparametric single and two-sample hypotheses testing approaches
required to: compare site location (e.g., mean, median) to a specified cleanup standard; perform site
versus background comparisons; or compare of two or more AOCs. These hypotheses testing methods
can handle both full (uncensored data sets without NDs) and left-censored (with nondetects) data sets.
Specifically, two-sample tests such as t-test, Wilcoxon Mann-Whitney (WMW) Rank Sum test, quantile
test, and Gehan's test are  available in ProUCL 4.0 to compare concentrations of two populations.

Single sample parametric (Student's t-test) and nonparametric (sign test, Wilcoxon Signed Rank (WSR)
test, tests for proportions and percentiles) hypotheses testing approaches are also available in ProUCL 4.0.
The single sample hypotheses tests are useful when the environmental parameters such as the clean
standard, action level, or compliance limits are known, and the objective is to compare site concentrations
with those known threshold values. Specifically, a t-test (or a sign test) may be used to verify the
attainment of cleanup levels at an AOC after a remediation activity; and a test for proportion may be used
to verify if the proportion of exceedances of an action level (or a compliance limit) by sample
concentrations collected from the AOC (or a MW) exceeds a certain specified proportion (e.g., 1%, 5%,
10%). As  mentioned before, ProUCL 4.0 can perform these hypotheses on data sets with and without
nondetect observations.

Note: It should be noted that as  cited in the literature, some of the hypotheses testing approaches (e.g.,
nonparametric two-sample WMW) deal with the single detection limit scenario.  If multiple detection
limits are present, all NDs below the largest detection limit need to be considered as NDs (Gilbert, 1987,
andHelsel, 2005). This in turn may reduce the power and increase uncertainty associated with test. As
mentioned before, it is always desirable to supplement the test statistics and test conclusions with
graphical displays such as the multiple Q-Q plots and side-by-side box plots. ProUCL 4.0 can graph box
plots and Q-Q plots for data sets with nondetect observations. Gehan test as available in Pro UCL 4.0
should be  used in case multiple detection limits are present. ProUCL 4.0 can draw Q-Qplots and box
plots for data sets with and without nondetect observations.
 10

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It should be pointed out that when using two-sample hypotheses approaches (WMWtest, Gehan test, and
quantile test) on data sets with NDs, both samples and variables (e.g., site-As, Back-As) should be
specified as having nondetects. This means, a ND column (0 = ND, and 1 = detect) should be provided
for each variable (here D_site-As, andD_Back-As) to be used in this comparison. If a variable (e.g., site-
As) does not have any nondetects, still a column with label D_site-As should be included in the data set
with all entries = 1 (detected values).

Moreover, in single sample hypotheses tests (e.g., sign test, proportion test) used to compare site
mean/median concentration level with a cleanup standard, Q or compliance limit (e.g., proportion test),
all NDs (if any) should lie below the cleanup standard, Q.

The differences between these tests should be noted and understood. Specifically, a t-test or a Wilcoxon
Signed Rank (WSR) test are used to compare the measures of location and central tendencies (e.g., mean,
median) of a site area (e.g., AOC) to a cleanup standard, Cs, or action level also representing a measure of
central tendency (e.g., mean, median); whereas,  a proportion test compares if the proportion of site
observations  from an AOC exceeding a compliance limit (CL) exceeds a specified proportion, P0 (e.g.,
5%, 10%). The percentile test compares a specified percentile (e.g., 95th) of the site data to a pre-
specified upper threshold (e.g., reporting limit, action level). All of these tests have been incorporated in
ProUCL 4.0. Most of the single sample and two-sample hypotheses tests also report associated p-values.
For some of the hypotheses tests (e.g., WMWtest, WSR test, proportion test), large sample approximate
p-values are computed using continuity correction factors.

Graphical Capabilities

ProUCL 4.0 has useful exploratory graphical methods that may be used to visually compare the
concentrations of:

           1.  A site area of concern (AOC) with an action level. This can be done using a box plot of
               site data with action level superimposed on that graph,
           2.  Two or more populations, including site versus background populations, surface versus
               subsurface concentrations, and
           3.  Two or more AOCs.

The graphical methods include double and multiple quantile-quantile (Q-Q) plots, side-by-side box plots,
and histograms. Whenever possible, it is desirable to supplement statistical test results and statistics with
visual graphical displays of data sets. There is no substitute for graphical displays of a data set as the
visual displays often provide useful information about a data set, which cannot be revealed by simple test
statistics such as t-test, SW test, Rosner test, WMW test. For example, in addition to providing
information about the data distribution, a normal Q-Q plot can also help identify outliers and multiple
populations that might be present in a data set. This kind of information cannot be revealed by simple test
statistics such as a Shapiro-Wilk (SW) test or Rosner's outlier test statistic. Specifically, the SW test may
lead to the conclusion that a mixture data set (representing two or more populations) can be modeled by a
normal (or lognormal) distribution, whereas the  occurrence of obvious breaks and jumps in the associated
Q-Q plot may suggest the presence of multiple populations in the mixture data set. It is suggested that the
user should use exploratory tools to gain necessary insight into a data set and the underlying assumptions
(e.g., distributional, single population) that may  not be revealed by simple test statistics.
                                                                                               11

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Note: On a Q-Q plot, observations well separated from the majority of the data may represent potential
outliers, and obvious jumps and breaks of significant magnitude may suggest the presence of observations
from multiple populations in the data set.

The analyses of data categorized by a group ID variable such as:  1) Surface vs. Subsurface;
2) AOC1 vs. AOC2; 3) Site vs. Background; and 4) Upgradient vs. Downgradient monitoring wells are
quite common in many environmental applications. ProUCL 4.0 offers this option for data sets with and
without nondetects. The Group Option provides a powerful tool to perform various statistical tests and
methods (including graphical displays) separately for each of the group (samples from different
populations) that may be present in a data set. For an example, the same data set may consist of samples
from the various groups or populations representing site, background, two or more AOCs, surface,
subsurface, monitoring wells. The graphical displays (e.g., box plots, Q-Q plots) and statistics
(computations of background statistics, UCLs, hypotheses testing approaches) of interest can be
computed separately for each group by using this option.

Technical Guide

In addition to this User Guide, a Technical document also accompanies ProUCL 4.0, providing useful
technical details of the graphical and statistical methods as incorporated in ProUCL 4.0. Most of the
mathematical algorithms and formulas (with references) used in the development of ProUCL 4.0 are
summarized in the Technical Guide.

Minimum Hardware Requirements

           •  Intel Pentium 1.0 GHz
           •  45 MB of hard drive space
           •  512 MB of memory (RAM)
           •  CD-ROM drive
           •  Windows 98 or newer. ProUCL was thoroughly tested on NT-4, Windows 2000, and
              Windows XP Operating Systems (limited testing on Windows ME).

Software Requirements

ProUCL 4.0 has been developed in the Microsoft .NET Framework using the C# programming language.
As such, to properly run ProUCL 4.0, the computer using the program must have the .NET Framework
pre-installed. The downloadable .NET files can be found at one of the following two Web sites:

           •  http://msdn.microsoft.com/netframework/downloads/updates/default.aspx
              Note: Download .Net version 1.1

           •  http://www.microsoft.com/downloads/details.aspx?FamilyId=262D25E3-F589-4842-
              8157-034DlE7CF3A3&displavlang=en

The first Web site lists all of the downloadable .NET Framework files, while the second Web site
provides information about the specific file (s) needed to run ProUCL 4.0. Download times are estimated
at 57 minutes for a dialup connection (56K), and 13 minutes on a DSL/Cable connection (256K).
 12

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Installation Instructions

           •   Download the file SETUP.EXE from the EPA Web site and save to a temporary location.
              Note: This download is pending release - beta-testers: See text file on CD.

           •   Run the SETUP.EXE program. This will create a ProUCL directory and two folders:
              1) the USER GUIDE (this document), and 2) DATA (example data sets).

           •   To run the program, use Windows Explorer to locate the ProUCL application file, and
              double click on it, or use the RUN command from the start menu to locate the
              ProUCL.exe file, and run ProUCL.exe.

           •   To uninstall the program, use Windows Explorer to locate and delete the ProUCL folder.

Caution: If you have previous versions of the ProUCL, which were installed on your computer, you
should remove or rename the directory in which earlier ProUCL versions are currently located.

Getting Started

The functionality and the use of the methods and options available in ProUCL 4.0 have been illustrated
using Screen Shots of output screen generated by ProUCL 4.0. ProUCL 4.0 uses a pull-down menu
structure, similar to a typical Windows program.

The screen below appears when the program is executed.

       Navigation
       Panel
        4
Main
Window
                                                                                  Log Panel
The screen consists of three main window panels:

           •   The MAIN WINDOW displays data sheets and outputs from the procedure used.
                                                                                        13

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           •   The NAVIGATION PANEL displays the name of data sets and all generated outputs.
               o   At present, the navigation panel can hold at most 20 outputs. In order to see more
                   files (data files or generated output files), one can click on Widow Option.

           •   The LOG PANEL displays transactions in green, warnings in orange, and errors in red.
               For an example, when one attempts to run a procedure meant for censored data sets on a
               full-uncensored data set, ProUCL 4.0 will print out a warning message in orange in this
               panel.
               o   Should both panels be unnecessary, you can click —J or choose Configure ^-
                   Panel ON/OFF

The use of this option will give extra space to see and print out the statistics of interest. For an example,
one may want to turn off these panels when multiple variables (e.g., multiple Q-Q plots) are analyzed and
GOF statistics and other statistics may need to be captured for all of the variables.
P?J>roUeL-f .0
 Navigation Fsmsr-T'
                 Summary Statistics  ROS Est, NDs  Graphs  Outlier Tests Goodness-of-Fit Hypothesis Testing  Background  LJCL  Window  Help
                   JM
                   , On/Off
1
                                            7
 Name
 ft) Worksheet wst
                       1
14

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                                       Chapter 1


  Guidance on the Use of Statistical Methods and Associated
                   Minimum Sample Size Requirements


This chapter briefly describes the differences between the various statistical limits (e.g., UCLs, UPLs,
UTLs) often used to estimate the environmental parameters of interest including exposure point
concentration (EPC) terms and background threshold values (BTVs). Suggestions are provided about the
minimum sample size requirements needed to use statistical inferential methods to estimate the
environmental parameters: EPC terms, BTVs and not-to-exceed values, and to compare site data with
background data or with some pre-established reference limits (e.g., preliminary remediation goals
(PRGs), action levels, compliance limits). It  is noted that several EPA guidance documents (e.g., EPA
1997, 2002a, 2006) discuss in details about data quality objectives (DQOs) and sample size
determinations based upon those DQOs needed for the various statistical methods used in environmental
applications.

Also, appropriate sample collection methods (e.g., instruments, sample weights, discrete or composite,
analytical methods) depend upon the medium (e.g., soil, sediment, water) under consideration. For an
example, Gerlach and Nocerino (EPA, 2003) describe optimal soil sample (based upon Gy theory)
collection methods. Therefore, the topics of  sample size determination based upon DQOs, data validation,
and appropriate sample collection methods for the various environmental media are not considered in
ProUCL 4.0 and its associated Technical Guide. It is assumed that data sets to be used in ProUCL are of
good quality, and whenever possible have been obtained using the guidance provided in various EPA
(2003, 2006) documents. It is the users' responsibility to assure that adequate amount of data have been
collected, and the collected data are  of good  quality.

Note: In Pro UCL 4.0 and its associated guidance documents, emphasis is given on the practical
applicability and appropriate use of statistical methods needed to address statistical issues arising in risk
management, background versus site evaluation studies, and various other environmental applications.
Specifically, guidance on minimum sample size requirements as provided in this chapter is useful  when
data have already been collected, or it is not possible (e.g., due to resource limitations) to collect  the
number of samples obtained using DQO processes as described in EPA (2006).

Decisions based upon statistics obtained using data sets of small sizes (e.g., 4 to 6 detected observations)
cannot be considered reliable enough to make a remediation decision that affects human health and the
environment. For an example, a background data set of size 4 to 6 is not large enough to characterize
background population, to  compute BTV values, or to perform background versus site comparisons. In
order to perform reliable and meaningful statistical inference (estimation and hypothesis testing), one
should determine the sample sizes that need  to be collected from the populations under investigation
using appropriate DQO processes and decision error rates (EPA, 2006). However, in some cases, it may
not be possible (e.g., resource constraints) to collect the same number of samples recommended by the
DQO process. In order to address such cases, minimum sample size requirements for background  and site
data sets are described.

The use of an appropriate statistical method  depends upon the environmental parameter(s) being
estimated or compared. The measures of central tendency (e.g., means, medians, or their upper confidence
limits (UCLs)) are often used to compare site mean concentrations (e.g., after remediation activity) with a
                                                                                         15

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cleanup standard, Cs, representing some central tendency measure of a reference area or other known
threshold representing a measure of central tendency. The upper threshold values, such as the compliance
limits (e.g., alternative concentration limit (ACL), maximum concentration limit (MCL)), or not-to-
exceed values, are used when individual point-by-point observations are compared with those not-to-
exceed values or other compliance limit.  It should be noted that depending upon whether the
environmental parameters (e.g., BTVs, not-to-exceed value, EPC term, cleanup standards) are known or
unknown, different statistical methods with different data requirements are needed to compare site
concentrations with pre-established (known) or estimated (unknown) cleanup standards and BTVs.

ProUCL 4.0 has been developed to address issues arising in exposure assessment, risk assessment, and
background versus site comparison applications.  Several upper limits, and single- and two-sample
hypotheses testing approaches, for both full uncensored and left-censored data sets, are available in
ProUCL 4.0. The details of the statistical and graphical methods included in ProUCL 4.0 can be found in
the ProUCL Technical Guidance. In order to make sure that the methods in ProUCL 4.0 are properly
used, this chapter provides guidance on:

            1.  analysis of site and background areas and data sets,
           2.  collection of discrete or composite samples,
           3.  appropriate use of the various upper limits,
           4.  guidance regarding minimum sample sizes,
           5.  point-by-point comparison of site observations with BTVs,
           6.  use of hypotheses testing approaches,
           7.  using small data sample sets,
           8.  use of maximum detected value,  and
           9.  discussion of ProUCL usage for  special cases.
1.1     Background Data Sets

The project team familiar with the site should identify and chose a background area. Depending upon the
site activities and the pollutants, the background area can be site-specific or a general reference area.  An
appropriate random sample of independent observations should be collected from the background area. A
defensible background data set  should represent a "single" background population (e.g., representing
pristine site conditions before any of the industrial site activities) free of contaminating observations such
as outliers. In a background data set, outliers may represent potentially contaminated observations from
impacted site areas under study or possibly from other polluted site(s). This scenario is common when
background samples are obtained from the various onsite areas (e.g., large federal facilities). Outlying
observations should not be included in the estimation (or hypotheses testing procedures) of the BTVs.
The presence of outliers in the background data set will yield distorted estimates of the BTVs and
hypothesis testing statistics. The proper disposition of outliers to include or not include them in the data
set should be decided by the project team.

Decisions based upon distorted statistics can be incorrect, misleading, and expensive. It should be noted
that the objective is to compute background statistics based upon the majority of the data set representing
the dominant background population, and not to accommodate a few low probability outliers that may
also be present in the background data set. A couple of simple classical outlier tests (Dixon and Rosner
tests) are available in ProUCL 4.0. Since these classical tests suffer from masking effects (e.g., extreme
outliers may mask the occurrence of other intermediate  outliers), it is suggested that these classical outlier
tests should always be supplemented with graphical  displays such as a box plot or a Q-Q plot. The use of
16

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robust and resistant outlier identification procedures (Singh and Nocerino, 1995, Rousseeuw and Leroy,
1987) is recommended when multiple outliers may be present in a data set. Those methods are beyond the
scope of ProUCL 4.0.

An appropriate background data set of a reasonable size (preferably computed using DQO processes) is
needed to characterize a background area including computation of upper limits (e.g., estimates of BTVs,
not-to-exceed values) based upon background data sets and also to compare site and background data sets
using hypotheses testing approaches. As mentioned before, a small background data set of size 4 to 6 is
not large enough to compute BTVs or to perform background versus site comparisons. At the minimum, a
background sample should have at least 8 to 10 (more observations are preferable) detected observations
to estimate BTVs or to use hypotheses testing approaches.

1.2    Site Data Sets

A defensible data set from a site population (e.g., AOC, EA, RU, group of monitoring wells) should be
representative of the site area under investigation. Depending upon the site areas under investigation,
different soil depths and soil types may be considered as representing different statistical populations. In
such cases, background-versus-site comparisons may have to be  conducted separately for each of those
site sub-populations (e.g., surface and sub-surface layers of an AOC, clay and sandy site areas). These
issues, such as comparing depths and soil types, should also be considered in a planning and sampling
design before starting to collect samples from the various site areas under investigation. Specifically, the
availability of an adequate amount of representative site data is required from each of those site sub-
populations defined by sample depths, soil types, and the various other characteristics. For detailed
guidance on soil sample collections, the reader is referred to Gerlach and Nocerino (EPA (2003)).

The site data collection requirements depend upon the objective of the study. Specifically, in background-
versus-site comparisons, site data are needed to perform:

           •   Individual point-by-point site observation comparisons with pre-established or estimated
               BTVs, PRGs, cleanup standards, and not-to-exceed-values. Typically, this approach is
               used when only a small number (e.g., < 4 to 6) of detected site observations (preferably
               based upon composite samples) are available which need to be compared with BTVs and
               not-to-exceed values.

           •   Single sample hypotheses tests to compare site data with pre-established cleanup
               standards, Cs (e.g., representing a measure of central tendency); or with BTVs and not-to-
               exceed values (used for tests for proportions and percentiles). The hypotheses testing
               approaches are used when enough site data are available. Specifically, when at least 8 to
               10 detected (more are desirable) site observations are available, it is preferable to use
               hypotheses testing approaches to compare  site observations with specified threshold
               values. The use of hypotheses testing approaches can control the two types (Type 1 and
               Type 2) of error rates more efficiently than the point-by-point individual observation
               comparisons. This is especially true as the number of point-by-point comparisons
               increases. This issue is illustrated by the following table summarizing the probabilities  of
               exceedances (false positive error rate) of the background threshold value (e.g., 95th
               percentile) by site observations, even when the site and background populations have
               comparable distributions. The probabilities of these chance exceedances increase as the
               sample size increases.
                                                                                              17

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                              Sample Size       Probability of Exceedance
                                    1                 0.05
                                    2                 0.10
                                    5                 0.23
                                    8                 0.34
                                    10                0.40
                                    12                0.46
                                    64                0.96

           •  Two-sample hypotheses testing to compare site data distribution with background data
              distribution to determine if the site concentrations are comparable to background
              concentrations. Adequate amount of data need to be made available from the site as well
              as the background populations. It is preferable to collect at least 8 to 10 detected
              observations from each of the population under comparison.

1.3    Discrete Samples or Composite Samples?

In a data set (background or site), collected samples should be either all discrete or all composite. In
general, both discrete and composite site samples may be used for individual point-by-point site
comparisons with a threshold value, and for single and two-sample hypotheses testing applications.

           •  If possible, the use of composite site samples is preferred when comparing individual
              point-by-point site  observations from an area (e.g., area of concern (AOC), remediation
              unit (RU), exposure area (EA)) with pre-established or estimated BTV, compliance limit
              (CL), or other not-to-exceed value. This comparison approach is useful when few (< 4 to
              6) detected site observations are compared with a pre-established or estimated BTV or
              other not-to-exceed threshold.

           •  When using a single sample hypothesis testing approach, site data  can be obtained by
              collecting all discrete or all composite samples. The hypothesis testing approach is used
              when many (e.g., exceeding 8 to 10) detected site observations are available. Details of
              the single sample hypothesis approaches are widely available in EPA documents  (1989,
              1997, and 2006). Selected single sample hypotheses testing procedures are available in
              ProUCL 4.0.

           •  If a two-sample hypotheses testing approach is used to perform site versus background
              comparisons, then samples from both of the populations should be either all discrete or
              all composite samples. The two-sample hypothesis testing approach is used when many
              (e.g., exceeding 8 to 10) site, as well as background, observations are available. For better
              and more accurate results with higher statistical power, the availability of more
              observations (e.g., exceeding 10-15) from each of the two populations is desirable,
              perhaps based upon an appropriate DQO process, as described in an EPA guidance
              document (2006).

1.4    Upper  Limits and  Their Use

The appropriate computation and use of statistical limits depend upon their applications and the
parameters (e.g., EPC term, not-to-exceed value) they are supposed to be estimating. Depending upon the
objective of the study, a pre-specified cleanup standard, Cs, or a risk-based cleanup  (RBC) can be viewed
18

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as to represent: 1) as average contaminant concentration; or 2) a not-to-exceed upper threshold value.
These two threshold values, an average value, jU0, and a not-to-exceed value, A0, represent two
significantly different parameters, and different statistical methods and limits are used to compare the site
data with these two different parameters or threshold values. Statistical limits, such as an upper
confidence limit (UCL) of the population mean, an upper prediction limit (UPL) for an independently
obtained "single" observation, or independently obtained k observations (also called future k
observations, next k observations, or k different observations), upper percentiles, and upper tolerance
limits (UTLs), are often used to estimate the environmental parameters, including the EPC terms,
compliance limits (e.g., ACL, MLC), BTVs, and other not-to-exceed values. Here, UTL95%-95%
represents a 95% confidence limit of the 95th percentile of the distribution of the contaminant under study.

It is important to understand and note the differences between the uses and numerical values of these
statistical limits so that they can be properly used. Specifically, the differences between UCLs and UPLs
(or upper percentiles), and UCLs and UTLs should be clearly understood and acknowledged. A UCL with
a 95% confidence limit (UCL95) of the mean represents an estimate of the population mean (measure of
the central tendency of a data distribution), whereas a UPL95, a UTL95%-95%, and an upper 95th
percentile represent estimates of a threshold value in the upper tail of the data distribution. Therefore, a
UCL95  should represent a smaller number than an upper percentile or an upper prediction limit. Also,
since a UTL 95%-95% represents a 95% UCL of the upper 95th percentile, a UTL should be > the
corresponding UPL95 and the 95th upper percentile. Typically, it is expected that the numerical values of
these limits should follow the order given  as follows:

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

It should also be pointed out that as the sample size increases, a UCL95 of the mean approaches
(converges to) the population mean, and a UPL95 approaches the 95th percentile. The differences among
the various upper limits are further illustrated in Example 1-1 below. It should be noted that, in some
cases, these limits might not follow the natural order described above. This is especially true when the
upper limits are computed based upon a lognormal distribution  (Singh, Singh, and Engelhardt, 1997). It  is
well known that a lognormal distribution-based H-UCL95 (Land's UCL95) often yields unstable and
impractically large UCL values. An H-UCL95 often becomes larger than UPL95 and even larger than a
UTL 95%-95%. This is especially true when dealing with skewed data sets of smaller sizes. Moreover, it
should also be noted that in some cases, a  H-UCL95 becomes smaller than the sample mean,  especially
when the data are mildly skewed to moderately skewed and the sample size is large (e.g., > 50, 100).

Example 1-1: Consider a  simple site-specific background data set associated with a Superfund site. The
data set  (given in Appendix 5 of the revised Guidance for Comparing Background and Chemical
Concentrations in Soil for CERCLA Sites (EPA, 2002b)) has several inorganic contaminants of potential
concern, including aluminum, arsenic, chromium, iron, and lead. It is noted that iron concentrations
follow a normal distribution. Upper limits for the iron data set are summarized in Table 1 -1. It is noted
that the  upper limits do follow the order as described above.

Table 1-1. Computation of Upper Limits for Iron (Normally Distributed)
Mean
9618
Median
9615
Min
3060
Max
18700
UCL95
11478
UPL95 for a
Single
Observation
18145
UPL95for4
Observations
21618
UTL95/95
21149
95% Upper
Percentile
17534
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A 95% UCL (UCL95) of the mean is the most commonly used limit in environmental applications. For an
example, a 95% UCL of mean is used as an estimate of the EPC. A UCL95 should not be used to estimate
a background threshold value (a value in the upper tail of the background data distribution) to be
compared with individual site observations. There are many instances in background evaluations and
background versus site comparison studies, when it is not appropriate to use a 95% UCL. Specifically,
when point-by-point site observations are to be compared with a BTV, then that BTV should be estimated
(or represented) by a limit from the upper tail of the reference set (background) data distribution.

A brief discussion about the differences between the applications and uses of the various statistical limits
is provided below. This will assist a typical user in determining which upper limit (e.g., UCL95 or
UPL95) to use to estimate the parameter of interest (e.g., EPC or BTV).

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

           •   When site averages (and not individual site observations) are compared with a threshold
               value (pre-determined or estimated), such as a PRO or a RBC, or with some other
               cleanup standard, Cs, then that threshold should represent an average value, and not a not-
               to-exceed threshold value for individual observation comparisons.

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

           •   A UPL, an upper percentile, or an UTL represents an upper limit to be used for point-by-
               point individual site observation comparisons. UPLs and UTLs are computed based upon
               background data sets, and individual site observations are compared with those limits. A
               site observation for a contaminant exceeding a background UTL or UPL may lead to the
               conclusion that the contaminant is a contaminant of potential concern (COPC) to be
               included in further risk evaluation and risk management studies.

           •   When individual point-by-point site observations are compared with a threshold value
               (pre-determined or estimated) of a background population or some other threshold and
               compliance limit value, such as a PRG, MLC, or ACL, then that threshold value should
               represent a not-to-exceed value. Such BTVs or not-to-exceed values are often estimated
               by a 95% UPL, UTL 95%-95%, or by an upper percentile. ProUCL 4.0 can be used to
               compute any of these upper limits based upon uncensored data sets as well as data sets
               with nondetect values.

           •   As the sample size increases, a UCL approaches the sample mean, and a UPL95
               approaches the corresponding 95th upper percentile.
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           •  It is pointed out that the developers of ProUCL 4.0 prefer the use of a 95% UPL (UPL95)
              as an estimate of BTV or a not-to-exceed value. As mentioned before, the option of
              comparing individual site observations with a BTV (specified or estimated) should be
              used when few (< 4 to 6) detected site observations (preferably composite values) are to
              be compared with a BTV.

           •  When enough (e.g., > 8 to 10) detected site observations are available, it is preferred to
              use hypotheses testing approaches. Specifically, single sample hypotheses testing
              (comparing site to a specified threshold) approaches should be used to perform site
              versus a known threshold comparison; and two-sample hypotheses testing (provided
              enough  background data are also available) approaches should be used to perform site
              versus background comparison. Several parametric and nonparametric single and two-
              sample hypotheses testing approaches are available in ProUCL 4.0.

It is re-emphasized that only averages should be compared with the averages or UCLs, and individual site
observations should be compared with UPLs, upper percentiles, or UTLs. For an example, the comparison
of a 95% UCL of one population (e.g., site) with a 90% or 95% upper percentile of another population
(e.g., background) cannot be considered fair and reasonable as these limits (e.g., UCL and UPL) estimate
and represent different parameters. It is hard to justify comparing a UCL of one population with a UPL of
the other population. Conclusions (e.g., site dirty or site clean) derived by comparing UCLs and UPLs, or
UCLs and upper percentiles as suggested in Wyoming DEQ, Fact Sheet #24  (2005), cannot be considered
fair and reliable. Specifically, the decision error rates associated with such comparisons can be
significantly different from the specified (e.g., Type I error = 0.1, Type II error = 0.1) decision errors.

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

Point-by-point observation comparison method is used when a small number (e.g., 4 to 6 locations) of
detected site observations are  compared with pre-established or estimated BTVs, screening levels, or
preliminary remediation goals (PRGs). In this case, individual point-by-point site observations (preferably
based upon composite samples from various  site locations) are compared with estimated or pre-
established background  (e.g., USGS values) values, PRGs, or some other not-to-exceed value. Typically,
a single exceedance of the BTV, PRG, or of a not-to-exceed value by a site (or from a monitoring well)
observation may be considered as an indication of contamination at the site area under investigation. The
conclusion of an exceedance by  a site value is some times confirmed by re-sampling (taking a few more
collocated samples) that site location (or a monitoring well) exhibiting contaminant concentration in
excess of the BTV or PRG. If all collocated (or collected during the same time period) sample
observations collected from the same site location (or well) exceed the PRG (or MLC) or a not-to-exceed
value, then it may be concluded  that the location (well) requires further investigation (e.g., continuing
treatment and monitoring) and cleanup.

When BTV contaminant concentrations are not known or pre-established, one has to collect, obtain, or
extract a data set of an appropriate size that can be considered as representative of the site related
background. Statistical upper limits are computed using the data set thus obtained, which are used as
estimates of BTVs and not-to-exceed values. It should be noted that in order to compute reasonably
reliable and accurate estimates of BTVs and not-to-exceed values based upon a background (or reference)
data set, enough background observations (minimum of 8 to 10) should be collected, perhaps using an
appropriate DQO process as described in EPA (2006). Typically, background samples are collected from
a comparable general reference area or site-specific areas that are known to be free of contamination due
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to any of the site related activities. Several statistical limits can be used to estimate the BTVs based upon
a defensible data set of an adequate size. A detailed description of the computation and estimation of
BTVs is given in Chapter 5 (for uncensored data sets) and in Chapter 6 for data sets with nondetects of
the revised background guidance document. Once again, the use of this point-by-point comparison
method is  recommended when not many (e.g., < 4 to 6) site observations are to be compared with
estimated BTVs or PRGs. An exceedance of the estimated BTV by a site value may be considered as an
indication of the existing or continuing contamination at the site.

Note: When BTVs are not known, it is suggested that at least 8 to 10 (more are preferable) detected
representative background observations be made available to compute reasonably reliable estimates of
BTVs and other not-to-exceed values.

The point-by-point comparison method is also useful when quick turnaround comparisons are required.
Specifically, when the decisions have to be made in real time by a sampling or screening crew, or when
few detected site samples are available, then individual point-by-point site concentrations are compared
either with pre-established PRGs, cleanup goals and standards, or with estimated BTVs and not-to-exceed
values. The crew can use these comparisons to make the following informative decisions:

           1.   Screen and identify the COPCs,
           2.  Identify the polluted site AOCs,
           3.  Continue or stop remediation or excavation at a site AOC or a RU, or
           4.  Move the cleanup apparatus and crew to the next AOC or RU.

During the screening phase, an exceedance of a compliance limit, action level, a BTV, or a PRO by site
values for a contaminant may declare that contaminant as a COPC. Those COPCs are then included in
future site remediation and risk management studies. During the remediation phase, an exceedance of the
threshold value such as a compliance limit (CL) or a BTV by sample values collected from a site area (or
a monitoring well  (MW)) may declare that site area as a polluted AOC, or a hot spot requiring further
sampling and cleanup. This comparison method can also be used to verify if the site concentrations (e.g.,
from the base or side walls of an excavated site area) are approaching or meeting PRO, BTV, or a cleanup
standard after excavation has been conducted at that site area.

If a larger  number of detected samples (e.g., greater than 8 tolO) are available from the site locations
representing the site area under investigation (e.g.,  RU, AOC, EA), then the use of hypotheses testing
approaches (both single sample and two-sample) is preferred. The use of a hypothesis testing approach
will control the error rates more tightly and efficiently than the individual point-by-point site observations
versus BTV comparisons, especially when many site observations are compared with a BTV or a not-to-
exceed value.

Note: In background versus site comparison evaluations, scientists usually prefer the use of hypotheses
testing approaches to point-by-point site observation comparisons with BTVs or not-to-exceed values.
Hypotheses testing approaches require the availability of larger data sets from the populations under
investigation. Both single sample (used when BTVs, not-to-exceed values, compliance limits, or cleanup
standards  are known and pre-established) and two-sample (used when BTVs and compliance limits are
unknown)  hypotheses testing approaches are available in ProUCL 4.0.
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1.6    Hypothesis Testing Approaches and Their Use

Both single sample and two-sample hypotheses testing approaches are used to make cleanup decisions at
polluted sites, and also to compare contaminant concentrations of two (e.g., site versus background) or
more (several monitoring wells (MWs)) populations. The uses of hypotheses testing approaches in those
environmental applications are described as follows.

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

When pre-established BTVs and not-to-exceed values are used, such as the USGS background values
(Shacklette and Boerngen (1984)), thresholds obtained from similar sites, or pre-established not-to-exceed
values, PRGs, or RBCs, there is no need to extract, establish, or collect a background or reference data
set. When the BTVs and cleanup standards are known, one-sample hypotheses are used to compare site
data (provided enough site data are available) with known and pre-established threshold values. It is
suggested that the project team determine (e.g., using DQO) or decide (depending upon resources) about
the number of site observations that should be collected and compared with the "pre-established"
standards before coming to a conclusion about the status (clean or polluted) of the site area (e.g., RU,
AOC) under investigation. When the number of available detected site samples is less than 4 to 6, one
might perform point-by-point site observation comparisons with a BTV; and when enough detected site
observations (> 8 to 10, more are preferable) are available, it is desirable to use single sample hypothesis
testing approaches.

Depending upon the parameter (e.g., the average value, fi0, or a not-to-exceed value, A0), represented by
the known threshold value, one can use single sample hypothesis tests for population mean (t-test, sign
test) or single sample tests for proportions and percentiles. The details of the single sample hypotheses
testing approaches can be found in EPA (2006) and the Technical Guide for ProUCL 4.0. Several single
sample tests listed as follows are available in ProUCL 4.0.

One-Sample t-Test: This test is used to compare the site mean, ju, with a specified cleanup standard, Cs,
where the cleanup standard, Cs, represents an average threshold value, fi0- The Student's t-test (or a UCL
of mean) is often used (assuming normality of site data or when site sample size is large such as larger
than 30, 50) to determine the attainment of cleanup levels at a polluted site after some remediation
activities.

One-Sample Sign Test or Wilcoxon Signed Rank (WSR) Test: These tests are nonparametric tests and can
also handle nondetect observations provided all nondetects (e.g.,  associated detection limits) fall below
the specified threshold value. Q. These tests are used to compare the  site location (e.g., median, mean)
with a specified cleanup standard, Cs, representing a similar location measure.

One-Sample Proportion Test or Percentile Test: When a specified cleanup standard, A0, such as a
PRG or a BTV represents an upper threshold value of a contaminant concentration distribution
(e.g., not-to-exceed value, compliance limit) rather than the mean threshold value, JLIO, of the
contaminant concentration distribution, then a test for proportion or a test for percentile (or
equivalently a UTL 95%-95%) can be used to compare site proportion or site percentile with the
specified threshold or action level, A0. This test can also handle ND observations provided all
NDs are below the compliance limit.
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In order to obtain reasonably reliable estimates and test statistics, an adequate amount of representative
site data (8 to 10 detected observations) is needed to perform the hypotheses tests. As mentioned before,
in case only a few (e.g., < 4 to 6) detected site observations are available, then point-by-point site
concentrations may be compared with the specified action level, A0.

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

When BTVs, not-to-exceed values, and other cleanup standards are not available, then site data are
compared directly with the background data. In such cases, a two-sample hypothesis testing approach can
be used to perform site versus background comparisons. Note that this approach can be used to compare
concentrations of any two populations including two different site areas or two different monitoring wells
(MWs). In order to use and perform a two-sample hypothesis testing approach, enough data should be
available (collected) from each of the two populations under investigation. Site and  background data
requirements (e.g., based upon DQOs) to perform two-sample hypothesis test approaches are described in
EPA (1989b, 2006), Breckenridge and Crockett (1995), and the VSP (2005) software package. While
collecting site and background data, for better representation of populations under investigation, one may
also want to account for the size of the background area (and site area for site samples) into sample size
determination. That  is, a larger number (> 10 to 15) of representative background (or site) samples should
be collected from larger background (or site) areas. As mentioned before, every effort should be made to
collect as many samples as determined using DQO processes as described in  EPA documents (2006).

The two-sample (or  more) hypotheses approaches are used when the site parameters (e.g., mean, shape,
distribution) are being compared with the background parameters (e.g., mean, shape, distribution). The
two-sample hypotheses testing approach is also used when the cleanup standards or screening levels are
not known a priori, and they need to be estimated based upon a data set from a background or reference
population. Specifically, two-sample hypotheses testing approaches are used  to compare 1) the average
contaminant concentrations of two or more populations such as the background population and the
potentially contaminated site areas, or 2) the proportions of site and background  observations exceeding a
pre-established compliance limit, A0 In order to derive reliable conclusions with  higher statistical power
based upon hypothesis testing approaches, enough data (e.g., minimum of 8 to 10 samples) should be
available from all of the populations under investigation. It is also desirable to supplement statistical
methods with graphical displays, such as the double Q-Q plots, or side-by-side multiple box plots, as
available in ProUCL 4.0. Several parametric and nonparametric two-sample hypotheses testing
approaches, including Student's t-test, the Wilcoxon-Mann-Whitney (WMW) test, Gehan's test, and
quantile test are included in ProUCL 4.0. Details of those methods are described in the ProUCL 4.0
Technical Guide. It should be noted that the WMW, Gehan, and quantile tests are also available for data
sets with NDs. Gehan's test is specifically meant to be used on data sets with multiple detection limits. It
is also suggested that for best and reliable conclusions, both the WMW and quantile tests should be used
on the same data set. The details of these two tests with examples are given in EPA  (1994, 2006).

The samples collected from the two (or more) populations should all be of the same type obtained  using
similar analytical methods and apparatus. In other words, the collected site and background samples
should be all discrete or all composite (obtained using the  same design and pattern), and be collected from
the same medium (soil) at similar depths (e.g., all surface samples or all subsurface  samples) and time
(e.g., during the same quarter in groundwater applications) using comparable (preferably same) analytical
methods. Good sample collection methods and sampling strategies are given  in EPA (1996, 2003)
guidance documents.
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1.7    Minimum Sample Size Requirements

Due to resource limitations, it may not be possible (nor needed) to sample the entire population (e.g.,
background area, site area, areas of concern, exposure areas) under study. Statistics is used to draw
inference(s) about the populations (clean, dirty) and their known or unknown parameters (e.g.,
comparability of population means, not-to-exceed values, upper percentiles, and spreads) based upon
much smaller data sets (samples) collected from those populations under study. In order to determine and
establish BTVs, not-to-exceed values, or site-specific screening levels, defensible data set(s) of
appropriate size(s) needs to be collected from background areas (e.g., site-specific, general reference or
pristine area, or historical data). The project team and site experts should decide what represents a site
population and what represents a background population. The project team should determine the
population size and boundaries based upon all current and future objectives for the data collection. The
size and area of the population (e.g., a remediation unit, area of concern, or an exposure  unit) may be
determined based upon the potential land use, and other exposure and risk management objectives and
decisions. Moreover, appropriate effort should be made to properly collect soil samples (e.g., methods
based upon Gy sampling theory), as described in Gerlach and Nocerino (2003).

Using the collected site and background data sets, statistical methods supplemented with graphical
displays are used to perform site versus background comparisons. The test results and statistics obtained
by performing such site versus background comparisons are used to determine if the site and background
level contaminant concentration are  comparable; or if the site concentrations exceed the background
threshold concentration level; or if an adequate amount of cleanup and remediation approaching the  BTV
or a cleanup level have been performed at polluted areas (e.g., AOC, RU) of the site under study.

In order to perform statistical inference (estimation and hypothesis testing), one needs to determine the
sample sizes that need to be collected from the populations (e.g., site and background) under investigation
using appropriate DQO processes (EPA 2006). However, in some cases, it may not be possible to collect
the  same number of samples as determined by using a DQO process. For example, the data might have
already been collected (often is the case in practice) without using a DQO process, or due to resource
constraints, it may not be possible to collect as many samples as determined by using a DQO-based
sample size formula. It is observed that, in practice, the project team and the decision makers may not
collect enough background samples, perhaps due to various resource constraints. However, every effort
should be made to collect at least 8 to 10 (more are desirable) background observations before using
methods as incorporated  in ProUCL 4.0. The minimum sample size recommendations as described here
are  useful when resources are limited (as often is the case), and it may not be possible to collect as many
background and site (e.g., AOC, EU) samples as computed using DQOs and the sample  size
determination formulae given in the EPA (2006).  Some minimum sample size requirements are also given
in Fact Sheet #24, prepared by Wyoming Department of Environmental Quality (June 2005).

As mentioned before, the topics of DQO processes and the sample size determination are described in
detail in the EPA (2006)  guidance document. Therefore, the sample size determination formulae based
upon DQO processes are not included in ProUCL 4.0 and its Technical Guide. It should be noted that
DQO-based sample size determination routines are available in DataQUEST (EPA, 1997) and VSP
(2005) software packages. Guidance and suggestions are provided on the minimum number of
background and site samples needed to be able to use statistical methods for the computation  of upper
limits, and to perform single sample tests, two-sample tests such as t-test, and the Wilcoxon-Mann-
Whitney (WMW) test. The minimum sample size recommendations (requirements) as described here are
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made so that reasonably reliable estimates of EPC terms and BTVs, and defensible values of test statistics
for single or two-sample hypotheses tests (e.g., t-test, WMW test), can be computed.

1.7.1  Minimum Sample Size for Estimation and Point-by-Point Site Observation Comparisons

           •   Point-by-point observation comparison method is used when a small number (e.g., 4 to 6
               locations) of detected site observations are compared with pre-established or estimated
               BTVs, screening levels, or PRGs. In this case, individual point-by-point site observations
               (preferably based upon composite samples from various site locations) are compared with
               estimated or pre-established background (e.g., USGS values) values, PRGs, or some
               other not-to-exceed value.

           •   When BTV contaminant concentrations are not known or pre-established, one has to
               collect, obtain, or extract a data set of an appropriate size that can be considered as
               representative of the site related background. Statistical upper limits are computed using
               the data set thus obtained; which are used as estimates of BTVs and not-to-exceed values.
               It should be noted that in order to compute reasonably reliable and accurate estimates of
               BTVs and not-to-exceed values based upon a background (or reference)  data set, enough
               background observations (minimum of 8 to 10) should be collected perhaps using an
               appropriate DQO process as described in EPA (2006). Typically, background samples are
               collected from a comparable general reference area or a site-specific area.

           •   When enough (e.g., > 8 to 10) detected site observations are available, it is preferred to
               use hypotheses testing approaches. Specifically, single sample hypotheses testing
               (comparing site to a specified threshold) approaches should be used to perform site
               versus a known threshold comparison and two-sample hypotheses testing (provided
               enough background data are also available) approaches should be used to perform site
               versus background comparison.

1.7.2  Minimum Sample Size Requirements for Hypothesis Testing Approaches

Statistical methods (as in ProUCL 4.0) used to estimate EPC terms, BTVs, PRGs, or to compare the site
contaminant concentration data distribution  with the background data distribution can be computed based
upon small site and background data sets (e.g., of sizes 3, 4, 5, or 6). However, those statistics cannot be
considered representative and reliable enough to make important cleanup and remediation decisions. It is
recommended not to use those statistics to draw cleanup and remediation decisions potentially impacting
the human health and the environment. It is  suggested that the estimation and hypothesis testing methods
as incorporated in ProUCL 4.0 may not be used on background data sets with fewer than 8 to 10 detected
observations.  Also, when using hypotheses testing approaches, it is suggested that the site and
background data be obtained using an appropriate DQO process as described in EPA (2006). In case that
is not possible, it is suggested that the project team at least collect 8 to 10 observations from each of the
populations (e.g., site area, MWs, background area) under investigation.

Site versus background comparisons and computation of the BTVs depend upon many factors, some of
which cannot be controlled. These factors include the site conditions,  lack of historical information, site
medium, lack of adequate resources, measurement and analytical errors, and accessibility of the site areas.
Therefore, whenever possible, it is desirable to use more than one statistical method to perform site versus
background comparison. The use of statistical methods should always be supplemented with appropriate
graphical displays.
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1.8    Sample Sizes for Bootstrap Methods

Several parametric and nonparametric (including bootstrap methods) UCL, UPL, and other limits
computation methods for both full-uncensored data sets (without nondetects) and left-censored data sets
with nondetects are available in ProUCL 4.0. It should be noted that bootstrap resampling methods are
useful when not too few (e.g., < 10-15) and not too many (e.g., > 500-1000) detected observations are
available. For bootstrap methods (e.g., percentile method, BCA bootstrap method, bootstrap t method), a
large number (e.g., 1000, 2000) of bootstrap resamples (with replacement) are drawn from the same data
set. Therefore, in order to obtain bootstrap resamples with some distinct values (so that statistics can be
computed from each resample), it is suggested that a bootstrap method should not be used when dealing
with small data sets of sizes less than 10-15. Also, it is not required to bootstrap a large data set of size
greater than 500 or 1000; that is when a data set of a large size (e.g., > 1000) is available, there is no need
to obtain bootstrap resamples to compute statistics of interest (e.g., UCLs).  One can  simply use a
statistical method on the original large data set. Moreover, bootstrapping a large data set of size greater
than  500 or 1000 will be time consuming.

1.9    Statistical Analyses by a Group ID

The analyses of data categorized by a group ID variable such as: 1) Surface vs. Subsurface;
2) AOC1 vs. AOC2; 3) Site vs. Background; and 4) Upgradient vs Downgradient monitoring wells are
quite common in many environmental applications. ProUCL 4.0 offers this option for data sets with and
without nondetects. The Group Option provides a powerful tool to perform various statistical tests and
methods (including graphical displays) separately for each of the group (samples from different
populations) that may be present in a data set. For an example, the same data set may consist of samples
from the various groups or populations representing site, background, two or more AOCs, surface,
subsurface, monitoring wells. The graphical displays (e.g., box plots, Q-Q plots) and statistics
(computations  of background statistics, UCLs, hypotheses testing approaches) of interest can be
computed separately for each group by using this option.

It should be pointed out that it is the users' responsibility to provide adequate amount of detected data to
perform the group operations. For an example, if the user desires to produce a graphical Q-Q plot (using
only detected data) with regression lines displayed, then there should be at least two  detected points (to
compute slope, intercept, sd) in the data set. Similarly if the graphs are desired for each of the group
specified by the group ID variable, there should be at least two detected observations in each group
specified by the group variable. ProUCL 4.0 generates a warning message (in orange color) in the lower
panel of the ProUCL 4.0 screen. Specifically, the user should make sure that a variable with nondetects
and categorized by a group variable should have enough detected data in each group to perform the
various methods (e.g., GOF tests, Q-Q plots with regression lines) as incorporated in ProUCL 4.0.

1.10  Use of Maximum Detected Values as Estimates of Upper Limits

Some practitioners tend to use the maximum detected value as an estimate of the EPC term. This is
especially true when the sample size is small such as < 5, or when a UCL95 exceeds the maximum
detected values (EPA, 1992b). Also, many times in practice, the BTVs and not-to-exceed values are
estimated by the maximum detected value. This section discusses the appropriateness of using the
maximum detected value as estimates of the EPC term, BTVs, or other nor-to-exceed values.
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1.10.1 Use of Maximum Detected Values to Estimate BTVs and Not-to-Exceed Values

It is noted that BTVs and not-to-exceed values represent upper threshold values in the upper tail of a data
distribution; therefore, depending upon the data distribution and sample size, the BTVs and other not-to-
exceed values may be estimated by the maximum detected value. As described earlier, upper limits, such
as UPLs, UTLs, and upper percentiles, are used to estimate the BTVs and not-to-exceed values. It is noted
that a nonparametric UPL or UTL is often estimated by higher order statistics such as the maximum value
or the second largest value (EPA  1992a, RCRA Guidance Addendum). The use of higher order statistics
to estimate the UTLs depends upon the sample size. For an example:  1) 59 to 92 samples, a
nonparametric UTL95%-95 is given by the maximum detected value; 2) 93 to 123 samples, a
nonparametric UTL95%-95 is given by the second largest maximum detected value; and 3) 124 to 152
samples, a UTL95%-95 is given by the third largest detected value in the sample.

Note: Therefore, when a data set does not follow a discernable distribution, the maximum observed value
(or other high order statistics) may be used as an  estimate ofBTV or a not-to-exceed value, provided the
maximum value does not represent an outlier or a contaminating observation perhaps representing a hot
location.

1.10.2 Use of Maximum Detected Values to Estimate EPC Terms

Some practitioners tend to use the maximum detected value as an estimate of the EPC term. This is
especially true when the sample size is small such as < 5, or when a UCL95 exceeds the maximum
detected values (EPA, 1992b). Specifically, a RAGS document (EPA, 1992) suggests the use of the
maximum detected value as a default value to estimate the EPC term when a 95% UCL (e.g., the H-UCL)
exceeded the maximum value.  ProUCL 4.0 can compute a 95% UCL of mean using several methods
based upon normal, Gamma, lognormal, and non-discernable distributions. In past (e.g., EPA, 1992b),
only two methods were used to estimate the EPC term based upon: 1) Student's t-statistic and a normal
distribution, and 2) Land's H-statistic (1975) and a lognormal model. The use of H-statistic often yields
unstable and unpractically large UCL95 of the mean (Singh, Singh, and laci, 2002). For skewed data sets
of smaller sizes (e.g., < 30, < 50), H-UCL often exceeds the maximum detected value. This is especially
true when extreme high outliers may be present in the data set. Since the use of a lognormal distribution
has been quite common (e.g., suggested as a default model in a RAGS document (EPA, 1992)), the
exceedance of the maximum detected value by H-UCL95 is frequent for many skewed data sets of
smaller sizes (e.g., < 30, < 50).  It is also be noted that for highly skewed data sets, the sample mean
indeed can even exceed the upper 90%, 95%, etc., percentiles, and consequently, a 95% UCL of mean can
exceed the maximum observed value of a data set.

All of these occurrences result in the possibility of using the maximum detected value as an estimate of
the EPC term. It should be pointed out that in some cases, the maximum observed value actually might
represent a highly polluted outlying observation. Obviously, it is not desirable to use a highly polluted
value as an estimate of average exposure (EPC term) for an exposure area. This is especially true when
one is dealing with lognormally distributed data sets of small sizes. As mentioned before, for such highly
skewed data sets that cannot be modeled by a gamma distribution, a 95% UCL of the mean should be
computed using an appropriate  distribution-free nonparametric method.

It should be pointed out that the EPC term represents the average exposure contracted by an individual
over an exposure area (EA) during a long  period of time; therefore, the EPC term should be estimated by
using an average value (such as an appropriate 95% UCL of the mean) and not by the maximum observed
concentration. One needs to compute an average exposure and not the maximum exposure. It is unlikely
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that an individual will visit the location (e.g., in an EA) of the maximum detected value all of the time.
One can argue that the use of this practice results in a conservative (higher) estimate of the EPC term. The
objective is to compute an accurate estimate of the EPC term. Several other methods (instead of H-UCL)
as described in EPA (2002), and included in ProUCL 4.0 (EPA 2007), are available to estimate the EPC
terms. It is unlikely (but possible with outliers) that the UCLs based upon those methods will exceed the
maximum detected value, unless some outliers are present in the data set. ProUCL 4.0 displays a warning
message when the recommended 95% UCL (e.g.,  Hall's or bootstrap t UCL with outliers) of the mean
exceeds the observed maximum concentration. When a 95% UCL does exceed the maximum observed
value, ProUCIAO recommends the use of an alternative UCL computation method based upon the
Chebyshev inequality. The detailed recommendations (as  functions of sample size and skewness) for the
use of those UCLs are summarized in ProUCL 3.0 User Guide (EPA, 2004).

Singh and Singh (2003) studied the performance of the max test (using the maximum observed value as
an estimate of the EPC term) via Monte Carlo simulation experiments. They noted that for skewed data
sets of small sizes (e.g., < 10-20), the max test does not provide the specified 95% coverage to the
population mean, and for larger data sets, it overestimates the EPC term, which may require unnecessary
further remediation. The use of the maximum value as an  estimate of the EPC term also ignores most
(except for maximum value) of the information contained in the data set. With the availability of so many
UCL computation methods (15 of them), the developers of ProUCL 4.0 do not recommend using the
maximum observed value as an estimate of the EPC term representing an average exposure by an
individual over an EA. Also, for the distributions considered, the maximum value is not a sufficient
statistic for the unknown population mean.

Note: It is recommended that the maximum observed value NOT be used as an estimate of the EPC term
representing average exposure contracted by an individual over an EA. For the sake of interested users,
ProUCL displays a warning message when the recommended95% UCL (e.g., Hall's bootstrap UCL etc.)
of the mean exceeds the observed maximum concentration. For such scenarios (when a 95% UCL does
exceed the maximum observed value), an alternative 95% UCL computation method is recommended by
ProUCL 4.0.

1.10.3 Samples with Nondetect Observations

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

1.10.4 Avoid the Use of DL/2 Method to Compute UCL95

Based upon the results of the report by Singh, Maichle, and Lee (EPA, 2006), it is strongly recommended
to avoid the use of the DL/2 method to perform GOF  test, and to compute  the summary statistics and
various other limits  (e.g., UCL, UPL) often used to estimate the EPC terms and BTVs. Until recently, the
DL/2 method has been the most commonly used method to compute the various statistics of interest for
data sets with BDL observations. The main reason of its common  use has been the lack of the availability
of other defensible methods and associated programs  that  can be used to estimate the various
                                                                                          29

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environmental parameters of interest. Today, several other methods (e.g., KM method, bootstrap
methods) with better performances are available that can be used to compute the various upper limits of
interest. Some of those parametric and nonparametric methods are available in ProUCL 4.0. Even though
the DL/2 method (to compute  UCLs, UPLs, and for goodness-of-fit test) has also been incorporated in
ProUCL 4.0, its use is not recommended due to its poor performance. The DL/2 method is included in
ProUCL 4.0 only for historical reasons as it had been the most commonly used and recommended method
until recently  (EPA, 2006).  Some of the  reviewers of ProUCL 4.0 suggested and requested the inclusion
of DL/2 method in ProUCL for comparison purposes.

Note: The DL/2 method has been incorporated in ProUCL 4.0 for historical reasons only. NERL-EPA,
Las Vegas strongly recommends avoiding the use of DL/2 method even when the percentage (%) ofNDs
is as low as 5%-10%. There are other methods available in ProUCL 4.0 that should be used to compute
the various summary statistics and upper limits based upon data sets with multiple detection limits.

1.10.5  Samples with Low Frequency of Detection

When all of the sampled data values are  reported as nondetects, the EPC term should also be reported as a
nondetect value, perhaps by the maximum reporting limit (RL) or maximum RL/2. Statistics (e.g.,
UCL95) computed based upon only a few detected values (e.g., < 4 to 6)  cannot be considered reliable
enough to estimate the EPC terms having potential impact on the human heath and the environment.
When the number of detected data is small, it is preferable to use simple ad hoc methods rather than using
statistical methods to compute the EPC terms and other upper limits. Specifically, it is suggested that in
cases when the detection frequency is low  (e.g., < 4%-5%) and the number of detected observations is
low, the project team and the decision makers together should make a decision on site-specific basis on
how to estimate the average exposure (EPC term) for the contaminant and area under consideration. For
such data sets with low detection frequencies, other measures  such as the median or mode represent better
estimates (with lesser uncertainty) of the population measure of central tendency.

Additionally,  it is also suggested that when most (e.g., > %95) of the observations for a contaminant lie
below the detection limit(s) or reporting limits  (RLs), the sample median  or the sample mode (rather than
the sample average which cannot be computed  accurately) may be used as an estimate the EPC term. Note
that when the  majority of the data are nondetects, the median and the mode will also be a nondetect. The
uncertainty associated with such estimates will be high. It is noted that the statistical properties, such as
the bias, accuracy, and precision of such estimates, would remain unknown. In order to be able to
compute defensible estimates, it is always desirable to collect more samples.

Note: In case  the number of available detected samples is small (< 5), it is suggested that the project
team decide about the estimation of the EPC term on site-specific basis. For such small data sets with
very few detected values (< 5), the final  decision  ("policy decision ") on how to estimate the EPC term
should be determined by the project team and decision makers.

1.11   Other Applications of Methods  in ProUCL 4.0

In addition to  performing background versus site  comparisons for CERCLA and RCRA sites,  and
estimating the EPC terms in exposure and risk  evaluation studies, the statistical methods as incorporated
in ProUCL 4.0 can be used  to address other issues dealing with environmental investigations that are
conducted at Superfund or RCRA sites.
30

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1.11.1 Identification of COPCs

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

In order to identify the COPCs, point-by-point site observations (preferably composite samples) are
compared with pre-established screening levels, SSL, or estimated BTVs. This is especially true when the
comparisons of site concentrations with screening levels or BTVs are conducted in real time by the
sampling or cleanup crew right there in the site field. The project team should decide about the type of
site samples (discrete or composite) and the number of detected site observations (not more than 4  to 6)
that should be collected and compared with the screening levels or the BTVs. In case BTVs, screening
levels, or not-to-exceed values are not known, the availability of a defensible background or reference
data set of reasonable size (e.g., > 8 to 10, more are preferable) is required to obtain reliable estimates of
BTVs and screening levels. When a reasonable number of detected site observations are available,  it is
preferable to use hypotheses testing approaches. The contaminants with concentrations exceeding the
respective screening values or BTVs may be considered as COPCs, whereas contaminants with
concentrations (in all  collected samples) lower than the screening value, PRG, or an estimated BTV may
be omitted from all future evaluations including the risk assessment and risk management investigations.

1.11.2 Identification of Non-Compliance Monitoring Wells

In monitoring well  (MW) compliance assessment applications, individual (often discrete) contaminant
concentrations from a MW are compared with pre-established ACL, MCL, or an estimated compliance
limit (CL) based upon a group  of upgradient wells representing the background population. An
exceedance of the MCL or the  BTV by a MW concentration may be considered as an indication of
contamination in that MW. In such individual concentration comparisons, the presence of contamination
(determined by an exceedance) may have to be confirmed by re-sampling from that MW. If
concentrations of contaminants in both the original sample and the re-sample(s) exceed the MCL or BTV,
then that MW may  require closer scrutiny, perhaps triggering the remediation remedies as determined by
the project team. If the concentration data from a MW for about 4 to 5 continuous quarters (or another
designated time period determined by the project team) are below the MCL or BTV level, then that MW
may be considered  as complying with (achieving) the pre-established or estimated standards. Statistical
methods as described in Chapters 5 and 6 of the revised background guidance document (EPA, 2002b)
can be used to estimate the not-to-exceed values or BTVs based upon background or upgradient wells in
case the ACLs or MCLs are not pre-determined.

1.11.3 Verification of the Attainment of Cleanup Standards, Cs

Hypothesis testing approaches  may be used to verify the attainment of the cleanup standard, Cs, at
polluted site areas of concern after conducting remediation and cleanup at the site AOC (EPA, 2006). In
order to properly address this scenario, a site data set of adequate size (minimum of 8 to 10 detected site
observations) needs to be made available from the remediated or excavated areas of the site under
investigation. The sample size  should also account for the size of the remediated site area; meaning that

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larger site areas should be sampled more (with more observations) to obtain a representative sample of the
site under investigation.

Typically, the null hypothesis of interest is H0: Site Mean, |o,s>=Cs versus the alternative hypothesis, HI:
Site Mean, |o,s < Cs> where the cleanup standard, Cs, is known a priori. The sample size needed to perform
such single sample hypotheses tests can be obtained using the DQO process-based sample size formula as
given in the EPA (2006) documents. In any case, in order to use this test, a minimum of 8 to 10 detected
site samples should be collected. The details of statistical methods used to perform single sample
hypothesis as described above can be found in EPA (2006).

1.11.4  Using BTVs (Upper Limits) to Identify Hot Spots

The use of upper limits (e.g., UTLs) to identify hot spot(s) has also been mentioned in the Guidance for
Comparing Background and Chemical Concentrations in Soil for CERCLA Sites (EPA, 2002b). Point-by-
point site observations (preferably using composite samples representing a site location) are compared
with a pre-established or estimated BTV. Exceedances of the BTV by site observations may be
considered as representing locations with elevated concentrations (hot spots). Chapters 5 and 6 of the
revised background guidance document (EPA, 2002b) describe several methods to estimate the BTVs
based upon full data sets without nondetects and left-censored data sets with nondetect observations.

The rest of the chapters of this User Guide illustrate the use of the various procedures as incorporated in
ProUCL 4.0. Those methods are useful  to analyze environmental data sets with and without the nondetect
observations. It is noted that ProUCL 4.0 is the first software package equipped with  single sample and
two-sample hypotheses testing approaches that can be used on data sets with nondetect observations.
32

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                                      Chapter 2

                      Entering and Manipulating Data
2.1    Creating a New Data Set

By simply executing the ProUCL 4.0, a new worksheet is generated and displayed for the user to enter
data.
To create a new worksheet: click Js-i or choose File ^- New

2.2    Opening an Existing Data Set

If your data sets are stored in the ProUCL data format (* .wst), then click  x   or choose File ^ Open

           •   If your data sets are stored in the Microsoft Excel format (* .xls), then choose File
              Other Files... ^Import Excel...  OR File ^ Load Excel Data
               Edit Configure Summary Statistics R05 Est NDs Graphs  Outiier Tests Goodness -of -Fit Hypothesis Testng Background  UCL Window Help
Ni Load Excel Data < (I 0123456789;

Close Export Excel...
Save |
Save As ... i
Print i
Print Preview i

Exit I
2
3
4
5
6
7
              Possible Error Messages:

              o  When you import an Excel file, make sure that you have an empty worksheet. If there
                 is no empty worksheet, then you must create a new worksheet before importing an
                 Excel file. Otherwise, there will be an error message in the Log Panel: "[Error]
                 Worksheet must be empty."
              o  First open a new worksheet and then import the Excel file.
              o  Make sure that the file you trying to import is not currently open. Otherwise, there
                 will be the following warning message in the Log panel:
              o  "[Information] Unable to open C:\***.xls." Check the validity of this file.
                                                                                         33

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2.3    Input File Format
               Note that ProUCL 4.0 does not require that data in each column must end with a nonzero
               value (ProUCL 3.0 requires this). Therefore, all zero values (in the beginning, middle, or
               end of data columns) are treated as valid zero values as part of the data set.

               The program can read Excel files. The user can perform typical Cut, Paste, and Copy
               operations.

               The first row in all input data files consist of alphanumeric (strings of numbers and
               characters) names representing the header row. Those header names may represent
               meaningful variable names such as Arsenic, Chromium, Lead, Group-ID, and so on.

               o   The Group-ID column has the labels for the groups (e.g., Background, AOC1,
                   AOC2, 1, 2, 3, a, b, c, Site 1, Site 2, and so on) that might be present in the data set.
                   The alphanumeric strings (e.g., Surface, Sub-surface) can be used to label the various
                   groups.
               o   The data file can have multiple variables (columns) with unequal number of
                   observations.
               o   Except for the header row  and columns representing the group labels, only numerical
                   values should appear in all other columns.
               o   All alphanumeric strings and characters (e.g., blank, other characters, and strings),
                   and all other values (that do not meet the requirements above) in the data file are
                   treated as missing values.
               o   Also, a large value denoted by 1E31 (=  IxlO31) can be used to represent missing data
                   values. All entries with this value are ignored from the computations. These values
                   are counted under number of missing values.
2.4   Number Precision
You may turn "Full Precision" on or off by choosing Configure
On/OFF
                                                                            Full Precision
                    r pVorkS.hert.wstI.
            [J File Edit Kll Summary Statistics RQS Est. NDs  Graphs Outlier Tests  Goodness-of-Fit Hypothesis Testing  Background IJCL Window Help
                       °n '
           •   By leaving "Full Precision" turned off, ProUCL will display numerical values using an
               appropriate (default) decimal digit option. However, by turning "Full Precision" off, all
               decimal values will be rounded to the nearest thousandths place.

           •   "Full Precision" on option is specifically useful when one is dealing with data sets
               consisting of small numerical values (e.g., < 1) resulting in small values of the various
               estimates and test statistics. These values may become so small with several leading zeros
34

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              (e.g., 0.00007332) after the decimal. In such situations, on may want to use "Full
              Precision" option to see nonzero values after the decimal.

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

2.5    Entering and Changing  a  Header Name

1.      Highlight the column whose header name (variable name) you want to change by clicking either
       the column number or the header as shown below.
2.
3.
Right-click and then clicl
0 1 2
Arsenic
1 I 4.5|
2 5.6
3 4.3
4 5.4
5 9.2
c "Header Name"
0 1 2
^^^^^^^^^^^^^^^^^^^^^^E2^l Header Name l^^^H

1 ! 4-b!
2 5.6
3 4.3
4 5.4
5 9.2
Change the Header Name.
^^^^^^^^^^^^1 HI

^^^^^^^^^^^^^^H
Header Name: Arsenic Site 1
OK Cancel

4.      Click the OK button to get the following output with the changed variable name.
                                                                                        35

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                                            0
                                       Arsenic Site 1
1 i
2
3
4
5
4.5]
5.6
4.3
5.4
9.2
2.6    Saving Files
                                        9 Edit  Configure
                                         New
                                         Open ,,,
                                         Load Excel Data
                                         Other Files ...    >
                                         Close
                                         Save As ,,,

                                         Print
                                         Print Preview

                                         Exit
            •   Save option allows the user to save the active window.

            •   Save As option allows the user to save the active window. This option follows typical
               Windows standards, and saves the active window to a file in Excel 95 (or higher) format.
               All modified/edited data files, and output screens (excluding graphical displays)
               generated by the software can be saved as Excel 95 (or higher) spreadsheet.

2.7    Editing

Click on the Edit menu item to reveal the following drop-down options.
     File |gw Configure Summary Statistics  ROS Est. NDs  Graphs Outlier Tests  Goodness-of-Fit Hypothesis Testing Background UCL  Window  Help

     I C
          Copy Ctri+C    L
   Na-.igal  Paste ctf+v    jl
0
       1
The following Edit drop-down menu options are available:

            •   Cut option: similar to a standard Windows Edit option, such as in Excel. It performs
               standard edit functions on selected highlighted data (similar to a buffer).

            •   Copy option: similar to a standard Windows Edit option, such as in Excel. It performs
36

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               typical edit functions on selected highlighted data (similar to a buffer).

           •   Paste option: similar to a standard Windows Edit option, such as in Excel. It performs
               typical edit functions of pasting the selected (highlighted) data to the designated
               spreadsheet cells or area.

           •   It should be noted that the Edit option could also be used to Copy Graphs. This topic
               (copying and pasting graphs) is illustrated in detail in Chapter 13.

2.8    Handling Nondetect Observations

           •   ProUCL 4.0 can handle data sets with single and multiple detection limits.

           •   For a variable with nondetect observations (e.g., arsenic), the detected values, and the
               numerical values of the associated detection limits (for less than values) are entered in the
               appropriate column associated with that variable.

           •   Specifically, the data for variables with nondetect values are provided in two columns.
               One column consists of the detected numerical values with less than (< DLj) values
               entered as the corresponding detection limits (or reporting limits), and the second column
               represents their detection status consisting of only 0 (for less than values) and 1 (for
               detected values) values. The name of the corresponding variable representing the
               detection status should start with d_,  or D_ (not case sensitive) and the variable name.
               The detection status column with variable name starting with a D_ (or a d_) should have
               only two values: 0 for nondetect values, and 1 for detected observations.

           •   For an example, the header name, D_Arsenic is used for the variable, Arsenic having
               nondetect observations. The variable D_Arsenic contains a 1 if the corresponding Arsenic
               value represents a detected entry, and contains a 0 if the corresponding entry for variable,
               Arsenic, represents a nondetect. An example data set illustrating these points is given as
               follows.
                                                                                              37

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^taampte.wst


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
jj]
0 1
Arsenic D_Arsenic
4.5 0
5.6 1
4.3 0
5.4 1
9.2 1
6.2 1
6.7 1
5.8 1
8.5 1
5.65 1
5.4 1
5.5 1
5.9 1
5.1 1
5.2 1
4.5 0
6.1 1
6.1 1
6.8 1
5 1





2
Mercury
0.07
0.07
0.11
0.2
0.61
0.12
0.04
0.06
0.88
0.125
0.18
0.21
0.29
0.44
0.12
0.055
0.055
0.21
0.67
0.1
0.8
0.26
0.97
0.05
0.26
3 4
D_Mercury Vanadium
1 16.4
1 16.8
0 17.2
0 18.4
1 15.3
1 30.8
1 29.4
1 13.8
1 18.8
1 17.25
1 17.2
1 16.3
1 16.8
1 17.1
1 10.3
1 15.1
1 24.3
1 18
1 16.8
1 12
1
1
1
1
1
5 6
Zinc Group — ^
89.3 Surface
90.7 Surface
95.5 Surface
113 Surface
266 Surface
80.9 Surface
80.4 Surface
89.2 Surface
182 Surface
80.4 Surface
91.9 Subsurface
112 Subsurface
172 Subsurface
88 Subsurface
80.7 Subsurface
66.3 Subsurface
75 Subsurface
185 Subsurface
184 Subsurface
68.4 Subsurface




^r1
2.9    Caution
           •   Care should be taken to avoid any misrepresentation of detected and nondetected values.
               Specifically, it is advised not to have any missing values (blanks, characters) in the
               D_column (detection status column). If a missing value is located in the D_ column (and
               not in the associated variable column), the corresponding value in the variable column is
               treated as a nondetect, even if this might not have been the intention of the user.

           •   It is mandatory that the user makes sure that only a 1 or a 0 are entered in the detection
               status D_column. If a value other than a 0 or a 1 is entered in the D_ column (the
               detection column), results may become unreliable, as the software defaults to any number
               other than 0 or 1  as a nondetect value.

           •   When computing statistics for full data sets without any nondetect values, the user should
               select only those  variables (from the list of available variables) that contain no nondetect
               observations. Specifically, nondetect values found in a column chosen for the summary
               statistics (full-uncensored data set) will be treated as a detected value; whatever value
               (e.g., detection limit) is entered in that column will be used to compute summary
               statistics for a full-uncensored data set without any nondetect values.

           •   Two-Sample Hypotheses: It should be noted that at present, when using two-sample
               hypotheses approaches (WMW test, Gehan test, and quantile test) on data sets with NDs,
38

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              both samples or variables (e.g., site-As, Back-As) should be specified as having
              nondetects, even though one of the variables may not have any ND observations. This
              means, a ND column (with 0 = ND, and 1 = detect) should be provided for each variable
              (here D_site-As, and D_Back-As) to be used in this comparison. If a variable (e.g., site-
              As) does not have any nondetects, still a column with label D_site-As should be included
              in the data set with all entries = 1 (detected values).

           •  The following sample (not from a Superfund site) data set given on the next page
              illustrates points related to this option and issues listed above. The data set considered
              contains some nondetect measurements for Arsenic and Mercury. It should also be noted
              that the Mercury concentrations are used to illustrate the points related to nondetect
              observations only. Arsenic and Zinc concentrations are used to illustrate the use of the
              group variable, Group (Surface, Subsurface).

           •  If for mercury, one computes summary statistics (assuming no nondetect values) using
              "Full" data set option, then all nondetect values (with "0" entries in D_Mercury column)
              will be treated as detected values, and summary statistics will be computed accordingly.

2.10  Summary Statistics for Data Sets with Nondetect Observations

           •  In order to compute simple summary statistics or to compute other statistics of interest
              (e.g., background statistics, GOF test, UCLs, outliers) for variables with nondetect
              values, one should choose the nondetect option, "With NDs" from the various available
              menu options such as  Outliers, Background Statistics, UCLs, Goodness-of-Fit test, Q-Q
              plot, Box Plot.

           •  It should be noted that "summary" statistics for a data set with nondetect observations
              represent (at least in ProUCL 4.0) simple summary statistics based upon the data set
              without using nondetect observations. All other parametric and nonparametric statistics
              and estimates of population mean, variance, percentiles (e.g., MLEs, KM, and ROS
              estimates) for variables with nondetect observations are given in other menu options such
              as background statistics and UCL. The simple "Summary Statistics/With NDs" option
              only provides simple statistics (e.g., % NDs, max ND, Min ND, and Mean of detected
              values) based upon detected values. These statistics (e.g., sd of log -transformed detected
              values) may help a user to determine the degree of skewness (e.g., mild, moderate, high)
              of the data set consisting of detected values. These statistics may also help the user to
              choose the most appropriate method (e.g., KM (BCA) UCL or KM (t) UCL) to compute
              UCLs, UPLs, and other limits.

           •  As mentioned before,  various  other statistics and estimates of interest (e.g., mean, sd,
              UCLs, UTLs,  MLEs, and KM estimates) for data sets with nondetect observations are
              computed in other the menu options (UCLs, Outliers, Background Statistics, GOF tests)
              available in ProUCL 4.0.
                                                                                            39

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2.11   Warning Messages and  Recommendations for Datasets with
        Insufficient Amount of Data
             •    ProUCL 4.0 provides some warning messages and recommendations for datasets with
                 insufficient amount of data to calculate meaningful estimates and statistics of interest. For
                 an example, it is not desirable to compute an estimate the EPC term based upon a data set
                 of size less than 5, especially when nondetects may be present in the data set. In such
                 cases, it is suggested to use site specific values (perhaps determined by the Project Team)
                 to estimate environmental parameters (e.g., EPC term, not-to-exceed background value)
                 of interest.

             •    Some examples of datasets with insufficient amount of data include datasets with less
                 than 4 distinct observations, datasets with only one detected observation, and datasets
                 consisting of all nondetects.

             •    Some of the warning messages given by ProUCL 4.0 are shown below.
                                  G eneral B ackground S tattslics f or D ala S ets wih NorrOetects
                    U ser S elected 0 ptions
                            From File C:\Doeuments and Settings\narmbya\Desktop\ProUCL-Test\EVlLDATA3.wst
                          Full Precision OFF
                     Confidence Coefficient S5%
                           Coverage 90%
                  Different or Future K Values 1
                Number of Bootstrap Operations 2000
              Var
                                                 General Statistics
                               Total Number of Observations  15

                              Raw Statistics
                                          Minimum  1
                                          Maximum  3
                                       Second Largest  3
                                         First Quartile  1
                                           Median  2
                                        Third Quartile  3
                                            Mean  2
                                             SD  0.845
                                   Coefficient of Variation  0.423
                                          Skewness  0
                                      Warning: T here are ernl^ 3 D is&inct Values in this 
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                             ,j -^ ene(a| u CL S tat jst jcs f OI p ata s ets with N onfietecte
          User Selected Option:
                     From File  CADocurnents and Seltings\narmbya\DesklopSProUCL-Test\EVILDATA3.wst
                  Full Precision  OFF
           Confidence Coefficient  35%
   Number of Bootstrap Operations  2000
 Var_x3
                                 Number of Valid Data
                       Number of Distinct Detected Data
                                                      General Statistics
                                                         15
  Number of Detected Data             1
Number of Non-Detect Data             14
     Percent Non-Detects         93.33K
         Warning: Onif orse distinct data value ¥ia$ detected! ProOCL for arej? othef software) shoyld not be used on such a date se$
  It is syggiested to use alternative site specific values determined by the Project Team to estimate environmental paraineters fe.@., EPC, BTV3-

                                       The data set fof variable Var_K3 was nol processed!
                              ! General Background Statistics for Data Sets with Non-Detects
          User Selected Options
                      From File  C: SD ocuments and S ettings\narmbya\D esktopVPtoU CL-T est\EVI LD AT A3. wst
                  Full Precision  OFF
          Confidence Coefficient  95%
                     Coverage  30%
      Different or Future K Values  1
   Number of Bootstrap Operations  2000
Vaf_x10
                                                      General Statistics
                                  Number of Valid Data    15
                        NumberofDistinctDetectedData    0
      NurnberofDetectedData     0
     NumbetofNon-DetectData    15
             Warning: AH observations are Non-Detects (MDs], therefore all statistics ar®d esteates shoukl also bs Wsi
           S pacifically, sample mean, U CLs, U PLs, and otherstatisticsaiealsoNDs lying below the largest detedkn EmK
      T he Proiecl T earn may decide lo use alternative site specific values to estimate environmental parameters (e.g., EPC, BTVT-

                                      T he data set for variable Var_s1Swasnol processed!
                                                                                                                                     41

-------
                                   Number of Valid Data
                           Number of Distinct Detected Data
                           Raw Statistics
                                     Minimum Detected
                                     Maximum Detected
                                     Mean of Detected
                                       3D of Detected
                                   Minimum Non-Detect
                                   Maximum Non-Detect
General Statistics
     15
     2
     1
     4
    2.5
  1643
     2
     5
         Note: Data have multiple OLs • Use of KM Method is recommended
         For all methods (except KM, DL/2, and ROS Methods),
         Observations < Largest ND are treated asNDs
         NumberofDetectedData       6
       NumberofNon-DetectData       9
            Percent Non-D elects   60.00%

Log-transformed Statistics
             Minimum Detected       0
             Maximum Detected    1.386
             Mean of Detected    0.693
               SD of Detected    0.759
            M inimum N on-D elect    0.693
            M aximum N on-D elect    1.609

      Number treated as Non-Detect       15
       Number treated as Detected       0
   SingleDLNon-DetectPercentage   100.00%
                                     Warning: Data set lias orii^ 2 District Detested Vakses.
                      This niaf not be adequate enough lo compute meaningly) and reliable lest statistics ami estimates.
             T he Project T earn may decide io use alternative site specific values to estimate environmental parameters fag., EPC, BTV}

                    U nless D ata Q uality 0 bjectives (D Q 0 s) have been met, it is suggested to collect additional observations.

                 The number of detected data may not be adequate enough to perfoim GOF tests, bootstrap, and ROS methods.
                                   Those methods will letuin a WA' value onyour output display

                               It is necessary to have 4 or more Distinct Values for bootstrap methods.
                   It is recommended to have 10 to 15 or more observations for accurate and meaningful results and eslroates
2.12   Handling Missing  Values
             •   ProUCL 4.0 can  handle missing values within a data set.

             •   All blanks, alphanumeric strings (except for group variables), or the specific large
                 number value Ie31 are considered as missing values.
             •   A group variable (representing two or more groups, populations, AOCs, MWs) can have
                 alphanumeric values (e.g., MW1, MW2,...).
             •   ProUCL 4.0 ignores all missing values in all mathematical operations it performs.
                 Missing values are therefore not treated as being part of a data set.
             •   Number of Valid Samples or Number of Valid Observations represents the Total Number
                 of Observations less the Number of Missing Values. If there are missing values, then
                 number of valid samples = total number of observations.
                      Valid  Samples = Total Number of Observations - Missing Values.
42

-------
           •   It is important to note, however, that if a missing value not meant (e.g., a blank, or Ie31)
               to represent a group category is present in a "Group" variable, ProUCL 4.0 will treat that
               blank value (or Ie31 value) as a new group. Any variable that corresponds to this missing
               value will be treated as part of a new group and not with any existing groups. It is
               therefore very important to check the consistency and validity of all data sets before
               performing complex mathematical operations.

           •   ProUCL 4.0 prints out the number of missing values (if any) associated with each
               variable in the data sheet. This information is provided in several output sheets (e.g.,
               summary statistics, background statistics, UCLs) generated by ProUCL 4.0.

For further clarification of labeling of missing values, the following example illustrates the terminology
used for the number of valid samples, number of unique and distinct samples on the various output sheets
generated by ProUCL 4.0.

Example: The following example illustrates the notion of Valid Samples, Unique or Distinct Samples,
and Missing Values. The data set also has nondetect values. ProUCL 4.0 computes these numbers and
prints them on the UCLs and background statistics output.
X
2
4
2.3
1.2
w34
l.OE+031

anm
34
23
0.5
0.5
2.3
2.3
2.3
34
73
D x
1
1
1
0
0
0
0
0
1
1
0
0





Valid Samples: Represents the total number of observations (censored and uncensored) excluding the
missing values. If a data set has no missing value, then the total number of data points equals number of
valid samples.

Missing Values: All values not representing a real numerical number are considered as missing values.
Specifically, all alphanumeric values including blanks are considered as missing values. Also unrealistic
big numbers such as 1 .Oe31 are also considered as missing values and are considered as not valid
observations.

Unique or Distinct Samples: The number of unique samples or number of distinct samples represents all
unique (or distinct) detected values. Number of unique or distinct values is computed for detected values
                                                                                             43

-------
only. This number is especially useful when using bootstrap methods. As well known, it is not desirable
and advisable to use bootstrap methods, when the number of unique samples is less than 4-5.

2.13  User Graphic Display Modification

Advanced users are provided two sets of tools to modify graphics displays. A graphics tool bar is
available above the graphics display and the user can right-click on the desired object within the graphics
display, and a drop-down menu will appear. The user can select an item from the drop-down menu list by
clicking on that item. This will allow the user to make desired modifications as available for the  selected
menu item. An illustration is given as follows.

2.13.1 Graphics Tool Bar
                    BUHisto.Group.gst
                          Histograms for Arsenic, NROS_Arsenic
                            Arsenic
                                                   NROS  Arsenic
The user can change fonts, font sizes, vertical and horizontal axis's, select new colors for the various
features and text. All these actions are generally used to modify the appearance of the graphic display.
The user is cautioned that these tools can be unforgiving and may put the user in a situation where the
user cannot go back to the original display. Users are on their own in exploring the robustness of these
tools. Therefore, less experienced users may want to stay away from using these  drop-down menu graphic
tools.

2.13.2  Drop-Down Menu Graphics Tools

These tools allow the user to move the mouse to a specific graphic  item like an Axis Label or a display
feature.  The user then right-clicks their mouse and a drop-down menu will appear. This menu presents the
user with available options for that particular control or graphic object. There is less of chance of making
an unrecoverable error but that risk is always present. As a cautionary note, the user can always delete the
graphics window and redraw the graphical displays by repeating their operations from the datasheet and
menu options available in ProUCL 4.0. An example of a drop-down menu obtained by right-clicking the
mouse on the background area of the graphics display is given as follows.
44

-------
    Histograms for Arsenic, NROS Arsenic
                                       Gallery
                                       Color
                                       Properties...
                                    *  Statistical Studies
NROS Arsenic  1   1  1
                                                    45

-------
                                       Chapter 3

                            Select Variables Screen

3.1    Select Variables Screen

           •   Variables need to be selected to perform statistical analyses.

           •   When the user clicks on a drop-down menu for a statistical procedure, the following
              window will appear.
            Select Variables
             Variables
                 Selected
              Name
                          ID
                                  Count
              Mercury
              Vanadium
              Zinc
              Group
30
20
20
20
                                                    Name
                                                                ID
                                                                         Count
                                                    Group by variable:
                                                       OK
                                                                  Cancel
              The Options button is available in certain menus. The use of this option leads to a
              different pop-up window.

              Multiple variables can be processed simultaneously in ProUCL 4.0. Note that this option
              was not available in ProUCL 3.0. ProUCL 4.0 can generate graphs, compute UCLs, and
              background statistics simultaneously for all selected variables.

              Moreover, if the user wants to perform statistical analysis on a variable (e.g.,
              contaminant) by a Group variable, click the arrow below the Group by variable to get a
              drop-down list of available variables to select an appropriate group variable. For an
              example, a group variable (e.g., Site Area) can have alphanumeric values such as AOC1,
46

-------
               AOC2, AOC3, and Background. Thus in this example, the group variable name, Site
               Area, takes four values such as AOC1, AOC2, AOC3, and Background.

           •   The Group variable is particularly useful when data from two or more samples need to be
               compared.

           •   Any variable can be a group variable. However, for meaningful results, only a variable,
               that really represents a group variable (categories) should be selected as a group variable.

           •   The number of observations in the group variable and the number observations in the
               selected variables (to be used in a statistical procedure) should be the same. In the
               example below, the variable "Mercury" is not selected because the number of
               observations for Mercury is 30; in other words mercury values have not been grouped.
               The group variable and each of the selected variables have 20 data values.
            Select Variables
              Variables
               Name
                           ID
                                    Count
              Mercury
              Group
30
20
                  Selected
                                                       Name
                                                                   ID
Arsenic
Vanadium
Zinc
                                                      Group by variable:
                                                      Arsenic ( Count = 20 )
                                                       vlercury ( Count = 30 )
                                                      Vanadium  ( Count = 20
                                                      Zinc (Count = 20
                                                                            Count
20
20
20
           •   It is recommended not to assign any missing value such as a "Blank" for the group
               variable. If there is a missing value (represented by blanks, strings or 1E31) for a group
               variable, ProUCL 4.0 will treat those missing values as a new group. As such, data values
               corresponding to the missing Group will be assigned to a new group.

Caution: Once again, care should  be taken to avoid misrepresentation and improper use of group
variables. It is recommended not to assign any missing value for the group variable.
                                                                                              47

-------
More on Group Option
           •   The Group Option provides a powerful tool to perform various statistical tests and
               methods (including graphical displays) separately for each of the group (samples from
               different populations) that may be present in a data set. For an example, the same data set
               may consist of samples from the various groups (populations). The graphical displays
               (e.g., box plots, Q-Q plots) and statistics of interest can be computed separately for each
               group by using this option.

           •   In order to use this option, at least one group variable (with alphanumeric values) should
               be included in the data set. The various values of the group variable represent different
               group categories that may be present in the data set. This can be seen in the example.wst
               data set used earlier in Chapter 2.

           •   At this time, the number of values (representing group membership) in a Group variable
               should equal to the number of values in the variable (e.g., Arsenic) of interest that needs
               to be partitioned into various groups (e.g., monitoring wells).

           •   Typically, the  data for the various groups (categorized by the group variable) represent
               data from the various site areas (e.g., background, AOC1, AOC2, ...), or from monitoring
               wells (e.g., MW1, MW2, ...).

3.1.1   Graphs by Groups

           •   Individual or multiple graphs (Q-Q  plots, box plots, and histograms) can be displayed on
               a graph by selecting the "Graphs by Groups" option.

           •   Individual graph for each group (specified by the selected group variable) is produced by
               selecting the "Individual Graph" option.

           •   Multiple graphs (e.g., side-by-side box plots, multiple  Q-Q plots on the same graph) are
               produced by selecting the "Group Graph" option for a variable categorized by a group
               variable. Using this "Group Graph" option, multiple graphs can be displayed for all sub-
               groups included in the Group variable. This option is useful when  data to be compared
               are given in the same column and are  classified by the group variable.

           •   Multiple graphs (e.g., side-by-side box plots, multiple  Q-Q plots) for selected variables
               are produced by selecting the "Group Graph" option. Using the "Group Graph" option,
               multiple graphs can be displayed for all selected  variables. This option is useful when
               data (e.g.,  lead) to be compared are  given in different columns, perhaps representing
               different populations.

Note: It should be noted that it is the users' responsibility to provide adequate amount of detected data to
perform the group operations.  For an example, if the user desires to produce a graphical Q-Q plot (using
only detected data) with regression lines displayed,  then there should be at least two detected points (to
compute slope, intercept, sd) in the data set. Similarly if the graphs are desired for each of the group
specified by the group ID variable, there should be at least two detected observations in each group
specified by the group variable. ProUCL 4.0 generates a warning message (in orange color) in the lower
panel of the ProUCL 4.0 screen. Specifically, the  user should make sure that a variable with nondetects
48

-------
and categorized by a group variable should have enough detected data in each group to perform the
various methods (e.g., GOF tests, Q-Qplots with regression lines) as incorporated in ProUCL 4.0.

As mentioned before, the analyses of data categorized by a group ID variable such as:
1) Surface vs. Subsurface; 2) AOC1 vs. AOC2; 3) Site vs. Background; and 4) Upgradient vs.
Downgradient monitoring wells are quite common in many environmental applications.

The usefulness of the group option is illustrated throughout the User Guide using various methods as
incorporated in ProUCL 4.0.
                                                                                             49

-------
                                        Chapter 4


                                Summary Statistics


This option is used to compute general summary statistics for all variables in the data file. Summary
statistics can be generated for full data sets without nondetect observations, and for data sets with
nondetect observations. Two Menu options: Full and With NDs are available.

           •   Full - This option computes summary statistics for all variables in a data set without any
               nondetect values.

           •   With NDs - This option computes simple summary statistics for all variables in a data set
               that have nondetect (ND) observations. For variables with ND observations, only simple
               summary statistics are computed based upon detected observations only.
               o  For this option, no attempt is made to compute estimates of population parameters
                  (e.g., mean, sd, SE) using parametric (e.g., MLE) or nonparametric (e.g., KM,
                  bootstrap) estimation methods. Those statistics are generated in other estimation
                  modules (e.g., Background and UCL) of ProUCL 4.0.

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

           •   Raw Statistics

           •   Log-Transformed

           •   In ProUCL, log-transformation means natural logarithm (In)

           •   When computing summary statistics for raw data, a message will be displayed for each
               variable that contains non-numeric values.

           •   The Summary Statistics option computes log-transformed data only if all of the data
               values for the selected variable(s) are positive real numbers. A message will be displayed
               if non-numeric characters, zero, or negative values are found in the column
               corresponding to the  selected variable.

4.1     Summary Statistics with Full Data Sets

1.       Click Summary Statistics >• Full
P? Prp,UCL4..q r [Worksheet.wstj
   File Edit  Configure
Summary Statistics
 RQS Est, NDs  Graphs Outiier Tests  Goodness-of-Fit  Hypothesis Testing  Background  UCL Window Help
Ray; Statistics
2.     Select either Log-Transformed or Raw Statistics option.
50

-------
3.     The Select Variables Screen (see Chapter 3) will appear.

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

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

           •   Click on the OK button to continue or on the Cancel button to cancel the Summary
               Statistic option.

Raw Statistics
11 FuLl_Raw_Stats.ost
From File: DAexample.wst
Variable NumObs
Arsenic (subsurface)
Arsenic (surface)
Vanadium (subsurface)
Vanadium (surface)
Zinc (subsurface)
Zinc (surface)
MJ
10
10
10
10
10
10

Minimum
4.5
4.3
10.3
13.8
66.3
80.4


S ummary S tatistics for R aw Full D ata S ets
Maximum
6.8
9.2
24.3
30.8
185
266

Mean
5.56
6.185
16.4
19.53
114.4
116.7

Median
5.45
5.725
16.85
17.23
95.45
90

Variance
0.449
2.5
13.86
33.72
2271
3680

SD MAD/0.67; Skewness
0.67
1.581
3.723
5.807
47.66
60.66

0.667
1.075
1.26
2.669
35.21
13.86

0.344
0.969
0.495
1.46
0.745
2.111

Kurtosis
-0.047
0.295
2.092
0.896
-1.288
4.065

cv
0.121
0.256
0.227
0.297
0.416
0.52
	 >J
Log-Transformed Statistics
EH Full_Log_Stats.ost


From File: D: \exarnple. wst
Variable NumObs
Arsenic [subsurface)
Arsenic (surface)
Vanadium (subsurface)
Vanadium (surface)
Zinc [subsurface)
Zinc (surface)
NJJ
10
10
10
10
10
10

!








*.
	 rd
Summary Statistics for Log-Transformed Full Data Sets
Minimum Maximum
1.504
1.459
2.332
2.625
4.194
4.387

1.917
2.219
3.19
3.428
5.22
5.583

Mean
1.709
1.795
2.774
2.938
4.666
4.673

Median Variance
1.696
1.745
2.824
2.846
4.558
4.5

0.0144
0.0591
0.0538
0.0699
0.159
0.162

SD MAD/O.G7E Skewness
0.12
0.243
0.232
0.264
0.399
0.403

0.123
0.176
0.0735
0.157
0.425
0.163

0.0692
0.527
-0.394
1.194
0.452
1.734

Kurtosis
-0.14
-0.153
1.351
0.504
-1.394
2.184

CV
0.0702
0.135
0.0836
0.09
0.0855
0.0862
_LJ
4.     The resulting Summary  Statistics screen as shown above can be saved as an Excel file. Click
       Save from the file menu.
                                                                                              51

-------
5.      On the output screen, the following summary statistics are displayed for each selected variable in
       the data file.

              NumObs = Number of Observations
              Minimum = Minimum value
              Maximum = Maximum value
              Mean = Sample average value
              Median = Median value
              Variance = Classical sample variance
              SD = Classical sample standard deviation
              MAD = Median absolute deviation
              MAD/0.675 = Robust estimate of variability, population standard deviation, a
              Skewness = Skewness statistic
              Kurtosis = Kurtosis statistic
              CV = Coefficient of Variation

The details of these summary statistics are described in an EPA (2006) guidance document and also in the
     ProUCL Technical Guide (A. Singh and A.K. Singh (EPA, 2007).

4.2    Summary  Statistics with NDs

1.      Click Summary Statistics ^ With NDs
 Navigation Panel
                             ROSEst. NDs  Graphs Outlier Tests  Goodness-of-Fit Hypothesis Testing Background UCL  Window  Help


                            Rav Statistics
 Name
 •_y Worksheet wst

2.      Select either Log-Transformed or Raw Statistics option.

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

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

           •   Only those variables that have nondetect values will be shown.

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

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

Note: It should be noted that in ProUCL 4.0,  "Summary Statistics "for a data set with nondetect
observations represent simple summary statistics based upon the data set without using nondetect
observations. All other parametric and nonparametric statistics and estimates of population mean,
52

-------
variance, percentiles (e.g., MLEs, KM, andROS estimates) for variables with nondetect observations are
given in other estimation menu options such as background statistics and UCL. The simple "Summary
Statistics/With NDs " option only provides simple statistics (e.g., % NDs, max ND, Min ND, Mean of
detected values) based upon detected values. These statistics (e.g., sd of log-transformed detected values)
may help a user to determine the degree ofskewness (e.g., mild,  moderate, high) of the data set consisting
of detected values. These statistics may also help the user to choose the most appropriate method (e.g.,
KM(BCA) UCL or KM ft) UCL) to compute UCLs, UPLs, and other limits.

Raw Statistics - Data Set with NDs
  1WND Raw Stats.ost
 From File: D:\example,wst
                               Summary Statistics for Raw Data Sets with NDs
     Variable      NumObs  NumNDs  2 NDs
           Arsenic   17       3    15.00%
           Mercury   25       5    16.67%
                Raw Statistics using Detected Observations
Maximum Minimum   Mean   Median    SD   MAD/0.67E Skewness   CV
  5       9.2      6.126     5.8     1.15     0.593    1.783    0.188
  0.04     0.99     0.312     0.18     0.315    0.163    1.202    1.01
 liU
Log-Transformed Statistics - Data Set with NDs
 HI WND_Log_Stats.ost

 From File: D:Vexamplawst
                               Summary Statistics for Log-transformed Data Sets with NDs
                                                    Statistics using Detected Log-transformed Observations
      Variable      NumObs  NumNDs  % NDs  Maximum Minimum   Mean   Median    SD   MAD/Q.67J Skewness   CV
           Arsenic   17       3    15.00%      1.609    2.219    1.798    1.758    0.168    1.458    0.0935   0.106
           Mercury   25       5    16.67%    -3.219    -0.0101    -1.66     -1.715    1.032    0.238   -0.622     1.4
 MJ
                The Summary Statistics screen shown above can be saved as an Excel file. Click the save
                from the file menu.

                On the results screen, the following summary statistics are displayed for each selected
                variable from the data file.
                                                                                                        53

-------
              Num Obs = Number of Observations
              NumNDs = Number of Nondetects
              % NDs = Percentage of Nondetect observations
              Minimum = Minimum value
              Maximum = Maximum value
              Mean = Sample average value
              Median = Median value
              SD = Classical sample standard deviation
              MAD = Median Absolute Deviation
              MAD/0.675 = Robust estimate of variability (standard deviation)
              CV = Coefficient of Variation
54

-------
55

-------
                                      Chapter 5


             Estimating  Nondetects Using ROS Methods


Regression on order statistics (ROS) can be used to extrapolate nondetect observations using a normal,
lognormal, or gamma model. ProUCL 4.0 has three ROS estimation methods that can be used to estimate
or extrapolate nondetect observations. The use of this option generates additional columns consisting of
all extrapolated nondetects and detected observations. These columns are appended to the existing open
spreadsheet. The user should save the updated file if they want to use the generated data for their other
application(s).
1.
Click ROS Est. NDs ^ Gamma ROS
P3 ProUCL 4.0 .- [C:\Narain\ProUCL-pata\Data\Oahu.wstl
M*]JBJMSJJ3J Normal ROS j

0 1 Lognormal ROS [ 3 A 5 67 8
Arsenic - L> Arsenic
1 ! 1! °
^^m
Window Help


9 10


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

          •   Select one or more variable(s) from the Select Variables screen.
                 Select Variables
                  Variables
                                          Selected
                   Name
                                   Count
                                                  Name
                                                                  Count
                                                  Arsenic
                                                 Group by Variable

                                                 !	3
                                                    OK
                                                             Cancel
       •  Click on the OK button to continue or on the Cancel button to cancel the option.
56

-------
Output Screen for ROS Est. NDs (Gamma) Option


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0
.Arsenic
I 1!
1
I 1-7
	 j 1
| 1
	 I 2
\ 3.2
	 ; 2
2
2.8
I 2
	 j 2
I 2
	 i 2
• 2
0.7
0.9
	 i 0.5
: 0.5
	 j 0.9
i 0.5
	 i 0.7
; 0.6
1.5
1
D_Arsenic
' 0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
1
1
1
1
0
1
1
1
1
2
GROS_Arsertic
1.031 51 31 074651 303
1.1993230BOSS37iO
1.7
1.33761048919618
1.46028213343334
0.957885781528449
3.2
1.08600369175045
1.19098252772635
2.S
1.28363461914326
1.368290930400230
1.44721892249559
1.52178645574469
1.59289577776005
0.7
0.9
0.5
0.5
1.226870494768360
0.5
0.7
O.G
1.5
Note: Columns with similar naming convention are generated for each selected variable and distribution
using this ROS option.
                                                                                              57

-------
                                        Chapter 6


                          Graphical Methods (Graph)


    Three commonly used graphical displays are available under the Graphs option:

              o   Box Plot
              o   Histogram
              o   Multi-QQ

           •  The box plots and multiple Q-Q plots can be used for Full data sets without nondetects
              and also for data sets with nondetect values.

           •  Three options are available to draw Q-Q plots with nondetect (ND) observations.
              Specifically, Q-Q plots are displayed only for detected values, or with NDs replaced by 1A
              DL values, or with NDs replaced by the respective detection limits. The statistics
              displayed on a Q-Q plot (mean, sd, slope, intercept) are computed according to the
              method used. The NDs are displayed using the smaller font and in red color.

           •  ProUCL 4.0 can display box plots for data sets with NDs. This kind of graph may not be
              very useful when many NDs may be present in a data set.

              o   A few choices are available to construct box plots for data sets with NDs. For an
                  example, some texts (e.g., Helsel) display box plots only for the detected
                  observations. Specifically, all nondetects below the largest detection limit (DL), and
                  portion of the box plot (if any) below the largest DL are not shown on the box plot. A
                  horizontal line is displayed at the largest detection limit level.
              o   ProUCL 4.0 constructs a box plot using all detected and nondetect (using DL values)
                  values. ProUCL 4.0 shows the  full box plot. However, a horizontal line is displayed
                  at the largest detection limit.

           •  When multiple variables are selected, one can choose to: 1) produce a multiple graphs on
              the same display by  choosing the Graph by group variable option, or 2) produce separate
              graphs for each selected variable.

           •  The Graph by group variable option produces side-by-side box plots, or multiple Q-Q
              plots, or histograms  for the groups  of the selected variable representing samples obtained
              from multiple populations (groups). These multiple graphs are particularly useful to
              perform two (background versus site) or more sample visual comparisons.

              o   Additionally, Box Plot has an optional feature, which can be used to draw lines at
                  statistical (e.g., upper limits of background data set) limits computed from one
                  population on the box plot obtained using the data from another population (a site
                  area of concern). This type of box plot represents a useful visual comparison of site
                  data with background threshold values (background upper limits).
              o   Up to four (4) statistics can be added (drawn) on a box plot. If the user inputs a value
                  in the value column, the check box in that row will get activated. For example, the
58

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                    user may want to draw horizontal lines at 80th percentile, 90th percentile, 95th
                    percentile, or a 95% UPL) on a box plot.
6.1     Box Plot

1.      Click Graphs ^ Box Plot
P? ProUCL4.0 - pWoricSheet.wst]
ay File  Edit  Configure  Summary Statistics  ROS Est. NDs
olc>  '
                Outlier Tests Goodness -of-Fit  Hypothesis Testing  Background  LJCL  VYindor.'  help
 Navigation Panel
0
Histogram
Mu!t-QQ
                       Full ('.v/oNDsj
                       \Yith fjDs
 Name
 ^WorkSheet.wst      1   '

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

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

            •   If graphs have to be produced by using a Group variable, then select a group variable by
                clicking the arrow below the Group by variable button. This will result in a drop-down
                list of available variables. The user should select an appropriate variable representing a
                group variable.

            •   When the Group by variable button is clicked, the following window is shown.
                                   Graph by Groups •

                                     f* Individual Graphs
                                             Label
                         f" Group Graphs
                               Value
                               1.  r  r
                               2.  r  r
                               3.  r  f
                               4.  r  r
                                    r Graphical Display Options'

                                    |   *•* Color Gradient

                                    |   r For Export (BW Printers)
                                                            Cancel
                                                                                                     59

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           •   The default option for Graph by Groups is Individual Graphs. This option will
               produce one graph for each selected variable. If you want to put all the selected variables
               into a single graph, then select the Group Graphs option. This Group Graphs option is
               used when multiple graphs categorized by a Group variable have to be produced on the
               same graph.

           •   The default option for Graphical Display Options is Color Gradient. If you want to use
               and import graphs in black and white into a document or report, then check the radio
               button next to For Export (BW Printers).

           •   Click  on the OK button to continue or on the Cancel button to cancel the Box Plot (or
               other selected graphical) option.

Box Plot Output Screen (Single Graph)
Selected options: Label (Background UPL), Value (103.85), Individual Graphs, and Color Gradient.
                                            Box Plol for Zinc
60

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Box Plot Output Screen (Group Graphs)
Selected options: Group Graphs and Color Gradient.
                                        Box Plots for X (1). X (2). X (3)
6.2     Histogram

1.       Click Graphs ^- Histogram
3? ProUCL 4.0 - [Worksheet. wst]
•g File Edit Configure Summary Statistics ROS Est. NDs [ J Outlier Tests Goodness -of-Fit Hypothesis Testing Background UCL Window Help
E]b&jja]_H]_ml_n]
Navigation Panel
Name |j_
©Worksheet. wst 1
0

1
Box Plot ^
Multi-QQ K
I I

3456789


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

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

           •   If graphs have to be produced by using a Group variable, then select a group variable by
               clicking the arrow below the Group by variable button. This will result in a drop-down
               list of available variables. The user should select and click on an appropriate variable
               representing a group variable.

           •   When that option button is clicked, the following window will be shown.
                                                                                              61

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                               B! Graphs (Histograms)
                                     Graph by Groups
                                       f Individual Graphs

                                       '•' Group Graphs
                                     Graphical Display Options

                                       (f Color Gradient

                                       C For Export (BW Printers)
                                                          Cancel
                The default selection for Graph by Groups is Individual Graphs. This option produces
                a histogram (or other graphs) separately for each selected variable. If multiple graphs or
                graphs by groups are desired, then check the radio button next to Group Graphs.

                The default option for Graphical Display Options is Color Gradient. If you want to use
                and import graphs  in black and white into a document or report, then check the radio
                button next to For Export (BW Printers).

                Click on the OK button to continue or on the Cancel button to cancel the Histogram (or
                other selected graphical) option.
Histogram Output Screen
Selected options: Group Graphs and Color Gradient.
                                 Histograms for Arsenic (subsurface). Arsenic (surface)
62

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6.3    Multi-QQ

6.3.1   Multi-QQ (Full)

1.      Click Graphs ^ Multi-QQ

2.      Multi-QQ can be obtained for data sets with (With NDs) and without NDs (Full).

            •  When that option button is clicked, the following window will be shown.
P? ProUCL 4,0 r |WorJcSheet.vwt]
   File  Edit Configure  Summary Statistics ROS Est. NDs
   oi aisle FC|
 Navigation Panel
    Outlier Tests  Goodness-of-Fit Hypothesis Testing  Background UCL Window Help
Box Plot  > I
Histogram   f              ,                     IL
  Name
  £> Worksheet wst
3 .       Select either Full or With NDs.

4.       The Select Variables Screen (Chapter 3) will appear.

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

            •    If graphs have to be produced by using a Group variable, then select a group variable by
                clicking the arrow below the Group by variable button. This will result in a drop-down
                list of available variables. The user should select and click on an appropriate variable
                representing a group variable.

            •    When the Group by variable option button is clicked, the following window will appear.
                               9 Multi QQ Options
                                   Display Regression Lines

                                     (•  Do Not Display

                                     f**  Display Regression Lines
                                   Graphical Display Options
                                     f**  Color Gradient

                                     r  For Export (BW Printers)

                                        OK
                Cancel
                                                                                                   63

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            •  The default option for Display Regression Lines is Do Not Display. If you want to see
               regression lines on graphs, then check the radio button next to Display Regression
               Lines.

            •  The default option for Graphical Display Options is Color Gradient. If you want to see
               the graphs in black and white, then check the radio button next to For Export (BW
               Printers).

            •  Click on the OK button to continue or on the Cancel button to cancel the selected Multi-
               QQ option.

Note: For Multi-QQ plot option, for both "Full" as well as for data sets  "With NDs, " the values along
the horizontal axis represent quantiles of a standardized normal distribution (Normal distribution with
mean 0 and standard deviation 1). Quantiles for other distributions (e.g., Gamma distribution) are used
when using Goodness-of-Fit (GOF) test option.

Output Screen for Multi-QQ (Full)
Selected options: Group Graph, Do Not Display Regression Lines, and Color Gradient.
Multiple Q-Q Plots
for Arsenic (subsurface), Arsenic (surface)
9.90
9.50
930
900
0
a. 40
8.10
™ 7.SO
O
•fi 7.50
I™
.Q 690
0 a
•0 6.60
| 630
O 6.00
5.70
540 fl
5.10 fl
4.30
450
420
390
-1 0 1
Theoretical Quantiles (Standard Normal)


















Arsenic (subsurface)
N = 10
Mean = 5 5600
Sd = 0.6703
Slope - 0.7023
Intercept - 5.5SOO
Correlation. R = 0.9351
Arsenic (surface)
Mean = 6. 1850
Sd = 1 .581 1
Slope - 1 .5930
Intercept = 6.1 350
Correlation.!? = 09474










• Arsenic (subsurface) Arsenic (surface)
6.3.2   Multi-QQ (with NDs)

1.       Click Graphs >• Multi-QQ
   ProUCL 4.0 - [Worksheet. wstj
•P File  Edit  Configure  Summary Statistics  ROS Est. NDs [   S Outlier Tests  Goodness-of-Fit  Hypothesis Testing  Background  UCL  Window  Help
                                         Box Plot
                                         Histogram
       e| H| ml
64

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Select With NDs option by clicking on it.

The Select Variables Screen (Chapter 3) will appear.

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

    •   If graphs have to be produced by using a Group variable, then select a group variable by
        clicking the arrow below the Group by variable button. This will result in a drop-down
        list of available variables. The user should select and click on an appropriate variable
        representing a group variable.

    •   When the Group by variable option button is clicked, the following screen appears.
                      HMMtttftafi.llIio.ns.
                           Display Non-Delects

                             ("  Do not Display Non-D elects

                             <•"  Display Non-D elect Values

                             <~  D isplay 112 N on-D elect Values

                           Display Regression Lines

                             f*  Do Not Display

                             f"  Display Regression Lines


                           Graphical Display Options
                             (*  Color Gradient

                             f"  For Export (BW Printers)
                                OK
Cancel
        The default option for Display Regression Lines is Do Not Display. If you want to see
        regression lines, then check the radio button next to Display Regression Lines.

        The default option for Display Nondetects is Display Nondetect Values.

        o  Do not Display Nondetects: Selection of this option excludes the NDs detects and
           plots only detected values on the associated Q-Q plot. The statistics are computed
           using only detected data.
        o  Display Nondetect Values: Selection of this option treats detection limits as detected
           values and plots those detection limits and detected values on the Q-Q plot. The
           statistics are computed accordingly.
                                                                                          65

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                o   Display Vz Nondetect Values: Selection of this option replaces the detection limits
                    with their half values, and plots half detection limits and detected values on the Q-Q
                    plot. The statistics are computed accordingly.

            •   The default option for Graphical Display Options is Color Gradient. If you want to see
                the graphs in black and white, then check the radio button next to For Export (BW
                Printers).

            •   Click on the OK button to continue or on the Cancel button to cancel the Multi-QQ
                option.

Output Screen for Multi-QQ (without NDs)
Options: Do Not Display Regression Lines, Do not display Nondetects, and Color Gradient.
                                    Q-Q Plot with without NDs for Mercury
                                      Theoretic*) Quantiles (Standard Normal]
66

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Output Screen for Multi-QQ (with NDs)
Options: Do not Display Regression Line, Display Nondetect Values, and Color Gradient.
Q-Q Plot with NDs for Mercury
1.00 H
a

0.90 J

0.30 H
0.70
H
g 0.60 '
1
£ 0-50
a
o
•o
-S
6
0.30 M
0.20
0.10 a B
i a H J
j H
0.00
-2-101 2

















Theoretical Quantiles (Standard Normal)
• Mercury
Mercury
Mean - 0.3055
3d -0.2937
Slope -0.271 7
Intercept = 0 3055
Correlation, R = 0.9001














Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red.
The legend size is made smaller for BWprinters.
Output Screen for Multi-QQ (with NDs)
Selected options: Do not Display Regression Lines, Display Vz, Nondetect Values, and Color Gradient.
                             a   -mm
                                      Theoretical Quantiles (Standard Normal)
                                                                                      N-30
                                                                                      Mean = 0.2829
                                                                                      Sd - 0.2962
                                                                                      Slope = 0.2S56
                                                                                      Intercept - 0.2329
                                                                                      CuirelFilion.R-08721
Note: The legend size of nondetect values is smaller than that of the detected values and is shown in red.
                                                                                                         67

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                                         Chapter 7


                         Simple Classical  Outlier Tests


Outliers are inevitable in data sets originating from environmental applications. There are many graphical
(Q-Q plots, Box plots), classical (Dixon, Rosner, Welch), and robust methods (biweight, Huber, PROP)
available to identify outliers. It is well known that the classical outlier tests (e.g., Dixon test, Rosen test,
EPA, 2006) suffer from masking (e.g., extreme outliers may mask intermediate outliers) effects. The use
of robust outlier identification procedures is recommended to identify multiple outliers, especially when
dealing with multivariate (having multiple contaminants) data sets. However, those preferred and more
effective robust outlier identification methods are beyond the scope of ProUCL 4.0. Several robust outlier
identification methods (e.g., based upon biweight, Huber, and PROP influence functions) are available in
the  Scout software package (EPA, 1999).

The two simple classical outlier tests (often cited in environmental literature): Dixon and Rosner tests are
available in ProUCL 4.0. These tests can be used on data sets with and without nondetect observations.
These tests also require the assumption of normality of the data set without the outliers. It should be noted
that in  environmental applications, one of the objectives  is to identify high outlying observations that
might be present in the right tail of a data distribution as  those observations often represent contaminated
locations of a polluted site. Therefore, for data sets with nondetects, two options are available in ProUCL
4.0  to deal with data sets with outliers. These options are: 1) exclude nondetects and 2) replace NDs by
DL/2 values. These options are used only to identify outliers and not to compute any estimates and limits
used in decision-making process.

It is suggested that these two classical outlier identification procedures be supplemented with graphical
displays such as Q-Q plots, Box and Whisker plot (called box plot), and IQR (= upper quartile, Q3 -
lower quartile, Ql).  These  graphical displays are available in ProUCL 4.0. Box plots with whiskers are
often used to identify outliers (e.g., EPA, 2006). Typically, a box plot gives a good indication of extreme
(outliers) observations that may present in a data set. The statistics (lower quartile, median, upper quartile,
and IQR) used in the construction of a box plot do not get distorted by outliers. On a box plot,
observations beyond the two whiskers may be considered as candidates for potential outliers.

Q-Q plots are also quite useful to identify outliers in a data set. For an example, on a normal Q-Q plot,
observations that are well separated from the bulk (central part) of the data typically represent potential
outliers needing further investigation. Also, significant and obvious jumps and breaks in a Q-Q plot (for
any distribution) are indications of the presence of more  than one population. Data sets exhibiting such
behavior of Q-Q plots should be partitioned out in component sub-populations before estimating an  EPC
Term or a background threshold value (BTV). It is strongly recommended that both graphical and formal
outlier identification tests should be used on the same data set to identify potential outliers that may  be
present in a data set under  study.  More details about the construction of graphical displays and outliers
test can be found in the Technical Guide for ProUCL 4.0.

    Dixon's Test (Extreme Value Test)

           •   This test is used to identify statistical outliers when the sample size is less than or equal
               to 25.
68

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            •   This test can be used to identify outliers or extreme values in both the left tail (Case 1)
                and the right tail (Case 2) of a data distribution. In environmental data sets, extremes
                found in the right tail may represent potentially contaminated site areas needing further
                investigation or remediation. The extremes in the left tail may represent ND values.

            •   This test assumes that the data without the suspected outlier are normally distributed;
                therefore, it is necessary to perform a test for normality on the data without the suspected
                outlier before applying this test.

            •   This test may suffer from masking in the presence of multiple outliers. This means that if
                more than one outlier is suspected, this test may fail to identify all of the outliers.
                Therefore, if you decide to use the Dixon's test for multiple outliers, apply the test to the
                least extreme value first. Alternatively, use more effective robust outlier identification
                procedures. Those outlier identification procedures will be available in Scout (EPA,
                1999) software.

    Rosner's Test

            •   This test can be used to identify and detect up to 10 outliers in data sets of sizes 25 and
                higher.

            •   This test also assumes that the data are normally distributed; therefore, it is necessary to
                perform a test for normality before applying this test.

Depending upon the selected variable(s) and the number of observations associated with them, either the
Dixon's test or the Rosner's test will be performed.

NOTE: Throughout this User Guide, and in ProUCL 4.0, it is assumed that the user is dealing with a
single population. If multiple populations are present in a data set, it is recommended to separate them
out using appropriate population partitioning methods and techniques. Appropriate tests and statistics
(e.g., goodness-of-fit tests, 95% UCLs, 95% UPLs) should be computed separately for each of the
identified populations. Also, outliers if any should be identified and thoroughly investigated. The presence
of outliers distorts all statistics including the all of the upper limits (UCLs, UPLs, upper percentiles). The
use of distorted statistics and limits may lead to incorrect conclusions having potential adverse effects on
the human health and the environment. Decisions about the disposition of outliers:  inclusion or exclusion
in the data set to be  used to  compute the UCLs, UPLs, and other statistics should be made by all parties
involved. Statistical methods supplemented with graphical displays (e.g.,  Q-Q plot and box plots) can
only help identify statistical outliers that may be present in a data set. The project team and experts
familiar with the site should interpret and assign physical meaning and significance to those identified
outliers. The entire project team should be involved in taking decisions about the appropriate disposition
(include or not include) of outliers.
                                                                                                69

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7.1     Outlier Test for Full  Data Set

1.      Click Outlier Tests ^ Full *• Compute
P3 ProUCL 4.0 - [WorkSheet.wst]
 •g File Edit Configure Summary Statistics  ROSEst. NDs  Graphs EElSljS3 Goodness-of-Fit  Hypothesis Testing Background  UCL Window Help

                                            I
                                              With NDs
 Navigation Panel
  [•lame
                      1
                                    1


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

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

           •   If graphs have to be produced by using a Group variable, then select a group variable by
               clicking the arrow below the Group by variable button. This will result in a drop-down
               list of available variables. The user should select and click on an appropriate variable
               representing a group variable.

           •   If at least one of the selected variables has 25 or more observations, then click the option
               button for the Rosner Test.
                      Select Number Outliers (Rosner Test)
                                                              - ri  x
                            Number of Outliers for Rosner Test   [l

                        Applicable to Rosrrer's test (N >= 25) Only
                        Dixon's test (N < 25) only tests for one outlier.
                                OK
                                                      Cancel
7.2
    •  The default option for the number of suspected outliers is 1. In order to use this test, the
       user has to obtain an initial guess about the number of outliers that may be present in the
       data set. This can be done by using graphical displays such as a Q-Q plot. On this
       graphical Q-Q plot, higher observations that are well separated from the rest of the data
       may be considered as potential or suspected outliers.

    •  Click on the OK button to continue or on the Cancel button to cancel the Outlier Tests.

Outlier Test for Data Set with  NDs
Typically, in environmental applications, one is interested in identifying high outliers (perhaps
representing contaminated parts of a site area, hot spots) that might be present in the right tail of the data
70

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distribution. Therefore, one may want to use the same outlier identification procedures (e.g., Dixon test,
Rosner test) that are used on full-uncensored data sets (without any NDs). The processes to perform such
tests using ProUCL 4.0 are described as follows.
1.
Click Outlier Tests ^ With NDs ^  Exclude NDs
 P? ProUCL 4.0 - jWorlcSheet.wstJ
    File  Edit Configure  Summary Statistics  ROS Est. NDs  Graphs gffljjMJ^^S Goodness -of-Fit  H>pothesis Testing  Background  UCL  Window  Help
                                                        Full
 Navigation Panel
  Name
                                   D        1
                                 Arsenic  D_Arsenic
                                                           DL/2 Estimates
                                                                                                 7
  W WorkSheet.wst      f|    '

Output Screen for Dixon's Outlier Test
                                                            Outlier Tests for Selected Variables
                                          User Selected Options
                                                   From File  WorkSbeetwst
                                                Full Precision  OFF
                          Test for Suspected Outliers with Dixon test  1
                          Test for Suspected Outliers for Rosner test  1
                                  Dixon's O utlier Test for Aisenic

                        Nbrrberofdata = 11
                        10% critical value: 0,517
                        5% critical value: 0.576
                        1', critical value: 0.679

                        1. 3 2 is a Potential Oudier (UpperTail)

                        Test Statistic: 0.556

                        fcr 10'i significance level, 3.2 is an outlier.
                        For 5% sicjnifieance level, 3.2 is not an outlier.
                        For 1 % sigrtificance level, 3.2 is not an outlier.

                        2.0.5 is a Potential Outiier (LowerTail)
                                                                                                                    71

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Output Screen for Rosner's Outlier Test

There are many observations in this data set that may represent potential outliers. The potential outliers
can also be seen in the graphical displays associated with this data set.
                    Selected Options: Number of Suspected Outliers for the Rosner Test = 4
                                                               i Outlier Tests for Selected Variables
                                            User S elected 0 ptions
                                                       From File   DANarairAProUCL 4.0\DataWoclor 1254.wsl
                                                    Full Precision   OFF
                             Test for Suspected Outliers with Dixon test   1
                             Test for Suspected Outliers for Rosner test   2
                                           Rosner's Outlier Test for Aroclt»1254
                         Number of data: 44
                         Number of suspected outliers: 2
                                                        Potential
                                 tt     Mean        sd     outlier
                                 1    1531.88   3316.58   19000.00
                                 2    1125.65   1998.60    8300.00
 Test     Critical    Critical
value  value (5%)  value (1 %)
 5.27       3.08      3.43
 3.58       3.07      3.41
                         For 5% significance level, there are 2 Potential Outliers
                         Therefore, Potential Statistical Outliers are
                         18000.00,8300.00

                         For 1% Significance Level, there are 2 Potential Outliers
                         Therefore, Potential Statistical Outliers are
                         18000.00,8300.00
Box plot of the Aroclorl254 Data Set
                                                     Box Plot for Aroclorl 254
          20000.00

          18000.00

          16000.00

          14000.00
        W  BC'OD.D:
        .Q
        o
           6000.00

           4000.00

           2000.00

             0.00 -
72

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Q-Q Plot of the Aroclorl254 Data Set
                                                 Q-Q Plot with NDs for ArocloM 254
           ucoo.oo
         H
         c
         •J3 12000.00
         (0
         5 10000.00

         ^  aooo.oo
         •
         ^  EO:O oo
         6
            4000.00

            2000.00

              0.00
Arock>r1254
  Total Number of Data = 53
  Number of Non-Detects = 9
  Number of Detects = 44
  Mean =1271.7646
  Sd =3105.5855
  Slope = 2117.6299
  Intercept = 1271.7646
  Correlation, R = 0.6693
                                             a aflaaBaa^jjjjj^jj^^^^iiiriBJydlri
                                                    Theoretical Quantiles (Standard Normal)
                                                                                                                                                        73

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                                         Chapter 8

                        Goodness-of-Fit  (G.O.F.) Tests
Several goodness-of-fit (G.O.F.) tests for full data sets (without nondetects) and for data sets with NDs
are available in ProUCL 4.0. Details of those tests are described in the ProUCL 4.0 Technical Guide. In
this User Guide, those tests and available options have been illustrated using screen shots generated by
ProUCL 4.0.

Two choices are available for Goodness-of-Fit menu: Full and With NDs.

           •  Full

               o   This option is used to analyze full data sets without any nondetect observations.
                   Throughout this User Guide and in ProUCL 4.0, "Full" represents data sets without
                   nondetect observations.
               o   This option tests for normal, gamma, or lognormal distribution of the variable(s)
                   selected using the Select Variables option.
               o   G.O.F. Statistics: This option is available for both full data sets and for data sets with
                   NDs. This option simply generates output log of GOF test statistics and derived
                   conclusions about the data distributions of all selected variables. This option is also
                   available for variables categorized by a group variable.
P?iRroUCL .4.0 - fp»rkSheet.v*fstJ
 •B File Edit  Configure  Summary Statistics  ROS Est. NDs  Graphs Outtier Tests
 ^jgjjii
 Navigation Panel
             Hypothesis Testing Background  UCL  Window Help
  Name
                            0       1      2
                          Arsenic  D Arsenic
_  With NDs  >   Gamma
3  ""*   *"	•>   Lognormal
              G.O.F. Statistics
 -^Worksheet wst
                      1
               With NDs

               o  Analyzes data sets that have both nondetected and detected values.
               o  Six sub-menu items listed and shown below are available for this option.

               1.  Exclude NDs: tests for normal, gamma, or lognormal distribution of the selected
                  variable(s) using only the detected values.

               2.  ROS Estimates: tests for normal, gamma, or lognormal distribution of the selected
                  variable(s) using the detected values and the extrapolated values for the nondetects.

                  o  Three ROS methods for normal, lognormal, and gamma distributions are
                      available. This option is used to estimate or extrapolate the NDs based upon the
                      specified distribution.

                  o  By using the menu item ROS Est. NDs, ProUCL 4.0 actually generates additional
                      column(s) of ROS estimated NDs based upon the selected distribution. This
74

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           option should be used for variables with NDs. This is further illustrated by a
           screen shot given in the following

    3.  DL/2 Estimates: tests for normal, gamma, or lognormal distribution of the selected
       variable(s) using the detected values and the ND values replaced by their respective
       DL/2 values. This option is included for historical reasons and also for curious users.

       Note: The use of fabricated data obtained using DL/2 (DL or 0 values) values is not
       a recommended method. At best, the user may use the fabricated data (e.g.,  DL/2
       option) for exploratory reasons. It is suggested that these substitution methods should
       not be used for estimation and for hypotheses testing approaches.

    4.  G.O.F. Statistics: As for the full data sets, this option simply generates output log of
       GOF test statistics and other relevant statistics for data sets with nondetects. The
       conclusions about the data distributions for all selected variables are also listed on the
       generated output file (Excel-type spreadsheet). This option is also available for
       variables categorized by a group variable.
10 File Edit Configure Summary Statistics ROSEst.
ojoj BJ H] mini
Navigation Panel
Name
^ Worksheet wst
0
Arsenic . D.
1 ! 1i
2 1
3 1 7
NDs Graphs Gutter Tests l^^ra
Ful
1 2 3 —
.Arsenic
0
0
1
^^S^^ Hypothesis Testing Background UCL Window
^ s
HHH Normal-ROS Estimates
Gamma-ROS Estimates
Log-ROS Estimates
DL/2 Estimates
G.O.F. Statistics
k
y| Normal i
^ Gamma <
^ Lognormal
I
^
Help

9

•   When multiple variables are selected from the Select Variables screen, you can choose
    either:

    o  Group Graph option to produce multiple Q-Q plots for all selected variables in a
       single graph. The relevant statistics (e.g., slope, intercept, correlation, test statistic
       and critical value) associated with the selected variables are shown on the right panel
       (in tan color; see page 77). In order to capture all the graphs and results shown on the
       window screen, it is preferable to print the graph using the Landscape option.  The use
       of this option is recommended when a selected variable has data coming from two or
       more groups or populations.
    o  Group Graph option is particularly useful to generate multiple Q-Q plots for the
       groups associated with a selected variable. In order to capture all the graphs and
       results shown on the window screen, it is preferable to print the graph using the
       Landscape option. The user may also want to turn off the Navigation Panel and Log
       Panel.

•   Individual Graph option is used to generate individual Q-Q plots and the associated
    statistics separately for each of the selected variable, one variable at a time.

    o  The linear pattern displayed by a Q-Q plot suggests an approximate goodness-of-fit
       for the selected distribution.
                                                                                    75

-------
               o   The program computes the intercept, slope, and the correlation coefficient for the
                   linear pattern displayed by the Q-Q plot. A high value of the correlation coefficient
                   (e.g., > 0.95) is an indication of a good fit for that distribution. This high correlation
                   should exhibit a definite linear pattern in the Q-Q plot. Specifically, when data are
                   sparse and correlation is high,  the use of correlation statistic to determine data
                   distribution is not desirable. Note that these statistics are displayed on the Q-Q plot.
               o   On a Q-Q plot, observations that are well separated from the bulk (central part) of the
                   data typically represent potential outliers needing further investigation.
               o   Significant and obvious jumps and breaks in a Q-Q plot (for any distribution) are
                   indications of the presence of more than one population. Data sets exhibiting such
                   behavior of Q-Q plots should be partitioned out in component sub-populations before
                   estimating an EPC term or a background threshold value (BTV). It is strongly
                   recommended that both graphical and formal goodness-of-fit tests should be used on
                   the same data set to determine the distribution of the data set under study.

               Normality or Lognormality Tests: In addition to informal graphical normal and
               lognormal Q-Q plots, a formal Goodness-of-Fit (GOF) test is also available to test the
               normality or lognormality of the data set.

               o   Lilliefors Test: a test typically used for samples of size larger than 50 (> 50). When
                   the sample size is greater than  50, the program defaults to the Lilliefors test.
                   However, the Lilliefors test (generalized Kolmogorov Smirnov test) is available for
                   samples of all sizes. There is no applicable upper limit for sample size for the
                   Lilliefors test.
               o   Shapiro and Wilk (SW) Test: a test used for samples of size smaller than or equal to
                   50 (<= 50). In ProUCL 4.0, the SW test is available only for samples of size 50 or
                   less. It should be noted that the critical values for SW test are now available for
                   sample of sizes up to  2000 (Royston, 1982). These values are not as yet available in
                   ProUCL 4.0. This extension of SW test will be available in Scout (EPA, 1999)
                   software package.
               o   It should be noted that sometimes these two tests might lead to different conclusions.
                   Therefore, the user should exercise caution interpreting  the results. Specifically, the
                   user should the pattern exhibited by the associated Q-Q plot.

               GOF test for Gamma Distribution: In addition to the graphical gamma Q-Q plot, two
               formal empirical distribution function (EOF) procedures are also available to test the
               gamma distribution of a data set. These tests are the Anderson-Darling test and the
               Kolmogorov-Smirnov test.

               o   It is noted that these two tests might lead to different conclusions. Therefore, the user
                   should exercise caution interpreting the results.
               o   These two tests may be used for samples of sizes in the  range of 4-2500. Also, for
                   these two tests, the value of the shape parameter, k (k hat) should lie in the interval
                   [0.01, 100.0]. Consult the ProUCL 4.0 Technical Guide (A. Singh and A.K.  Singh
                   (EPA, 2007)) for a detailed description of gamma distribution and its parameters,
                   including k. Extrapolation beyond these sample sizes and values of k is not
                   recommended.
76

-------
           •   ProUCL computes the relevant test statistic and the associated critical value, and prints
               them on the associated Q-Q plot. On this Q-Q plot, the program informs the user if the
               data are gamma, normally, or lognormally distributed.

           •   Even though, the G.O.F. Statistics option prints out all GOF test statistics for all selected
               variables, it is suggested that the user should look at the graphical Q-Q plot displays to
               gain extra insight (e.g., outliers, multiple population) into the data set.

Note: It is highly recommended not to skip the use of a graphical Q-Q plot to determine the data
distribution as a Q-Q plot also provides a useful information about the presence of multiple populations
or outliers.

8.1     ROS Estimated (Est.)  NDs - Saving Extrapolated  NDs

           •   As mentioned before, for a variable with NDs, ProUCL 4.0 can generate additional
               column(s) consisting of detected data and the estimated (extrapolated) values of NDs
               using the ROS method assuming a normal, lognormal, or a gamma distribution.

           •   The user may want to use the resulting full data set (detected and estimated NDs) thus
               obtained to compute the  statistics of interest such as a bootstrap BCA UCL95 or a gamma
               95% upper percentile.

           •   This option of saving estimated NDs is provided only for experienced users and
               researchers. It is expected that the  user knows and understands the theory behind these
               methods. Therefore, it is suggested that this option be used with care. For an example,
               often, the use of a ROS method yields infeasible (e.g., negative, exceeding the DLs)
               estimates of NDs, and therefore, the associated estimates of EPC terms and of BTVs may
               be biased and not reliable. This is especially true when the data set contains potential
               outlier(s).
        P?; PrUCL 4.0
        •0 File  Edit Configure  Summary Statistics  ROS Est. NDs  Graphs  Outlier Tests  Goodness-of-Fit
        oh&l	
Navigation Panel
Name
@WorkSheet.wst












1
2
3
4
5
6
T
•'
&
9
10
11
0
Arsenic
1
1
1.7
1
1
2
3.2
2
2
2.8
2
1 2 3
D_Arsen.ic NRQS_Arsenic j
0 0.123366712938579
0 0.563125541541462
1 1.7
00.903911763702151! j
0 1.24633976124866
0).0994872439069666
1 3.2
0 0.273006451203256
0 0.542296528947428
1 2.8
0 0.773119303054192
                                                                                            11

-------
8.2    Goodness-of-Fit Tests with Full Data Sets

1.      Click Goodness-of-Fit ^ Full
RJ ProUCL4,§ -
   File  Edit Configure  Summary Statistics  ROS Est. NDs Graphs  Outlier Tests
                                                      Goodness-of-fit
       a
            ml n
 Navigation Panel
  Name
             Hypothesis Testing Background UCL Window Help

              Normal         i
__  With NDs >!  Gamma         I	
   3^™........,^*-...™,.,        .          ~T       Q      Q
   ^   4-^1  Lognormal       I   '       °      y
              G.O.F. Statistics   I
 0 Worksheet wst     I   '   1	!

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

            •   To test your variable for normality, click on Normal from the drop-down menu list.

            •   To test variable for lognormality, click on Lognormal from the drop-down menu list.

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

8.2.1   GOF Tests for Normal and Lognormal Distribution

1.      Click Goodness-of-Fit ^- Full ^-  Normal or Lognormal
P?jProUCL 4.0 - tWprkSheet-wstJ
•0 File  Edit  Configure  Summary Statistics ROS Est. NDs  Graphs Outlier Tests
f*q
tJ
 Navigation Panel
                                                      Goodness-of-fit
             I Hypothesis Testing Background UCL Window Help
          _  Normal         i
    With NDs >  Gamma         1	—
                              0
                                     1
  Name
                                                                  G.O.F. Statistics
                                                                                 7
 CsJWorkSheet.wst

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

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

           •   If graphs have to be produced by using a Group variable, then select a group variable by
               clicking the arrow below the Group by variable button. This will result in a drop-down
               list of available variables. The user should select and click on an appropriate variable
               representing a group variable.

           •   When the option button is clicked, the following window will be shown.
78

-------

                                      • Select Confidence Level -
                                            r 90%

                                            <*• 35%

                                            r 99%


                                       Method

                                         <*' Shapiro Wilk

                                         '-"" Lilliefors


                                       Display Regression Lines

                                         <~ Do Not Display

                                         (• Display Regression Liees


                                       Graphs by Group

                                         (* Individual Graphs

                                         i " Group Graphs


                                       Graphical Display Options

                                         <"* Color Gradient

                                         f" For Export (BW Printers)
                                        OK
                                                          Cancel
               o  The default option for the Confidence Level is 95%.
               o  The default GOF Method is Shapiro Wilk. If the sample size is greater than 50, the
                   program automatically uses the Lilliefors test.
               o  The default method for Display Regression Lines is Do Not Display. If you want to
                   see regression lines on a Q-Q plot, then check the radio button next to Display
                   Regression Lines.
               o  The default option for Graphs by Group is Individual Graphs. If you want to see
                   the plots for all selected variables on a single graph, then check the radio button next
                   to Group Graphs.

Note: This option for Graphs by Group is specifically provided when the user wants to display multiple
graphs for a variable by a group variable (e.g., site AOC1, site AOC2, background). This kind of display
represents a useful visual comparison of the values of a variable (e.g., concentrations ofCOPC-Arsenic)
collected from two or more groups (e.g., upgradient wells, monitoring wells, residential wells).

               o  The default option for Graphical Display Options is Color Gradient. If you want
                   to see the graphs in black and white to be included in reports for later use, then check
                   the radio button next to For Export (BW Printers).
                                                                                                 79

-------
               •    Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Normal Distribution (Full)
Selected Options: Shapiro Wilk, Display Regression Line, and For Export (BW Printers).
                                         Normal Q-Q Plot for Arsenic
9.90
9.60
9.30
9.00
8.70
8.40
8.10
7.80
7.50
7.20
6.90
6.60
6.30
6.00
5.70
5.40
5.10
4.80
4.50
4.20
3.90
                                       1                   0                   1
                                      Theoretical Quantiles (Standard Normal)
                                                                                                      Arsenic
                                                                                                        N-20
                                                                                                        Mean - 5.8725
                                                                                                        Sd - 1.2247
                                                                                                        Slope-1.1798
                                                                                                        Intercept = 5.8725
                                                                                                        Correlation, R 0.9281
                                                                                                        Shapiro-Wilk Test
                                                                                                        Test Value - 0.868
                                                                                                        Critical Val(0.05) - 0.905
                                                                                                        Data not Normal
                -J- Arsenic
Output Screen for Lognormal Distribution (Full)
Selected options: Shapiro Wilk, Display Regression Lines, and Color Gradient.
         > 1.90
         0)
         I
         l
         I
                                         Lognormal Q-Q Plot for Arsenic
                                          Theoretical Quantiles (Standard Normal)
                                                                                                          Mean = 1.7519
                                                                                                          Sd = 0.1917
                                                                                                          Slope-0.1917
                                                                                                          Intercept = 1.7519
                                                                                                          Correlation, R = 0.9636
                                                                                                          Shapiro-Wilk Test
                                                                                                          Test Statistic = 0.932
                                                                                                          Critical ValueiO.05) = 0.905
                                                                                                          Data appear Lognormal
80

-------
8.2.2   GOF Tests for Gamma Distribution

1.      Click Goodness-of-Fit ^ Full ^ Gamma
P? RroUCL 4.0..- (Worksheet.wsfl
•f File Edit Configure Summary Statistics ROS Est. NDs  Graphs Outlier Tests
ojo
 Navigation Panel
 Name
 © Worksheetwst
       Q       1
     Arsenic  D_Arsenic
1   I	Ti "    o
                                        gj Hypothesis Testing  Background UCL Window Help
                                           Normal
                                                                  Lognormal
                                                                  G.O.F. Statistics
2.      The Select Variables Screen (described in Chapter 3) will appear.

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

            •    If graphs have to be produced by using a Group variable, then select a group variable by
                clicking the arrow below the Group by variable button. This will result in a drop-down
                list of available variables. The user should select and click on an appropriate variable
                representing a group variable.

            •    When the option button is clicked, the following window will be shown.
                                                                                                  81

-------

                                      Select Confidence Level
                                           r 90%

                                           ff 95 %

                                           r 39%


                                      Method

                                       •'•' Anderson Darling
                                       f  Kol mogorov Stni rnov

                                      Display Regression Lines

                                       C Do Not Display

                                       '•  Display Regression- Lines

                                      Graph by Groups
                                       •'•  Individual Graphs
                                       1 " Group Graphs


                                     'Graphical Display Options

                                       '"•'" Color Gradient

                                       <~ For Export (BW Printers]
                                                        Cancel
               o   The default option for the Confidence Level is 95%.
               o   The default GOF method is Anderson Darling.
               o   The default option for Display Regression Lines is Do Not Display. If you want to
                   see regression lines on the Gamma Q-Q plot, then check the radio button next to
                   Display Regression Lines.
               o   The default option for Graph by Groups is Individual Graphs. If you want to see
                   the graphs for all the selected variables into a single graph, then check the radio
                   button next to Group Graphs.
               o   The default option for Graphical Display Options is Color Gradient. If you want
                   to see the graphs in black and white, check the radio button next to For Export (BW
                   Printers).

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

           •   Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
82

-------
Output Screen for Gamma Distribution (Full)
Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient.
                                   Gamma Q-Q Plot for Mercury
                                                                                    Mean = 0.3055
                                                                                    k star = 1.1722
                                                                                      = 1 0468
                                                                                    lntercept = -O.Q111
                                                                                    Correlation, R = 09713

                                                                                    Test Statistic = 0.730
                                                                                    Critical ValueCO.05)-0.7S9
                                                                                    Dots appear Gamma Distributed
                                 Theoretical Quantiles of Gamma Distribution
8.3    Goodness-of-Fit Tests Excluding NDs

1.      Click Goodness-of-Fit ^ With NDs ^ Exclude NDs
 •0 File  Edit  Configure  Summary Statistics  ROS Est, NDs  Graphs  Outlier Tests
                                                                    Hypothesis Testing Background UCL Window  Help
 Navigation Panel
                                                                      Normal-ROS Estimates
                                                                      Gamma-ROS Estimates
                                                                      Log-ROS Estimates
                                                                      DL/2 Estimates
                                                                      G.O.F. Stafetcs
i? Worksheet wst
2.      Select distribution to be tested: Normal, Gamma, or Lognormal.

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

            •   To test for lognormality, click on Lognormal from the drop-down menu list.

            •   To test for gamma distribution, click on Gamma from the drop-down menu list.
                                                                                                       83

-------
8.3.1   Normal and Log normal Options

1.      Click Goodness-of-Fit ^  With NDs ^ Excluded NDs ^ Normal or Lognormal
P?jRrpUCL 4..0 - Qf prkSheet.wstf
   File  Edit Configure  Summary Statistics  ROS Est. NDs Graphs Outlier Tests
                                                       Goodness-rffit
                                                         Full
                                           Hypothesis Testing  Background  UCL Window  Help
 Navigation Panel
  Name
 .^WorkSheet.wst
                                                         Wtn.AOs>|  BtebdeWs
       0       1       2
     Arsenic  D_Arsenic,
1   T	i]      o'
2   :	l"      0
3         1.7      1
Normal-ROS Estimates
Gamma-ROS Estimates
Log-ROS Estimates
DL/2 Estimates
G.O.F. Statistics
2.      The Select Variables Screen (Chapter 3) will appear.

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

            •   If graphs have to be produced by using a Group variable, then select a group variable by
                clicking the arrow below the Group by variable button. This will result in a drop-down
                list of available variables. The user should select and click on an appropriate variable
                representing a group variable.

            •   When the option button (Normal or Lognormal) is clicked, the following window will be
                shown.
84

-------

                                     "Select Confidence Level  •
                                            r 90%

                                            (f 95%

                                            r 99 %


                                      Method

                                        (• ShapiroWilk

                                        f~ Lilliefofs


                                      Display Regression Lines

                                        <~ Do Not Display

                                        '-*" Display Regression Lines


                                      Graphs by Group	

                                        (* Individual Graphs

                                        f*~ Group Graphs


                                      Graphical Display Options

                                        (* Color Gradient

                                        r For Export (EW Printers)
                                        OK
Cancel
                o  The default option for the Confidence Level is 95%.
                o  The default GOF Method is Shapiro Wilk. If the sample size is greater than 50, the
                   program defaults to Lilliefors test.
                o  The default for Display Regression Lines is Do Not Display. If you want to see
                   regression lines on the associated Q-Q plot, check the radio button next to Display
                   Regression Lines.
                o  The default option for Graphs by Group is Individual Graphs. If you want to see
                   the plots for all selected variables on a single graph, check the radio button next to
                   Group Graphs.

Note: This option for Graphs by Group is specifically useful when  the user wants to display multiple
graphs for a variable by a group variable (e.g., site AOC1, Site AOC2, background). This kind of display
represents a useful visual comparison of the values of a variable (e.g., concentrations ofCOPC-Arsenic)
collected from two or more groups (e.g., upgradient wells, monitoring wells, and residential wells).
                                                                                                  85

-------
                  o   The default option for Graphical Display Option is Color Gradient. If you want to
                       see the graphs in black and white, check the radio button next to For Export (BW
                       Printers).

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

              •   Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Normal Distribution (Exclude NDs)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and For Export (BW Printers).
                              Normal Q-Q Plots (Statistics using Detected Data)
                                  forArsenic (subsurface), Arsenic (surface)
Arsenic (subsurface)
 Total Number of Data = 1C
 Number treated as ND = 1
 DL = 4.5
 N = 9
 Percent NDs = 10%
 Mean = 5.6778
 Sd-0.5911
 Slope - 0.1431
 Intercept = 5.6778
 Correlation, R = 0.2265
 Shapiro Wilk Test
 Test Statistic = 0.927
 Critical Value |0.05> = 0.829
 Data appear Normal
Arsenic (surface)
 Total Number of Data = 10
 Number treated as ND = 2
 DL - 4.5
 N = 8
 Percent NDs-20%
 Mean-6.6313
 Sd = 1.4400
 Slope - 0.1952
 Intercept = 6.6313
 Correlation, R = 0.1262
 Shapiro-WilkTest
 Test Statistic = 0.807
 Critical Value(0.05) - 0.818
 Data not Normal
                - Arsenic (subsurface)
                                      Theoretical Quantiles (Standard Normal)
                                                              -o- Arsenic (surface)
86

-------
Output Result for Lognormal Distribution (Exclude NDs)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.
                                Lognormal Q-Q Plots (Statistics using Detected Data)
                                    forArsenic (subsurface), Arsenic (surface)
             • Arsenic (subsurface)
                                         Theoretical Quantiles (Standard Normal)
                                                             Arsenic (surface)
                     Arsenic (subsurftce)
                      Total Number ot Data = 10
                      Number treated as ND = 1
                      DL -1.5040774
                      N = 8
                      Percent NDs > 10%
                      Uteri = 1.7319
                      SO-0.1019
                      Slope - 0.1059
                      Intercept -1.7319
                      Cor relation ,R * 0.9726
                      Shapiro-Wife Test
                      Test Statistic - 0.938
                      Critical Val(0.05) = 0.829
                      Data appear Lognormal
                     Arsenic (surface)
                      Total Number of Data = 10
                      Number treated as ND - 2
                      DL -1 5040774
                      N-8
                      Percent NDs - 20%
                      Mean = 1.3731
                      Sd- 0.2018
                      Slope = 0.1997
                      Intercept -1.8731
                      Correlaticn,R = 0.9211
                      Shapirc-Wilk Test
                      Test Statistic - 0.335
                      Critical Val(0 05) = O.S1 8
                      Data appear Lognormal
8.3.2   Gamma Distribution Option
1.       Click Goodness-of-Fit ^ With NDs ^ Excluded NDs ^  Gamma
PP ProUCL 4.0 - [Worksheet.wst]
    File  Edit  Configure  Summary Statistics  ROS Est. NDs  Graphs  Cutter Tests
                                                               Goodness-of-Rt
Hypothesis Testing  Background  UCL  Window  Help
 Navigation Panel
  Name
 £JWorkSheet.wst
                                Arsenic
                                        D_Arsenic
                                    	i"
                                     1.7
                      Normal
                 ^^	
 Normal-ROS Estimates
 Gamma-ROS Estimates  >   Lognormal
 Log-ROS Estimates
 DL/2 Estimates
 G.O.F. Statistics
2.       The Select Variables Screen (Chapter 3) will appear.
              •    Select one or more variable(s) from the Select Variables screen.
              •    If graphs have to be produced by using a Group variable, then select a group variable by
                  clicking the arrow below the Group by variable button. This will result in a drop-down
                  list of available variables. The user should select and click on an appropriate variable
                  representing a group variable.
              •    When the option button (Gamma) is clicked, the following window is  shown.
                                                                                                                  87

-------

                                  •Select Confidence Level
                                        T 90 %

                                        ^ 35%

                                        T 99%


                                  Method •

                                    f* Anderson Darling

                                    '"" tolmogorovSmirncw

                                  'Display Regression Lines

                                    r Do Not Display

                                    (• Display Regression Lines

                                  Graph by Groups
                                    f* Individual Graphs

                                    f Group Graphs


                                  Graphical Display'Options

                                    (* Color Gradient

                                    r For Export (EM Printers)
                                     OK
Cancel
                   o  The default option for the Confidence Level is 95%.
                   o  The default GOF test Method is Anderson Darling.
                   o  The default method for Display Regression Lines is Do Not Display. If you
                      want to see regression lines on the normal Q-Q plot, check the radio button next
                      to Display Regression Lines.
                   o  The default option for Graph by Groups is Individual Graphs. If you want to
                      display all selected variables on a single graph, check the radio button next to
                      Group Graphs.
                   o  The default option for Graphical Display Options is Color Gradient. If you
                      want to see the graphs in black and white, check the radio button next to For
                      Export (BW Printers).

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

           •   Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
88

-------
Output Screen for Gamma Distribution (Exclude NDs)
Selected options: Anderson Darling, Do Not Display, Individual Graphs, and For Export (BW Printers).
                           Gamma Q-Q Plot for Mercury with NDs
                               Statistics using Detected Data
         4)
         01 0.50
         5
         O 0.30
Mercury
 Total Number of Data = 30
 Number treated as ND - 5
 DL - 0.5
 N-25
 Percent NDs-17%
 Mean - 0.3124
 kstar = 1.0533
 Slope- 1.0294
 Intercept- 4.0048
 Correlation, R-0.9590
 Anderson-Darling Test
 Test Statistic = 0.861
 Critical V,iliicl0.05l - 0.770
 Data not Gamma Distributed
                            Theoretical Quantiles of Gamma Distribution
               --i- Mercury
8.4    Goodness-of-Fit Tests with Log-ROS Estimates
1.       Click Goodness-of-Fit ^ With NDs ^ Log-ROS Estimates
3?VMI!K!VV|W9T|lff6NffilRPfj^l
•2 File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests
elel tftlHlml n|
Navigation Panel

Name



1
2
3
0 1
Arsenic D_Arsenic
11 0
1 0
1.7 1
2




3
Q Hypothesis Testing Background UCL Window Help
Full >
H2I<33S3
^mmntmf






1
Exclude NDs >
Normal -ROS Estimates >
Gamma-ROS Estimates *
• ^^^^^•1 Loa-ROS Estimates >



DL/2 Estimates K
G.O.F. Statistics



5 ,
i
Normal
Gamma
Lognormal




2.      Select the distribution to be tested: Normal, Lognormal, or Gamma
            •   To test your variable for normality, click on Normal from the drop-down menu list.

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

            •   To test your variable for lognormality, click on Lognormal from drop-down menu.
                                                                                                   89

-------
8.4.1   Normal or Log normal Distribution (Log-ROS Estimates)

1.       Click Goodness-of-Fit ^ With NDs ^ Log-ROS Estimates ^ Normal, Lognormal
P? ProUCL 4,0 - ||ft)rkSheet,wstI
File  Edit Configure  Summary Statistics  ROS Est. NDs Graphs  Gutter Tests
h&lBlHlmlEl                      "            ™FJ
                                                                Hypothesis Testing Background UCL  Window Help
                                                               *
 Navigation Panel
  Name
                          0       1       2
                        Arsenic  D_Arservic
                    1   [~~J]      °


                    1   \     1-7      1
                                                3  "
mK  Exdude NDs        >
      Normal-ROS Estimates  >
      Gamma-ROS Estimates  >
                                                                LoQ*OSEslmates
                                                                 DL/2 Estimates      >  Gamma
                                                                 G.O.F. Statistics
2.      The Select Variables Screen (Chapter 3) will appear.

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

           •   If graphs have to be produced by using a Group variable, then select a group variable by
               clicking the arrow below the Group by variable button. This will result in a drop-down
               list of available variables. The user should select and click on an appropriate variable
               representing a group variable.

           •   When the option button (Normal or Lognormal) is clicked, the following window will be
               shown.
90

-------
                    '•Select Confidence Level
                           T 90%

                           <¥ 95%

                           r 99%


                     Method

                       <• ShapiroWilk

                       T Lilliefors


                    'Display Regression Lines

                       f" Do Not Display

                       '** Display Regression Lines


                    - Graphs by Group •

                       & Individual Graphs

                       f" Group Graphs


                    - Graphical Display Options -

                       f*" Color Gradient

                       T For Export (BW Printers)
                      OK
Cancel
o   The default option for the Confidence Level is 95%.
o   The default GOF test Method is Shapiro Wilk. If the sample size is greater than 50,
    the program defaults to use the Lilliefors test.
o   The default method for Display Regression Lines is Do Not Display. If you want to
    see regression lines on the normal Q-Q plot, check the radio button next to Display
    Regression Lines.
o   The default option for Graphs by Group is Individual Graphs. If you want to
    display all selected variables into a single graph, check the radio button next to
    Group Graphs.
o   The default option for Graphical Display Options is Color Gradient. If you want
    to see the graphs in black and white, check the radio button next to For Export (BW
    Printers).
                                                                                91

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

             •    Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Normal Distribution (Log-ROS Estimates)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and For Export (BW Printers).
                           Normal Q-Q Plots using Robust ROS Method
                                        for Arsenic, Mercury
       5 6.00
      1
       aj 5.00
       
      O
      •O 4.00
             -J- Arsenic
   1                 o                 1
Theoretical Quantiles (Standard Normal)
                          -o Mercury
                                                           Arsenic
                                                             N = 20
                                                             Mean - 5.8307
                                                             Sd-1.2799
                                                             Slope-1.2517
                                                             Intercept - 5.8307
                                                             Correlation, R = 0.9421
                                                             Shapiro-WilkTest
                                                             Tea Value-0.895
                                                             Critical Val(0.05| - 0.905
                                                             Data not Normal
                                                           Mercury
                                                             N-30
                                                             Mean-0.2767
                                                             Sd - 0.2984
                                                             Slope - 0.2643
                                                             Intercept = 0.2767
                                                             Correlation, R - 0.8618
                                                             Shapiro-WilkTest
                                                             Test Value-0.733
                                                             Critical Val(0.05| - 0.927
                                                             Data not Normal
Note: The legend size of nondetect values is smaller than that of the detected values.
92

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Output Screen for Lognormal Distribution (Log-ROS Estimates)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.
                                    Lognormal Q-Q Plot for Group
                                 Statistics using Robust ROS Method
      I17
            Arsenic i :ul:i:urrai:e i
                                  Theoretical Ouantiles of Gamma Distribution
                                                          Arsenic (surface)
                                                          Arsenic (subsurface)
                                                           N = 1Q
                                                           Mean = 1.7063
                                                           Sd = 0.1246
                                                           Slope = 0.1308
                                                           Intercept = 1 .7068
                                                           Correlation, R = 0.9366
                                                           Shapiro-VMIk Test
                                                           Test Statistic = 0.961
                                                           Critical Value(O.OS) = 0.842
                                                           Data appear Lognormal
                                                          Arsenic (surface)
                                                           N = 1Q
                                                           Mean = 1.7731
                                                           Sd = 0.2678
                                                           Slope = 0.2758
                                                           Intercept -1.7731
                                                           Correlation, R = 0.9682
                                                           Shapiro-Will; Test
                                                           Test Statistic = 0.930
                                                           Critical ValueC0.05)-0.842
                                                           Data appear Lognormal
Note: The legend size ofnondetect values is smaller than that of the detected values and is shown in red.
8.4.2   Gamma Distribution (Log-ROS Estimates)
1.       Click Goodness-of-Fit ^  With NDs  ^  Log-ROS Estimates ^  Gamma
P3 ProUCL 4.0 - [Worksheet. wst]
 •g File Edit Configure Summary Statistics ROS Est, NDs  Graphs  Outiier Tests [         J Hypothesis Testing Background  UCL  Window  Help
                                                              Full      TT
 Navigation Panel
  Name
 ^WorkSheet.wst
                               Arsenic
       1
                                      D_Arsenic
  II
	l"
 1.7
 Oj
~0~
                              Exdude NDs
                              Normal-ROS Estimates
                              Gamma-ROS Estimates
it
                                                                8
2.       The Select Variables Screen (Chapter 3) will appear.

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

             •   If graphs have to be produced by using a Group variable, then select a group variable by
                 clicking the arrow below the Group by variable button. This will result in a drop-down
                 list of available variables. The user should select and click on an appropriate variable
                 representing a group variable.
                                                                                                            93

-------
               When the option button (Gamma) is clicked, the following window will be shown.

                                  'Select Confidence Level
                                        T 30 %
                                        <* 95%

                                        r 99 %


                                  -Method 	

                                    (• Anderson Darling

                                    T" KblmogorovSmirrtov

                                  'Display Regression Lines	

                                    r Do Not Display

                                    t**" Display Regression Lines

                                  -Graph by Groups	
                                    f* Individual Graphs

                                    <^~ Group Graphs


                                  'Graphical Display Options	

                                    f* Color Gradient

                                    r For Export (EM Printers)
                                     OK
Cancel
               o   The default option for the Confidence Level is 95%.
               o   The default GOF test Method is Anderson Darling.
               o   The default method for Display Regression Lines is Do Not Display. If you want to
                   see regression lines on the normal Q-Q plot, check the radio button next to Display
                   Regression Lines.
               o   The default option for Graph by Groups is Individual Graphs. If you want to put
                   all of the selected variables into a single graph, check the radio button next to Group
                   Graphs.
               o   The default option for Graphical Display Options is Color Gradient. If you want
                   to see the graphs in black and white, check the radio button next to For Export (BW
                   Printers).

           •   Click the OK button to continue or the  Cancel button to cancel the Goodness-of-Fit tests.
94

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Output Screen for Gamma Distribution (Log-ROS Estimates)
Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient.
                                    Gamma Q-Q Plot for Mercury
                                 Statistics using Robust ROS Method
                                                                                      Mean = 0.2767
                                                                                      K Star = 1.0653
                                                                                      Slope = 1.1071
                                                                                      Intercept = -0.0262
                                                                                      Correlation, R = 0.9579
                                                                                      Anderson-Darling Test
                                                                                      Test Statistic = 1.425
                                                                                      Critical Value(O.OS) = 0.772
                                                                                      Data not Gamma Distributed
                                                                                      0.0053 - 0.0400
                                                                                      00165-0.0500
                                                                                      0.0326 - 0.0550
                                                                                      0.0326 - 0.0550
                                                                                      0.0492-0.0600
                                                                                      0.0607 - 0.0645
                                                                                      0.0765 - 0.0700
                                                                                      0.07B5 - 0.0700
                                                                                      00974-0.0367
                                                                                      0.1107-0.0922
                                                                                      1.1000-0.9900
                                  Theoretical Quantiles of Gamma Distribution
Note: The legend size ofnondetect values is smaller than that of detected values and is shown in red.
8.5    Goodness-of-Fit Tests with DL/2  Estimates
1       Click Goodness-of-Fit ^ With NDs ^ DL/2 Estimates
P? ProUCL 4.0 - [WorkSheet.wst]
•S File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests f
E|
Normal-ROS Estimates >
Gamma-ROS Estimates >
Log-ROS Estimates >
G.O.F. Statistics








UCL Window Help

8 9




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

            •   To test the variable for normality, click on Normal from the drop-down menu list.
            •   To test the variable for lognormality, click on Lognormal from the drop-down menu list.
                                                                                                        95

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

8.5.1   Normal or Log normal Distribution (DL/2 Estimates)

1.       Click Goodness-of-Fit ^ With NDs ^ DL/2 Estimates ^ Normal or Lognormal
B? ' ProUCL 4.0 - {Worksheet, wsJJ
   File Edit Configure Summary Statistics  ROS Est. NDs Graphs  Gutter Tests
Hypothesis Testing  Background  UCL  Window  Help
                                                       Full
 Navigation Panel
 Name
 -if Worksheet .wst


1
2
3
4

Arsenic D_Arsen-ie
! 1! 0
	 T o
1.7 1
: 1 0
 Exdude NDs       >
 Normal-ROS Estimates >
 Gamma-ROS Estimates > J
 Log-ROS Estimates   >
I23^S^H
 G.O.F. Statistics
                                                                                   Normal
                                                                                   Gamma
2.      The Select Variables Screen (Chapter 3) will appear.

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

           •   If graphs have to be produced by using a Group variable, then select a group variable by
               clicking the arrow below the Group by variable button. This will result in a drop-down
               list of available variables. The user should select and click on an appropriate variable
               representing a group variable.

           •   When Normal or Lognormal button is clicked, following window is displayed.
96

-------
                    '•Select Confidence Level
                           T 90%

                           <¥ 95%

                           r 99%


                     Method

                       <• ShapiroWilk

                       T Lilliefors


                    'Display Regression Lines

                       f" Do Not Display

                       '** Display Regression Lines


                    - Graphs by Group •

                       & Individual Graphs

                       f" Group Graphs


                    - Graphical Display Options -

                       f*" Color Gradient

                       T For Export (BW Printers)
                      OK
Cancel
o   The default option for the Confidence Level is 95%.
o   The default Method is Shapiro Wilk. If the sample size is greater than 50, the
    program defaults to the Lilliefors test.
o   The default method for Display Regression Lines is Do Not Display. If you want to
    see regression lines on the normal Q-Q plot, check the radio button next to Display
    Regression Lines.
o   The default option for Graphs by Group is Individual Graphs. If you want to put
    all of the selected variables into a single graph, check the radio button next to Group
    Graphs.
o   The default option for Graphical Display Options is Color Gradient. If you want
    to see the graphs in black and white, check the radio button next to  For Export (BW
    Printers)
                                                                                97

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

              •    Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.

Output Screen for Normal Distribution (DL/2 Estimates)
Selected options: Shapiro Wilk, Display Regression Lines, Group Graphs, and Color Gradient.
                                          Normal Q-Q Plot for Group
                                         Statistics using DL/2 Method
            -* Arsenic (subsurface)
                                        Theoretical Quantiles (Standard Normal)
                                                               •j Arsenic (surface)
Arsenic (subsurface)
 N-10
 Mean = 5.3350
 3d = 12180
 Slope = 1.1396
 Intercept = 5.3350
 Correlation, R = 0.8792
 Shapiro-VMIk Statistic
 Test Value = 0.803
 Critical Val(0.05) = 0.842
 Data not Normal
Arsenic (surface)
 N = 10
 Mean = 5.7450
 Sd = 2.2592
 Slope-2.2938
 Intercept = 5.7450
 Correlaiion, R = 0.9547
 Shapiro-VMIk Statistic
 Test Value - 0.909
 Critical Val(0.05) = 0.842
 Data appear Normal
Note: The legend size ofnondetect values is smaller than that of the detected values and is shown in red.
98

-------
Output Screen for Lognormal Distribution (DL/2 Estimates)
Selected options: Shapiro Wilk, Display Regression Lines, Individual Graphs, and For Export (BW Printers).
        0.00
        4.20
        J1.40
        -0.60
        -0.80
        -1.00
      flS -1.40
      
           Normal-ROS Estimates  >
           Gamma-ROS Estimates  >
           Log-ROS Estimates    >
2.      The Select Variables Screen (Chapter 3) will appear.

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

             •    If graphs have to be produced by using a Group variable, then select a group variable by
                 clicking the arrow below the Group by variable button. This will result in a drop-down
                                                                                                          99

-------
               list of available variables. The user should select and click on an appropriate variable
               representing a group variable.

           •   When the Gamma option button is clicked, the following window will be shown.
                                                                    . in..
                                                                    E3

                                 r Select Confidence Level	
                                        T 90%

                                        P 95 %

                                        r m %


                                 ."Method	

                                    (•" Anderson Darling

                                    '*"** KolmogGravSmirnew

                                 i Display Regression Lines	

                                    r Do Not Display

                                    {•" Display Regression Lines

                                 .-•"Graph by Groups	
                                    <•* Individual Graphs

                                    f" Group Graphs


                                  Graphical Display Options	

                                    (* Color Gradient

                                    r For Export (BW Printers)
                                     OK
Cancel
                   The default option for the Confidence Level is 95%.
                   The default Method is Anderson Darling.
                   The default method for Display Regression Lines is Do Not Display. If you want to
                   see regression lines on the normal Q-Q plot, check the radio button next to Display
                   Regression Lines.
                   The default option for Graph by Groups is Individual Graphs. If you want to put
                   all of the selected variables into a single graph, check the radio button next to Group
                   Graphs.
100

-------
                 o   The default option for Graphical Display Options is Color Gradient. If you want
                     to see the graphs in black and white, check the radio button next to For Export (BW
                     Printers).

            •    Click the OK button to continue or the Cancel button to cancel the Goodness-of-Fit tests.
Output Screen for Gamma Distribution (DL/2 Estimates)
Selected options: Anderson Darling, Display Regression Lines, Individual Graphs, and Color Gradient.
                                     Gamma Q-Q Plot for Mercury
                                Statistics using DL/2 Substitution Method
          " 060
          O

          w 05°
          a
          o
          TJ
          £ 040
                                                     Mercury
                                                      hi = 30
                                                      Mean = 0 2829
                                                      k star - 1.0875
                                                      Slope = 1.0897
                                                      Intercept = -0.0220
                                                      Correlation, R = 0.9617
                                                      Anderson-Darling Test
                                                      Test Statistic = 1.126
                                                      Critical Value(0.05) = 0.771
                                                      Data not Gamma Distributed
                                   Theoretical Quantiles of Gamma Distribution
Note: The legend size ofnondetect values is smaller than that of the detected values and is shown in red.
8.6    Goodness-of-Fit Tests Statistics
    1   Click Goodness-of-Fit >  With NDs > G.O.F. Statistics
P3 ProUCL 4.0 - [WorkSheet.wst]
   File  Edit  Configure  Summary Statistics ROS Est. NDs Graphs  Outiier Tests
                                     Hypothesis Testing  Background  UCL  Window Help
 Navigation Panel
 Name
   WorkSheet.wst
 Hi GOF_Stats_wND ost
                              Arsenic
                                        1
                                     D Arsenic
  i
	i"
 1.7
Exdude NDs
Normal-ROS Estimates
Gamma-ROS Estimates
Log-ROS Estimates
DL/2 Estimates
    2.  The Select Variables Screen (Chapter 3) will appear.
            •    Select one or more variable(s) from the Select Variables screen.
                                                                                                        101

-------
                When the option button is clicked, the following window will be shown.
                                          Select Confidence Level
                                          .Select Confidence Level
                                                    957,

                                                    99%
                                               OK
Cancel
            •   The default confidence level is 95%.

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

Sample Output Screen for G.O.F. Test Statistics
                          User Selected Options
                                From File WorkSheetwst
                              Full Precision OFF
                          Confidence Coefficient 0.95

Arsenic Date

Statistics (Non-Detects Only)
Statistics (Detects Only)
Statistics (All: NDs treated as DL value)
Statistics (All: NDs treated as DL2 value)
Statistics (Normal ROS Estimated Data)
Statistics (Gamma ROS Estimated Data)
Statistics (Lognormal ROS Estimated Data)

Statistics (Detects Only)
Statistics (NDs-DL)
NumObs
24
Number
13
11
24
24
24
24
24
KHat
2.257
3.538
Num Miss
0
Minimum
0.9
0.5
0.5
0.45
-0.0995
0.5
0.349
KStar
2.002
3.124
Num Valid
24
Maximum
2
3.2
3.2
3.2
3.2
3.2
3.2
TNeta Hat
0.548
0.406
Detects
11
Mean
1.608
1.236
1.438
1.002
0.997
1.263
0.972
Log Mean
-0.0255
0.215
NDs
13
Median
2
0.7
1.25
0.95
0.737
1.213
0.7
Log Stdv
0.694
0.574
% NDs
54.17%
SO
0.517
0.965
0.761
0.699
0.776
0.652
0.718
LogC.V.
-27.26
2.669
102

-------
Output Screen for the G.O.F. Test Statistics - (continued)
                                                                    Normal Distri Indian Test Results
                                                                         Test value  Crit. (0.95)
                                                                                                     Conclusion with Alpha(0.05)
                                                Shapiro-Wilks (Detects Only)
                                                    Lilliefors (Detects Only)
                                                  Shapiro-Wilks (NDs = DL)
                                                       Lilliefors (NDs-DL)
                                                 Shapiro-Wilks (NDs = Db'2)
                                                     Lilliefors (NDs .DL^)
                                       Shapiro-Wilks (Normal ROS Estimates)
                                            Lilliefors (Normal ROS Estimates)
0.777      0.85    Data Not Normal
0.273      0.267   Data Not Normal
0.89       0.916   Data Not Normal
0.217      0.181   Data Not Normal
0.701      0.916   Data Not Normal
0.335      0.181   Data Not Normal
0.868      0.916   Data Not Normal
0.17       0.181   Data Appear Normal
                                                                    Gamma Distri tuition Test ResuHs
                                                                         Test value  Crit. (0.95)
                                                                                                     Conclusion with Alpha(0.05)
                                             Anderson-Darling (Detects Only)
                                          Kolmogorov-Smirnov (Detects Only)
                                               Anderson-Darling (NDs - DL)
                                             Kolmogorov-Smirnov (NDs = DL)
                                              Anderson-Darling (NDs = Db'2)
                                           Kolrrwgorov-Smirnov (NDs = DU2)
                                    Anderson-Darling (Gamma ROS Estimates)
                                      Kolmoporoy-Smirnov (Gamma  ROS Est.)
0.787      0.738
0.254      0.258   Data appear Approximate Gamma Distribution
0,98       0.75
0.214      0.179   Data Not Gamrra Distributed
1.492      0.751
0.261      0.179   Data Not Gafrn-a Distributed
0.524      0.747
0.127      0.179   Data Appear Gamma Distributed
                                                                                                                                                               103

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                                       Chapter 9


             Single Sample and Two-Sample Hypotheses
                               Testing Approaches


This chapter illustrates single sample and two-sample parametric and nonparametric hypotheses testing
approaches as incorporated in ProUCL 4.0. ProUCL 4.0 can perform these hypotheses tests on data sets
with and without nondetect observations. It should be pointed out that, when one wants to use two-sample
hypotheses tests on data sets with NDs, ProUCL 4.0 assumes that samples from both of the groups have
nondetect observations. All this means that, a ND column (with 0 or 1 entries only) needs to be provided
for the variable in each of the two groups. This has to be done even if one of the groups has all detected
entries; in this case the associated ND column will have all entries equal to "1." This will allow the user
to compare two groups (e.g., arsenic in background vs. site samples) with one of the groups having some
NDs and the other group having all detected data.

9.1    Single Sample Hypotheses  Tests

In many environmental applications, single sample hypotheses tests are used to compare site data
(provided enough site data are available) with pre-specified cleanup standards or compliance limits.
ProUCL 4.0 contains single sample parametric and nonparametric tests including Student's t-test, sign
test, Wilcoxon Signed Rank (WSR) test, and test for proportion. The single sample hypotheses tests are
useful when the environmental parameters such as the clean standard, action level, or compliance limits
(CLs) are known, and the objective is to compare site concentrations with those known threshold values.
Specifically, a t-test (or a sign test) may be used to verify the attainment of cleanup levels at an AOC after
remediation activity; and a test for proportion may be used to verify if the proportion of exceedances of an
action level (or a compliance  limit) by sample concentrations collected from an AOC (or a MW) exceeds
a certain specified proportion (e.g., 1%, 5%,
ProUCL 4.0 can perform these hypotheses on data sets with and without nondetect observations.
However, it should be noted that for single sample hypotheses tests (e.g., sign test, proportion test) used
to compare site mean/median concentration level with a cleanup standard, Cs, or a compliance limit (e.g.,
proportion test), all NDs (if any) should lie below the cleanup standard, Cs. For proper use of these
hypotheses testing approaches, the differences between these tests should be noted and understood.
Specifically, a t-test or a WSR test are used to compare the measures of location and central tendencies
(e.g., mean, median) of a site area (e.g., AOC) to a cleanup standard, Cs, or action level also representing
a measure of central tendency (e.g., mean, median); whereas, a proportion test compares if the proportion
of site observations from an AOC exceeding a compliance limit (CL) exceeds a specified proportion, P0
(e.g., 5%, 10%). ProUCL 4.0 has useful graphical methods that may be used to visually  compare the
concentrations of a site area of concern (AOC) with an action level. This can be done using a box plot of
site data with action level superimposed on that graph. The details of the various single sample
hypotheses testing approaches can be found in EPA guidance documents (1989, 2006). A brief discussion
of these methods is also given in the ProUCL 4.0 Technical Guide.
104

-------
9.1.1   Single Sample Hypothesis Testing for Full Data without Nondetects

1.      Click Hypothesis Testing^- Single Sample
P? ProUCL 4.0 - [WorkSheet.wstJ
   File Edit  Configure  Summary Statistics ROS Est. NDs Graphs  Outlier Tests Goodness-of-Fit
 Navigation Panel
 Name
  1 Sample-Prop
i	"	TiS
                                                             HjpoSiessTesSng
                                                               Single Sample  M  Full (w/o NOsJ
                                                 Background UCL Window  Help
                                                              t-Test
                                      Two Sample  >  With NDs    >'  Proportion
                                      _,,5—t—-8 «™ ' !	-7—  sjgn test

                                                              Wilcoxon Signed Rank
                                                                                                   r
 £J WorkSheet.wst     |

2.      Select Full (w/o NDs) - This option is used for full data sets without nondetects.

            •   To perform a t-test, click on t-Test from the drop-down menu as shown above.

            •   To perform a proportion test, click on Proportion from the drop-down menu.

            •   To run a sign test, click on Sign test from the drop-down menu.

            •   To run a Wilcoxon Signed Rank test, click on Wilcoxon Signed Rank from the drop-
                down menu.

9.1.1.1 Single Sample t-Test

1.      Click Hypothesis Testing ^ Single Sample ^ Full (w/o NDs) ^ t-Test
R ProUCL 4.0 - [Worksheet.wst]
File Edit  Configure  Summary Statistics ROS Est, NDs Graphs  Outlier Tests Goodness-of-Fit KlCT^^^SB-l Background UCL  Window  Help

                                                            Two Sample  ^  With NDs    >  Proportion

                                                                                    Wilcoxon Signed Rank
 Navigation Panel
 Name
 @ WorkSheet.wst
                      1
     3         1
  1Sample-Prop
        Tl9i
2.      The Select Variables Screen (see page 130) will appear.

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

            •   When the Options button is clicked, the following window will be shown.
                                                                                                      105

-------
                         II Single, Saiifple t Test pptipns
                                                            0.95
                                                             I
          Confidence Level  |

    Substantial Difference. S T~
      (Used with Test Form 2)

          Compliance Limit  j

^Select Null Hypothesis Form

 ***"  Mean <= Compliance Limit (Form 1)


 f"  Mean >= Compliance Limit (Form 2)
                              Mean >= Compliance Limit + S (Form 2)
                              Mean = Compliance Limit (2 Sided Alternative)
                                    OK
                              Cancel

               o   Specify the Confidence Level; default is 0.95.
               o   Specify meaningful values for Substantial Difference, S and the Compliance Limit.
                   The default choice for S is "0."
               o   Select the form of Null Hypothesis; default is Mean <= Compliance Limit (Form 1).
               o   Click on OK button to continue or on Cancel button to cancel the test.
106

-------
Output for Single Sample t-Test (Full Data without NDs)
                          1 Sample-1
                                               Si ngle Sample t-Test

                                                 Raw Statistics
                                             N umber of Va I id Samples     9
                                           Number of Distinct Samples     9
                                                          Minimum    82.39
                                                          Maximum   113.2
                                                             Mean    99.38
                                                            Median.   103.5
                                                               SD    10.41
                                                        SE of Mean     3.468

                          HO: Site Mean = 100

                                                        Test Value   -0.178
                                        Two Sided Critical Value (0.05)     2.306
                                                           P-Value     0.863

                          Concl usion with Alpha = 0.05
                            Do Not Reject HO. Conclude Mean = 100
                            P-Value>Alpha(O.Og
9.1.1.2 Single Sample Proportion Test
1.      Click Hypothesis Testing ^  Single Sample ^- Full (w/o NDs) ^- Proportion
P? jPrffUg. $.Qr- fWockSheet,wst j.
«B File Edit  Configure  Summary Statistics ROS Est, NDs Graphs Outlier Tests  Goodness-of-Fit
    0           1
1Samp!e-Prop
        413
       5.33S6
                                   HwotteasTesBuo:-
                                                                 SoSe:Sample M :ft»ft*/ofC>sJ '*
 Navigation Panel
 Name
Background UCL  Window  Help
        	|  t-Test            j

           f   Sign test           !
              Wilcoxon Signed Rank   |
 v. Worksheet wst
2.      The Select Variables Screen will appear.

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

            •    When the Options button is clicked, the following window will be shown.
                                                                                                       107

-------
                                    ^'••"••i-"""""--^--?-''--;;^^
                                                           3.95
                                                            0.3
Confidence Level
      Proportion
                              Action/Compliance Limit
                              Select Null Hypothesis Form
                               C*  P <= Porportion (Form 1)
                                  P >= Proportion (Form 2)
                                  P = Proportion (2 Side Alternatived)
OK
Cancel

in
               o  Specify the Confidence level; default is 0.95.
               o  Specify the Proportion level and a meaningful Action/Compliance Limit.
               o  Select the form of Null Hypothesis; default is P <= Proportion (Form 1).
               o  Click on OK button to continue or on Cancel button to cancel the test.
108

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Output for Single Sample Proportion Test (Full Data without NDs)
                                         O ne-Sample Proportion Test

                                                Raw Statistics
                                           N umber of Val id Samples    B5
                                        Number of Distinct Samples    83
                                                         Minimum     0.598
                                                        Maximum     7.676
                                                            Mean     5.183
                                                          Median     5.564
                                                              SD     1.588
                                                      SE of Mean     0.172
                                           Number of Exceedances    27
                                  Sample Proportion of Exceedances     0.318

                        HO: Site Proportion <= 0.3  (Form 1)

                                         Large Sample z-Test Val ue     0.237
                                               Critical Value (0.05)     1.645
                                                         P-Value     0.406

                        Conclusion with Alpha =0.06
                          Do Not Reject H0. ConcIude Site Proportion <= 0.3

                          P-Vali*e> Alpha (0.05)
9.1.1.3 Single Sample Sign Test
1.      Click Hypothesis Testing ^- Single Sample ^- Full (w/o NDs) ^- Sign test
   File Edit  Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests  Goodness-of-Rt
                                    ^pottffisis: Testog.
                                                               Sfrrife'Sflrkfc M
 Navigation Panel
 Name
  D      1
Sigr Test
                                          234
                                        Arsenic  D_Arsenic
          Background UCL  Window Heip
                  	  t-Test            |
                  -	i—mnir                 |
Two Sample  *   With NDs    *  Proportion         j................

                        Wilcoxon Signed Rank   !
 ^JWorkSheet-wst
2.      The Select Variables Screen will appear.

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

            •    When the Options button is clicked, the following window will be shown.
                                                                                                        109

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                                                               0.95
             Confidence Level   j

       Substantial Difference. S  I
          [Used with Test Form 2)    '

      Action/Compliance Limit   )

Select Null Hypothesis Form

I*" Median <= Compliance Limit [Form 1)


»*"* Median >= Compliance Limit  (Form 2)
                          C~* Median >= Compliance Limit + S (Form 2)
                          l*"* Median = Comtpl iartce Li mit (2 Sided Alternative)
                                  OK
                                Cancel
               o   Specify the Confidence Level; default choice is 0.95.
               o   Specify meaningful values for Substantial Difference, S and Action/Compliance
                   Limit.
               o   Select the form of Null Hypothesis; default is Median <= Compliance Limit (Form
                   1).
               o   Click on OK button to continue or on Cancel button to cancel the test.
110

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Output for Single Sample Proportion Test (Full Data without NDs)
                                         Si ngle Sample Sign Test

                                             Raw Statistics
                                         N umber of Val id Samples    10
                                       Number of Distinct Samples    10
                                                      Minimum   750
                                                      Maximum  1161
                                                         Mean   925.7
                                                        Median   888
                                                           SD   136.7
                                                    SE of Mean    43.24
                                             Number Above Limit     3
                                             Number Equal Limit    0
                                             Number Below Limit     7
                       HO: Site Median >= 1000 (Form 2)

                                                     Test Value
                                        Lower Critical Value (0.05)
                                                       P-Value

                       Conclusion with Alpha =0.05
                        Do Hot Reject HO. Conclude Median >= 1000
                        P-Value>AJpbaCQ05)
                                                          3
                                                          1
                                                          0.172
9.1.1.4 Single Sample Wilcoxon Signed Rank (WSR) Test
1.      Click Hypothesis Testing ^- Single Sample ^-  Full (w/o NDs)  ^- Wilcoxon Signed Rank
        F? ProUCL 4.0 - [WorkShest_a,wst]
           File  Edit Configure  Summary Statistics ROS Est, NDs  Graphs  Outlier Tests  Goodness -of-Fit
                                                          jrtwethesasTesfcg,
                                                                    SngteSannle
         Navigation Panel
         Name


1
2
0
WSR1
! ' 97i
1Wi
1 2 3
WSR2
52B
5.89
4
Arsenic D
1
1
          Background  UCL Window Help
                       t-Test
Two Sample   >   With NDs    >  Proportion
2.
         iy Worksheet wst
         i,?WorkSheet_a.wst
The Select Variables Screen will appear.

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

    •   When the Options button is clicked, the following window will be shown.
                                                                                                    Ill

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                    3H Single Sample Wilcoxon Signed Rank Test Options    X
                                    Confidence Level ]      095
                             Substantial Difference. S
                               (Used with Test Form 2)
                             Act i on/Com pliance Limit

                         Select Null Hypothesis Form —
                           Mean/Median <= Compliance Limit (Form 1)
                           Mearv'Median >= Compliance Limit (Form 2)
                           Mearv'Median >= Compliance Limit + S  (Form 2)
                           Meanj'Median = Compliance Limit (2 Sided Alternative)
                                   OK
Cancel
               o  Specify the Confidence Level; default is 0.95.
               o  Specify meaningful values for Substantial Difference, S, and Action/Compliance
                  Limit.
               o  Select the form of Null Hypothesis; default is Mean/Median <= Compliance Limit
                  (Form 1).
               o  Click on OK button to continue or on Cancel button to cancel the test.
112

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

                                             Raw Statistics
                                          Number of Valid Samples 10
                                        Number of Distinct Samples 10
                                                       Minimum 75S
                                                       Maximum 1161
                                                         Mean'925.7
                                                        Median SSS
                                                           SD 136.7
                                                     SE of Mean 4124
                                              Number Above Limit 3
                                               Number Equal Limit 0
                                              Number Below Limit 7
                                                         T-plus 11.5
                                                        T-minus 43.5

                       HO: SHe Median <= 1000  (Form  1)

                                                     Test Value 11.5
                                              Critical Value (0.05).45
                                                        P-Value'O.S47

                       Conclusion with Alpha = 0,05
                         Do Not Reject HO. Conclude Mean/Median <= 1000
                         P-Value > Alpha (0.05)
                                                                                                     113

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9.12   Single Sample Hypothesis Testing for Data Sets with Nondetects

Most of the one-sample tests such as the Proportion test and the Sign test on data sets with nondetect
values assume that all nondetect observations lie below the compliance limit (CL) or an action level, A0.
The single sample tests cannot be performed if ND observations exceed the CL or action levels.

1.      Click on Hypothesis Testing^- Single Sample
            {WprkSheet.wstJ
i File Edit Configure  Summary Statistics ROS Est. NDs  Graphs  Outlier Tests  Goodness-of-Fit
    ejLj|m| ml
                                                           Hypothesis Testing
 Navigation Panel
| Background IICL  Window Help
  Full [w/o NDs) > t
 Name
                                                                                   Wilcoxon Signed Rank
 "ir Worksheet wst
2.       Select the With NDs option

           •   To perform a proportion test, click on Proportion from the drop-down menu.

           •   To perform a sign test, click on Sign test from the drop-down menu.

           •   To perform a Wilcoxon Signed Rank test, click on Wilcoxon Signed Rank from the drop-
               down menu list.

9.1.2.1 Single Proportion Test on Data Sets with NDs

1.       Click Hypothesis Testing >• Single Sample >• With NDs ^ Proportion
E? ProUCL 4.0- [C:\Narain\ProUCL-Data\Data\WSREPA (2006).wst]
D o s|a|mj uj
Navigation Panel
Name
-i WoikSheet >ist

1
2

0
V/SR1
9 4
1 544

1 2 3
WSR2
5BB
5S9
KfaBBSSSlffla HwxrthessTesfcrw?
^^E^^S^B
Arsenic D_Arsenic
1 0
1 0
^^^^^^^^
Background UCL Window Help
••npHH^i^^l
"'"* "^ ''" V S>gn test 11
wilcoxop Signed Rarf |
2.      The Select Variables screen will appear.

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

           •   When the Options button is clicked, the following window will be shown.
114

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              Single Sample Proportion Test Options   X
                      Confidence Level  [    0-95
                            Proportion
               Action/Compliance Limit
   0.3
                Select Null Hypothesis Form —

                 <•" P <= Porportion [Form 1)


                 C' P >= Proportion [Form 2)


                 C P = Proportion (2 Side Alternatived)
                     OK
Cancel
o   Specify the Confidence Level; default is 0.95.
o   Specify meaningful values for Proportion and the Action/Compliance Limit.
o   Select the form of Null Hypothesis; default is P <= Proportion (Form 1).
o   Click on OK button to continue or on Cancel button to cancel the test.
                                                                               115

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Output for Single Sample Proportion Test (with NDs)
                  Arsenic
                                  Single Sample Proportion Test

                                         Raw Statistics
                                    Number of Valid Samples   24
                                  N umber of Disti net Samples   10
                                   N umber of Non-Detect Data   13
                                     Number of Detected Data   11
                                         Percent Norv Detects 54.17%
                                         Minimum Non-d«tect    0.9
                                         Maximum Men-detect    2
                                          Minimum Detected    0.5
                                          Maximum Detected    3.2
                                       Mean of Detected Data    1.236
                                     Median of Detected Data    0.7
                                         SD of Detected Data    0.965
                                     N urnber of Exceedances    2
                            Sample Proportion of Exceedances    0.0833

                                 Some Men- Detect Values Exceed
                            The User          Aefan/Compiance limit
                         Unable to do Proportion Test wit i such
9.1.2.2 Single Sample Sign Test with NDs
1.      Click Hypothesis Testing ^- Single Sample ^- With NDs ^-  Sign test
           - BrY0rkSljeet.wstJ
  > File Edit Configure Summary Statistics  ROS Est, NDs  Graphs  Outiier Tests Goodness-of-Fit n^gy^^^gM Background  UCL Window Help
               Fp|                                           H^S|rKij5im   Full (w/o NDs)  t s
                                                              Two Sample   > K]32E2^M1   Proportion
 Navigation Panel
 Name
 iiJWortcSheet.wst
  01234
Sign Test         Arsenic  D_Arsemc
   'T'S             1      0
   1 D44              10
                                                             -•9
2.      The Select Variables screen will appear.

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

            •   When the Options button is clicked, the following window will be shown.
116

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         1 Single Sample Sign Test Options
                        Confidence Level       0.95
                  Substantial Difference. S
                     (Used with Test Form 2}
                 Action/Compliance Limit   |

           Select Null Hypothesis Form -
           (*  Median <= Compliance Limit (Form 1)
           ("" Median >= Compliance Limit [Form 2)
           (*" Median >= Compliance Limit + S (Form 2)
              Median = Compliance Limit [2 Sided Alternative)
                   OK
Cancel
o   Specify the Confidence Level; default is 0.95.
o   Specify meaningful values for Substantial Difference, S and Action/Compliance
    Limit.
o   Select the form of Null Hypothesis; default is Median <= Compliance Limit (Form
    1).
o   Click on OK button to continue or on Cancel button to cancel the test.
                                                                               117

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Output for Single Sample Sign Test (Data with Nondetects)
                             Arsenic
                                               Single Sample Sign Test

                                                   Raw Statistics
                                               Number of Valid Samples    24
                                             N umber of Disti net Samples    10
                                             Number of Non-Detect Data    13
                                               N umber of Detected Data    11
                                                   Percent Non-Detects 54.17%
                                                   Minimum Non-detect     0.9
                                                   Maximum Non-detect     2
                                                    Minimum Detected     0.5
                                                    Maximum Detected     3.2
                                                 Mean of Detected Data     1,23€
                                                Median of Detected Data     0.7
                                                   SD of Detected Data     0.965
                                                   Number Above Limit    0
                                                   Number Equal Limit    0
                                                   Number Below Limit    24
                             HO: Site Median <= 5 (Forml)

                                                          Test Value    0
                                              Upper Critical Value (0.05)    17
                                                             P-Value     1

                             "oncl its ion with Alpha = 0.06
                              Do Not Reject HO. Conclude Median <= 5
                              P-Value> Alpha (005)
9.1.2.3 Single Sample Wilcoxon Signed Rank Test with NDs
1.       Click Hypothesis Testing ^  Single Sample ^- With NDs ^- Wilcoxon Signed Rank
P?iPrpUCL 4,0 - [V^orkSheet_a.wstJ
  ? FileEdit Configure  Summary Statistics ROS Est, NDs  Graphs Outlier Tests Goodness-of-Fit I

	,.!	A	1_J	1 ^±—I ^                                           —  jwo 3amp|e
 Navigation Panel
 Name
 43 Worksheet, wst
 OWorkSheet_a.wst


1
2
0 1
V/SR1
! 	 ^
1.044
2 3
WSR2
5.BB
589
4
Arsenic D
1
1
Background UCL Window Help
  Full Q../Q »Ds) > i
 [2222^^^B   Proportion
  '  T   ''    V   sign test
118

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2.      The Select Variables Screen will appear.

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

           •   When the Options button is clicked, the following window will be shown.
                    BH Single Sample Wilcoxon Signed Rank Test Options
                                    Confidence Level

                             Substantial Difference. S
                                (Used with Test Form 2)

                             Action/Compliance Limit

                         Select Null Hypothesis Form -
   0.55
                         <* Mean/Median C= Compliance Limit (Form 1)
                           Mean/Median >= Compliance Limit (Form 2)
                           Mean/Median >= Compliance Limit + S (Form 2)
                           Meanj'Median = Compliance Limit [2 Sided Alternative)
                                   OK
Cancel
               o   Specify the Confidence Level; default is 0.95.
               o   Specify meaningful values for Substantial Difference, S and Action/Compliance
                   Limit.
               o   Select the form of Null Hypothesis; default is Mean/Median <= Compliance Limit
                   (Form 1).
               o   Click on OK button to continue or on Cancel button to cancel the test.
                                                                                             119

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Output for Single Sample Wilcoxon Signed Rank Test (Data with Nondetects)
                               Arsenic
                                          Single Sample WiIcoxon Signed Rank Test

                                                      Raw Statistics
                                                 Number of Valid Samples   24
                                               N umber of Disti net Samples   10
                                               Number of Non-Defect Data   13
                                                 N umber of Detected Data   11
                                                     Percent Non-Defects 54.17%
                                                     Minimum Non-detect    0.9
                                                     Maximum Non-detect    2
                                                       Minimum Detected    0.5
                                                      Maximum Detected    3.2
                                                   Mean of Detected Data    1.236
                                                  Median of Detected Data    0.7
                                                     SD of Detected Data    Q.9S5
                                                     N umber Above Li mil    0
                                                      Number Equal Limit    0
                                                     Number Below Limit   24
                                                                T-plus    0
                                                               T-minus  300

                               HO: Site Median <=6  (Forml)

                                               Large Sample z-Test Value  -4.293
                                                     Critical Value (0.05)    1.645
                                                               P-Value    1

                               Corel us ion with Alpha = 0.05
                                Do Not Reject HO. Conclude Mean/Median <= 6
                                P-Val we > Alpha (0.05)
                               Data set con tains multiple Nan-Detect values!
                                 All Observations <2 ate treated as Nan-Detects
120

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9.2   Two-Sample Hypotheses Testing Approaches

In this section, the two-sample hypotheses testing approaches as incorporated in ProUCL 4.0 have been
illustrated. These approaches are used to compare the parameters and distributions of the two populations
(e.g., Background vs. AOC) based upon data sets collected from those populations. Both forms (Form 1
and Form 2, Form 2 with Substantial Difference, S) of two-sample hypothesis testing approaches have
been included in ProUCL 4.0. The methods are available for full data sets as well as for data sets with
below detection limit (BDL) values.

           •   Full - analyzes data sets consisting of all detected values. The following parametric and
               nonparametric tests are available:

               o  Student's t and Satterthwaite tests to compare the means of two populations (e.g.
                  Background versus AOC).
               o  F-test to the check the equality of dispersions of two populations.
               o  Two-sample nonparametric Wilcoxon-Mann-Whitney (WMW) test. This test is
                  equivalent to Wilcoxon Rank Sum (WRS) test.
               o  Quantile test is often used to compare upper tails of two data distributions. This test
                  is normally performed in parallel with WMW test.

           •   With NDs - analyzes data sets consisting of both nondetected and detected values. The
               following tests are available:

               o  Wilcoxon-Mann-Whitney test. All observations (including detected values) below the
                  highest detection limit are treated as ND (less than the highest DL) values.
               o  Quantile test is used to compare upper tails of two  data distributions. This test is
                  performed in parallel with WMW test.
               o  Gehan's test, useful when multiple detection limits may be present.

The details of these methods can be found in the ProUCL 4.0 Technical Guide and are also available in
EPA (1997, 2006). It is re-stated that the use  of informal graphical displays (e.g., side-by-side box plots,
multiple Q-Q plots) should always accompany the formal hypothesis testing approaches listed above.
This is especially warranted when the data sets may consist of observations from multiple populations
(e.g., mixture samples collected from various onsite locations) and outliers.

Note: As mentioned before, it is pointed out that, when one wants to use two-sample hypotheses tests  on
data sets with NDs, ProUCL 4.0 assumes that samples from both of the groups have nondetect
observations. This may not be the case,  as data from a polluted site may not have any ND observations.
ProUCL  can handle such data sets. However, the user will have to provide a ND column (with 0 or 1
entries only) for the selected variable of each of the two groups. Thus when one of the groups (e.g., site
arsenic) has no ND value, the user supplies an associated ND column with all entries equal to  "1. " This
will allow the user to compare two groups (e.g., arsenic in background vs. site samples) with one of the
group having some NDs and the other group having all detected data.
                                                                                           121

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9.2.1   Two-Sample Hypothesis Tests for Full Data

Full - This option is used to analyze data sets consisting of all detected values. The following two-sample
tests are available in ProUCL 4.0.

           •   Student's t and Satterthwaite tests to compare the means of two populations (e.g.,
               Background versus AOC).

           •   F-test is also available to test the equality of dispersions of two populations.

           •   Two-sample nonparametric Wilcoxon-Mann-Whitney (WMW) test.

           •   Two-sample quantile test.

           •   Student's t-Test

               o  This test can be used to compare the site mean concentration of a COPC with that of
                  the background mean concentration provided the populations are normally
                  distributed. The data sets are given by independent random observations, Xu X2, . . .,
                  Xn collected from a site, and independent random observations, YI, Y2, . . ., Ym
                  collected from a background population. The same terminology is used for all other
                  two-sample tests in ProUCL 4.0.
               o  Student's t-test also assumes that the spread (variance) of the two populations are
                  approximately equal.
               o  The F-test can be used to the check the equality of dispersions of two populations.

           •   Satterthwaite t-Test

               o  This test is used to compare the population means of two populations when the
                  variances or Spreads of those populations may not be equal. As mentioned before, the
                  F-distribution based test can be used to verify the equality  of dispersions of two
                  populations.

           •   Test for Equality of two Dispersions (F-test)

               o  This test is used to determine whether the true underlying variances of two
                  populations are equal. Usually the F-test is employed as  a preliminary test, before
                  conducting the two-sample t-test for testing the equality  of means of two populations.
               o  The assumptions underlying the F-test are that the two-samples represent
                  independent random samples from two normal populations. The F-test for equality of
                  variances is highly sensitive to departures from normality.

           •   Two-Sample Nonparametric WMW Test

               o  This test is used to determine the approximate equality of the two continuous data
                  distributions. This test also assumes that the shapes  (e.g., as determined by spread,
                  skewness, and graphical displays) of the two populations are roughly equal. The test
                  is often used to determine if the measures of central locations of the two populations
122

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                   are significantly different. Specifically, the test can be used to determine if the site
                   concentrations exceed the background concentrations.
               o   The Wilcoxon-Mann-Whitney test does not assume that the data are normally or log-
                   normally distributed. For large samples (e.g., > 20), the distribution of the WMWtest
                   statistic can be approximated by a normal distribution.
               o   This test is used to determine if measurements from one population consistently tend
                   to be larger (or smaller) than those from the other population.

           •   Two-Sample Quantile Test

               o   The nonparametric quantile test does not  assume that the data are normally or log-
                   normally distributed. For large samples (e.g., > 20), the distribution of the quantile
                   test statistic can  be approximated by a normal distribution.
               o   This test is used in parallel with the WMW test. This test is often used in Background
                   Test Form 1 to determine if the concentrations from the upper tail of site data
                   distribution are comparable to (lower than or equal to) that of the background data
                   distribution. The critical values for this Form 1 test are available in EPA, 1994. The
                   details of the test are given in EPA (1994, 2006).

Note: The use of the tests listed above is not recommended on log-transformed data sets, especially when
the parameters of interests are the population means. In practice, the cleanup and remediation decisions
have to be made in original scale based upon statistics and estimates computed in the original scale. The
equality of means in log-scale does not necessarily imply the equality of means in the original scale. This
topic is discussed in detail in Chapter 3 of the revised background document (EPA, 2002) for CERCLA
sites (currently under revision).
1.
Click on Hypothesis Testing ^ Two Sample
P? ProJJCLJ,0 -
 if File  Edit Configure Summary Statistics ROS Est, NDs Graphs Outlier Tests  Goodness-of-Fit
 Navigation Panel
                                                        HwatiesSTesIno
                                                  Single Sample

                                                 *"5T	*'   b
                                                                   Background UCL Window  Help
 Name
tTest
Wilcoxon-Mann-Whitney
Qusntile test
 ® Worksheet wst
 0 Worksheet a.wst
2.      Select the Full (w/o NDs) option

           •   To perform a t-test, click on t Test from the drop-down menu.

           •   To perform a Wilcoxon-Mann-Whitney, click on Wilcoxon-Mann-Whitney from the
               drop-down menu list.

           •   To perform a quantile test, click on Quantile test from the drop-down menu.
                                                                                               123

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9.2.1.1 Two-Sample t-Test without NDs

1.      Click Hypothesis Testing ^ Two Sample ^- Full (w/o NDs) ^- t Test
P? ProUCL 4.0 - [C:\Narain\PraUCL-Data\Data\WSR EPA (2006).wsQ.
   Fi!e  Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit Ja^*E^ijt[^|i^ya!i|| Background  UCL Window Help
                                                          S,nge Sample
 Navigation Panel
 Name
 -i?','<'ort-Sheet >,st
 -V WSR EPA (2006) *st
                                                                  :>I  fiJfwfcNOsS M  tTat
                          0
                        V/SR1
1 044
1093
 2      3   r   4    --r  •
WSR2         Arsenic  D_Arsemc
  5.88             1      0
  5.8S             1      0
  1.4fi            1.7      1
                                                      Wiicoxon^lann-Whitney
                                                      Quantiie test
2.      The Select Variables screen will appear.

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

           •   Without Group Variable: This option is used when the data values of the variable
               (COPC) for the site and the background are given in separate columns.

           •   With Group Variable: This option is used when data values of the variable (COPC) for
               the site and the background are given in the same column. The values are separated into
               different populations (groups) by the values of an associated Group Variable. The group
               variable may represent several populations (e.g., several AOCs, MWs). The user can
               compare two groups at a time by using this option.

           •   When using this option, the user should select a group variable by clicking the arrow next
               to the Group Var option for a drop-down list of available variable. The user selects an
               appropriate (meaningful) variable representing groups such as Background and AOC.
               The user is allowed to use letters, numbers,  or alphanumeric labels for the group names.

               o   When the options button is clicked, the  following window will be shown.
124

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Site vs Background Comparison fx]


Substantial Difference. S c
(Used with Test Form 2)
Confidence Coefficient
<~ 99.9% <~ 97.5%
<~ 99.5% ^ 95%
<~ 99% <~ 90%

<•" AOC <= Background (Form 1)
C AOC >= Background (Form 2}
f AOC >= Background + S (Form 2)
C AOC = Background [2 Sided)

OK Cancel


    o  Specify a useful Substantial Difference, S value. The default choice is 0.
    o  Choose the Confidence level. The default choice is 95%.
    o  Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).
    o  Click on OK button to continue or on Cancel button to cancel the option.

•   Click on the OK button to continue or on the Cancel button to cancel the Site versus
    Background Comparison.
                                                                                125

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Output for Two-Sample t-Test (Full Data without NDs)
                             Raw Statistics
                                         Site
                 Number of Valid Samples     77
               Number of Distinct Samples     57
                               Minimum
                              Maximum     1S8.S
                                  Mean
                                 Median       0.43
                                    SD     20.02
                             SE of Mean       2.281
       Background
         47
         40
0.09       0.22
          1.33
          0.599
                                                              3.915
                                                                         0.54
                                                                         0.2B4
                                                                         0.0414
                                    Sitevs Background Two-Sample t-Test
                  HO: Mu of Site- Mu of Background <= 0
                                                             t-Test     Critical
                                                             Value    t [0.050)    P-Value
                                                             1.134     1.S57      0.129
                                                             1.454     1.SG5      0.075
Method                            DF
Pooled (Equal Variance)              122
Satterthwaite (Unequal Variance)      76.1
Pooled SD 15.799
Conclusion with Alpha = 0.050
 * Student t (Pooled] Test: Do Not Reject HO. Conclude Site <= Background
 * Satterthwaite Test: Do Not Reject HO. Conclude Site <= Background
9.2.1.2 Two-Sample Wilcoxon-Mann-Whitney (WMW) Test without NDs
1.      Click Hypothesis Testing ^- Two Sample ^- Full (w/o NDs) ^- Wilcoxon-Mann-Whitney
        Test
    P? ProUCL 4.0 - [V/prkSheet-wstJ
      f File Edit  Configure Summary Statistics  ROS Est. NDs  Graphs  Outiier Tests Goodness-of-Fit EE5ElMEil^MSal Background  UCL Window Help
                                                                  Single Sample
     r»   K, t ^  -jt- .|, ^

     Navigation Panel
     Name
     v> Worksheet wst
126

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2.
The Select Variables Screen will appear.
            Select Variables
             Variables
              Name
                                Count
              Site         1
              Background    2
              Site         4
                         10
                         15
                         15
'• Without Group Variable

        Background / Ambient

        Area of Concern / Site
                                        r With Group Variable

                                        	|  Variabte

                                                Group Var

                                                Background / Ambient

                                                Area of Concern /Site
                                            Optior^s
                                                           OK
                                                                          Cancel
           •   Select variable (variables) from the Select Variables screen.

           •   Without Group Variable: This option is used when the data values of the variable
               (COPC) for the site and the background are given in separate columns.

           •   With Group Variable: This option is used when data values of the variable (COPC) for
               the site and the background are given in the same column. The values are separated into
               different populations (groups) by the values of an associated Group Variable. When using
               this option, the user should select a group variable by clicking the arrow next to the
               Group Var option for a drop-down list of available variables.

           •   ProUCL 4.0 has been written using environmental terminology such as performing
               background versus site comparisons. However, all the tests and procedures in ProUCL
               4.0 can be used for any other application if used properly. The  user selects an appropriate
               group variable representing groups such as Background and AOC. For other applications
               such as comparing a new treatment drug versus older treatment drug, the group variable
               may represent the  two groups:  old drug group and new drug group. The user is allowed to
               use letters, numbers, or alphanumeric labels for the group names.

           •   When the Options button is clicked, the following window is shown.
                                                                                            127

-------

                             Substantial Difference. S        Q
                                 (Used with Test Form 2)
                             I ""Confidence Coefficient
99.9%
 9.5%
                                                     r 97.5%

                                                     fS" 95%

                                                     r 90%
                               Select Null Hypothesis Form
                                     ADC <= Background (Form 1)
                                     ADC >= Background (Form 2)
                                  f" AOC >= Background + S (Form 2)
                                     AOC = Background [2 Sided)

OK
Cancel


               o   Specify a Substantial Difference, S value. The default choice is 0.
               o   Choose the Confidence level. The default choice is 95%.
               o   Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).
               o   Click on OK button to continue or on Cancel button to cancel the selected options.

               Click on the OK button to continue or on the Cancel button to cancel the Site versus
               Background Comparison.
128

-------
Output for Two-Sample Wilcoxon-Mann-Whitney Test (Full Data)
Area of Concern Data: Site
Background Data: Background
Raw Statist

Number of Valid Samples
Number of Distinct Samples
Minimum
Maximum
Mean
Median
SD
SE of Mean


ics
Site
10
9
15
100
48.7
34.5
33.36
10.55



Background
10
8
23
79
49.5
50.5
16.76
5.3
                                     Wilcoxon-Mann-VAwlney (WMW) Test

                     HO: Mean/Median of Site or AOC <= Mean/Median of Background

                                        Site Rank Sum W-Stat   97.5
                                            WMW Test U-Stat   42.5
                                    WMW Critical Value (0.050)    72
                                         Approximate P-Value    0.727

                     Concl its ion with Alpha = 0.05
                       Do Not Reject HO. Conclude She <= Background
9.2.1.3 Two-Sample Quantile Test for Full Data without NDs
As mentioned before, the quantile test is often used in parallel with the WMW test. Typically, both tests
are performed on the same data set before coming to the conclusion about comparability (or non-
comparability) of the data distributions of the two populations.
1.
Click Hypothesis Testing ^ Two Sample ^ Full (w/o NDs) ^ Quantile test
  °? File Edit  Configure Summary Statistics  ROS Est, NDs  Graphs  Outlier Tests Goodness-of-Fit
        Background
   1   1	23j
   2             36
                       1
                       Site
                    15
                    15
                           2           3
                        Background    D_Background
                                                              Single Sample
                                                                                      tTest
 4    -   •»    ••   With NDs    t   Wilcoxon-Mann-Whitnev   JO
Site      D_Site  '       "
     5         0
    10         0
                                                                                                      129

-------
2.      The Select Variables Screen shown below will appear.
             Variables
              Name
                         ID
I Count     (? Without Group Variable
              Site         1
              Background     2
              Site         4
 10
 15
 15
»    Background / Ambient

»    Area of Concern / Site
                                           With Group Variable

                                               |  Variable

                                                 Group Var

                                                 Background / Ambient

                                                 Area of Concern / Site
                                            Options
                                                             OK
                                                                           Cancel
           •   Select variable (variables) from the Select Variables screen.

           •   Without Group Variable: This option is used when the data values of the variable
               (COPC) for the site and the background are given in different columns.

           •   With Group Variable: This option is used when data values of the variable (COPC) for
               the site and the background are given in the same column. The values are separated into
               different groups by using the values of the associated Group Variable. When using this
               option, the user should select a group variable by clicking the arrow next to the Group
               Var  option for a drop-down list of available variables. The user selects an appropriate
               group variable representing groups such as Background and AOC. The user is allowed to
               use letters, numbers, or alphanumeric labels for the group names.

           •   When the Options button is clicked, the following window will be shown.
                                  Quantile Test Options
                                  Select Confidence Coefficient

                                      r 99%        r  97.5%

                                      ff 95%        r  90%
                                      OK
                                                        Cancel
130

-------
               o  Choose the Confidence level; the default choice is 95%.
               o  Click on OK button to continue or on Cancel button to cancel the option.

               Click on the OK button to continue or on the Cancel button to cancel the Site versus
               Background Comparison.
Output for Two-Sample Quantile Test (Full Data)

                 Area of Concern D ata: Site
                 Background Data: Background
Raw Statistics

Site
Number of Valid Samples
Number of Distinct Samples
Minimum
Maximum
Mean
Median
SD
SE of Mean
10
9
15
1CC<
48.7
34.5
33.3S
10.55

Background
10
8
23
79
49.5
50.5
16.76
5.3
                                            Quantile Test

                 HO: Site Concentration <= Back ground Concentration (Form 1)

                               Appraxi mate R Val ue [0.043)    4
                               Approxi mate K Val ue (0.043)    4
                     Number of Site Observations in 'R' Largest    3
                                         Calculated Alpha    0:0433

                 C one I us ion with Alpha = 0,013
                  Do Not Reject H 0. Perform Wi I c oxon-Marm-YVrtitney Ranked Sum Test
                                                                                                131

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9.2.2   Two-Sample Hypothesis Testing for Data Sets with Nondetects

1.      Click Hypothesis Testing^- Two Sample
P? ProUCL 4,0 - [C;\Narain\ProUCL-Data\Data\{iuantile,wstI
  I File Edit Configure Summary Statistics  ROS Est. NDs Graphs  Outiier Tests  Goodness-of-Fit EESSgjljgHsl Background UCL  Window  Help
               r---|                                             Single Sample > (
0
Background
1 - ' 2$
1
Site
15
2 3
background D_Background
3 0
4
Site

                                                                         Full {w/o NDs) ^
                                                                                     Wilcoxon-Mann-Whitney
                                                                                     Gehan
                                                                                     Quantile Test
                                                             5        0
                                        3           0        10        0
2.      Select the With NDs option. A list of available tests will appear (shown above).

            •   To perform a Wilcoxon-Mann-Whitney test, click on Wilcoxon-Mann-Whitney from the
                drop-down menu list.

            •   To perform a Gehan test, click on Gehan from the drop-down menu.

            •   To perform a quantile test, click on Quantile Test from the drop-down menu.

9.2.2.1 Two-Sample Wilcoxon-Mann-Whitney Test with Nondetects

1.      Click Hypothesis Testing ^- Two Sample ^- With NDs ^- Wilcoxon-Mann-Whitney
   ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\Quairtile,wstj
   File Edit Configure Summary Statistics  ROS Est. NDs Graphs  Outlier Tests  Goodness-of-Fit |jggg^gl|   Background UCL  Window  Help
     | a|BJmj d                                             Single Sample
0
Background
1 ' 21
2 ' 36
1
Ste




15
15
2
Background
3
3
3
^Background
0
0
4
Ste




5
10
5
D_Ste
0
0
                                                                         Full (w/o NDs)
                                                                         With NDs    H  Wlcoxon-Mann-WfhlSiey
                                                                                     Gehan
                                                                                     Quantile Test
132

-------
2.      The Select Variables Screen shown below will appear.
             Variables
              Name
                         ID
I Count     (? Without Group Variable
              Site         1
              Background     2
              Site         4
 10
 15
 15
»    Background / Ambient

»    Area of Concern / Site
                                          With Group Variable

                                               |  Variable

                                                 Group Var

                                                 Background / Ambient

                                                 Area of Concern / Site
                                            Options
                                                            OK
                                                                          Cancel
           •   Select variable (variables) from the Select Variables screen.

           •   Without Group Variable: This option is used when the data values of the variable
               (COPC) for the site and the background are given in separate columns.

           •   With Group Variable: This option is used when data values of the variable (COPC) for
               the site and the background are given in the same column. The values are separated into
               different populations (groups) by the values of an associated Group Variable. When using
               this option, the user should select a group variable by clicking the arrow next to the
               Group Var option for a drop-down list of available variables. The user selects an
               appropriate variable representing groups such as Background and AOC. The user is
               allowed to use letters, numbers, or alphanumeric labels for the group names.

           •   When the Options button is clicked, the  following window will be shown.
                                                                                             133

-------
                         Site vs Background Comparison
                              Substantial Difference: S
                                 (Used with Test Form 2)
                              ^Confidence Coefficient
        0
                                  r 99.9%
                                  r 99.5%
                                  r 99%
r 97.5%
(? 95%
r 90%
                               Select Null Hypothesis Form
                                  (•  AOC <= Background (Form
                                  C AOC >= Background (Form 2)
                                  C AOC >= Background + S (Form 2)
                                  C AOC = Background (2 Sided)
                                    OK
    Cancel
               o   Specify a meaningful Substantial Difference, S value. The default choice is 0.
               o   Choose the Confidence level. The default choice is 95%.
               o   Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).
               o   Click on the OK button to continue or on the Cancel button to cancel the selected
                   options.

           •   Click on OK button to continue or on Cancel button to cancel the Site versus Background
               Comparison.
134

-------
Output for Two-Sample Wilcoxon-Mann-Whitney Test (with Nondetects)

                     Area of Concern Data: Site
                     Background Data: Background

                                               Raw Statistics
                                                          Site      Background
                                    Number of Valid Samples   15        15
                                   Number of Non-Detect Data   8        12
                                      N u mber of Detect D ata   7        3
                                       Minimum Non-Detect      5        3
                                       Maximum Non-Detect    300       25
                                        Percent Non detects   53.33%    80.00%
                                         Minimum Detected     11        8
                                         Maximum Detected    200       22
                                      Mean of Detected Data     74.43     15
                                     Median of Detected Data     70       15
                                        SD of Detected Data     6S.42      7

                                 WilcQior-Manr-VVhrtney Srtevs Background Test
                               Allobservations <=300(Max DL}
                                      Wi Icoxon- Man n-Whitney (WMW) Test

                     HO: Mean/Median of Site or AOC <= Mean/Median of Background

                                        Site Rank Sum W-Stat  232.5
                                           WMWTestU-Stat  112.5
                                    WMW C ritical Va I ue (0.050)  152
                                         Approxi mate P-Va I ue    0.508

                     Conclusion with Alpha = 0.05
                       Do Not Reject HO. Conclude Site <= Background

Note: In the WMW test, all observations below  the largest detection limit are considered as NDs
(potentially including some detected values) and hence they all receive the same average rank. This
action may reduce the associated power of the WMW test considerably. This in turn may lead to incorrect
conclusion. As mentioned before, all hypotheses testing approaches should be supplemented with
graphical displays such as Q-Q plots and box plots.  When multiple detection limits are present, the use of
the Gehan test is preferable.
                                                                                                      135

-------
9.2.2.2 Two-Sample Gehan Test for Data Sets with Nondetects

1.      Click Hypothesis Testing ^  Two Sample ^- With NDs ^- Gehan
P? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\Quantile.w5t]
 •g File  Edit Configure  Summary Statistics  ROS Est, NDs Graphs Gutter Tests Goodness-of
                                                          Hypothesis Testing
                                                                     Background  UCL Window Help
           mjjpj
        Background
                      Site
                               Background
                           15
                           15
                                         D_Background
                                                                       Jitn NDs    ^ I  Wilcoxon-Mann-Whitnev
                                                       Site
                                                               D Site
                                                 Gehan
                                                 QuanflleTest
2.      The Select Variables Screen will appear.
              Variables
               Name
                          ID
I Count     (? Without Group Variable
              Site          1
              Background     2
              Site          4
 10
 15
 15
»     Background / Ambient

»     Area of Concern / Site
                                             With Group Variable

                                                 |  Variable

                                                   Group Var

                                                   Background / Ambient

                                                   Area of Concern / Site
                                              Options
                                                               OK
                                                                              Cancel
            •   Select variable (variables) from the Select Variables screen.

            •   Without Group Variable: This option is used when the data values of the variable
                (COPC) for the site and the background are given in separate columns.

            •   With Group Variable: This option is used when data values of the variable (COPC) for
                the site and the background are given in the same column. The values are separated into
                different populations (groups) by the values of an associated Group Variable. When using
                this option, the user should select a group variable by clicking the arrow next to the
                Group Var option for a drop-down list of available variables. The user selects a group
                variable representing groups  such as Background and AOC.

            •   When the Options button is clicked, the following window will be shown.
136

-------
           Site vs Background Comparison
               Substantial Difference: S
                  (Used with Test Form 2)
               ^Confidence Coefficient
        0
                   r 99.9%
                   r 99.5%
                   r 99%
r 97.5%
(? 95%
r 90%
                Select Null Hypothesis Form
                   (• AOC <= Background (Form
                   C AOC >= Background (Form 2)
                   C AOC >= Background + S (Form 2)
                   C AOC = Background (2 Sided)
                     OK
    Cancel
o   Specify a Substantial Difference, S value. The default choice is 0.
o   Choose the Confidence level. The default choice is 95%.
o   Select the form of Null Hypothesis. The default is AOC <= Background (Form 1).
o   Click on OK button to continue or on Cancel button to cancel selected options.

Click on the OK button to continue or on the Cancel button to cancel the Site versus
Background Comparison.
                                                                               137

-------
Output for Two-Sample Gehan Test (with Nondetects)
                 Area of Concern Data: Site
                 Background Date Background
                                           Raw Statistics

                                Number of Valid Samples
                              Number of Non-Detect Data
                                  Number of Detect Data
                                   Minimum Non-Detect
                                   Maximum Non-Detect
                                    Percent Non detects
                                     Minimum Detected
                                     Maximum Detected
                                  Mean of Detected Data
                                 Median of Detected Data
                                    SD of Detected Data
                                    Site vs Background Gehan Test

                 H0: M u of Site or AO C > = M u of background

                                      Gehan z Test Value    1.769
                                         Critical z (0,05)  -1.645
                                                P-Value    0.962

                 Conclusion with Alpha = 0.05
                   Do Not Reject HO. Conclude Site >= Background
                   P-Value >= alpha (Q.05)
Site
10
2
8
4
35
20.00%
2
43
23.63
22.5
14.74
Background
10
4
6
4
25
40.00%
1
27
12.17
11
3.S42
9.2.2.3 Two-Sample Quantile Test for Data Sets with Nondetects
Quantile test as described in EPA (1994) has been included in ProUCL 4.0. The detailed power of the test
with many ND values is not well studied. The conclusion of this test should also be supplemented with
graphical displays. The use of the Gehan test is preferred when the data set may consist of many NDs
with multiple detection limits.
138

-------
1.      Click Hypothesis Testing ^- Two Sample ^- With NDs ^- Quantile Test
f? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\Quantile.wst]

PI 1 Ch u=l I H i IT! 1 ^Tl Single Sample ^ (
0
1 234 a WffJSt!l^fD Wilcoxon-Mann-Whitney 0
Background Site Background DJtackground Site D_Site Gehan
1 n 15 3 050 K^^X^^^^^H
2 36 15| 3 0 10 0
2. The Select Variables Screen will appear.
^^^^H Select Variables ^^^^H






Variables
Name ID | Count * Without Group Variable

Background 2 IS » I Background / Ambient T
Site 4 15
» Area of Concern / Site
f~ With Group Variable
» | Variable |~~
Group Var
Background / Ambient
Area of Concern / Site
Options
OK Cancel




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

           •   Without Group Variable: This option is used when the data values of the variable
              (COPC) for the site and the background are given in separate columns.

           •   With Group Variable: This option is used when data values of the variable (COPC) for
              the site and the background are given in the same column. The values are separated into
              different populations (groups) by the values of an associated Group Variable. When using
              this option, the user should select a group variable by clicking the arrow next to the
              Group Var option for a drop-down list of available variables. The user selects an
              appropriate group variable representing groups such as Background and AOC.

           •   When the Options button is clicked, the following window will be shown.
                                                                                          139

-------
                                       •JJuarrtile Test.{|jjtiori5.
                                        Select Confidence Coefficient
                                             r 99%          r 97.5%
                                             <• 95%          C 90%
                                             OK
                                                                  Cancel
                  o   Choose the Confidence level; the default choice is 95%.
                  o   Click on OK button to continue or on Cancel button to cancel the option.

             •    Click on the OK button to continue or on the Cancel button to cancel the Site versus
                  Background Comparison.

Output for Two-Sample Quantile Test (with Nondetects)
                           Area of Concern Date: Site
                           Background Data: Background

                                                    Raw Statistics

                                         Number of Valid Samples
                                        Number of Non-Detect Data
                                           Number of Detect Data
                                             Minimum Non-Detect
                                            Maximum Non-Detect
                                             Percent Non detects
                                              Minimum Detected
                                              Maximum Detected
                                           Mean of Detected Data
                                          Median of Detected Data
                                             SD of Detected Data
                                                    Quantile Test

                           HO: Site Concentration <= Background Concentration (Form 1)

                                         Approximate R Value (0.05)    4
                                         Approximate K Value (0.05)    4
                               Number of Site Observations in 'FT Largest    4

                             Non-Detect Values in the'R' Largest- Cannot eanfjleieQuanljfe Test
Site
15
8
7
5
300
53.33%
11
200
74.43
70
6S.42
Background
15
12
3
3
25
80.00%
8
22
15
15
-
140

-------
141

-------
                                      Chapter 10

                              Background Statistics
This chapter illustrates the computations of various parametric and nonparametric statistics and upper
limits that can be used as estimates of background threshold values (BTVs) and other not-to-exceed
values. The BTV estimation methods are available for all data sets with and without nondetect (ND)
observations. The details of those methods are given in Chapter 5 (full data sets without NDs) and
Chapter 6 (data sets with NDs) of the revised background document for CERCLA sites (EPA, 2002).
Technical details can also be found in the Technical Guide associated with ProUCL 4.0. For each selected
variable, this option computes various upper limits such as UPLs, UTLs, and upper percentiles to estimate
the background threshold values (BTVs) and other compliance limits that are used in site versus
background evaluations.

As before, two choices for data sets are available to compute background statistics:

           •   Full - computes background statistics for a Full data set without any NDs.

           •   With NDs - computes background statistics for a data set with nondetected as wells as
              detected values. Multiple detection limits are allowed.

The user specifies the confidence level (probability) associated with each interval estimate. The
reasonable confidence level as  incorporated in ProUCL 4.0 represents a number in the interval [0.5, 1),
0.5 inclusive. The default choice is 0.95.

For data sets with and without NDs, ProUCL 4.0 can compute the following statistics that can be used as
estimates of BTVs and not-to-exceed values.

           •   Parametric and nonparametric upper percentiles.

           •   Parametric and nonparametric upper prediction limits (UPLs) for a single observation,
              future or next k (> 1) observations, mean of next k observations. Here future k, or next k
              observations may also represent k observations from another population (e.g., site)
              different from  the sampled (background) population (used to compute UPLs, UTLs).

           •   Parametric and nonparametric upper tolerance Limits (UTLs).

           •   Nonparametric IQR-based upper limits.
142

-------
10.1   Background Statistics for Full Data Sets without Nondetects

1.       Click Background ^ Full (w/o NDs) Background Statistics
P? PrpUCL 4.0 - {C:\Narain\ProUCL-Oata\Data\Cadraium,(wstJ
aQ File Edit Configure Summary Statistics  ROSEst, NDs Graphs Outlier Tests  Goodness-of-Fit  Hypothesis Testing |
                                                                       I LICL  Window  Help
 Navigation Panel [
 Name
 Q Worksheet wst
 i$ Cadmium, vvst
                        D     1
                       Pop-ID  Cadmium
                      	"	T|    0.24
                      	i"    0.26
                                                                  With NDs Background Statistics
                                                                                       Normal
                                                                                       Gamma
                                                                Non-Parametric
                                                                AH
2.      Select Full Background Statistics.

            •   To compute the background statistics assuming the normal distribution, click on Normal
                from the drop-down menu list.

            •   To compute the background statistics assuming the gamma distribution, click on Gamma
                from the drop-down menu list.

            •   To compute the background statistics assuming the lognormal distribution, click on
                Lognormal from the drop-down menu list.

            •   To compute the background statistics using distribution-free nonparametric methods,
                click on Non-Parametric from the drop-down menu list.

            •   To compute and see all background statistics available in ProUCL 4.0, click on the All
                option from the drop-down menu list.

10.1.1  Normal or Lognormal Distribution

1.      Click Background ^- Full (w/o NDs) Background Statistics ^- Normal or Lognormal
P? PrpUCL 4.0 -JC:VNarain\ProUCL-pata\pata\qadroium,yretI
•y File Edit Configure  Summary Statistics ROS Est, NDs Graphs  Outlier Tests Goodness-of-Fit Hypothesis Testing
C'lvl ta|"|rr:| n:
 Navigation Panel
                                                                 Background
                                                                  RiwnfunSaag'iundStattsticB M
 Name
 c; WorkSheetwst
 ^Cadmium.wst
 0      1
Pop-ID  Cadrriurr
    V   0 24
    1    026
                                                LICL Window  Help


                                           With NDs Background Statistics
                                                                                       Gamma
                                                                                       Lognormal     '(12
                                                                                       Non-Parametric
                                                                                       All
2.      The Select Variables Screen (Chapter 3) will appear.

            •   Select a variable (variables) from the Select Variables screen.

            •   If needed, select a group variable by clicking the arrow below the Group by variable to
                obtain a drop-down list of available variables and select an appropriate group variable.
                                                                                                  143

-------
              When the option button is clicked, the following window will be shown.
                      1 Background Statistics Options
                                     Confidence Level
                                             Coverage
     !Baa
     0.9
                           Different or Future K Values  I       1
                       Number of Bootstrap Operations
                                  OK
Cancel
              o   Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                  default choice is 0.95.
              o   Specify the Coverage coefficient (for a percentile) needed to compute UTLs.
                  Coverage represents a number in the interval (0.0, 1). The default choice is 0.9.
                  Remember, a UTL is an upper confidence limit (e.g., with confidence level = 0.95)
                  for a 90% (e.g., with coverage = 0.90) percentile.
              o   Specify the Different or Future K Values. The default choice is 1. It is noted that
                  when K = 1, the resulting interval will be a UPL for a single future (or site)
                  observations. In the example shown above, a value of K = 1 has been used.
              o   Specify the Number of Bootstrap Operations (resamples). The default choice is
                  2000.
              o   Click on OK button to continue or on Cancel button to cancel this option.

           •   Click on OK button to continue or on Cancel button to cancel the Background  Statistics
              Options.
144

-------
Output Screen for Normal Distribution (Full)
                         Cadmium
                                                Raw Statistics
                                                      Number of Valid Samples         33
                                                    Number of Unique Samples         28
                                                                   Minimum      0.094
                                                                   Maximum         21
                                                              Second Largest      6.967
                                                                      Mean      3.024
                                                               FirstQuantile       0.35
                                                                    Median        2.5
                                                              Third Quantile        3.8
                                                                        SD      3.751
                                                        Coefficient of Variation       1.24
                                                                  Skewness      3.607

                                            Normal Distribution Test
                                                     Shapiro Wilfc Test Statistic      0.642
                                                 5% Shapiro Wilk Critical Value      0.931
                                    Data not Normal at 5Z Sgnifcanoe Lesel
                                            Normal Distribution Test
                                                     Shapiro Wilk Test Statistic     0.642
                                                 5% Shapi ro Wi I k C ritical Val ue      0.931
                                    Data not Normal at 5% Significance

                                Background Statistics Assuming Normal Distribution
                                                            90% Pereentile (z)       7.83
                                                           95% Percerrtile (z)      9.193
                                                            99% Pereentile (z)      11.75

                                                  95% UTL with 90% Coverage      9.549

                                                                 95% UPL (t)      9.472

                                Note: UPL (or upper perc en tile for gamma distributed
                                    data) represents a preferred estimate of BTV
                                                                                                                    145

-------
Output Screen for Lognormal Distribution (Full)
                                               Log-Transformed Statistics
                                                     Number of Valid Samples   33
                                                   Number of Unique Samples   28
                                                                  Minimum   -2.364
                                                                 Maximum    3.045
                                                            Second Largest    1.941
                                                                     Mean    0.459
                                                              First Quantile    1.653
                                                                   Median    0.316
                                                             Third Quantile    0.709
                                                                       SD    1.308

                                               Lognormal Distribution Test
                                                    Shapiro Wilk Test Statistic    Q.9Q9
                                                5% Shapiro Wilk Critical  Value    0.931
                                        Data not Log normal at 551 Significance Lewei

                                    Background Statistics Assuming Lognormal Distribution
                                                          30% Percent! le (z)    S.45B
                                                          95% Percentile (z)   13.6
                                                          99% Percent! le (z)   33.18
                                                                 95% UPL   15
                                                 95% UTL with 90% Coverage   15.4
                                   Background Statistics Assuming Lognormal Distribution
                                                         30% Percerttile (z)    8.45S
                                                         95% Percentile (z)   13.6
                                                         99% Percentile (z)   33.18
                                                                 95% UPL   15
                                                95% UTL with  90% Coverage   15.4
                                        Some Nonparametric Background Statistics
                                                       95% Chebyshev UPL   19.62
                                    95% Bootstrap BCA UTL with 90% Coverage    6.737
                               95% Percentile Bootstrap UTL with 90% Coverage    6.967

                                    Note: UPL (or upper percentile tor gamma distributed
                                        data) represents a preferred estimate of BTV
146

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10.1.2  Gamma Distribution

1.       Click Background ^- Full (w/o NDs) Background Statistics ^- Gamma
P3 ProUCL4.0 - [C:\Narain\ProUCL-Data\Data\Cadmium.wst]
  1 File Edit Configure Summary Statistics ROS Est. NDs Graphs  Gutter Tests Goodness-of-Fit Hypothesis Testing
                                                                      UCL Window Help
 Navigation Panel I
 Name
 OWorkSheet.wst
 •_;; Cadmium.wst
                       Pop-ID
    0.24
1    0.26
2.      The Select Variables Screen (Chapter 3) will appear.

           •   Select a variable (variables) from the Select Variables screen.

           •   If needed, select a group variable by clicking the arrow below the Group by variable to
               obtain a drop-down list of available variables, and select a proper group variable.

           •   When the option button is clicked, the following window will be shown.
                       j§l Background Statistics Options
                                        Confidence Level
                                               Coverage
                             Different or Future K Values  [       1
                         Number of Bootstrap Operations
                                    OK
                                 Cancel
               o   Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                   default choice is 0.95.
               o   Specify the Coverage level; a number in interval (0.0, 1). Default choice is 0.9.
               o   Specify the next K. The default choice is 1.
               o   Specify the Number of Bootstrap Operations. The default choice is 2000.
               o   Click on OK button to continue or on Cancel button to cancel the option.

               Click on OK button to continue or on Cancel button to cancel the Background Statistics
               Options.
                                                                                                147

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Output Screen for Gamma Distribution (Full)
                          Cadmium

                                                 Raw Statistics
                                                      N umber of Val id Samples 33
                                                     Number of Unique Samples 28
                                                                    Minimum     0.094
                                                                    Maximum    21
                                                              Second Largest     6,967
                                                                       Mean     3.024
                                                                First Quarrtile     0.35
                                                                     Median     2.5
                                                               Third Quantile     3.8
                                                                         SD     3.751

                                             Gamma Distribution Test
                                                                       k hat     0.902
                                                                    Theta hat     3.352
                                                                      nu hat    59.54
                                                                       k star     0.84
                                                                   Theta star     3.598
                                                                      nu star    55.46
                                                95% Percentile of Chisquare (2k)     5.356
                                                            A-D Test Statistic     0.794
                                                         5% A-D Critical Value     0.78
                                                             K-S Test Statistic     0.143
                                                         5% K-S Critical Value     0.158
                            Data fol low Appr. Gamma Distribution at5X Significance Level

                                Background Statistics Assuming Gamma Distribution
                                                               90% Percentile     7.265
                                                               95% Percentile     9.637
                                                               99% Percentile    15.22

                                       Nonparametic Background Statistics
                                                         95% Chebyshev UPL    19.62
                                      95% BCA Bootstrap UTL with 90% Coverage     6.737
                                       95% Bootstrap (%) UTL with 90% Coverage     6.737

                                Note: UPL (or upper perceri tile far gamma distributed
                                     data) represents a preferred estimate of BTV
148

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10.1.3  Nonparametric Methods

1.       Click Background ^- Full (w/o NDs) Background Statistics ^- Non-Parametric
E? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
my File Edit Configure Summary Statistics  ROS Est. NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing
                                                                      UCL Window Help
jgJjaJ.jBlBl.gil el
 Navigation Panel I
 Name
 •rf Worksheet, wst
 O MW89.wst
                          Normal
     With NDs Background Statistics   >\  Gamma
                          Lognormal
                         ^^
                          All
2.      The Select Variables Screen (Chapter 3) will appear.

           •   Select a variable (variables) from the Select Variables screen.

           •   If needed, select a group variable by clicking the arrow below the Group by variable to
               obtain a drop-down list of available variables, and select a proper group variable.

           •   When the option button is clicked, the following window will be shown.
                       al Background Statistics Options
                                        Confidence Level
        -   n  x
                                                Coverage  [       0.9
                             Different or Future K Values  [        1
                         Number of Bootstrap Operations  f
                                     OK
Cancel
               o  Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                   default choice is 0.95.
               o  Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9.
               o  Specify the next K. The default choice is 1.
               o  Specify the Number of Bootstrap Operations. The default choice is 2000.
               o  Click on the OK button to continue or on the Cancel button to cancel the option.

            •   Click on OK button to continue or on Cancel button to cancel the Background Statistics
               Options.
                                                                                                149

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Output Screen for Nonparametric Option (Full)
                                     Cadmium
                                                    Some Non-Parametric Statistics
                                                                 Number of Valid Samples        33
                                                               Number of Unique Samples        28
                                                                              Minimum      0.054
                                                                              Maximum        21
                                                                         Second Largest      6.967
                                                                                 Mean      3.024
                                                                          First Quarttile       0.35
                                                                               Median        2.5
                                                                          Third Quantile        3.8
                                                                                   SD      3.751
                                                                              Variance      14.07
                                                                   Coefficient of Variation       1.24
                                                                             Skewness      3.607
                                                             Mean of Log-Transformed data      0.459
                                                               SD of Log-Transformed data      1.308
                                       Data Follow Appr. Gamma Distribution al 5% Spaieaice Lewd)

                                                  Non-Parametric Badcpound Statistics
                                                                         90% Percentile      5.27
                                                                         95% Percentile     6.468
                                                                         99% Percentile     16.37

                                                       95% UTL with 90% Coverage
                                                                          Order Statistic        32
                                                                           Achieved CC     0.969
                                                                                  UTL     S.9S7

                                                 95% BCA Bootstrap LITL with 90% Coverage     6.737
                                             95% Percentile Bootstrap UTL with 90% Coverage     6.967

                                                                              95%UPL     11.18
                                                                    35% Chebyshev UPL     19.62

                                                               Upper Limit Based upon IQR     8.975

                                           Note: UPL (or upper per c en tile far gamma distributed
                                                data) represents a preferred estimate at BTV
150

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10.1.4 All Statistics Option

1.       Click Background ^ Full Background Statistics ^ All
  ProllCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
  File Edit Configure Summary Statistics ROS Est. NDs  Graphs Outlier Tests Goodness-of-fit Hypothesis Testing |jj    Q UCL  Window Help

                                                                •
                                                                  With NDs Background Statistics
 Navigation Panel
 Name
 •cJWorkSheet.wst
 •3 MW89-WSI
                       Well ID
                               1
    460
1	527
                                                            Gamma
                                                            Lognormal
                                                            Non-Parametric
2.      The Select Variables Screen (Chapter 3) will appear.

            •    Select a variable (variables) from the Select Variables screen.

            •    If needed, select a group variable by clicking the arrow below the Group by variable to
                obtain a drop-down list of available variables, and select a proper group variable.

            •    When the option button is clicked, the following window will be shown.
                          Background Statistics Options
                                        Confidence Level


                                                Coverage  I
                              Different or Future K Values          1
                         Number of Bootstrap Operations
                                     OK
                                 Cancel
                o   Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                    default choice is 0.95.
                o   Specify the Coverage level; a number in the interval (0.0, 1). Default is 0.9.
                o   Specify the next K. The default choice is 1.
                o   Specify the Number of Bootstrap Operations. The default choice is 2000.
                o   Click on OK button to continue or on Cancel button to cancel the option.

            •   Click on OK button to continue or on Cancel button to cancel the Background Statistics
                Options.
                                                                                                  151

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Output Screen for All Statistics Option (Full)
              Cadmium
                                                                 General Statistics
                                          Total Number of Samples    33
                                                                                                   Number of Unique Samples   28
                                     Raw Statistics
                                                       Minimym    0.094
                                                       Maximum   21
                                                 Second Largest    6.967
                                                   First Quantile    0.35
                                                        Median    2.5
                                                   Third Quantile    3.S
                                                          Mean    3.024
                                                            SD    3.751
                                            Coefficient of Variation    1.24
                                                      Skewness    3.607
                                     Log-Transformed Statistics
                                                            Minimum   -2.3S4
                                                            Maximum    3.045
                                                       Second Largest    1.941
                                                         First Quantile   -1.13
                                                              Median    0.916
                                                        Third Quantile    1.334
                                                               Mean    0.459
                                                                 SD    1.308
                                                               Background Statistics
                                 Normal Distribution Test
                                         Shapiro Wilk Test Statistic    0.642
                                         Shapiro Wilk Critical Value    0.931
                         Oats not Normal si 5% Sjf ^ei
                             Assuming Normal Distribution
                                    95V. UTL with 90% Coverage    9.549
                                                    95% UPL (t)    9.472
                                              90%  Percerttile (z)    7.83
                                              95%  Percentile (z)    9.193
                                              99%  Percentile (z)   11.75
                                     Lognormal D istribution Test
                                              Shapiro Wilk Test Statistic    0.909
                                              Shapiro Wilk Critical Value    0.931
                             Data imt Lognccmal al§X Sifwieanee Lwd

                                   Assuming Lognormal Distribution
                                            95% UTL with  90% Coverage    15.4
                                                           95% UPL (t)    15
                                                      90% Percentile (z)     8.458
                                                      95% Percentile (z)    13.6
                                                      99% Percentile (z)    33.1S
                               Gamma Distribution Test
    k star     0.84
Theta Star     3.598
   mi star    55.46
                                                                                            Data Distribution Tests
                                                                          Data Follow Appr. Gamma Distribution at 5% Significance Level
                                               A-D Test Statistic    0.794
                                            5% A-D Critical Value    0.78
                                               K-S Test Statistic    0.143
                                            5% K-S Critical Value    0.158
              Data follow Appx. Gamma Distribution at 5% Significance Level
                           Assuming Gamma Distribution
                                                90% Percentile    7.265
                                                95% Percentile    9.637
                                                99% Percentile   15.22
                                      Nonparametric Statistics
                                                         90% Percentile     5.27
                                                         95% Percentile     6.468
                                                         99% Percentile    16.37

                                             95% UTL with 90% Coverage     6.967
                           95% Percentile Bootstrap UTL with 90% Coverage     6.967
                                95% BCA Bootstrap UTL with 90% Coverage     6.737
                                                               95% UPL    11.18
                                                     95% Chebyshev UPL    19.62
                                     Upper Threshold Limit Based upon IQR     8.975
                           Note: UPL (or upper percentile for gamma distributed data) represents a preferred estimate of BTV
152

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10.2   Background Statistics with NDs

1.       Click Background ^ With NDs Background Statistics
P? ProUCL 4.0 - [C:\Narain\ProUCL Data\Data\MW89.wst]
•y File Edit Configure Summary Statistics ROSEst, NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing K CT UCL Window Help
Ell's! OlHlffl! D| Full (w/o NDs) Background Statistics >
Navigation Panel

Name |
ijj Worksheet wst
& MW89.wst

1
2
3
0
Well ID
1
Mn
1 460
1!" 527
1
579
2


3


4


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12




2.      Select the With NDs Background Statistics option.

           •   To compute the background statistics assuming the normal distribution, click on Normal
               from the drop-down menu list.

           •   To compute the background statistics assuming the gamma distribution, click on Gamma
               from the drop-down menu list.

           •   To compute the background statistics assuming the lognormal distribution, click on
               Lognormal from the drop-down menu list.

           •   To compute the background statistics using distribution-free methods, click on Non-
               Parametric from the drop-down menu list.

           •   To compute all available background statistics in ProUCL 4.0, click on the All option
               from the drop-down menu list.

10.2.1  Normal or Lognormal Distribution

1.      Click Background ^- With NDs Background Statistics  ^- Normal or Lognormal
t? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
•5 File Edit Configure Summary Statistics ROSEst. NDs Graphs Outlier Tests Goodness-of-Fit Hypothesis Testing ESS  J| UCL Window Help
                                                            Full (w/o NDs) Background Statistics T~L
 Navigation Panel
 Name
 •3WorkSh.8flt.wst
 <•) MW89.wst
1!	527]
2.     The Select Variables Screen (Chapter 3) will appear.

           •   Select a variable (variables) from the Select Variables screen.

           •   If needed, select a group variable by clicking the arrow below the Group by variable to
               obtain a drop-down list of available variables, and select a proper group variable.

           •   When the option button is clicked, the following window will be shown.
                                                                                             153

-------
                      ? Background Statistics Options
                                     Confidence Level


                                            Coverage
                           Different or Future K Values |       1
                       Number of Bootstrap Operations
                                  OK
Cancel
              o   Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                  default choice is 0.95.
              o   Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9.
              o   Specify the next K. The default choice is 1.
              o   Specify the Number of Boostrap Operations. The default choice is 2000.
              o   Click on the OK button to continue or on the Cancel button to cancel the option.

              Click on OK button to continue or on Cancel button to cancel the Background Statistics
              Options.
154

-------
Output Screen for Normal Distribution (with NDs)
                          Arsenic
                                                        Raw Statistics
                                                          Total Number of Data 24
                                                     Number of Non-Deteet Data 13
                                                       Number of Detected Data 11
                                                            Minimum Detected     0.5
                                                            Maximum Detected     3.2
                                                          Percent Non-Detects 54.17%
                                                          Minimum Norv-detect     0.3
                                                          Maximum Non-detect     2
                                                        Mean of Detected Data     1.236
                                                          SD of Detected Data     0.965
                                        Normal Distribution Test with Detected Values Oirfy
                                                      Shapiro Wilk Test Statistic     0.777
                                                  5% Shapiro Wilk Critical Value     0.85
                                            Data not Normal at 5% Significance Level
                                        Normal Distri but ion Test with Delected Values Only
                                                     Shapiro Wilk Test Statistic    0.777
                                                  5% Shapiro Wilk Critical Value    0.85
                                            Data mot Normal at SX Signtirance Lewd

                                        Background Statistics Assuming  Normal Distribution

                                                    DU2 Substitution Method
                                                                      Mean    1.002
                                                                         SD    0.633
                                                       95% UTL 90% Coverage    2.2%
                                                                 95% UPL (t)    2.224
                                                           90% Percentile (z)    1.897
                                                           95% Percentile (z)    2.151
                                                           93% Percentile (z)    2.627
                          Note: DU2 is not a recommended melhod.

                                           Maximum Likelihood Estimate (MLE) Method
                                        MLE Method is Not Appliable for This Data
                                                                                                                         155

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                                          Maximum Likelihood Estimate (MLE) Method
                                        MLE Method is Not Appliable for This Data

                                                  Kaplan-Meier (KM) Method
                                                                     Mean    0.949
                                                                       SD    0.713
                                                                SE of Mean    0.165
                                                  95% UTL with 30% Coverage    2.27
                                                             95% KM U PL (t)    2.196
                                                     35% KM Chebyshev UPL    4.121
                                                           90% Pereentiie (z)    1.863
                                                           95% Percentile (z)    2.122
                                                           99% Pereentiie (z)    2.608

                               Note: UPL(or upperpercentilefor gamma distributed data) represents a
                                   preferred esti mate of BTV  For ma Example: KM-UPL may be used
                                             when muIti pie Detection limrts are present
156

-------
Output Screen for Lognormal Distribution (with NDs)
                                                 Log Transformed Statistics
                                                           Number of Valid Samples    24
                                                          Number of Unique Samples    10
                                                                        Minimum   -0.693
                                                                        Maximum     1.163
                                                                   Second Largest     1.03
                                                                           Mean     0.215
                                                                     First Quantile     0.693
                                                                          Median     0.203
                                                                    Third Quantile   -0.693
                                                                             SD     0.574

                                                 Lognormal Distribution Test
                                                          Shapiro Wilk Test Statistic     0.906
                                                       5% Shapiro Wilk Critical Value     0.916
                                         D ata    Log normal at 5X           Lori

                                    Background Statistics Assuming Lognormal Distribution
                                                                90% Percentile (z)     2.587
                                                                95% Percent! le (z)     3.186
                                                                99% Percentile (z)     4.711
                                                                        95% LI PL     3.3S3
                                                       95% UTL with  90% Coverage     3.59

                                         Some Nonparametric Background Statistics
                                                              95% Chebyshev UPL     4.825
                                           95% Bootstrap BCA UTL with  90% Coverage     1.8
                                      95% Percentiie Bootstrap UTL with  90% Coverage     3.04

                                     Note:  UPL (or upper per c en tile far gamma distributed
                                         data] represents a preferred estimate of BTV
                                                                                                                    157

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10.2.2 Gamma Distribution

1.      Click Background ^- With NDs Background Statistics ^- Gamma
E? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
•B File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness-of-F
t Hypothesis Testing 1
^^^^^^2]
UCI Window Hftp
nlCll Pl|HlrTll Fll Fu" 'w/0 NDs' Background Statistics > 1 	
Navigation Panel 1
Name |
QWorkShaet.wst
•3 MW89.wst


1
2
3
D
Well ID
1
1
1
1
Mr
460
	 527
579
2


3


4




5




6




^m^m




^Ki^^Ki^^



Lognormal
Non -Parametric
All 1

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

    •   Select a variable (variables) from the Select Variables screen.

    •   If needed, select a group variable by clicking the arrow below the Group by variable to
       obtain a drop-down list of available variables and select a proper group variable.

    •   When the option button is clicked, the following window will be shown.
                         Background Statistics (Gamma)
                             Confidence Level
                                     Coverage
                                      OK
                                                         0.9
                                                 Cancel
               o  Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                  default choice is 0.95.
               o  Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9.
               o  Click on the OK button to continue or on the Cancel button to cancel option.

               Click on OK button to continue or on Cancel button to cancel the Background Statistics.
158

-------
Output Screen for Gamma Distribution (with NDs)
                                                                                                    24
                                                                                                    13
                                                                                                    11
                                                                                                    0.5
                                                                                                    3.2
                                                                                                54.17%
                                                                                                    0.9
                                                                                                     2
                                                                                                  1.236
                                                                                                  0.9S5
Arsenic

                       Raw Statistics
                                Total Number of Data
                           Number of Non-Deiect Data
                             Number of Detected Data
                                   Minimum Detected
                                  Maximum Detected
                                 Percent Non-Deteots
                                 Minimum Non-detect
                                 Maximum Non-detect
                               Mean of Detected Data
                                 SD of Detected Data

       Gamma Distribution Test with Detected Values Only
                                              kstar      1.702
                                         Theta star      0.727
                                            rm star      37.44
                      95% Percentile of Chisquare (2k)      8.503
                                   A-D Test Statistic      0.7S7
                                5% A-D Critical Value      0.738
                                    K-S Test Statistic      0.254
                                5% K-S Critical Value      0.258
  Data follow Appr. Gamma Distribution at 5% Significance Levd
        Background Statistics Assuming Gamma Dtsiribeiiioci
          Gamma ROS Statistics with Extrapolated Data
                                            Mean        1.263
                                           Median        1.213
                                              SD        O.S52
                                            k Star        3.974
                                        Theta Star        0.318
                                           mi Star        190.8
                      95% Percenti le of Chisquare (2k)        15.43
                                    90% Percentile        1.861
                                    95% Percentile        2.16
                                    99% Percentile        2.801

                   Kaplan Merer (KM) Method
                                            Mean        0.949
                                              SD        0.713
                                       SE of Mean        0.165
                             95% UTL 90% Coverage        2.27
                            95% KM Chebyshev UPL        4.121
                                    95%KMUPL(t)        2.196
                                    90% Percentile        1.863
                                 95% Percentile (z)        2.122
                                 99% Percentile (z)        2.608
Note: UPL(or upperpercentileforgamma distributed data) represents a
    preferred estimate of BTV. For an Example: KM-UPL may be used
              when multiple detection limits are present
                                                                                                                                           159

-------
10.2.3 Nonparametric Methods (with NDs)

1.      Click Background ^- With NDs Background Statistics ^- Non-Parametric
  File Edit Configure Summarv Statistics ROS Est. NDs Graphs Gutter Tests Gtradness-ot-Fit  Hypothesis Testing
2.     The Select Variables Screen (Chapter 3) will appear.

           •   Select a variable (variables) from the Select Variables screen.

           •   If needed, select a group variable by clicking the arrow below the Group by variable to
               obtain a drop-down list of available variables and select a proper group variable.

           •   When the option button is clicked, the following window will be shown.
                          Background Statistics (Gamma)
                              Confidence Level

                                      Coverage
Ifflaa
                                       OK
                                                           c.s
Cancel
               o   Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                   default choice is 0.95.
               o   Specify the Coverage level; a number in interval (0.0, 1). Default choice is 0.9.
               o   Click on the OK button to continue or on the Cancel button to cancel the option.

           •   Click on OK button to continue or on Cancel button to cancel the Background Statistics.
160

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Output Screen for Nonparametric Methods (with NDs)
                         Arsenic
                                                         Total Nymberof Data 24
                                                    Number of NorvDetect Data 13
                                                      Number of Detected Data 1 1
                                                           Minimum Detected    0.5
                                                           Maximum Detected    3.2
                                                          Percent Norr Detects 54.17%
                                                          Minimum Non-detect    0.9
                                                          Maximum Non-{fetect    2
                                                        Mean of Detected Data    1 .236
                                                          SD of Detected Data    0.965
                                         Mean of Log-Transformed Detected Data  -0.0255
                                           SD of Log-Transformed Detected Data    0.694

                               Data Follow Appr. Gamma Distribution at 5%. Significance Level

                                           Nonparametric Background Statistics
                                                  X UTLwith 30%. towerage
                                                               Order Statistic   23
                                                                Achieved CC   92.02
                                                                        UTL    2.8
                                                    Largest Non-detect at 0 rtier   22

                                                Kaplan-Meter (KM) Method
                                                                      Mean    0.949
                                                                        SD    0.713
                                                        Standard Error of Mean    0. 1 65
                                                       95% UTL 90% Coverage    2.27
                                                      95% KM Chebyshev  UPL    4.121
                                                              95% KM LI PL (t)    2.1%
                                                        90% KM Percentile (z)    1.863
                                                        95% KM Percent! le (z)    2.122
                                                        99% KM Percentile (z)    2.608

                            Note: U PL (or upper percent! le for gamma distributed data) represents m
                                preferred esti mate of BTV. For an Example: KM-UPL may be used
                                           when multi pie detection limits are present
                                                                                                                  161

-------
10.2.4 All Statistics Option
1.       Click Background >• With NDs Background Statistics >• All
(? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
•P File  Edit  Configure  Summary Statistics ROS Est. NDs Graphs Outter Tests Goodness-of-Fit  Hypothesis Testing
                                                                        UCL Window  Help
 Navigation Panel
 Name
 <3 Worksheet wst
 © MW89.wst
                       WslllD
                               Mn
                                460
                                579
                                                                   Full (w/o NDs) Background Statistics  f f
2.      The Select Variables Screen (Chapter 3) will appear.

            •   Select a variable (variables) from the Select Variables screen.

            •   If needed, select a group variable by clicking the arrow below the Group by variable to
                obtain a drop-down list of available variables, and select a proper group variable.

            •   When the option button is clicked, the following window will be shown.
                             Background Statistics
                                 Confidence Level         0.95
                                         Coverage         09
                                            Next K
                                       OK
  11
Cancel
                o   Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                    default choice is 0.95.
                o   Specify the Coverage level; a number in the interval (0.0, 1). Default choice is 0.9.
                o   Specify the Next K. The default choice is 1.
                o   Click on the OK button to continue or on the Cancel button to cancel the option.

                Click on OK button to continue or on Cancel button to cancel the Background Statistics.
162

-------
Output Screen for All Statistics Option (with NDs)
            Ar:
                                                                General Statistics
                                         N umber of Val id Samples    24
                                       Number of Unique Samples     8
                                   Raw Statistics
                                              Minimum Detected    0.5
                                              Maximum Detected    3.2
                                               Mean of Detected    1.236
                                                 SD of Detected    0.965
                                            Minimum Non-Deteet    0.9
                                            Maximum Non-Detect    2

                          Data with Multiple Detection Limits
            Mote: Data have multiple DLs - Use of KM Method ss recommended
            For all rrelbods (except KM, DL'2. and ROS Methods).
            Observations < Largest ND aie treated 33 NDs
                                    N umber of Detected Data   11
                                  Number of Non-Detect Data   13
                                       Percent Nan-Detects 54.17%

                         Log-transformed Statistics
                                         Minimum Detected   -0.693
                                         Maximum Detected    1.163
                                          Mean of Detected  -0.0255
                                            SD of Detected    0.694
                                       Minimum Non-Detect   -0.105
                                       Maximum Non-Detect    0.693

                       Single Detection Limit Scenario
                   Number treated as Non-Detect with Single DL 22
                     Number treated as Detected with Single DL 2
                             Single DL Non-Detect Percentage 91.67*4
                                                              Background Statistics
                   Normal Distribution Test with Detected Values Only              Lognormal Distribution Test with Detected Values Only
                                       Shapiro Wilk Test Statistic     0.777                              Shapiro Wilk Test Statistic     0.86
                                    5% Shapiro Wilk Critical Value     0.85                            5% Shapiro Wilk Critical Value     0.85
                       Data pot ffenrai at 5% Sifplic^Mse Lw^                    Data appear Lognormal at5X Significance Level
                            Assuming Normal Distribution
                                         DL/2 Substitution Method
                                                         Mean
                                                           SD
                                        95%UTL  90% Coverage
                                                   35% UPL (t)
                                              90% Percentile (z)
                                              95% Percenttle (z)
                                              99% Percentile (z)
                      Assuming Lognormal Distribution
                                    DL/2 Substitution Method
1.002                                      Mean (Log Scale)   -0.16
0.699                                        SD (Log Scale)     0.542
2.296                               95%UTL  90% Coverage     2.327
2.224                                          95% UPL (t)     2.2
1.897                                     90% Percentile (z)     1.707
2.151                                     95% Percentile (z)     2.079
2.627                                     99% Percentile (z)     3.009
                                                                                                                                           163

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                          Maximum Likelihood EstimBte(MLE) Method N/A
                                Log ROS Method
                           Mean in Original Scale
                             SD in Original Scale
                               Mean in Log Scale   -0.209
                                 SD in Log Scale    0.571
                        95%UTL 90% Coverage
                                    95% UPL (t)
                               90S Percentile (z)
                               95% Percentile (2)
                               99% Percentile (z)
                                                                                                                                2.337
                                                                                                                                2.202
                                                                                                                                1.SS6
                                                                                                                                2.075
                                                                                                                                3.062
                    Gamma Distribution Test with Detected Values Only
                                             k star (bias corrected)     1 .702
                                                       Theta Star     0727
                                                         riu star   37.44
    Data Distribution Tests with Delected Values Only
Data follow Appr. Gamma Distribution at 5% Significance Level
                                                A-D Test Statistic    0.787
                                             5% A-D Critical Value    0.73B
                                                K-S Test Statistic    D.254
                                             5% K-S Critical Value    0.258
               DatafollowAppx. Gamma Distribution at 5% Significance Level

                             Assuming Gamma Distribution
                        Gamma ROS Statistics with extrapolated Data
                                                          Mean    1.2S3
                                                        Median    1.213
                                                            SD    0.652
                                                          k star    3.974
                                                      Thetastar    0.318
                                                        Muster  190.8
                                   95% Percentile of Chisquare (2k)   1 5.43
                                                  90% Percertble    1.861
                                                  95% Percentile    2.16
                                                  39% Percentile    2.801
               Non para metric Statistics
                        Kaplan-Meier (KM) Method
                                          Mean    0.949
                                            SD    0.713
                                     SE of Mean    0.165
                 95% KMUTLwith   90% Coverage    2.27
                         95% KM Cbebyshev UPL    4.121
                                95% KM U PL (t)    2.196
                              90% Percentile (z)    1.863
                              35% Percentile (z)    2.122
                              99% Percentile (z)    2.608
                              Note: U PL (or upper percent! le for gamma distributed dab) represents a preferred estimate of BTV
                                     For an Example: KM-UPL may be used when multiple detection limits are present
                 : OU2 is not a ree«nme.Mled ntefsl
164

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165

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                                     Chapter 11


       Computing Upper Confidence  Limits (UCLs) of Mean


The UCL computation module of ProUCL 4.0 represents an update of the UCL module of ProUCL 3.0.
The detailed theory and formulae used to compute gamma and lognormal statistics are given by Land
(1971, 1975), Gilbert (1987), Singh, Singh, and Engelhardt (1997, 1999), Singh et al. (2002a), Singh et al.
(2002b), and Singh and Singh (2003).

Several parametric and nonparametric UCL computation methods for data sets with NDs have been
incorporated in ProUCL 4.0. Methods such as the Kaplan-Meier (KM) and regression on order statistics
(ROS) methods as incorporated in ProUCL 4.0 can handle multiple detection limits. For details regarding
the distributions and methods available in ProUCL 4.0, refer to the ProUCL 4.0 Technical Guide and
Singh, Maichle, and Lee (USEPA, 2006). Recommendations for the computations of UCLs for data sets
with NDs have been made based upon the findings of the simulation experiments performed by Singh,
Maichle, and Lee (USEPA, 2006).

In ProUCL 4.0, two choices are available to compute UCL statistics:

           •   Full - Computes UCLs for full data sets without any nondetected values.

           •   With NDs - Computes UCLs for data sets that have detected as well as BDL
              observations. It is pointed out that it is not desirable to use statistical methods as
              incorporated in ProUCL  4.0 on data sets consisting of all nondetect values.  Discussion
              about the detection sampling frequency is provided in Chapter 1 of this User Guide.
              Some of the available methods can handle multiple detection limits. The program
              provides a message to the user about the use of an appropriate  method when multiple
              detection limits may be present.

           •   For full data sets without NDs and also for data sets with NDs, the following options and
              choices are available to compute UCLs of the population mean.

              o   The user specifies the confidence level;  a number in  the interval [0.5, 1), 0.5
                  inclusive. The default choice is 0.95.
              o   The program computes several nonparametric UCLs using the central limit theorem
                  (CLT), Chebyshev inequality, jackknife, and bootstrap re-sampling methods.
              o   For the bootstrap method, the user can select the number of bootstrap runs (re-
                  samples). The default choice for the number of bootstrap runs is 2000.
              o   The user is responsible for selecting an appropriate choice for the data distribution:
                  normal, gamma, lognormal, or nonparametric. It is desirable that user determines data
                  distribution using the Goodness-of-Fit test option prior to using the UCL option. The
                  UCL option informs  the user if data are  normal, gamma, lognormal, or  a non-
                  discernable distribution. Program computes statistics depending on the  user selection.
              o   For data sets, which are not normal, one may try the  gamma UCL next. The program
                  will offer you advice if you chose the wrong UCL option.
              o   For data sets, which are neither normal nor gamma, one may try the lognormal UCL.
                  The program will offer you advice  if you chose the wrong UCL option.
166

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               o  Data sets that are not normal, gamma, or lognormal are classified as distribution-free
                  nonparametric data sets. The user may use nonparametric UCL option for such data
                  sets. The program will offer you advice if you chose the wrong UCL option.
               o  The program also provides the All option. By selecting this option, the UCLs are
                  computed using most of the relevant methods available in ProUCL 4.0. The program
                  informs the user about the distribution of the underlying data set, and offers advice
                  regarding the use of an appropriate UCL.
               o  For lognormal data sets, ProUCL can compute only a 90% or a 95% Land's statistic-
                  based H-UCL of the mean. For all other methods, ProUCL can compute a UCL for
                  any confidence coefficient in the interval [0.5,1.0), 0.5 inclusive.
               o  If you have selected a distribution, then ProUCL will provide a recommended UCL
                  computation method for 0.95, confidence coefficient. Even though ProUCL can
                  compute UCLs for confidence coefficients in the interval  [0.5, 1.0),
                  recommendations are provided only for 95% UCL; as EPC term is estimated by a
                  95% UCL of the mean.

Note: It is recommended that the user identify a few low probability outlying observations that may be
present in the data set. Outliers distort many statistics of interest including summary statistics,  data
distributions, test statistics, UCLs, and estimates ofBTVs. Decisions based upon distorted statistics may
be misleading and incorrect. The objective is to compute relevant statistics and estimates based upon the
majority of the data set(s) representing the dominant population(s). Those few low probability outlying
observations require separate attention and investigation. The project team should decide about the
proper disposition (to include or not to include) of outliers before computing the statistics to estimate the
EPC terms and BTVs. In order to determine and compare the improper and unbalanced influence of
outliers on UCLs and background statistics, the project team may want to compute statistics using data
sets with outliers and without outliers.

11.1   UCLs for Full Data Sets

11.1.1  Normal Distribution (Full Data Sets without NDs)

1.      Click UCL ^ Full ^ Normal
E? ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wst]
•^ File Edit Configure Summary Statistics ROS Est, NDs Graphs Outiier Tests Goodness-of-Fit Hypothesis Testing Background
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Gamma
Lognormal
Non -Parametric
All

11




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

           •   Select a variable (variables) from the Select Variables screen.

           •   If needed, select a group variable by clicking the arrow below the Group by variable to
               obtain a drop-down list of available variables, and select a proper group variable.

           •   When the option button is clicked, the following window will be shown.
                                                                                           167

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                           Confidence Level
                             Confidence Level          ItBEl
                                   OK
Cancel
               o  Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                  default choice is 0.95.
               o  Click on OK button to continue or on Cancel button to cancel the option.

           •   Click on OK button to continue or on Cancel button to cancel the UCL computation
               option.
168

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Output Screen for Normal Distribution (Full Data without NDs)
                                                      Normal UCL Statistics for Full Data Sets
                                  User Selected Options
                                            From File   C:\Narain\ProUCL-Data\Data\Aroclor 1254.wst
                                         Full Precision   OFF
                                  Confidence Coefficient   95%
                         Aroclor_V/ithout_NonDetects

                                                    Number of Valid Samples        44
                                                   Number of Unique Samples        41
                                                                 Minimum      0.21
                                                                 Maximum     19000
                                                                    Mean      1532
                                                                  Median      94.5
                                                                      SD      3355
                                                                 Variance  11255555
                                                      Coefficient of Variation      2.19
                                                                Skewness     3.756

                                                   Shapiro Wilk Test Statistic     0.526
                                                5% Shapi ro Wi I k C ritical Val ue     0.544
                                     Data not Normal at 5°i Significance Level

                                        §5%  UCL [Assuming Normal Distribution)
                                                                  Student's-t LCL
                                                                  Studenfs-t UCL

                                           Data appear Log normal (0.05)

                                          M ay want to try Log n ormal UCLs
681.6
 2382
                                                                                                                169

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11.1.2 Gamma, Lognormal, Nonparametric, All Statistics Option (Full Data without NDs)

1.      Click UCL ^ Full ^- Gamma, Lognormal, Non-Parametric, or All
B3 ProUCL 4.0 - [C:\Narain\ProUCL-Data\Data\MW89.wstj
•B File Edit Configure Summary Statistics ROS Est. NDs Graphs Gutter Tests Goodness-of-Fit Hypothesis Testing Background
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Normal
Gamma
Lognormal
Non -Parametric


11


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

           •   Select a variable (variables) from the Select Variables screen.

           •   If desired, select a group variable by clicking the arrow below the Group by variable to
              obtain a drop-down list of available variables, and select a proper group variable.

           •   When the option button is clicked, the following window will be shown.
                                     Confidence Level
                       Number of Bootstrap Operations  |     2000
                                    OK
Cancel
              o   Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                  default choice is 0.95.
              o   Specify the Number of Bootstrap Operations (runs). Default choice is 2000.
              o   Click on OK button to continue or on Cancel button to cancel the UCLs option.

           •   Click on OK button to continue or on Cancel button to cancel the selected UCL
              computation option.
170

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Output Screen for Gamma Distribution (Full)
                                                          Gamma UCL Statistics for Full Data Sets
                                       User Selected Options
                                                From File   C:\N3rain\PraUCL-Data\Datay\rodQr 1254.wst
                                             Full Precision   OFF
                                       Confidence Coefficient   95%
                               Number of Bootstrap 0perations   2000
                              Arodor_With-out_Non Detects

                                                        Number of Valid Samples   44
                                                       Number of Unique Samples   41
                                                                    Minimum    0-21
                                                                    Maximum 18000
                                                                       Mean  1532
                                                                      Median   94-5
                                                             Standard Deviation  3355
                                                                    Variance 11255595
                                                          k star (bias corrected)    0.247
                                                                   Theta Star  6208
                                                                      nystar   21.72
                                                Approximate Chi Square Value (.05)   12.12
                                                    Adjusted Level of Significance    0.0445
                                                   Adjusted Chi Square Value (.05)   11.88
                                                      Anderson-Darling Test Statistic     1.01
                                                      Anderson-Darling Critical Value     O.SE4
                                                   Kolmogorov-Smirnov Test Statistic     0.169
                                                   Kolmogorov-Smirnov Critical Value     0.146
                                      Data not Garr-ma distributed at 5% Significance Level

                                             95% UCLs (Adjusted for Skewness)
                                                            95% Adjusted-CLT UCL   2670
                                                                95% Modified-! UCL   2430

                                                  95% Non-Parametric UCLs
                                                               95% Bootstrap-t UCL   3010
                                                           95% Hall's Bootstrap UCL   5380

                                       95% Gamma UCLsfAssuming Gamma Distribution)
                                                      95% Approximate Gamma UCL   2744
                                                       95% Approximate Gamma LCL   991
                                                          95% Adjusted Gamma UCL   2800
                                                          95% Adjusted Gamma LCL   976.E

                                              Data appear Lognormal (0_Q5)

                                             May want to try LognormaJ UCLs
                                                                                                                               171

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Output Screen for Lognormal Distribution (Full)
                                                       I Log normal UCL Statistics fof Full Data Sets
                                   User Selected Options
                                             From File   C:\Narain\ProUCL-Data\Data\Aroclor 1254.wst
                                          Full Precision   OFF
                                   Confidence Coefficient   95%
                           Number of Bootstrap Operations   2000
                          Aroclor_W]thout_NonDetects

                                                      Number of Valid Samples        44
                                                    Number of Unique Samples        41
                                                          Minimum of log data     -1.561
                                                          Maximum of log data      9.852
                                                             Mean of log data      4.474
                                                               SD of log data      3.163
                                                          Variance of log data        10

                                                     Shapiro Wilfc Test Statistic      Q.94S
                                                 Shapiro Wilk 5% Critical Value      0.944
                                   Data appear Lognormal at 5% Significance Level

                                       95% UCL (Assuming- Normal Distribution)
                                                          95% Student's-t UCL      2332
                                    ML Estimates Assuming Lognormal Distribution
                                                                      Mean     13038
                                                                        SD   1938235
                                                        Coefficient of Variation      14S.7
                                                                  Skewness   3285837
                                                                    Median      87.7
                                                               80% Quantile      1256
                                                               S0% Quantile      5050
                                                               95% Quantile     15935
                                                               99% Quantile    137549

                                                       MVU Estimate of Median      78.25
                                                        MVU Esti mate of Mean      7626
                                                          MVU Estimate of SD    165089
                                          MVU Estimate of Standard Error of Mean      5553

                                      UCLs (Assuming Lognormal Distribution)
                                                                95% H-UCL      S7513
                                                 95% Chebyshev (MVUE) UCL      31832
                                                97.5% Chebyshev (MVUE) UCL      42307
                                                 99% Chebyshev (MVUE) UCL      62381

                                              Potential UCL to Use
                                              Use 99% Chebyshev (MVUE) UCL      62881
                              Recommended UCL exceeds the njajdimwi
172

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Output for Nonparametric Methods (Full Data without NDs)
                                                i Nonparimetric UCL Stafofcs far Full Data Sete
                            User Selected Options
                                       From File  C:\Narain\ProLICL-Data\Data\Aroclor 1254.wst
                                    Full Precision  OFF
                            Confidence Coefficient  95%
                    Number of Bootstrap Operations  2000
                   Aroclor Without NorDetects
                                                Number of Valid Samples        44
                                              Number of Unique Samples        41
                                                             Minimum       0.21
                                                             Maximum      19000
                                                                Mean       1532
                                                              Median       94.5
                                                                  SD       3355
                                                             Variance   11255595
                                                  Coefficient of Variation       2.19
                                                            Skewness      3.756
                                                       Mean of log data      4.474
                                                         SD of log data      3.163
                                      N on Parametric UCLs
                                                         95%CLTUCL     135.1
                                                    95%JackknifeUCL     136.1
                                             95% 5 tandard B ootstrap U CL     134.4
                                                    95% Bootstrap-l UCL     144.5
                                                95% Hall's Bootstrap UCL     139.7
                                            35% Percentile B ootstrap U CL     136.1
                                                 95% BCA Bootstrap UCL     141.3
                                          95% Chebyshev(M ean, S d) U CL     167.3
                                         97.5% Chebyshev(M ean, S d) U CL     189.7
                                          99% Chebyshev(M ean, 5 d) U CL     233.7

                                       Potential UCL to Use
                                                Use 95% Student's-* UCL     136.1
                                                  Or 95% Modified-* UCL     136.8
                                                                                                                 173

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Output Screen for All Statistics Option (Full Data without NDs)
                                          ! General UCL Statistics for Full Data Seb
                      User SeJM^ Options
                                From File  C:\Narain\ProUCL-Data\Data\Aroclor 1254.wst
                             Full Precision  OFF
                      Confidence Coefficient  95%
              Number of Bootstrap Operations  2000
             Aroclor_WithoutJ4onDetects

                                                                General Statistics
                                         N umber of Val id Samples   44

                                    Raw Statistics
                                                      Minimum    0-21
                                                      Maximum  19000
                                                         Mean  1532
                                                       Median,   94.5
                                                           SD  3355
                                           Coefficient of Variation    2.19
                                                     Skewness    3.756
                                 Number of Unique Samples   41

                        Log-transformed Statistics
                                      Minimym of Leg Data   -1.561
                                     Maximum of Log Data    9.852
                                         Mean of log Data    4.474
                                           SD of log Data    3.163
                                                             Relevant UCL Statistics
                                Normal Distribution Test
                                        Shapiro Wilk Test Statistic
                                        Shapiro Wilk Critical Value
                        Data eet Normal at 5% StgnBearsee Levd
0.526
0.944
                             Assuming Normal Distribution
                                             95% Siudent's-t UCL  2382
                           95% UCLs (Adjusted far Skewness)
                                          95% Adjusted-CLT UCL  2670
                                             95% Modified-t UCL  2430

                               Gamma Distribution Test
                                           k star (bias corrected)    0.247
                                                    ThetaStar  6208
                                                       nu star   21.72
                                Approximate Chi Square Value (.05)   12.12
                                    Adjusted Level of Significance    0.0445
                                       Adjusted Chi Square Value   11 .SB

                                   Anderson-Darling Test Statistic    1.01
                                Anderson-Darling 5% Critical Value    0.8S4
                                 Kolmogorov-Smirn-ov Test Statistic    0.163
                             Kdmogorov-Smirriov 5% Critical Value    0.146
                  Data not Gamma Distributed at 5% S§wieases Lwel

                            Assuming Gamma Distribution
                                   95% Approximate Gamma UCL  2744
                                       95% .Adjusted Gamma UCL  2SOO
         Log normal Distribution Test
                   Shapiro Wilk Test Statistic    0.948
                   Shapiro Wilk Critical Value    0.944
Data appear Log normal at 5% Significance Level

       Assuming Lognormal Distribution
                               95%H-UCL B7513
                95% Chebyshev (MVUE) UCL 31832
               97.5% Chebyshev (MVUE) UCL 42307
                99% Chebyshev (MVUE) UCL 62881

           Data Distributor Tests
Data appear Lognormal at 5% Significance Level
                        Non para metnc Statistics
                                           95%CLTUCL  2364
                                      95% Jackknife UCL  2382
                               35% Standard Bootstrap UCL  2352
                                     95% Bootstrap-t UCL  3035
                                 95% Hall's Bootstrap UCL  5521
                              95% Percentile Bootstrap UCL  2404
                                  95% BCA Bootstrap UCL  2727
                            95% Chebyshev(Mean, Sd) UCL  3737
                           97.5% ChebyshevjMean, Sd) UCL  4690
                            99% ChebyshevjMean, Sd) UCL  6564
                                 Potential UCLto Use
                                                    p mended UCL exceeds ttig raessrswn*
                                                                                            Use 99% Chebyshev (MVUE) UCL  62881
174

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Note: Once again, it should be noted that the number of valid samples represents the total number of
samples minus (-) the missing values (if any). The number of unique or distinct samples simply represents
number of distinct observations. The information about the number of distinct samples is useful when
using bootstrap methods. Specifically, it is not desirable to use bootstrap methods on data sets with only a
few (< 4-5) distinct values.

11.2   UCL  for Data Sets with NDs

1.      Click UCL ^ With NDs
   ProUCt 4,fl .- [C:\Narai,n\ProU|:L-Oata\l)ata\MW89,)TOtl
ay File Edit Configure Summary Statistics ROS Est. NDs Graphs Qutiier Tests  Goodness-of-Fit Hypothesis Testing Background ||H9 Window  Heip
                                                                          FULL   > t
                                                                                   Normal      &&.
                                                                                   Gamma      |1
                                                                                   Lognormal     ]
                                                                                   Non-Parametric
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2.      Choose the Normal, Gamma, Lognormal, Non-Parametric, or All option.

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

            •   Select a variable (variables) from the Select Variables screen.

            •   If desired, select a group variable by clicking the arrow below the Group by variable to
               obtain a drop-down list of available variables, and select a proper group variable. The
               selection of this option will compute the relevant statistics separately for each group that
               may be present in the data set.

            •   When the option button is clicked, the following window will be shown.
                                       Confidence Level
                         Number of Bootstrap Operations
                                                               jfig
                                                               2000
                                                         Cancel
                o  Specify the Confidence Level; a number in the interval [0.5, 1), 0.5 inclusive. The
                   default choice is 0.95.
                o  Specify the Number of Bootstrap Operations. The default choice is 2000.
                                                                                               175

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

                Click on OK button to continue or on Cancel button to cancel the selected UCL
                computation option.
Output Screen for Normal Distribution (with NDs)
                                     Normal UCL Statistics for Data Sets with Non-Detects
                User Selected Options
                            From File  D:\example.wst
                         Full Precision  OFF
                 Confidence Coefficient  95%
          Number of Bootstrap Operations  2000
       Arsenic

                                       Total Number of Data        20
                                  Number of Non-Detect Data         3
                                    NumberofDetectedData        17
                                          Minimum Detected         5
                                         Maximum Detected        9.2
                                       PercentNon-Detects    15.00%
                                        Minimum Non-detect        4.3
                                       Maximum Non-detect        4.5

                                      MeanofDetectedData      6.126
                                       SD of Detected Data       1.15

                                  -     of KM        is
       For all                     	17..
                           is      for all
176

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Output Screen for Normal Distribution (with NDs) - Continued
                 Normal Distribution Test with Detected Values Only
                                      S hapiro Wilk T est S tatistic     0.794
                                  5% S hapiro Wilk Critical Value     0.892
                      Data not Normal at 5% Significance Level

                                      DL/2 Substitution Method
                                                      Mean      5.54
                                                        SD     1.779
                                            35% DU2 (t) UCL     6.228

                                   Maximum Likelihood Method
                                                      Mean     5.774
                                                        SD     1.348
                                             95% MLE (t] UCL     6.295
                                         95% MLE (Tiku) UCL     6.301

                                      Kaplan Meier(KM) Method
                                                      Mean     5.958
                                                        SD     1.104
                                        Standard Error of Mean     0.255
                                              95% KM (t) UCL     6.398
                                              95% KM (z) UCL     6.376
                                           95% KM (BCA) UCL     6.393
                             95% KM (Percentile Bootstrap) UCL     6.388

               Data do not follow a Discernable Distribution (0.05)

                      May want to try Non-Parametric UCLs
                                                                                                       177

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Output Screen for Gamma Distribution (with NDs)
                                           Gamma UCL Statistics for Data Sets with Non-Detects
                      User Selected Options
                                  From File  DAexample.mt
                              Full Precision  OFF
                       Confidence Coefficient  95%
               Number of Bootstrap Operations  2000
             Arsenic

                                             Total Number of Data    20
                                          NumberofMissingValues    20
                                        Number of Non-Delect Data     3
                                          NumberofDetectedData    17
                                                Minimum Detected     5
                                                Maximum Detected     9.2
                                              Percent Non-Detects 15.00%
                                              Minimum Non-detect     4.3
                                              Maximum Non-detect     4.5

                                            MeanofDetectedData     6.126
                                          MedianofDetectedData     5.8
                                              SD of Detected Data     1.15
                                            k Star of Detected Data    28.99
                                        ThetaStarofDetectedData     0.211
                                          NuStarofDetectedData   985.5


             For ail
             the                 is      for
178

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Output Screen for Gamma Distribution (with NDs) - Continued
                    Gamma Distribution Test with Detected Values Only
                                                 A-D Test Statistic    1.027
                                               5% A-D Critical Value    0.737
                                                  K-S Test Statistic    0.214
                                               5% K-S Critical Value    0.209
                     Data not Gamma Distributed at 5% Significance Level

                        Gamma ROS Statistics with Extrapolated Data
                                                        Minimum    3.516
                                                        Maximum    9.2
                                                           Mean    5.804
                                                          Median    5.625
                                                             SD    1.325
                                                           kStar   18.24
                                                       ThetaStar    0.318
                                                         Nu Star  729.8
                                     95% Percentile of Chisquare (2k)   51.58
                                                         AppChi2  668.1
                                      95% Gamma Approximate UCL    6.34
                                          95% G amma Adjusted U CL    6.384

                                 Kaplan Meier(KM) Method
                                                           Mean    5.958
                                                             SD    1.104
                                             StandardErrorofMean    0.255
                                                   95% KM (t) UCL    6.398
                                                95% KM (BCA) UCL    6.455
                                   95% KM (Percentile Bootstrap) UCL    6.405
                                           95%  KM (Chebyshev) UCL    7.067

                    Data do not follow a Discernable Distribution (0 05]

                           May want to try Nonparametric UCLs
                                                                                                       179

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Output Screen for Lognormal Distribution (with NDs)
                              Lognormal UCL Statistics for Data Sets with Nan-Delects
         User Selected Options
                     From File   D: \example.wst
                 Full Precision   OFF
          Confidence Coefficient   95%
   Number of Bootstrap Operations   2000
Arsenic
                                         General Statistics
                   Number of Valid Samples    20
                  Number of Unique Samples    15
                        Maximum Detected
                    Mean of Detected Data
Raw Statistics
9.2
6.126
                         Number of Detected Data    17
                       Number of Non-Detect Data     3
                            Percent Non-Detects 15.00%
Maximum Non-detect    4.5
SD of Detected Data    1.15
                                     Log-T ransf ormed S tatistics
                    Mean of Detected Data     1.798                        SD of Detected Data    0.168
 Note: Data have multiple DLs • Use of KM Method is recomrn                Number treated as Non-Detect 3
 For all methods (except  KM, DL/2, i	lobust I	|OS, and Gamma                 Number treated as Detected 17
 those Observations < Largest ND are treated as NDs                   Single DL Non-Detect Percentage 15.00%

                         Lognormal Distribution Test with Detected Values Only
                  Shapiro Wilk Test Statistic     0.854                 5% Shapiro Wilk Critical Value    0.892
                                               at S5J S
Note: Once again, it should be noted that the number of valid samples represents total number of samples
minus (-) the missing values (if any). The number of unique or distinct samples simply represents number
of distinct observations. The information about the number of distinct samples is useful when using
bootstrap methods. Specifically, it is not desirable to  use bootstrap methods on data sets with only a few
(< 4-5) distinct values.
180

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Output Screen for Lognormal Distribution (with NDs) - Continued
                                                DL/2 Substitution Method
                                             Mean    5.54
                                               SD    1.779
                                 Mean (in Log Scale)    1.648
                                   SD (in Log Scale)    0.398

                                                  Robust ROS Method
                                             Mean    5.831
                                               SD    1.28
                                 Mean (in Log Scale)    1.742
                                   SD (in Log Scale)    0.207

                                                Kaplan Meier (KM) Method
                                             Mean    5.958
                                               SD    1.104
                                        SE of Mean    0.255
95% H-Stat (DL/2) UCL    6.522
95% Percentile Bootstrap UCL
    95% BCA Bootstrap UCL
           95% KM (t) UCL
        95% KM (BCA) UCL
  95% KM (% Bootstrap) UCL
   95%KM(Chebyshev)UCL
 97. 5% KM (Chebyshev) UCL
   99%KM(Chebyshev)UCL
                      6.31 8
                      6.357
                      6.398
                      6.43
                      6.373
                      7.067
                      7. 547
                      8.49
                                         Potential UCL to Use
                                                                Data do not follow a Discernable Distribution (0.05)
                                                                      May want to try Nonpararnetric UCLs
                                                                                                                  181

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Output Screen for Nonparametric Methods (with NDs)
                                       Nonparametric UCL Statistics for Data Sets with Non-Detects
                 User Selected Options
                             From File   D:\exannple.wst
                          Full Precision   OFF
                  Confidence Coefficient   95%
          Number of Bootstrap Operations   2000
        Arsenic

                                         Total Number of Data         20
                                    Number of Non-Detect Data          3
                                     Number of Detected Data         17
                                           Minimum Detected          5
                                           Maximum Detected        9.2
                                         Percent Non-Detects     15.00%
                                          Minimum Non-detect        4.3
                                         Maximum Non-detect        4.5

                                       Mean of Detected Data      6.126
                                      Median of Detected Data        5.8
                                     VarianceofDetectedData      1.323
                                         SD of Detected Data       1.15
                                         CV of Detected Data      0.188
                                    S kewness of D elected D ata      1.783
                                     MeanofDetectedlogdata      1.788
                                      SDofDetectedLogdata      0.168

                                DI	s -     of KM        is
           all                     DL/2,
                                Dl	      is      for
182

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Output for Nonparametric Methods (with NDs) - Continued
                Nonparametric Test with Detected Values Only
               Data do not follow a Discernable Distribution (0 06)

                                       WinsorizationMethod      0.168
                                                    Mean      5.738
                                                      SD      0.624
                                         95%Winsor(t)UCL      5.985

                                    Kaplan Meier (KM) Method
                                                    Mean      5.958
                                                      SD      1.104
                                      Standard Error of Mean      0.255
                                            95% KM (t) UCL      6.398
                                            95% KM (z) UCL      6.376
                                         95% KM (BCA) UCL      6.478
                            95% KM  (Percentile Bootstrap) UCL      6.438
                                    95% KM (Chebyshev) U CL      7.067
                                  97.5% KM (Cheby she v) U CL      7.547
                                    99% KM (Cheby she v) UCL       8.49

                            Potential UCL to Use
                                    95% KM (Chebyshev) UCL      7.067
                                                                                                    183

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 Output Screen for All Statistics Option (with NDs)
                            General UCL Statistics foi Data Sets with Non-Detects
          User Selected Options
                    From File  D:\example.wst
                 Full Precision  OFF
          Confidence Coefficient  95%
   Number of Bootstrap Operations  2000
 Arsenic
                            Number of Valid Samples
                          Number of Unique Samples
                      Raw Statistics
                                Minimum Detected
                                Maximum Detected
                                Mean of Detected
                                   SD of Detected
                               Minimum Non-Detect
                              Maximum Non-Detect
General Statistics
     20
     15
      5
     9.2
   6.126
    1.15
     4.3
     4.5
          Number of Detected Data        17
        NumberofNon-DetectData        3
             Percent Noil-Detects    15.00%

Log transformed Statistics
               MinimumDetected     1.609
               Maximum Detected     2.219
               Mean of Detected     1.799
                 SD of Detected     0.169
             Minimum Non-Detect     1.459
             Maximum Non-Detect     1.504

      Number treated as Non-Detect        3
        Number treated as Detected        17
    Single DL Non-Detect Percentage    15.00%
                                                 UCL Statistics
        Normal Distribution Test with Detected Values Only
                           Shapiro Wilk Test Statistic
                        5% Shapiro Wilk Critical Value
           Data not formal a! 5X Significance Level
             Lognormal Distribution Test with Detected Values Only
   0.7S4                          S hapiro Wilk T est S tatistic     0.854
   0.892                        5% S hapiro Wilk Critical Value     0.892
                Data no! Lognorraal at 5?£ Significance Level
 Note: Once again, it should be noted that the number of valid samples represents the total number of
 samples minus (-) the missing values (if any). The number of unique or distinct samples simply represents
 number of distinct observations.  The information about the number of distinct samples is useful when
 using bootstrap methods. Specifically, it is not desirable to use bootstrap methods on data sets with only a
few (< 4-5) distinct values.
 184

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Output Screen for All Statistics Option (with NDs) - Continued
Assuming Normal Distribution
DU2 Substitution Method
Mean
SD
95% DU2 (t] UCL
Maximum Likelihood Estirnate(MLE) Method
Mean
SD
95% MLE (t) UCL
95% MLE (Tiku) UCL


Gamma Distribution Test with Detected Values Only
k star (bias corrected)
Theta Star
nu star
A-D Test Statistic
5% A-D Critical Value
K-S Test Statistic
5% K-S Critical Value
D ala not G Distributed at 5% S

Assuming Gamma Distribution
Garnnna ROS Statistics using Extrapolated Data
Minimum
Maximum
Mean
Median
SD
kstar
Theta star
Nu star
AppChi2
95% Gamma Approximate UCL
95% Adjusted Gamma UCL


5.54
1.779
6.228

5.774
1.348
6.295
6.301



28.99
0.211
985.5
1.027
0.737
0.737
0.209




3.516
9.2
5.804
5.625
1.325
18.24
0.318
729.8
668.1
6.34
6.384
Assuming Lognormal Distribution
DL/2 Substitution Method
Mean
SD
95% H-Stat (DL/2) UCL
Robust ROS Method
Mean in Log Scale
SD in Log Scale
Mean in Original Scale
SD in Original Scale
95% Percentile Bootstrap UCL
95% EGA Bootstrap UCL
N onparametic T est with D elected Values Only
Data do not follow a Discernable Distribution (0.05)


Nonparametric Statistics
Kaplan-Meier (KM (Method
Mean
SD
SE of Mean
95% KM [t] UCL
95% KM (z) UCL
95% KM (jackknife) UCL
95% KM (bootstrap t) UCL
95% KM (BCA) UCL
95% KM (percentile) UCL
95% KM (Chebyshev) UCL
97.5% KM (Chebjishev) UCL
99% KM (Chebyshev) UCL

Potential UCLs to Use
95% KM (Chebyshev)UCL




1.648
0.398
6.522

1.742
0.207
5.831
1.28
6.314
6.37






5.958
1.104
0.255
6.398
6.376
6.387
6.673
6.43
6.398
7.067
7.547
8.49


7.067


                                                                                                      185

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                                       Chapter 12

                                        Windows
 y File  Edit Configure  Summary Statistics  ROS Est. NDs Graphs Clutter Tests  Goodness-of-Fit Hypothesis Testing  Background
                                                                                  Cascade
                                                                                  Tile Vertically
                                                                                  Tile Horizontally
 CJ Worksheet wst
Click on the Window menu to reveal the drop-down options shown above.

The following Window drop-down menu options are available:

           •   Cascade option: arranges windows in a cascade format. This is similar to a typical
               Windows program option.

           •   Tile option: resizes each window vertically or horizontally and then displays all open
               windows. This is similar to atypical Windows program option.

           •   The drop-down options list also includes a list of all open windows with a check mark in
               front of the active window. Click on any of the windows listed to make that window
               active. This is especially useful if you have more than 20 windows open, as the
               navigation panel only holds the first 20 windows.
186

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187

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                                       Chapter 13

                                           Help
When the Help menu is clicked, the following window will appear.
B? ProUCL 4.0 - [WorkSheet.wst]
•g File Edit Configure Summary Statistics ROS Est. NDs Graphs Outlier Tests Goodness -of-Fit Hypothesis Testing Background UCL Window
0|«&| olalml nl
Navigation Panel
Name
©WorkSheet.wst

1
2
0



1


2



3



4



5


6



7



8





O

About ProUCL
Technical Support












Three options are available under Help menu:

           •  About ProUCL: This option provides a brief description of ProUCL 4.0, and all
              improvements made compared to ProUCL 3.0.

           •  Statistical Help: This option executes an online help directory. This option provides
              information about the various algorithms and formulae (with references) used in the
              development of ProUCL 4.0. More information on the various topics covered under
              Statistical Help is provided below.

           •  Technical Support: This option will provide contact information for primary technical
              support via phone and e-mail.

Statistical Help provides online help notes for the methods and options available in ProUCL 4.0. A screen
(shown below), listing the topics containing help notes, appears after clicking on the Help menu.
          51
                         ProUCL 4.0 Online Help Directory
                         I i. 1.1 MI li,-l|i:
                               c/M>,
                         Suck!

188

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The online help directory is divided into the following submenus:

           •   Getting Started: This chapter provides basic information on the software, including
               software installation to various menu displays.

           •   Summary Statistics: This chapter provides information and examples on procedures for
               simple classical summary statistics for data sets with and without nondetects.

           •   ROS Estimates of NDs: This chapter briefly describes the estimation (extrapolation) of
               nondetects using regression on order statistics (ROS)  for normal, lognormal, and gamma
               distribution.

           •   Graphs: This chapter provides information and examples on the graphical displays that
               ProUCL 4.0 can produce: box plots, histograms, and Multi Q-Q plots.

           •   Outlier Tests: This chapter provides  information and examples for two classical outlier
               tests available in ProUCL 4.0: Dixon's and Rosner's tests for data sets with and without
               outliers.

           •   Goodness-of-Fit: This chapter provides information and examples for several goodness-
               of-fit tests available in ProUCL 4.0 for data sets with  and without NDs.

           •   Background Statistics: This chapter provides information and examples for the
               computation of Background Statistics needed to estimate the BTVs and not-to-exceed
               values. These statistics are sometimes used to compare point-by-point site data (not more
               than 4 to 5 site samples) with the BTVs.

           •   Hypotheses Testing: This chapter provides brief descriptions (with examples) of the
               various single sample and two-sample hypotheses testing approaches as incorporated in
               ProUCL 4.0.

               o   Single Sample Hypotheses Testing: This chapter provides brief description and
                   examples of single sample hypotheses testing approaches that are useful to compare
                   site concentrations with cleanup standards, compliance limits, or not-to-exceed
                   limits. The minimum sample size requirements for site data are briefly discussed in
                   Chapter 1 of this User Guide.
               o   Two-sample hypotheses are used to compare site  data with background data (or
                   upgradient and downgradient wells) provided enough site and background data are
                   available. The minimum sample size requirements for site and background data are
                   briefly discussed in Chapter 1 of this User Guide.

           •   Upper Confidence Limits (UCLs): This option provides information and examples for the
               various UCL computation methods as incorporated in ProUCL 4.0.
                                                                                           189

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                                        Chapter 14

                Handling the Output Screens and  Graphs

Copying Graphs

1.       Click the graph you want to copy or save in the Navigation Panel. The graphs can be saved using
        the copy option.
 P? ProUCL 4.0 - [Histo_Group.gst]
 •P File Edit Configure Window Help
   e| B|B|m| c|
                                                                                            - S X
 Navigation Panel
 Name
 ^worksheet.wst
 @ example, wst
 El Histo_Arsenic (sub...
 13 Histo_Arsenic (surf..
                                             Histogram Plot for Group
                  fr
                      I Arsenic (subsurface)  Arsenic (surface)
                   Log Panel
                 LOG: 2:03:21 PM ^Information] Box_Arsenic (4.5 ).gst closed!
                 LOG: 2:03:21 PM >[Information] Box_Arsenic (4.3).gst closed!
                 LOG: 2:03:33 PM >[Information] Histogram was generated!
                 LOG: 2:03:33 PM >[Information] Histogram was generated!
2.      Click Edit ^ Copy Graph.

                                  File I   I Configure  Window  Help




3.      Once the user has clicked "Copy Graph," the graph is ready to be imported (pasted) into most
        Microsoft Office applications (Word, Excel, and PowerPoint have been tested) by clicking
        Edit ^ Paste in these Microsoft applications.
190

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4.     It is important to note that the graph cannot be saved as its own file and must be imported into an
       application to be saved. This will save the graph, but the overall file attribute and properties will
       be that of the application in which the graph was saved in. For example, if the graph was saved
       within Microsoft Word, the graph will be saved in a document with a .doc extension.

Printing Graphs

1.     Click the graph you want to print in the Navigation Panel.

2.     Click File ^ Page Setup.
                                   Edit  Configure  Window  Help
                                New
                                Open ...
                                Load Excel Data
                                Close
                                Save
                                Save As ,,,
                                Page 5etup
                                Print
                                Print Preview
                                Exit
       Check the radio button next to Portrait or Landscape, and click OK. In some cases, with larger
       headings and captions, it may be desirable to use the Landscape printing option.
Page Setup




Size: Le
S_ource: AL

r Portrait



^^^

i
;


ter
o Select
ii • f i i
Left: [05
Jop: [05

OK | Cance

•EH




3
3

Bottom: [tl5



Jl











4.     Click File ^ Print to print the graph, and File ^ Print Preview to preview the graph before
       printing.
                                                                                            191

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                                                  Edit   Configure   Window   Help
                                              New
                                              Open ...
                                              Load Excel Data
                                              Close
                                              Save
                                              Save As ,,,
                                              Page 5etup
                                              Print
                                              Print Preview
                                              Exit
Printing  Non-graphical Outputs
1.         Click the output you want to copy or print in the  Navigation Panel.
 n=  File  Edit  Configure  Window  Help
       jalMniJjn]
 Navigation Panel
  Name
 & worksheet, wst
 £j$ example, wst
 S| output.ost
From File: D:\example_w
General UCL Statistics
 13 NormQQ_Arsenic.gst
                                          Raw Statistics
                                                  Total Number of Data      20
                                               Number of Detected Data      17
                                              Number of Non-Detect Data       3
                                                  Detection Percentage   65.00%
                                                    Minimum Detected
                                                    Maximum Detected
                                                    Mean of Detected
                                                      SD of Detected
                                                  Minimum Non-Detect
                                                  Maximum Non-Detect
                           Log Panel
                                                                Log-transformed Statistics
                                                                              Minimum Detected    1.6094
                                                                              Maximum Detected    2.2192
                                                                              Mean of Detected    1.7983
                                                                                SD of Detected    0.16S2
                                            5.0000
                                            9.2000
                                            6.1265
Minimum Non-Detect
Maximum Non-Detect
1.4586
1.5041
                                            1.1500 N ote: Data have multiple D Ls - U se of KM methods is recommended
                                                   - Use of KM methods is recommended
                                            4 3000 For all methods (Except KM, DL/2. and Helsers ROSJ
                                            4 5000 Observations < largest ND are treated as NDs
                                                                      Number treated as Non-Detect
                                                                        Number treated as Detected
                                                                     Single DL Detection Percentage
                                                                       UCL
                                                                                                                         3
                                                                                                                        17
                                                                                                                     85.00%
                       LOG: 6:1 7:37 PM >[Information] D:\exarnple.wst loaded!
                       LOG: 6:1 7:45 PM > [Information] Generated results!
                       LOG: 6:1 9:02 PM >[Informat!on] Generated results!
                       LOG: 6:30:50 PM >[Information] Q-Q Plot graph for Arsenic was generated!
192

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2.
Click File ^ Print.
                              ^^J Edit  Configure  Window  Help
                                  New
                                  Open .,.
                                  Load Excel Data
                                  Close

                                  Save
                                  Save As ...

                                  Page Setup
                                  Print
                                  Print Preview

                                  Exit

Saving Output Screens as Excel Files

1.       Click on the output you want to save in the Navigation Panel List.

2.       Click File ^- Other Files... ^-Export Excel (preserve formatting).

                      ^^J Edit  Configure  Window Help
                         New
                         Open ...
                         Load Excel Data
                 Other Files ...
                         Close

                         Save As ...

                         Print
                         Print Preview

                         Exit
                                           Export Excel (preserve formatting). . .
                                                                                                193

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3.      Enter the desired file name you want to use, and click Save, and save the file in the desired folder
       using your browser.
Save As ? X
Save in:
uL^J
My Recent
Documents
Desktop
*w^
My Computer
My Network
Places

@ Desktop _J <3=fcetilT
_J My Computer
*JMy Network Places





File name: II3TTH5BJ3 T| S_ave

Save as type: | Excel Workbook (K.xls) _^J Cancel

194

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195

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                                    Chapter 15


 Recommendations to Compute a 95% UCL (Estimate of EPC
    Term)  of the  Population Mean, [Ji,  Using Symmetric  and

   Positively Skewed Full Data Sets without any Nondetects


This chapter describes the recommendations for the computation of a 95% UCL of the unknown
population arithmetic mean, }JL\, of a contaminant data distribution based upon full data sets without any
nondetect observations. These recommendations are based upon the findings of Singh, Singh, and
Engelhardt (1997, 1999); Singh etal. (2002a); Singh, Singh, and laci (2002b); and Singh and Singh
(2003). These recommendations are applicable to full data sets without censoring and nondetect (ND)
observations. Recommendations to compute UCL95 based upon data sets with NDs are summarized in
Chapter 16.

For skewed parametric as well as nonparametric data sets, there is no simple solution to compute a 95%
UCL of the population mean, ^. Contrary to the general conjecture, Singh et al. (2002a); Singh, Singh,
and laci (2002b); and Singh and Singh (2003) noted that the UCLs based upon the skewness adjusted
methods, such as the Johnson's modified t and Chen's adjusted-CLrdo not provide the specified
coverage (e.g., 95 %) to the population mean even for mildly to moderately  skewed (e.g., a the sd of log-
transformed data in interval [0.5, 1.0)) data sets for samples of size as large as 100. The coverage of the
population mean by the skewness-adjusted UCLs becomes much smaller than the specified coverage of
0.95 for highly skewed data sets, where skewness is defined as a function of a or a (sd of log-
transformed data).

It is noted that even though, the simulation results for highly skewed data sets of small sizes suggest that
the bootstrap t and Hall's bootstrap methods do approximately provide the adequate coverage to the
population mean, sometimes in practice these two bootstrap methods yield erratic inflated values (orders
of magnitude higher than the  other UCL values) when dealing with individual highly skewed data sets of
small sizes. This is especially true when potential outliers may be present in the data set. ProUCL 4.0
provides warning messages whenever the recommendations are made regarding the use the bootstrap t
method or Hall's bootstrap method.

15.1   Normally or Approximately Normally Distributed Data Sets

          •   For normally distributed data sets, a UCL based upon the Student's t-statistic provides the
              optimal UCL of the population mean. Therefore, for normally distributed data sets, one
              should always use a 95% UCL based upon the Student's t-statistic.

          •   The 95% UCL of the mean based upon Student's t-statistic may also be used when the
              Sd, sy of the log-transformed data, is less than 0.5, or when the data set approximately
              follows a normal distribution. Typically, a data set is approximately normal when the
              normal Q-Q plot displays a linear pattern (without outliers and jumps of significant
              magnitude) and the resulting correlation coefficient is high (e.g., 0.95 or higher). The
              jumps and breaks in a Q-Q plot (even with a high correlation coefficient) suggest the
              presence of multiple populations in the data set under study.
196

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           •  Student's t-UCL may also be used when the data set is symmetric (but possibly not
              normally distributed). A measure of symmetry (or skewness) is k3. A value of k3 close
              to zero (e.g., if the absolute value of the skewness is roughly less than 0.2 or 0.3) suggests
              approximate symmetry. The approximate symmetry of a data distribution can also be
              judged by looking at the histogram of the data set.

15.2  Gamma Distributed Skewed Data Sets

In practice, many skewed data sets can be modeled both by a lognormal  distribution and a gamma
distribution, especially when the sample size is smaller than 70-100. As well known, the 95%H-UCL of
the mean based upon a lognormal model often behaves in an erratic manner. Specifically, 95%H-UCL
often results in an unjustifiably large and impractical 95% UCL value when the sample size is small (e.g.,
n <20, 50, ..) and skewness is high. Moreover, it is also observed that a 95% UCL based upon Land H-
statistic becomes even smaller than the sample arithmetic mean. This is especially true for mildly skewed
to moderately skewed data sets of large sizes (e.g., > 50-70). In such cases, a gamma model, G(k, 6),  may
be used to compute a reliable  95% UCL of the unknown population mean, }j.\.

           •  Many skewed data sets follow a lognormal as well as a gamma distribution. It should be
              noted that the population means based upon the two models could differ significantly. A
              lognormal model based upon a highly skewed (e.g., a > 2.5) data set will have an
              unjustifiably large and impractical population mean, pi,  and its associated UCL. The
              gamma distribution is better suited to model positively skewed environmental data sets.

           •  One should always first check if a given skewed data set follows  a gamma distribution. If
              a data set does follow a gamma distribution or an approximate gamma distribution, one
              should compute a 95% UCL based upon a gamma distribution. Use of highly skewed
              (e.g.,  a > 2.5-3.0) lognormal distributions should be avoided. For such highly skewed
              lognormally distributed data sets that cannot be modeled by a gamma or an approximate
              gamma distribution, nonparametric UCL computation methods based upon  the
              Chebyshev inequality may be used.

           •  The five bootstrap methods do not perform better than the two gamma UCL computation
              methods. It is noted that the performances (in terms of coverage probabilities) of the
              bootstrap t and Hall's bootstrap methods are very similar. Out of the five bootstrap
              methods, bootstrap t and Hall's bootstrap methods perform the best (with coverage
              probabilities for population mean closer to the nominal level of 0.95). This  is especially
              true when the skewness is quite high (e.g., k  < 0.1) and the sample size is small (e.g., n <
              10-15). Whenever the use  of Hall's UCL or bootstrap t UCL is recommended, an
              informative warning message about their use is also provided.

           •  Contrary to the conjecture, the bootstrap BCA method does not perform better than the
              Hall's method or the bootstrap t method. The  coverage for the population mean, }j.\,
              provided by the  BCA method is much lower than the specified 95% coverage. This is
              especially true when the skewness is high  (e.g., k<\) and the sample size is small (e.g.,
              Singh and Singh (2003), and Singh, Singh, and laci (2002b)).
                                                                                          197

-------
           •  From the results presented in Singh, Singh, and laci (2002b), and in Singh and Singh
              (2003), it is concluded that for data sets which follow a gamma distribution, a 95% UCL
              of the mean should be computed using the adjusted gamma UCL when the shape
              parameter, k, is: 0.1 < k < 0.5, and for values of £ > 0.5, a 95% UCL can be computed
              using an approximate gamma UCL of the mean, ^.

           •  For highly skewed gamma distributed data sets with k < 0.1, the bootstrap t UCL or
              Hall's bootstrap (Singh and Singh 2003) may be used when the sample size is smaller
              than 15, and the adjusted gamma UCL should be used when the sample size starts
              approaching or exceeding 15. The small sample size requirement increases as the
              skewness increases (that is, as k decreases, the required sample size, n, increases).

           •  The bootstrap t and Hall's bootstrap methods should be used with caution as these
              methods may yield erratic, unreasonably inflated, and unstable UCL values, especially in
              the presence of outliers. In case Hall's bootstrap and bootstrap t methods yield inflated
              and erratic UCL results, the 95% UCL of the mean should be computed based upon the
              adjusted gamma 95% UCL. ProUCL 4.0 prints out a warning message associated with the
              recommended  use of the UCLs based upon the bootstrap t method or Hall's bootstrap
              method. The recommendations for gamma distribution are summarized in Table  15-1.

Table 15-1. Computation of a UCL95 of the Unknown Mean, fih of a Gamma Distribution
k
k >0.5
0.1 < k<0.5
k <0.1
k <0.1
Sample Size, n
For all n
For all n
IK15
n>15
Recommendation
Approximate gamma 95% UCL
Adjusted gamma 95% UCL
95% UCL based upon bootstrap t
or Hall's bootstrap method*
Adjusted gamma 95% UCL if available,
otherwise use approximate gamma 95% UCL
*If bootstrap t or Hall's bootstrap methods yield erratic, inflated, and unstable UCL values (which often
happens when outliers are present), the UCL of the mean should be computed using the adjusted gamma
UCL.

15.3  Lognormally Distributed Skewed Data Sets

For lognormally, LN (ju, a2), distributed data sets, the H-statistic-based//-f/CL provides specified 0.95,
coverage for the population mean for all values of a. However, the H-statistic often results in unjustifiably
large UCL values that do not occur in practice. This is especially true when the skewness is high (e.g., a >
2.0). The use of a lognormal model unjustifiably accommodates large and impractical values of the mean
concentration and its UCLs. The problem associated with the use of a lognormal distribution is that the
population mean, //1; of a lognormal model becomes impractically large for larger values of a, which in
turn results in an inflated H-UCL of the population mean, ^\. Since the  population mean of a lognormal
model becomes too large, none of the other methods, except for the H-UCL, provides the specified 95%
coverage for that inflated population mean, n\.  This is especially true when the sample size is small and
198

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the skewness is high. For extremely highly skewed data sets (with a> 2.5-3.0) of smaller sizes (e.g., < 70-
100), the use of a lognormal distribution-based H-UCL should be avoided (e.g., see Singh et al. (2002a)
and Singh and  Singh (2003)). Therefore, alternative UCL computation methods, such as the use of a
gamma distribution or the use of a UCL based upon nonparametric bootstrap methods or the Chebyshev
inequality-based methods, are desirable. All skewed data sets should first be tested for a gamma
distribution. For lognormally distributed data (that cannot be modeled by gamma distribution), methods
summarized in Table  15-2 may be used to compute a 95% UCL of mean.

ProUCL can compute an H-UCL for samples  of sizes up to 1000. For highly skewed lognormally
distributed data sets of smaller sizes, alternative methods to compute a 95% UCL of the population mean,
/j.\, are  summarized in Table 15-2.  Since skewness is a function of o (or  a), the recommendations for the
computation of the UCL of the population mean are also summarized in terms of a and the sample size,
n. Here, a is an MLE of o, and is given by the Sd of log -transformed data. Note that Table 15-2 is
applicable only to the computation of a 95% UCL of the population mean based upon lognormally
distributed data sets without nondetect observations.

Table 15-2.  Computation of a 95% UCL of Mean, Hi of a Lognormal Population
/\
a
a < 0.5
0.5 < a <1.0
1.0< a <1.5
1.5< a <2.0
1.5< a <2.0
2.5 < a < 3.0
3.0 < a <3.5
a > 3.5
Sample Size, n
For all n
For all n
n<25
n>25
n<20
20 < n < 50
n>50
n<20
20 < n < 50
50 < n < 70
n>70
n < 30
30 < n < 70
70100
n<15
15 150
For all n
Recommendation
Student's t, modified t, or H-UCL
H-UCL
95% Chebyshev (MVUE) UCL
H-UCL
99% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
99% Chebyshev (MVUE) UCL
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Larger of 99% Chebyshev (MVUE) UCL or
99% Chebyshev (Mean, Sd)
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Hall's bootstrap method*
Larger of 99% Chebyshev (MVUE) UCL or
99% Chebyshev (Mean, Sd)
97.5% Chebyshev (MVUE) UCL
95% Chebyshev (MVUE) UCL
H-UCL
Use nonparametric methods*
                                                                                          199

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*If Hall's bootstrap method yields an erratic or unrealistically large UCL value, then the UCL of the mean
may be computed based upon the Chebyshev inequality.

15.4  Data Sets without a Discernable Skewed Distribution - Nonparametric
Methods for Skewed Data Sets

The use of gamma and lognormal distributions as discussed here cover a wide range of skewed data
distributions. For skewed data sets which are neither gamma nor lognormal, one can use a nonparametric
Chebyshev UCL or Hall's bootstrap UCL (for small samples) of the mean to estimate the EPC term.

          •   For skewed nonparametric data sets with negative and zero values, use a 95% Chebyshev
              (Mean, Sd) UCL for the population mean, y.\.

For all other nonparametric data sets with only positive values, the following method may be used to
estimate the EPC term.

          •   For mildly skewed data sets with a < 0.5, one can use the Student's t-statistic or
              modified t-statistic to compute a 95% UCL of mean, ^.

          •   For nonparametric moderately skewed data sets (e.g.,  a or its estimate, a, in the interval
              (0.5, 1]), one may use a 95% Chebyshev (Mean, Sd) UCL of the population mean, y.\.

          •   For nonparametric moderately to highly skewed data sets (e.g., a in the interval (1.0,
              2.0]), one may use a 99% Chebyshev (Mean, Sd) UCL or a 97.5% Chebyshev (Mean, Sd)
              UCL of the population mean, //1; to obtain an estimate of the EPC term.

          •   For highly skewed to extremely highly skewed data sets with a in the interval (2.0, 3.0],
              one may use Hall's UCL or a 99% Chebyshev (Mean,  Sd) UCL to compute the EPC term.

          •   Extremely skewed nonparametric data sets with a exceeding 3 provide poor coverage.
              For such highly skewed data distributions, none of the methods considered provide the
              specified 95% coverage for the population mean, ^. The coverage provided by the
              methods decrease as a increases. For such data sets of sizes less than 30, a 95% UCL can
              be computed based upon Hall's bootstrap method or bootstrap t method. Hall's bootstrap
              method provides the highest coverage (but less than 0.95) when the sample size is small.
              It is noted that the coverage  for the population mean provided by Hall's method (and
              bootstrap t method) does not increase much as the sample size, n, increases.  However, as
              the sample size increases, coverage provided by the 99% Chebyshev (Mean, Sd) UCL
              method also increases. Therefore, for larger samples, a UCL should be computed based
              upon the 99% Chebyshev (Mean, Sd) method. This large sample size requirement
              increases as a increases. These recommendations are summarized in Table  15-3.
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Table 15-3. Computation of 95% UCL of Mean, fih Based Upon a Skewed Data Set (with all positive values) without a
Discernable Distribution, where G issrfof Log-Transformed Data
/*,
a
o <0.5
0.5 < a < 1.0
1.0< a <2.0
2.0 < a <3.0
3.0 < a <3.5
a > 3.5
Sample Size, n
For all n
For all n
n<50
n>50
n<10
n>10
n < 30
n>30
n< 100
n>100
Recommendation
95% UCL based on Student's t or modified t-statistic
95% Chebyshev (Mean, Sd) UCL
99% Chebyshev (Mean, Sd) UCL
97.5% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
Hall's Bootstrap UCL*
99% Chebyshev (Mean, Sd) UCL
* If Hall's bootstrap method yields an erratic and unstable UCL value (e.g., happens when outliers are
present), a UCL of the population mean may be computed based upon the 99% Chebyshev (Mean, Sd)
method. The results as summarized in Tables 15-1 through 15-3 are summarized in Table 15-4, shown on
the next page.
                                                                                             201

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Table 15-4. Recommended UCL95 Computation Methods for Full-Uncensored Data Sets without Nondetect Observations
Skewness
Sample Size
95% Student t
95% modified t
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o
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s?
m
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95% Hall's Bootstrap
95% Chebyshev (MVUE)
97.5% Chebyshev (MVUE)
99% Chebyshev (MVUE)
95% Chebyshev (Mean, Sd)
97.5% Chebyshev (Mean, Sd)
99% Chebyshev (Mean, Sd)
95% Approx. Gamma
95% Bootstrap t
95% Adjusted Gamma
Normal or Approximate Normal (with a < 0.5) Distribution

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15.5  Should the Maximum Observed Concentration be Used as an
       Estimate of the EPC Term?

This topic has been discussed earlier in Chapter 1. It is included here only for the convenience of the user.
In practice, atypical user tends to use the maximum sample value as an estimate of the EPC term.  This is
especially true when the sample size is small or the data are highly skewed. The discussion and
suggestions as described in Chapter 1 apply to both Chapters 15 and 16. Singh and Singh (2003) studied
the max test (using the maximum observed value as an estimate of the EPC term) in their simulation
study. Previous (e.g., RAGS document (EPA, 1992)) use of the maximum observed value has been
recommended as a default option to estimate the EPC term when a 95% UCL (e.g., the H-UCL) exceeded
the maximum value. Only two 95% UCL computation methods, namely the Student's t-UCL and Land's
H-UCL, were used previously to estimate the EPC term (e.g., EPA,  1992). ProUCL 4.0 can compute a
95% UCL of the mean using several methods based upon the normal, gamma, lognormal, and
"nonparametric" distributions.  Furthermore, since the EPC term represents the average exposure
contracted by an individual over an exposure area (EA) during a long period of time, the EPC term should
be estimated by using an average value (such as an appropriate 95% UCL of the mean) and not by the
maximum observed concentration.
                                                                                      203

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Singh and Singh (2003) also noted that for skewed data sets of small sizes (e.g., < 10-20), the max test
does not provide the specified 95% coverage to the population mean, and for larger data sets, it
overestimates the EPC term which may require unnecessary further remediation. For the distributions
considered, the maximum value is not a sufficient statistic for the unknown population mean. The use of
the maximum value as an estimate of the EPC term ignores most (except for the maximum value) of the
information contained in a data set. It is, therefore, not desirable to use the maximum observed value as
an estimate of the EPC term representing average exposure by an individual over an EA.

It is recommended that the maximum observed value NOT be used as an estimate of the EPC term.
For the sake of interested users, ProUCL displays a warning message when the recommended 95% UCL
(e.g., Hall's bootstrap UCL) of the mean exceeds the observed maximum concentration. When a 95%
UCL exceeds the maximum observed value, ProUCL recommends the use of an alternative UCL method
based upon a 97.5% or 99% Chebyshev UCL.

It should also be noted that for highly skewed data sets, the sample mean indeed can even exceed the
upper 90%, or higher, etc., percentiles, and consequently, a 95% UCL of the mean can exceed the
maximum observed value of a data set. This is especially true when one is dealing with lognormally
distributed data sets of small sizes. For such highly skewed data sets which cannot be modeled by a
gamma distribution, a 95% UCL of the mean should be computed using an appropriate nonparametric
method as summarized in Table 15-4.
204

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205

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                                    Chapter 16


 Recommendations to Compute a 95% UCL of the Population
    Mean, fJi, Using Data Sets with Nondetects with Multiple
                                Detection  Limits


This chapter summarizes the recommendations based on the simulation experiments conducted by Singh,
Maichle, and Lee (USEPA, 2006) to compare the performances of the UCL computation methods based
upon data sets with BDLs and multiple detection limits (DLs). ProUCL 4.0 suggests the use of
appropriate UCLs based upon the findings of Singh, Maichle, and Lee (USEPA, 2006). For convenience,
the recommended UCL95 computation methods have been tabulated in Table 16 as functions of the
sample size, skewness, and censoring intensity. General observations and recommendations regarding the
difficulties associated with data sets with NDs are described first.

16.1   General Recommendations and Suggestions

          •  In practice, it is not easy to verify (perform goodness-of-fit) the distribution of a left-
             censored data set with NDs. Therefore, emphasis is given on the use of nonparametric
             UCL95 computation methods, which can also be used to handle multiple detection limits.

          •  This is specifically true when the percentage (%) of nondetects exceeds 40%-50%.

          •  Most of the parametric MLE methods assume that there is only one detection limit. It
             should also be noted that the MLEs behave in an unstable manner when the % of NDs
             exceeds 40%-50%. Moreover, as mentioned before, it is hard to verify and justify the
             conclusion of a GOF test for data sets with nondetects in excess of 40%-50%.

          •  Therefore, for data sets with many nondetects (> 40%-50%), it is suggested to use
             nonparametric methods to estimate the various environmental parameters (BTVs, EPC
             terms) of interest and to perform site versus background comparisons.

          •  In practice, a left-censored data set often has multiple detection limits. For such methods,
             the KM method can be used. ProUCL 4.0 provides UCL computation methods that can
             be used on data sets with multiple detection limits including the DL/2 method, KM
             method, and robust ROS method.

          •  As mentioned earlier, for reliable and accurate results, it is suggested that the user make
             sure that the data set under study represents a single statistical population (e.g.,
             background reference area, or an AOC) and not a mixture population (e.g., clean and
             contaminated site areas).

          •  It is recommended to identify all of the potential outliers and study them separately. The
             computation of the statistics such as UCL95 and background statistics should be based
             upon the majority of the data set representing a single dominant population. Decisions
             about the appropriate disposition (include or not include) of outliers should be made by
             all interested members of the project team. When in doubt, it is suggested to compute and
206

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•   Simple classical outlier identification methods (Dixon test and Rosner test) are also
    available in ProUCL 4.0. More effective robust outlier procedures (e.g., Rousseeuw and
    Leroy (1987) and Singh and Nocerino (1995)) are available in Scout (1999).

•   In case a data set represents a mixture sample (from two or more populations), one should
    partition the mixture sample into component sub-samples (e.g., Singh, Singh, and
    Flatman(1994)).

•   Avoid the use of transformations (to achieve symmetry) while computing the upper limits
    for various environmental applications, as all remediation, cleanup, background
    evaluation decisions, and risk assessment decisions have to be made using statistics in the
    original scale. Also, it is more accurate and easier to interpret the  results computed in the
    original scale. The results and statistics computed in the original scale do not suffer from
    an unknown amount of transformations bias.

•   Specifically, avoid the use of a lognormal model even when the data appear to be
    lognormally distributed. Its use often results in incorrect and unrealistic statistics of no
    practical purpose or importance or significance. Several variations of estimation methods
    (e.g., robust ROS and FP-ROS on log-transformed data, delta lognormal method) on log-
    transformed data have been developed and used by the practitioners. This has caused
    some confusion among the users of the statistical methods dealing with environmental
    data sets. The proper use of a lognormal distribution (e.g., how to properly back-
    transform UCL of mean in the log-scale to obtain a UCL of mean in original scale) is not
    clear to many users, which in turn may result in the incorrect use and computation of an
    estimate (=  UCL95) of the population mean.

•   The parameter in the transformed space may not be of interest to make cleanup decisions.
    The cleanup and remediation decisions are often made in the original raw scale;
    therefore, the statistics (e.g., UCL95) computed in transformed space need to be back-
    transformed in the original scale. It is not clear to a typical user how to back-transform
    results in the log-scale or any other scale obtained using a Box-Cox (BC)-type
    transformation to original raw scale. The transformed results often suffer from significant
    amount of transformation bias.

•   The question now arises-how one should back-transform results from a log-space (or any
    other transformed space) to the original space? Unfortunately, no  defensible guidance is
    available in the environmental literature to address this  question. Moreover, the back-
    transformation formula will change from transformation to transformation (BC-type
    transformations), and the bias introduced by such transformations will remain unknown.
    Therefore, in cases when a data set in the "raw" scale cannot be modeled by a parametric
    distribution, it is desirable to use nonparametric methods (many available in ProUCL 4.0)
    rather than testing or estimating a parameter in the transformed space.

•   On page (78) of Helsel (2005), the use of the robust ROS  MLE method (Kroll, C.N. and
    J.R. Stedinger (1996)) has been suggested to compute summary statistics. In this hybrid
                                                                                207

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              method, MLEs are computed using log-transformed data. Using the regression model as
              given by equation (3-21) of Section 3, the MLEs of the mean (used as intercept) and sd
              (used as slope) in the log-scale are used to extrapolate the NDs in the log-scale. Just like
              in the robust ROS method, all of the NDs are transformed back in the original scale by
              exponentiation. This results in a full data set in the original scale. One may then compute
              the mean and sd using the full data set. The estimates thus obtained are called robust ROS
              ML estimates (Helsel (2005), and Kroll and Stedinger (1996)). However, the
              performance of such a hybrid estimation method is not well known. Moreover, for higher
              censoring levels, the MLE methods sometimes behave in an unstable manner, especially
              when dealing with moderately skewed to highly skewed data sets (e.g., with a >1.0).

              o   It should be noted that the performance of this hybrid method is unknown.
              o   It is not clear why this method is called a robust method.
              o   The stability of the MLEs obtained using the log-transformed data is doubtful,
                  especially for higher censoring levels.
              o   The BCA and (% bootstrap) UCLs based upon this method will fail to provide the
                  adequate coverage for the population mean for moderately skewed to highly skewed
                  data sets.

           •   The DL/2 (t) UCL method does not provide adequate coverage (for any distribution and
              sample size) for the population mean, even for censoring levels as low as 5%, 10%, 15%.
              This is contrary to the conjecture and assertion (e.g., EPA (2000)) often made that the
              DL/2 method can be used for lower (< 20%) censoring levels. The coverage provided by
              the DL/2 (t) method deteriorates fast as the censoring intensity increases.

This DL/2 (t) UCL method is not recommended by the authors and developers of ProUCL 4.0; it is
included only for comparison or research purposes.

           •   The KM method is a preferred method as it can handle multiple detection limits.
              Moreover, the nonparametric UCL95 methods (KM (BCA), KM (z), KM (%), KM (t))
              based upon the KM estimates provide good coverages for the population mean (e.g.,
              Helsel (2005) and Singh et al. (2006)).

           •   For a symmetric distribution (approximate normality), several UCL95 methods provide
              good coverage (-95%) for the population mean, including the Winsorization mean,
              Cohen's MLE (t), Cohen's MLE (Tiku), KM (z), KM (t), KM (%) and KM (BCA) (e.g.,
              Helsel (2005) and Singh et al. (2006)).

16.2  Recommended UCL95 Methods for Normal (Approximate Normal)
       Distribution

           •   Fornormal and approximately normal (e.g., symmetric or with sd, a < 0.5) distribution:
              The most appropriate UCL95 computation methods for normal or approximately normal
              distributions are the KM (t) or KM (%) methods. For symmetric distributions, both of
              these methods perform equally well on left-censored data sets for all censoring levels (%
              nondetects) and sample sizes.
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16.3   Recommended UCL95 Methods for Gamma Distribution

          •  Highly skewed gamma distributions, G(k, 9), with shape parameter, k < 1:
             o   Use the nonparametric KM (Chebyshev) UCL95 method for censoring levels < 30%.
             o   Use the nonparametric KM (BCA) UCL95 method for censoring levels in the interval
                 [30%, 50%).
             o   Use the nonparametric KM (t) UCL95 method for censoring levels > 50%.

          •  Moderately skewed gamma distributions, G(k, 9), with shape parameter, K k < 2:
             o   For censoring level < 10%, use the KM (Chebyshev) UCL95 method.
             o   For higher censoring levels [10%, 25%), use the KM (BCA) UCL95 method.
             o   For censoring levels in [25%, 40%), use the KM (%) UCL95 method.
             o   For censoring levels > 40%, use the KM (t) UCL95 method.

          •  Mildly skewed gamma distributions, G(k, 9), with k > 2:
             o   Use the KM (BCA) UCL95 method for lower censoring levels (< 20%).
             o   For censoring levels in the interval [20%, 40%), use the KM (%) UCL95.
             o   For censoring > 40%, use the KM (t) UCL95 computation method.

16.4   Recommended UCL95 Methods for Lognormal Distribution

          •  Mildly skewed data sets with a < 1:
             o   For censoring levels (< 20%) and sample of sizes less than 50-70, use the KM
                 (Chebyshev) UCL95.
             o   For censoring levels (< 20%) and samples of sizes greater than 50-70, use the KM
                 (BCA)UCL95.
             o   For censoring levels in the interval [20%, 40%) and all sample sizes, use the KM
                 (BCA)UCL95.
             o   For censoring level > 40%, use the KM (%) or KM (t) UCL95 method.

          •  Data sets with a in the interval (1, 1.5]:
             o   For censoring levels < 50% and samples of sizes < 40, use the 97.5% KM
                 (Chebyshev) UCL.
             o   For censoring levels < 50% , samples of sizes > 40, use 95% KM (Chebyshev) UCL.
             o   For censoring levels > 50%, use the KM (BCA) UCL95 for samples of all sizes.

          •  Highly skewed data sets with a in the  interval (1.5, 2]:
             o   For sample sizes < 40, censoring levels  <50%, use 99% KM (Chebyshev) UCL.
             o   For sample sizes > 40, censoring levels  < 50%, use 97.5% KM (Chebyshev) UCL.
             o   For samples of sizes < 40-50 and censoring levels > 50%, use the 97.5% KM
                 (Chebyshev) UCL.
             o   For samples of sizes > 40-50, and censoring levels > 50%, use the 95% KM
                 (Chebyshev) UCL.

          •  Use a similar pattern for more highly skewed data sets with a > 2.0, 3.0:
             o   For extremely highly skewed data sets, an appropriate estimate of the EPC term (in
                 terms of adequate coverage) is given by a UCL based upon the Chebyshev inequality
                                                                                    209

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                 and KM estimates. The confidence coefficient to be used will depend upon the
                 skewness. For highly skewed data sets, a higher (e.g., > 95%) confidence coefficient
                 may have to be used to estimate the EPC.
              o  As the skewness increases, the confidence coefficient also increases.
              o  For such highly skewed distributions (with a > 2.0, 3.0), for lower sample sizes (e.g.,
                 < 50-60), one may simply use 99% KM (Chebyshev) UCL to estimate the population
                 mean, EPC term, and other relevant threshold (e.g., UPL, percentiles) values.
              o  For sample sizes greater than 60, one may use a 97.5% KM (Chebyshev) UCL as an
                 estimate of the population mean or mass.

16.5  Recommended Nonparametric UCL Methods

          •   For symmetric or approximately symmetric distribution-free, nonparametric data sets
              with a <  0.5: Use the same UCL computation methods as for the data sets coming from
              a normal or an approximate normal (symmetric) population. These methods are
              summarized above in the normal UCL computation section.

          •   For skewed distribution-free, nonparametric data sets with a >0.5: Most of the
              recommended UCL computation methods for a lognormal distribution, as described
              above in the lognormal UCL section, do not assume the lognormality of the data set.
              Therefore, the UCL computation methods, as described in the lognormal UCL
              computation section, can be used on skewed nonparametric data sets that do not follow
              any of the well-known parametric distributions.

The suggested parametric and nonparametric UCL95 computation methods for data sets with nondetect
observations are summarized in Table 16, shown on the next page.
210

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Table 16. Recommended UCL95 Computation Methods for Left-Censored Data Sets with Nondetect Observations
Skewness
Sample Size
%ND
+J
S.
^
g
o>
g
S.
^
g
o>
,
Si
9)
^
o_
s.
^
g
o>
1
(0
£

,
^
9)
.C
O_
s.
^
£
o>
<
O
OQ_
S.
^
g
o>
Normal or Approximate Normal (with a < 0.5) Distribution
<7 <0.5
Alln
>0%
•
•




Gamma Distribution
k <1
1 < k <2
k >2
Alln
Alln
Alln
Alln
Alln
Alln
Alln
Alln
Alln
Alln
< 30%
[30%, 50%)
> 50%
<10%
[10%, 25%)
[25%, 40%)
> 40%
< 20%
[20%, 40%)
> 40%


•



•


•





•


•

•


•



























•


•


•


Lognormal Distribution
<7 <1.0
1 < <7 <1.5
1.5< a <2.0
<7 > 2.0,3.0
n < 50-70
n > 50-70
Alln
Alln
n <40
n >40
Alln
n <40
n >40
n < 40-50
n > 40-50
n < 50-60
n>60
< 20%
[20%, 40%)
> 40%
< 50%
> 50%
< 50%
> 50%
>0%



•












•









•




•




•






•



•
•


•







•



•


•
•



•






                                                                                                  211

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Table 16. Recommended UCL95 Computation Methods for Left-Censored Data Sets with Nondetect Observations -
Continued
Skewness
Sample Size
%ND
+J
5
X.
s?
8
g
5
X.
s?
s
95% KM (Chebyshev)
,
Si
9)
.C
O_
s
*:
s?
m
Si
1
(0
£

Symmetric or Approximate Symmetric Non-Discernable Distribution
a < 0.5
Alln
>0%
•
•




Moderately Skewed to Highly Skewed Non-Discernable Distribution
0.5 < a <1.0
1 < a <1.5
1.5< a <2.0
a > 2.0,3.0
n < 50-70
n > 50-70
Alln
Alln
n <40
n >40
Alln
n <40
n >40
n < 40-50
n > 40-50
n < 50-60
n>60
< 20%
[20%, 40%)
> 40%
< 50%
> 50%
< 50%
> 50%
>0%



•












•









•




•




•






•



•
•


•







•



•


•
•



•






Note: In Table 16, phrase  "All n " represents only valid (e.g., n > 3) and recommended (n > 8 to 10)
values of the sample size, n. As mentioned throughout the report, it is not desirable to use statistical
methods on data sets of small sizes (e.g., with n < 8 to 10). However, it should be noted that the sample
size requirements and recommendations (n > 8 to 10) as described in this report are not the limitations of
the methods considered in  this report. One of the main reasons for the recommendation that the sample
size should be at least 8 to  10 is that the estimates and UCLs based upon small data sets, especially with
many below detection limit observations (e.g., 30%, 40%, 50%, and more), may not be reliable and
accurate enough to draw conclusions for environmental applications. It should be noted that in order to
be able  to use bootstrap re-sampling methods, it is  desirable to have a minimum of 10-15 observations
(e.g., n > 10-15). Therefore, the phrase "Alln" in Table 16, should be interpreted as that the sample size,
n, is least 8 to 10. The software, ProUCL 4.0, will provide appropriate warning messages when a user
tries to use a  method on data sets of small sizes.

Also, Hall's bootstrap and  bootstrap t methods to compute a UCL based upon a full data set (without
nondetects) should be used with caution. These two bootstrap methods may yield erratic and unstable
UCL results,  especially, when outliers are present. In such cases, it is desirable to use alternative UCL
212

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methods based upon Chebyshev inequality. ProUCL software provides a warning message for erratic
UCL results based upon Hall's bootstrap t methods.
                                                                                        213

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                                         Glossary
This glossary defines selected words in this User Guide to describe impractically large UCL values of the
unknown population mean, mi. In practice, the UCLs based upon Land's H-statistic (H-UCL), and
bootstrap methods such as the bootstrap t and Hall's bootstrap methods (especially when outliers are
present) can become impractically large. The UCLs based upon these methods often become larger than
the UCLs based upon all other methods by several orders of magnitude. Such large UCL values are not
achievable as they do not occur in practice. Words like "unstable" and "unrealistic" are used to describe
such impractically large UCL values.

UCL: upper confidence limit of the unknown population mean.

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

Optimum: An interval is optimum if it possesses optimal properties as defined in the statistical literature.
This may mean that it is the shortest interval providing the specified coverage (e.g., 0.95) to the
population mean. For example, for normally distributed data sets, the UCL of the population mean based
upon Student's t-distribution is optimum.

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

Reliable UCL: This is similar to a stable UCL.

Unstable UCL = Unreliable UCL = Unrealistic UCL: The UCL of a population mean is unstable,
unrealistic,  or unreliable if it is orders of magnitude higher than the other UCLs of population mean. It
represents an impractically large value that cannot be achieved in practice. For example, the use of Land's
H-statistic often results in impractically large inflated  UCL value.  UCLs such as the bootstrap t  UCL and
Hall's UCL can be inflated by outliers, resulting in an impractically large and unstable value. All such
impractically large  UCL values are called unstable, unrealistic, unreliable, or inflated UCLs  in this User
Guide.
214

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215

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                                      References


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Conover, W. J. (1999). Practical Nonparametric Statistics. Third Edition. John Wiley.

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Land, C. E. (1975), "Tables of Confidence Limits for Linear Functions of the Normal Mean and
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                                                                                         217

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USEPA. 1992. Supplemental Guidance to RAGS: Calculating the Concentration Term. Publication EPA
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