4>EPA United States Environmental Protection Agency ProUCL Version 5.1 Technical Guide Statistical Software for Environmental Applications for Data Sets with and without Nondetect Observations RESEARCH AND DEVELOPMENT ------- ------- ------- ------- EPA/600/R-07/041 October 2015 www.epa.gov ProUCL Version 5.1 Technical Statistical Software for Environmental Applications for Data Sets with and without Nondetect Observations Prepared for: Felicia Barnett, Director ORD Site Characterization and Monitoring Technical Support Center (SCMTSC) Superfund and Technology Liaison, Region 4 U.S. Environmental Protection Agency 61 Forsyth Street SW, Atlanta, GA 30303 Prepared by: Anita Singh, Ph.D.1 and Ashok K. Singh, Ph.D.2 1 Lockheed Martin/SERAS IS&GS-CIVIL 2890 Woodbridge Ave Edison NJ 08837 2Professor of Statistics, Department of Hotel Management University of Nevada Las Vegas Las Vegas, NV89154 U.S. Environmental Protection Agency Office of Research and Development Washington, DC 20460 Notice: Although this work was reviewed by EPA and approved for publication, it may not necessarily reflect official Agency policy. Mention of trade names and commercial products does not constitute endorsement or recommendation for use. 129cmb15 ------- NOTICE The United States Environmental Protection Agency (U.S. EPA) through its Office of Research and Development (ORD) funded and managed the research described in this ProUCL Technical Guide. It has been peer reviewed by the U.S. EPA and approved for publication. Mention of trade names or commercial products does not constitute endorsement or recommendation by the U.S. EPA for use. • All versions of the ProUCL software including the current version ProUCL 5.1 have been developed by Lockheed Martin, IS&GS - CIVIL under the Scientific, Engineering, Response and Analytical Services contract with the U.S. EPA and is made available through the U.S. EPA Technical Support Center (TSC) in Atlanta, Georgia (GA). • Use of any portion of ProUCL that does not comply with the ProUCL Technical Guide is not recommended. • ProUCL contains embedded licensed software. Any modification of the ProUCL source code may violate the embedded licensed software agreements and is expressly forbidden. • ProUCL software provided by the EPA was scanned with McAfee VirusScan version 4.5.1 SP1 and is certified free of viruses. With respect to ProUCL distributed software and documentation, neither the U.S. EPA nor any of their employees, assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed. Furthermore, software and documentation are supplied "as-is" without guarantee or warranty, expressed or implied, including without limitation, any warranty of merchantability or fitness for a specific purpose. ProUCL software is a statistical software package providing statistical methods described in various U.S. EPA guidance documents. ProUCL does not describe U.S. EPA policies and should not be considered to represent U.S. EPA policies. ------- Minimum Hardware Requirements ProUCL 5.1 will function but will run slowly and page a lot. • Intel Pentium 1.0 gigahertz (GHz) • 45 MB of hard drive space • 512 MB of memory (RAM) • CD-ROM drive or internet connection • Windows XP (with SP3), Vista (with SP1 or later), or Windows 7. ProUCL 5.1 will function but some titles and some Graphical User Interfaces (GUIs) will need to be scrolled. Definition without color will be marginal. • 800 by 600 Pixels • Basic Color is preferred Preferred Hardware Requirements • 1 GHz or faster Processor. • 1 gigabyte (GB) of memory (RAM) • 1024 by 768 Pixels or greater color display Software Requirements ProUCL 5.1 has been developed in the Microsoft .NET Framework 4.0 using the C# programming language. To properly run ProUCL 5.1 software, the computer using the program must have the .NET Framework 4.0 pre-installed. The downloadable .NET Framework 4.0 files can be obtained from one of the following websites: • http://msdn.microsoft.com/netframework/downloads/updates/default.aspx http://www.microsoft.com/en-us/download/details.aspx?id= 17851 Quicker site for 32 Bit Operating systems • http://www.microsoft.com/en-us/download/details.aspx?id=24872 Use this site if you have a 64 Bit operating system in ------- Installation Instructions when Downloading ProUCL 5.1 from the EPA Web Site • Download the file SETUP.EXE from the EPA Web site and save to a temporary location. • Run the SETUP.EXE program. This will create a ProUCL directory and two folders: 1) The TECHNICAL GUIDE (this document), and 2) DATA (example data sets). • To run the program, use Windows Explorer to locate the ProUCL application file, and Double click on it, or use the RUN command from the start menu to locate the ProUCL.exe file, and run ProUCL.exe. • To uninstall the program, use Windows Explorer to locate and delete the ProUCL folder. Caution: If you have previous versions of the ProUCL, which were installed on your computer, you should remove or rename the directory in which earlier ProUCL versions are currently located. Installation Instructions when Copying ProUCL 5.1 from a CD • Create a folder named ProUCL 5.1 on a local hard drive of the machine you wish to install ProUCL 5.1. • Extract the zipped file ProUCL.zip to the folder you have just created. • Run ProUCL.exe Note: If you have extension turned off, the program will show with the name ProUCL in your directory and have an Icon with the label ProUCL. Creating a Shortcut for ProUCL 5.1 on Desktop • To create a shortcut of the ProUCL program on your desktop, go to your ProUCL directory and right click on the executable program and send it to desktop. A ProUCL icon will be displayed on your desktop. This shortcut will point to the ProUCL directory consisting of all files required to execute ProUCL 5.1. Caution: Because all files in your ProUCL directory are needed to execute the ProUCL software, one needs to generate a shortcut using the process described above. Simply dragging the ProUCL executable file from Window Explorer onto your desktop will not work successfully (an error message will appear) as all files needed to run the software are not available on your desktop. Your shortcut should point to the directory path with all required ProUCL files. All ProUCL files should reside in one directory on your computer (and not on your Network System) and your shortcut should point to that directory. IV ------- ProUCL 5.1 Software ProUCL version 5.1 (ProUCL 5.1), its earlier versions: ProUCL version 3.00.01, 4.00.02, 4.00.04, 4.00.05, 4.1.00, 4.1.01, and ProUCL 5.0.00, associated Facts Sheet, User Guides and Technical Guides (e.g., EPA 2010a, 2010b, 2013a, 2013b) can be downloaded from the following EPA website: http://www.epa.gov/osp/hstl/tsc/software.htm http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm Material for ProUCL webinars offered in March 2011, and relevant literature used in the development of various ProUCL versions can also be downloaded from the above EPA website. Contact Information for all Versions of ProUCL Since 1999, the ProUCL software has been developed under the direction of the Technical Support Center (TSC). As of November 2007, the direction of the TSC is transferred from Brian Schumacher to Felicia Barnett. Therefore, any comments or questions concerning all versions of ProUCL software should be addressed to: Felicia Barnett, Director ORD Site Characterization and Monitoring Technical Support Center (SCMTSC) Superfund and Technology Liaison, Region 4 U.S. Environmental Protection Agency 61 Forsyth Street SW, Atlanta, GA 30303-8960 barnett.fericia@epa.gov (404)562-8659 Fax: (404) 562-8439 ------- EXECUTIVE SUMMARY The main objective of the ProUCL software funded by the United States Environmental Protection Agency (EPA) is to compute rigorous statistics to help decision makers and project teams in making good decisions at a polluted site which are cost-effective, and protective of human health and the environment. The ProUCL software is based upon the philosophy that rigorous statistical methods can be used to compute reliable estimates of population parameters and decision making statistics including: the upper confidence limit (UCL) of the mean, the upper tolerance limit (UTL), and the upper prediction limit (UPL) to help decision makers and project teams in making correct decisions. A few commonly used text book type methods (e.g., Central Limit Theorem [CLT], Student's t-UCL) alone cannot address all scenarios and situations occurring in environmental studies. Since many environmental decisions are based upon a 95 percent (%) UCL (UCL95) of the population mean, it is important to compute UCLs of practical merit. The use and applicability of a statistical method (e.g., student's t-UCL, CLT-UCL, adjusted gamma-UCL, Chebyshev UCL, bootstrap-t UCL) depend upon data size, data skewness, and data distribution. ProUCL computes decision statistics using several parametric and nonparametric methods covering a wide-range of data variability, distribution, skewness, and sample size. It is anticipated that the availability of the statistical methods in the ProUCL software covering a wide range of environmental data sets will help the decision makers in making more informative and correct decisions at Superfund and Resource Conservation and Recovery Act (RCRA) sites. It is noted that for moderately skewed to highly skewed environmental data sets, UCLs based on the CLT and the Student's t-statistic fail to provide the desired coverage (e.g., 0.95) to the population mean even when the sample sizes are as large as 100 or more. The sample size requirements associated with the CLT increases with skewness. It would be incorrect to state that a CLT or Student's statistic based UCLs are adequate to estimate Exposure Point Concentrations (EPC) terms based upon skewed data sets. These facts have been described in the published documents (Singh, Singh, and Engelhardt [1997, 1999]; Singh, Singh, and laci 2002; and Singh et al. 2006) summarizing simulation experiments conducted on positively skewed data sets to evaluate the performances of the various UCL computation methods. The use of a parametric lognormal distribution on a lognormally distributed data set yields unstable impractically large UCLs values, especially when the standard deviation (sd) of the log-transformed data becomes greater than 1.0 and the data set is of small size less than (<) 30-50. Many environmental data sets can be modeled by a gamma as well as a lognormal distribution. The use of a gamma distribution on gamma distributed data sets tends to yield UCL values of practical merit. Therefore, the use of gamma distribution based decision statistics such as UCLs, UPLs, and UTLs should not be dismissed by stating that it is easier to use a lognormal model to compute these upper limits. The suggestions made in ProUCL are based upon the extensive experience of the developers in environmental statistical methods, published environmental literature, and procedures described in many EPA guidance documents. These suggestions are made to help the users in selecting the most appropriate UCL to estimate the EPC term which is routinely used in exposure assessment and risk management studies of the USEPA. The suggestions are based upon the findings of many simulation studies described in Singh, Singh, and Engelhardt (1997, 1999); Singh, Singh, and laci (2002); and Singh et al. (2006). It should be pointed out that a typical simulation study does not (cannot) cover all real world data sets of various sizes and skewness from all distributions. When deemed necessary, the user may want to consult a statistician to select an appropriate upper limit to estimate the EPC term and other environmental parameters of interest. For an analyte (data set) with skewness (sd of logged data) near the end points of the skewness intervals presented in decision tables of Chapter 2 (e.g., Tables 2-9 through 2-11), the user VI ------- may select the most appropriate UCL based upon the site conceptual site model (CSM), expert site knowledge, toxicity of the analyte, and exposure risks associated with that analyte. The inclusion of outliers in the computation of the various decision statistics tends to yield inflated values of those decision statistics, which can lead to poor decisions. Often statistics that are computed for a data set which includes a few outliers tend to be inflated and represent those outliers rather than representing the main dominant population of interest (e.g., reference area). Identification of outliers, observations coming from population(s) other than the main dominant population is suggested, before computing the decision statistics needed to address project objectives. The project team may want to perform the statistical evaluations twice, once with outliers and once without outliers. This exercise will help the project team in computing reliable and defensible decision statistics which are needed to make cleanup and remediation decisions at polluted sites. The initial development during 1999-2000 and all subsequent upgrades and enhancements of the ProUCL software have been funded by U.S. EPA through its Office of Research and Development (ORD). Initially ProUCL was developed as a research tool for U.S. EPA scientists and researchers of the Technical Support Center (TSC) and ORD- National Exposure Research Laboratory (NERL), Las Vegas. Background evaluations, groundwater (GW) monitoring, exposure and risk management and cleanup decisions in support of the Comprehensive Environmental Recovery, Compensation, and Liability Act (CERCLA) and RCRA site projects of the U.S. EPA are often derived based upon test statistics such as the Shapiro-Wilk (S-W) test, t-test, Wilcoxon-Mann-Whitney (WMW) test, analysis of variance (ANOVA), and Mann-Kendall (MK) test and decision statistics including UCLs of the mean, UPLs, and UTLs. To address the statistical needs of the environmental projects of the USEPA, over the years ProUCL software has been upgraded and enhanced to include many graphical tools and statistical methods described in many EPA guidance documents including: EPA 1989a, 1989b, 1991, 1992a, 1992b, 2000 Multi-Agency Radiation Survey and Site Investigation Manual (MARSSIM), 2002a, 2002b, 2002c, 2006a, 2006b, and 2009. Several statistically rigorous methods (e.g., for data sets with nondetects [NDs]) not easily available in the existing guidance documents and in the environmental literature are also available in ProUCL 5.0/ProUCL 5.1. ProUCL 5.1/ProUCL 5.0 has graphical, estimation, and hypotheses testing methods for uncensored-full data sets and for left-censored data sets including ND observations with multiple detection limits (DLs) or reporting limits (RLs). In addition to computing general statistics, ProUCL 5.1 has goodness-of-fit (GOF) tests for normal, lognormal and gamma distributions, and parametric and nonparametric methods including bootstrap methods for skewed data sets for computation of decision making statistics such as UCLs of the mean (EPA 2002a), percentiles, UPLs for a pre-specified number of future observations (e.g., k with k=\, 2, 3,...), UPLs for mean of future k (>1) observations, and UTLs (e.g., EPA 1992b, 2002b, and 2009). Many positively skewed environmental data sets can be modeled by a lognormal as well as a gamma model. It is well-known that for moderately skewed to highly skewed data sets, the use of a lognormal distribution tends to yield inflated and unrealistically large values of the decision statistics especially when the sample size is small (e.g., <20-30). For gamma distributed skewed uncensored and left-censored data sets, ProUCL software computes decision statistics including UCLs, percentiles, UPLs for future k (>1) observations, UTLs, and upper simultaneous limits (USLs). For data sets with NDs, ProUCL has several estimation methods including the Kaplan-Meier (KM) method, regression on order statistics (ROS) methods and substitution methods (e.g., replacing NDs by DL, DL/2). ProUCL 5.1 can be used to compute upper limits which adjust for data skewness; specifically, for skewed data sets, ProUCL computes upper limits using KM estimates in gamma (lognormal) UCL and UTL equations provided the detected observations in the left-censored data set follow a gamma (lognormal) distribution. Some poor performing commonly used and cited methods such vn ------- as the DL/2 substitution method and H-statistic based UCL computation method have been retained in ProUCL 5.1 for historical reasons, and research and comparison purposes. The Sample Sizes module of ProUCL can be used to develop data quality objectives (DQOs) based sampling designs and to perform power evaluations needed to address statistical issues associated with a variety of site projects. ProUCL provides user-friendly options to enter the desired values for the decision parameters such as Type I and Type II error rates, and other DQOs used to determine the minimum sample sizes needed to address project objectives. The Sample Sizes module can compute DQO-based minimum sample sizes needed: to estimate the population mean; to perform single and two-sample hypotheses testing approaches; and in acceptance sampling to accept or reject a batch of discrete items such as a lot of drums containing hazardous waste. Both parametric (e.g., t-test) and nonparametric (e.g., Sign test, WMW test, test for proportions) sample size determination methods are available in ProUCL. ProUCL has exploratory graphical methods for both uncensored data sets and for left-censored data sets consisting of ND observations. Graphical methods in ProUCL include histograms, multiple quantile- quantile (Q-Q) plots, and side-by-side box plots. The use of graphical displays provides additional insight about the information contained in a data set that may not otherwise be revealed by the use of estimates (e.g., 95% upper limits) and test statistics (e.g., two-sample t-test, WMW test). In addition to providing information about the data distributions (e.g., normal or gamma), Q-Q plots are also useful in identifying outliers and the presence of mixture populations (e.g., data from several populations) potentially present in a data set. Side-by-side box plots and multiple Q-Q plots are useful to visually compare two or more data sets, such as: site-versus-background concentrations, surface-versus-subsurface concentrations, and constituent concentrations of several GW monitoring wells (MWs). ProUCL also has a couple of classical outlier test procedures, such as the Dixon test and the Rosner test which can be used on uncensored data sets as well as on left-censored data sets containing ND observations. ProUCL has parametric and nonparametric single-sample and two-sample hypotheses testing approaches for uncensored as well as left-censored data sets. Single-sample hypotheses tests: Student's t-test, Sign test, Wilcoxon Signed Rank test, and the Proportion test are used to compare site mean/median concentrations (or some other threshold such as an upper percentile) with some average cleanup standard, Cs (or a not-to-exceed compliance limit, Ao) to verify the attainment of cleanup levels (EPA 1989a; MARSSIM/EPA 2000; EPA 2006a) at remediated site areas of concern. Single-sample tests such as the Sign test and Proportion test, and upper limits including UTLs and UPLs are also used to perform intra- well comparisons. Several two-sample hypotheses tests as described in EPA guidance documents (e.g., 2002b, 2006b, 2009) are also available in the ProUCL software. The two-sample hypotheses testing approaches in ProUCL include: Student's t-test, WMW test, Gehan test and Tarone-Ware (T-W) test. The two-sample tests are used to compare concentrations of two populations such as site versus background, surface versus subsurface soils, and upgradient versus downgradient wells. The Oneway ANOVA module in ProUCL has both classical and nonparametric Kruskal-Wallis (K-W) tests. Oneway ANOVA is used to compare means (or medians) of multiple groups such as comparing mean concentrations of areas of concern and to perform inter-well comparisons. In GW monitoring applications, the ordinary least squares (OLS) regression model, trend tests, and time series plots are used to identify upwards or downwards trends potentially present in constituent concentrations identified in wells over a certain period of time. The Trend Analysis module performs the M-K trend test and Theil- Sen (T-S) trend test on data sets with missing values; and generates trend graphs displaying a parametric OLS regression line and nonparametric T-S trend line. The Time Series Plots option can be used to compare multiple time-series data sets. The use of the incremental sampling methodology (ISM) has been recommended by the Interstate Technology and Regulatory Council (ITRC 2012) for collecting ISM soil samples to compute mean Vlll ------- concentrations of the decision units (DUs) and sampling units (SUs) requiring characterization and remediation activities. At many polluted sites, a large amount of discrete onsite and/or offsite background data are already available which cannot be directly compared with newly collected ISM data. In order to provide a tool to compare the existing discrete background data with actual field onsite or background ISM data, a Monte Carlo Background Incremental Sample Simulator (BISS) module was incorporated in ProUCL 5.0 and retained in ProUCL 5.1 (currently blocked from general use) which may be used on a large existing discrete background data set. The BISS module simulates incremental sampling methodology based equivalent background incremental samples. The availability of a large discrete background data set collected from areas with geological conditions comparable to the DU(s) of interest is a pre-requisite for successful application of this module. For now, the BISS module has been blocked for use as this module is awaiting adequate guidance and instructions for its intended use on discrete background data sets. ProUCL software is a user-friendly freeware package providing statistical and graphical tools needed to address statistical issues described in many U.S. EPA guidance documents. ProUCL 5.0/ProUCL 5.1 can process many constituents (variables) simultaneously to: perform statistical tests (e.g., ANOVA and trend test statistics) and compute decision statistics including UCLs of mean, UPLs, and UTLs - a capability not available in several commercial software packages such as Minitab 16 and NADA for R (Helsel 2013). ProUCL also has the capability of processing data by group variables. Significant efforts have been made to make the software as user friendly as possible. For example, on the various GOF graphical displays, output sheets for GOF tests, OLS and ANOVA, in addition to critical values and/or ^-values, the conclusion derived based upon those values is also displayed. ProUCL is easy to use and does not require any programming skills as needed when using commercial software packages and programs written in R. Methods incorporated in ProUCL have been tested and verified extensively by the developers, researchers, scientists, and users. The results obtained by ProUCL are in agreement with the results obtained by using other software packages including Minitab, SAS®, and programs written in R Script. ProUCL 5.0/ProUCL 5.1 computes decision statistics (e.g., UPL, UTL) based upon the KM method in a straight forward manner without flipping the data and re-flipping the computed statistics for left-censored data sets; these operations are not easy for a typical user to understand and perform. This can become unnecessarily tedious when computing decision statistics for multiple variables/analytes. Moreover, unlike survival analysis, it is important to compute an accurate estimate of the sd which is needed to compute decision making statistics including UPLs and UTLs. For left-censored data sets, ProUCL computes a KM estimate of sd directly. These issues are elaborated by examples discussed in this User Guide and in the accompanying ProUCL 5.1 Technical Guide. ProUCL does not represent a policy software of the government. ProUCL has been developed on limited resources, and it does provide many statistical methods often used in environmental applications. The objective of the freely available user-friendly software, ProUCL is to provide statistical and graphical tools to address environmental issues of environmental site projects for all users including those users who cannot or may not want to program and/or do not have access to commercial software packages. Some users have criticized ProUCL and pointed out some deficiencies such as: it does not have geostatistical methods; it does not perform simulations; and does not offer programming interface for automation. Due to the limited scope of ProUCL, advanced methods have not been incorporated in ProUCL. For methods not available in ProUCL, users can use other statistical software packages such as SAS® (available to EPA personnel) and R script to address their computational needs. Contributions from scientists and researchers to enhance methods incorporated in ProUCL will be very much appreciated. Just like other government documents (e.g., U.S. EPA 2009), various versions of ProUCL (2007, 2009, 2011, 2013, 2016) also make some rule-of thumb type suggestions (e.g., minimum sample size requirement of 8-10) based upon professional judgment and experience of the developers. It is IX ------- recommended that the users/project team/agencies make their own determinations about the rule-of- thumb type suggestions made in ProUCL before applying a statistical method. x ------- XI ------- ACRONYMS and ABBREVIATIONS ACL A-D, AD AL AOC ANOVA Ao BC BCA BD BISS BTV CC,cc CERCLA CL CLT COPC Cs CSM Df DL DL/2 (t) DL/2 Estimates DOE DQOs DU EA EOF EM EPA EPC Alternative compliance or concentration limit Anderson-Darling test Action limit Area(s) of concern Analysis of variance Not to exceed compliance limit or specified action level Box-Cox transformation Bias-corrected accelerated bootstrap method Binomial distribution Background Incremental Sample Simulator Background threshold value Confidence coefficient Comprehensive Environmental Recovery, Compensation, and Liability Act Compliance limit Central Limit Theorem Contaminant/constituent of potential concern Cleanup standards Conceptual site model Degrees of freedom Detection limit UCL based upon DL/2 method using Student's t-distribution cutoff value Estimates based upon data set with NDs replaced by 1/2 of the respective detection limits Department of Energy Data quality objectives Decision unit Exposure area Empirical distribution function Expectation maximization United States Environmental Protection Agency Exposure point concentration xn ------- GA GB GHz GROS GOF, G.O.F. GUI GW Hi H0 H-UCL i.i.d. ISM ITRC k, K K K,k khat k star KM (%) KM (Chebyshev) KM(t) KM(z) K-M, KM K-S, KS K-W LCL LN,/w LCL LPL LROS LTL Georgia Gigabyte Gigahertz Gamma ROS Goodness-of-fit Graphical user interface Groundwater Alternative hypothesis Null hypothesis UCL based upon Land's H-statistic Independently and identically distributed Incremental sampling methodology Interstate Technology & Regulatory Council Positive integer representing future or next k observations Shape parameter of a gamma distribution Number of nondetects in a data set MLE of the shape parameter of a gamma distribution Biased corrected MLE of the shape parameter of a gamma distribution UCL based upon Kaplan-Meier estimates using the percentile bootstrap method UCL based upon Kaplan-Meier estimates using the Chebyshev inequality UCL based upon Kaplan-Meier estimates using the Student's t-distribution critical value UCL based upon Kaplan-Meier estimates using critical value of a standard normal distribution Kaplan-Meier Kolmogorov-Smirnov Kruskal Wallis Lower confidence limit Lognormal distribution Lower confidence limit of mean Lower prediction limit LogROS; robust ROS Lower tolerance limit Xlll ------- LSL M,m MARS SIM MCL MOD MDL MK, M-K ML MLE TV N MVUE MW NARPM ND, nd, Nd NERL NRC OKG OLS ORD OSRTI OU PCA PDF, pdf .pdf PRO PROP p-values QA QC Q-Q R,r Lower simultaneous limit Applied to incremental sampling: number in increments in an ISM sample Multi-Agency Radiation Survey and Site Investigation Manual Maximum concentration limit, maximum compliance limit Minimum detectable difference Method detection limit Mann-Kendall Maximum likelihood Maximum likelihood estimate Number of observations/measurements in a sample Number of observations/measurements in a population Minimum variance unbiased estimate Monitoring well National Association of Remedial Project Managers Nondetect National Exposure Research Laboratory Nuclear Regulatory Commission Orthogonalized Kettenring Gnanadesikan Ordinary least squares Office of Research and Development Office of Superfund Remediation and Technology Innovation Operating unit Principal component analysis Probability density function Files in Portable Document Format Preliminary remediation goals Proposed influence function Probability-values Quality assurance Quality Quantile-quantile Applied to incremental sampling: number of replicates of ISM samples xiv ------- RAGS RCRA RL RMLE ROS RPM RSD RV S SCMTSC SD, Sd, sd SND SNV SE SSL SQL su s-w, sw T-S TSC TW, T-W UCL UCL95 UPL UPL95 U.S. EPA, EPA UTL UTL95-95 USGS USL vs. WMW Risk Assessment Guidance for Superfund Resource Conservation and Recovery Act Reporting limit Restricted maximum likelihood estimate Regression on order statistics Remedial Project Manager Relative standard deviation Random variable Substantial difference Site Characterization and Monitoring Technical Support Center Standard deviation Standard Normal Distribution Standard Normal Variate Standard error Soil screening levels Sample quantitation limit Sampling unit Shapiro-Wilk Theil-Sen Technical Support Center Tarone-Ware Upper confidence limit 95% upper confidence limit Upper prediction limit 95% upper prediction limit United States Environmental Protection Agency Upper tolerance limit 95% upper tolerance limit with 95% coverage U.S. Geological Survey Upper simultaneous limit Versus Wilcoxon-Mann-Whitney xv ------- WRS Wilcoxon Rank Sum WSR Wilcoxon Signed Rank Xp pth percentile of a distribution < Less than > Greater than > Greater than or equal to < Less than or equal to A Greek letter denoting the width of the gray region associated with hypothesis testing Z Greek letter representing the summation of several mathematical quantities, numbers % Percent a Type I error rate (3 Type II error rate 0 Scale parameter of the gamma distribution Z Standard deviation of the log-transformed data A carat sign over a parameter, indicates that it represents a statistic/estimate computed using the sampled data xvi ------- GLOSSARY Anderson-Darling (A-D) test: The Anderson-Darling test assesses whether known data come from a specified distribution. In ProUCL the A-D test is used to test the null hypothesis that a sample data set, x\, ..., xn came from a gamma distributed population. Background Measurements: Measurements that are not site-related or impacted by site activities. Background sources can be naturally occurring or anthropogenic (man-made). Bias: The systematic or persistent distortion of a measured value from its true value (this can occur during sampling design, the sampling process, or laboratory analysis). Bootstrap Method: The bootstrap method is a computer-based method for assigning measures of accuracy to sample estimates. This technique allows estimation of the sample distribution of almost any statistic using only very simple methods. Bootstrap methods are generally superior to ANOVA for small data sets or where sample distributions are non-normal. Central Limit Theorem (CLT): The central limit theorem states that given a distribution with a mean, u, and variance, a2, the sampling distribution of the mean approaches a normal distribution with a mean (u) and a variance o2/N as N, the sample size, increases. Censored Data Sets: Data sets that contain one or more observations which are nondetects. Coefficient of Variation (CV): A dimensionless quantity used to measure the spread of data relative to the size of the numbers. For a normal distribution, the coefficient of variation is given by s/xBar. It is also known as the relative standard deviation (RSD). Confidence Coefficient (CC): The confidence coefficient (a number in the closed interval [0, 1]) associated with a confidence interval for a population parameter is the probability that the random interval constructed from a random sample (data set) contains the true value of the parameter. The confidence coefficient is related to the significance level of an associated hypothesis test by the equality: level of significance = 1 - confidence coefficient. Confidence Interval: Based upon the sampled data set, a confidence interval for a parameter is a random interval within which the unknown population parameter, such as the mean, or a future observation, x0, falls. Confidence Limit: The lower or an upper boundary of a confidence interval. For example, the 95% upper confidence limit (UCL) is given by the upper bound of the associated confidence interval. Coverage, Coverage Probability: The coverage probability (e.g., = 0.95) of an upper confidence limit (UCL) of the population mean represents the confidence coefficient associated with the UCL. Critical Value: The critical value for a hypothesis test is a threshold to which the value of the test statistic is compared to determine whether or not the null hypothesis is rejected. The critical value for any hypothesis test depends on the sample size, the significance level, a at which the test is carried out, and whether the test is one-sided or two-sided. xvn ------- Data Quality Objectives (DQOs): Qualitative and quantitative statements derived from the DQO process that clarify study technical and quality objectives, define the appropriate type of data, and specify tolerable levels of potential decision errors that will be used as the basis for establishing the quality and quantity of data needed to support decisions. Detection Limit: A measure of the capability of an analytical method to distinguish samples that do not contain a specific analyte from samples that contain low concentrations of the analyte. It is the lowest concentration or amount of the target analyte that can be determined to be different from zero by a single measurement at a stated level of probability. Detection limits are analyte and matrix-specific and may be laboratory-dependent. Empirical Distribution Function (EOF): In statistics, an empirical distribution function is a cumulative probability distribution function that concentrates probability \ln at each of the n numbers in a sample. Estimate: A numerical value computed using a random data set (sample), and is used to guess (estimate) the population parameter of interest (e.g., mean). For example, a sample mean represents an estimate of the unknown population mean. Expectation Maximization (EM): The EM algorithm is used to approximate a probability density function (PDF). EM is typically used to compute maximum likelihood estimates given incomplete samples. Exposure Point Concentration (EPC): The constituent concentration within an exposure unit to which the receptors are exposed. Estimates of the EPC represent the concentration term used in exposure assessment. Extreme Values: Values that are well-separated from the majority of the data set coming from the far/extreme tails of the data distribution. Goodness-of-Fit (GOF): In general, the level of agreement between an observed set of values and a set wholly or partly derived from a model of the data. Gray Region: A range of values of the population parameter of interest (such as mean constituent concentration) within which the consequences of making a decision error are relatively minor. The gray region is bounded on one side by the action level. The width of the gray region is denoted by the Greek letter delta, A, in this guidance. H-Statistic: Land's statistic used to compute UCL of mean of a lognormal population H-UCL: UCL based on Land's H-Statistic. Hypothesis: Hypothesis is a statement about the population parameter(s) that may be supported or rejected by examining the data set collected for this purpose. There are two hypotheses: a null hypothesis, (Ho), representing a testable presumption (often set up to be rejected based upon the sampled data), and an alternative hypothesis (HA), representing the logical opposite of the null hypothesis. Jackknife Method: A statistical procedure in which, in its simplest form, estimates are formed of a parameter based on a set of N observations by deleting each observation in turn to obtain, in addition to the usual estimate based on N observations, N estimates each based on N-l observations. xvin ------- Kolmogorov-Smirnov (KS) test: The Kolmogorov-Smirnov test is used to decide if a data set comes from a population with a specific distribution. The Kolmogorov-Smirnov test is based on the empirical distribution function (EDF). ProUCL uses the KS test to test the null hypothesis if a data set follows a gamma distribution. Left-censored Data Set: An observation is left-censored when it is below a certain value (detection limit) but it is unknown by how much; left-censored observations are also called nondetect (ND) observations. A data set consisting of left-censored observations is called a left-censored data set. In environmental applications trace concentrations of chemicals may indeed be present in an environmental sample (e.g., groundwater, soil, sediment) but cannot be detected and are reported as less than the detection limit of the analytical instrument or laboratory method used. Level of Significance (a): The error probability (also known as false positive error rate) tolerated of falsely rejecting the null hypothesis and accepting the alternative hypothesis. Lilliefors test: A goodness-of-fit test that tests for normality of large data sets when population mean and variance are unknown. Maximum Likelihood Estimates (MLE): MLE is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. Mean: The sum of all the values of a set of measurements divided by the number of values in the set; a measure of central tendency. Median: The middle value for an ordered set of n values. It is represented by the central value when n is odd or by the average of the two most central values when n is even. The median is the 50th percentile. Minimum Detectable Difference (MOD): The MDD is the smallest difference in means that the statistical test can resolve. The MDD depends on sample-to-sample variability, the number of samples, and the power of the statistical test. Minimum Variance Unbiased Estimates (MVUE): A minimum variance unbiased estimator (MVUE or MVU estimator) is an unbiased estimator of parameters, whose variance is minimized for all values of the parameters. If an estimator is unbiased, then its mean squared error is equal to its variance. Nondetect (ND) values: Censored data values. Typically, in environmental applications, concentrations or measurements that are less than the analytical/instrument method detection limit or reporting limit. Nonparametric: A term describing statistical methods that do not assume a particular population probability distribution, and are therefore valid for data from any population with any probability distribution, which can remain unknown. Optimum: An interval is optimum if it possesses optimal properties as defined in the statistical literature. This may mean that it is the shortest interval providing the specified coverage (e.g., 0.95) to the population mean. For example, for normally distributed data sets, the UCL of the population mean based upon Student's t distribution is optimum. xix ------- Outlier: Measurements (usually larger or smaller than the majority of the data values in a sample) that are not representative of the population from which they were drawn. The presence of outliers distorts most statistics if used in any calculations. Probability - Values (p-value): In statistical hypothesis testing, the p-value associated with an observed value, ^observed of some random variable T used as a test statistic is the probability that, given that the null hypothesis is true, T will assume a value as or more unfavorable to the null hypothesis as the observed value Observed. The null hypothesis is rejected for all levels of significance, a greater than or equal to the p- value. Parameter: A parameter is an unknown or known constant associated with the distribution used to model the population. Parametric: A term describing statistical methods that assume a probability distribution such as a normal, lognormal, or a gamma distribution. Population: The total collection of N objects, media, or people to be studied and from which a sample is to be drawn. It is the totality of items or units under consideration. Prediction Interval: The interval (based upon historical data, background data) within which a newly and independently obtained (often labeled as a future observation) site observation (e.g., onsite, compliance well) of the predicted variable (e.g., lead) falls with a given probability (or confidence coefficient). Probability of Type II (2) Error (P): The probability, referred to as (3 (beta), that the null hypothesis will not be rejected when in fact it is false (false negative). Probability of Type I (1) Error = Level of Significance (a): The probability, referred to as a (alpha), that the null hypothesis will be rejected when in fact it is true (false positive). pth Percentile or pth Quantile: The specific value, Xp of a distribution that partitions a data set of measurements in such a way that the p percent (a number between 0 and 100) of the measurements fall at or below this value, and (100-p) percent of the measurements exceed this value, Xp. Quality Assurance (QA): An integrated system of management activities involving planning, implementation, assessment, reporting, and quality improvement to ensure that a process, item, or service is of the type and quality needed and expected by the client. Quality Assurance Project Plan: A formal document describing, in comprehensive detail, the necessary QA, quality control (QC), and other technical activities that must be implemented to ensure that the results of the work performed will satisfy the stated performance criteria. Quantile Plot: A graph that displays the entire distribution of a data set, ranging from the lowest to the highest value. The vertical axis represents the measured concentrations, and the horizontal axis is used to plot the percentiles/quantiles of the distribution. Range: The numerical difference between the minimum and maximum of a set of values. xx ------- Regression on Order Statistics (ROS): A regression line is fit to the normal scores of the order statistics for the uncensored observations and is used to fill in values imputed from the straight line for the observations below the detection limit. Resampling: The repeated process of obtaining representative samples and/or measurements of a population of interest. Reliable UCL: see Stable UCL. Robustness: Robustness is used to compare statistical tests. A robust test is the one with good performance (that is not unduly affected by outliers and underlying assumptions) for a wide variety of data distributions. Resistant Estimate: A test/estimate which is not affected by outliers is called a resistant test/estimate Sample: Represents a random sample (data set) obtained from the population of interest (e.g., a site area, a reference area, or a monitoring well). The sample is supposed to be a representative sample of the population under study. The sample is used to draw inferences about the population parameter(s). Shapiro-Wilk (SW) test: Shapiro-Wilk test is a goodness-of-fit test that tests the null hypothesis that a sample data set, x\,..., x» came from a normally distributed population. Skewness: A measure of asymmetry of the distribution of the parameter under study (e.g., lead concentrations). It can also be measured in terms of the standard deviation of log-transformed data. The greater the standard deviation, the greater is the skewness. Stable UCL: The UCL of a population mean is a stable UCL if it represents a number of practical merit (e.g., a realistic value which can occur at a site), which also has some physical meaning. That is, a stable UCL represents a realistic number (e.g., constituent concentration) that can occur in practice. Also, a stable UCL provides the specified (at least approximately, as much as possible, as close as possible to the specified value) coverage (e.g., -0.95) to the population mean. Standard Deviation (sd, sd, SD): A measure of variation (or spread) from an average value of the sample data values. Standard Error (SE): A measure of an estimate's variability (or precision). The greater the standard error in relation to the size of the estimate, the less reliable is the estimate. Standard errors are needed to construct confidence intervals for the parameters of interests such as the population mean and population percentiles. Substitution Method: The substitution method is a method for handling NDs in a data set, where the ND is replaced by a defined value such as 0, DL/2 or DL prior to statistical calculations or graphical analyses. This method has been included in ProUCL 5.1 for historical comparative purposes but is not recommended for use. The bias introduced by applying the substitution method cannot be quantified with any certainty. ProUCL 5.1 will provide a warning when this option is chosen. Uncensored Data Set: A data set without any censored (nondetects) observations. xxi ------- Unreliable UCL, Unstable UCL, Unrealistic UCL: The UCL of a population mean is unstable, unrealistic, or unreliable if it is orders of magnitude higher than the other UCLs of a population mean. It represents an unpractically large value that cannot be achieved in practice. For example, the use of Land's H-statistic often results in an impractically large inflated UCL value. Some other UCLs, such as the bootstrap-t UCL and Hall's UCL, can be inflated by outliers resulting in an impractically large and unstable value. All such impractically large UCL values are called unstable, unrealistic, unreliable, or inflated UCLs. Upper Confidence Limit (UCL): The upper boundary (or limit) of a confidence interval of a parameter of interest such as the population mean. Upper Prediction Limit (UPL): The upper boundary of a prediction interval for an independently obtained observation (or an independent future observation). Upper Tolerance Limit (UTL): A confidence limit on a percentile of the population rather than a confidence limit on the mean. For example, a 95% one-sided UTL for 95% coverage represents the value below which 95% of the population values are expected to fall with 95 % confidence. In other words, a 95% UTL with coverage coefficient 95% represents a 95% UCL for the 95th percentile. Upper Simultaneous Limit (USL): The upper boundary of the largest value. xBar: arithmetic average of computed using the sampled data values xxn ------- ACKNOWLEDGEMENTS We wish to express our gratitude and thanks to our friends and colleagues who have contributed during the development of past versions of ProUCL and to all of the many people who reviewed, tested, and gave helpful suggestions throughout the development of the ProUCL software package. We wish to especially acknowledge EPA scientists including Deana Crumbling, Nancy Rios-Jafolla, Tim Frederick, Dr. Maliha Nash, Kira Lynch, and Marc Stiffleman; James Durant of ATSDR, Dr. Steve Roberts of University of Florida, Dr. Elise A. Striz of the National Regulatory Commission (NRC), and Drs. Phillip Goodrum and John Samuelian of Integral Consulting Inc. for testing and reviewing ProUCL 5.0 and its associated guidance documents, and for providing helpful comments and suggestions. We also wish to thank Dr. D. Beal of Leidos for reviewing ProUCL 5.0. Special thanks go to Ms. Donna Getty and Mr. Richard Leuser of Lockheed Martin for providing a thorough technical and editorial review of ProUCL 5.1 and also ProUCL 5.0 User Guide and Technical Guide. A special note of thanks is due to Ms. Felicia Barnett of EPA ORD Site Characterization and Monitoring Technical Support Center (SCMTSC), without whose assistance the development of the ProUCL 5.1 software and associated guidance documents would not have been possible. Finally, we wish to dedicate the ProUCL 5.1 (and ProUCL 5.0) software package to our friend and colleague, John M. Nocerino who had contributed significantly in the development of ProUCL and Scout software packages. xxin ------- XXIV ------- XXV ------- of NOTICE ii Software Requirements iii Installation Instructions when Downloading ProUCL 5.1 from the EPA Web Site iv ProUCL5.1 v Contact Information for all Versions of ProUCL v EXECUTIVE SUM MARY vi ACRONYMS and ABBREVIATIONS xii GLOSSARY xvii ACKNOWLEDGEMENTS xxiii INTRODUCTION OVERVIEW OF ProUCL VERSION 5.1 SOFTWARE 1 The Need for ProUCL Software 6 ProUCL 5.1 Capabilities 9 ProUCL 5.1 User Guide 16 CHAPTER 1 Guidance on the Use of Statistical Methods in ProUCL Software 17 1.1 Background Data Sets 17 1.2 Site Data Sets 18 1.3 Discrete Samples or Composite Samples? 19 1.4 Upper Limits and Their Use 20 1.5 Point-by-Point Comparison of Site Observations with BTVs, Compliance Limits and Other Threshold Values 22 1.6 Hypothesis Testing Approaches and Their Use 23 1.6.1 Single Sample Hypotheses (Pre-established BTVs and Not-to-Exceed Values are Known) 23 1.6.2 Two-Sample Hypotheses (BTVs and Not-to-Exceed Values are Unknown) 24 1.7 Minimum Sample Size Requirements and Power Evaluations 25 1.7.1 Why a data set of minimum size, n = 8 throughlO? 26 1.7.2 Sample Sizes for Bootstrap Methods 27 1.8 Statistical Analyses by a Group ID 28 1.9 Statistical Analyses for Many Constituents/Variables 28 1.10 Use of Maximum Detected Value as Estimates of Upper Limits 28 1.10.1 Use of Maximum Detected Value to Estimate BTVs and Not-to-Exceed Values 29 1.10.2 Use of Maximum Detected Value to Estimate EPC Terms 29 1.10.2. IChebyshev Inequality Based UCL95 30 1.11 Samples with Nondetect Observations 30 1.11.1 Avoid the Use of the DL/2 Substitution Method to Compute UCL95 30 1.11.2 ProUCL Does Not Distinguish between Detection Limits, Reporting limits, or Method Detection Limits 31 1.12 Samples with Low Frequency of Detection 31 1.13 Some Other Applications of Methods in ProUCL 5.1 32 1.13.1 Identification of COPCs 32 1.13.2 Identification of Non-Compliance Monitoring Wells 32 1.13.3 Verification of the Attainment of Cleanup Standards, Cs 33 1.13.4 Using BTVs (Upper Limits) to Identify Hot Spots 33 1.14 Some General Issues, Suggestions and Recommendations made by ProUCL 33 xxvi ------- 1.14.1 Handling of Field Duplicates 33 1.14.2 ProUCL Recommendation about ROS Method and Substitution (DL/2) Method 33 1.14.3 Unhandled Exceptions and Crashes in ProUCL 34 1.15 The Unofficial User Guide to ProUCL4 (Helsel and Gilroy 2012) 34 1.16 Box and Whisker Plots 41 CHAPTER 2 Goodness-of-Fit Tests and Methods to Compute Upper Confidence Limit of Mean for Uncensored Data Sets without Nondetect Observations 45 2.1 Introduction 45 2.2 Goodness-of-Fit (GOF) Tests 47 2.2.1 Test Normality and Lognormality of a Data Set 48 2.2.7.7 Normal Quantile-quantile (Q-Q) Plot 48 2.2.7.2 Shapiro-Wilk (S-W) Test 49 2.2.1.3 Lilliefors Test 49 2.2.2 Gamma Distribution 50 2.2.2.7 Quantile-Quantile (Q-Q) Plot for a Gamma Distribution 57 2.2.2.2 Empirical Distribution Function (EDF)-Based Goodness-ofFit Tests. 51 2.3 Estimation of Parameters of the Three Distributions Incorporated in ProUCL 53 2.3.1 Normal Distribution 53 2.3.2 Lognormal Distribution 54 2.3.2.1 MLEs of the Parameters of a Lognormal Distribution 54 2.3.2.2 Relationship between Skewness and Standard Deviation, a 54 2.3.2.3 MLEs of the Quantiles of a Lognormal Distribution 56 2.3.2.4 MVUEs of Parameters of a Lognormal Distribution 57 2.3.3 Estimation of the Parameters of a Gamma Distribution 57 2.4 Methods for Computing aUCL of the Unknown Population Mean 60 2.4.1 (1-a)* 100 UCL of the Mean Based upon Student's t-Statistic 61 2.4.2 Computation of the UCL of the Mean of a Gamma, G (k, 9), Distribution 61 2.4.3 (1- a)* 100 UCL of the Mean Based Upon H-Statistic (H-UCL) 64 2.4.4 (1 - a)* 100 UCL of the Mean Based upon Modified-t-Statistic for Asymmetrical Populations 72 2.4.5 (1 - a)* 100 UCL of the Mean Based upon the Central Limit Theorem 73 2.4.6 (1 - a)* 100 UCL of the Mean Based upon the Adjusted Central Limit Theorem (Adjusted-CLT) 74 2.4.7 Chebyshev (1 - a)* 100 UCL of the Mean Using Sample Mean and Sample sd. 74 2.4.8 Chebyshev (1 - a)* 100 UCL of the Mean of a Lognormal Population Using the MVUE of the Mean and its Standard Error 76 2.4.9 (1 - a)* 100 UCL of the Mean Using the Jackknife and Bootstrap Methods 77 2.4.9.1 (I- a) *1 00 UCL of the Mean Based upon the Jackknife Method 77 2.4.9.2 (1 - a) *100 UCL of the Mean Based upon the Standard Bootstrap Method 78 2.4.9.3 (1 - a) *100 UCL of the Mean Based upon the Simple Percentile Bootstrap Method 80 2.4.9.4 (1 - a) *100 UCL of the Mean Based upon the Bias-Corrected Accelerated (BCA) Percentile Bootstrap Method 80 2.4.9.5 (1- a) *100 UCL of the Mean Based upon the Bootstrap-t Method 81 2.4.9.6 (1- a) *7 00 UCL of the Mean Based upon Hall's Bootstrap Method ...82 2.5 Suggestions and Summary 89 xxvn ------- 2.5.1 Suggestions for Computing a 95% UCL of the Unknown Population Mean, (ii, Using Symmetric and Positively Skewed Data Sets 90 2.5.1.1 Normally or Approximately Normally Distributed Data Sets 90 2.5.1.2 Gamma or Approximately Gamma Distributed Data Sets 91 2.5.1.3 Lognormally or Approximately Lognormally Distributed Skewed Data Sets 92 2.5.1.4 Nonparametric Skewed Data Sets without a Discernible Distribution... 94 2.5.2 Summary of the Procedure to Compute a 95% UCL of the Unknown Population Mean, \a\t Based upon Full Uncensored Data Sets without Nondetect Observations 96 CHAPTER 3 Computing Upper Limits to Estimate Background Threshold Values Based Upon Uncensored Data Sets without Nondetect Observations 98 3.1 Introduction 98 3.1.1 Description and Interpretation of Upper Limits used to Estimate BTVs 101 3.1.2 Confidence Coefficient (CC) and Sample Size 103 3.2 Treatment of Outliers 104 3.3 Upper p* 100% Percentiles as Estimates of BTVs 105 3.3.1 Nonparametric p*l00% Percentile 105 3.3.2 Normal p* 100% Percentile 106 3.3.3 Lognormalp* 100% Percentile 106 3.3.4 Gamma p* 100% Percentile 106 3.4 Upper Tolerance Limits 107 3.4.1 Normal Upper Tolerance Limits 107 3.4.2 Lognormal Upper Tolerance Limits 108 3.4.3 Gamma Distribution Upper Tolerance Limits 108 3.4.4 Nonparametric Upper Tolerance Limits 109 3.4.4.1 Determining the Order, r, of the Statistic, X(r), to Compute UTLp,(l-a) 110 3.4.4.2 Determining the Achieved Confidence Coefficient, CC achieve, Associated WithX(r) 110 3.4.4.3 Determining the Sample Size 110 3.4.4.4 Nonparametric UTL Based upon the Percentile Bootstrap Method. ...Ill 3.4.4.5 Nonparametric UTL Based upon the Bias-Corrected Accelerated (BCA) Percentile Bootstrap Method 777 3.5 Upper Prediction Limits 112 3.5.1 Normal Upper Prediction Limit 112 3.5.2 Lognormal Upper Prediction Limit 112 3.5.3 Gamma Upper Prediction Limit 113 3.5.4 Nonparametric Upper Prediction Limit 113 3.5.4.1 Upper Prediction Limit Based upon the Chebyshev Inequality 114 3.5.5 Normal, Lognormal, and Gamma Distribution based Upper Prediction Limits for k Future Comparisons 114 3.5.6 Proper Use of Upper Prediction Limits 115 3.6 Upper Simultaneous Limits 115 3.6.1 Upper Simultaneous Limits for Normal, Lognormal and Gamma Distributions 116 CHAPTER 4 Computing Upper Confidence Limit of the Population Mean Based upon Left-Censored Data Sets Containing Nondetect Observations 125 4.1 Introduction 125 xxvin ------- 4.2 Pre-processing a Data Set and Handling of Outliers 127 4.2.1 Assessing the Influence of Outliers and Disposition of Outliers 127 4.2.2 Avoid Data Transformation 127 4.2.3 Do Not Use DL/2(t) UCL Method 128 4.2.4 Minimum Data Requirement 128 4.3 Goodness-of-Fit (GOF) Tests and Skewness for Left-Censored Data Sets 128 4.4 Nonparametric Kaplan-Meier (KM) Estimation Method 129 4.5 Regression on Order Statistics (ROS) Methods 131 4.5.1 Computation of the Plotting Positions (Percentiles) and Quantiles 132 4.5.2 Computing OLS Regression Line to Impute NDs 133 4.5.2.1 Influence of Outliers on Regression Estimates and Imputed NDs 134 4.5.3 ROS Method for Lognormal Distribution 134 4.5.3.1 Fully Parametric Log ROS Method 134 4.5.3.2 Robust ROS Method on Log-Transformed Data 135 4.5.3.3 Gamma ROS Method 138 4.6 A Hybrid KM Estimates and Distribution of Detected Observations Based Approach to Compute Upper Limits for Skewed Data Sets - New in ProUCL 5.0/ ProUCL 5.1 141 4.6.1 Detected Data Set Follows aNormal Distribution 142 4.6.2 Detected Data Set Follows a Gamma Distribution 142 4.6.3 Detected Data Set Follows a Lognormal Distribution 143 4.6.3.1 Issues Associated with the Use of Lognormal distribution to Compute a UCL of Mean for Data Sets with Nondetects 148 4.6.3.1.1 Impact of Using DL and DL/2 for Nondetects on UCL95 Computations 149 4.6.3.1.2 Impact of Outlier, 16.1 ppb on UCL95 Computations 150 4.7 Bootstrap UCL Computation Methods for Left-Censored Data Sets 151 4.7.1 Bootstrapping Data Sets with Nondetect Observations 152 4.7.1.1 UCL of Mean Based upon Standard Bootstrap Method 153 4.7.1.2 UCL of Mean Based upon Bootstrap-t Method 154 4.7.1.3 Percentile Bootstrap Method 154 4.7.1.4 Bias-Corrected Accelerated (BCA) Percentile Bootstrap Procedure.. 154 4.8 (l-a)*100% UCL Based upon Chebyshev Inequality 155 4.9 Saving Imputed NDs Using Stats/Sample Sizes Module of ProUCL 158 4.10 Parametric Methods to Compute UCLs Based upon Left-Censored Data Sets 158 4.11 Summary and Suggestions 158 CHAPTER 5 Computing Upper Limits to Estimate Background Threshold Values Based upon Data Sets Consisting of Nondetect (ND) Observations 163 5.1 Introduction 163 5.2 Treatment of Outliers in Background Data Sets with NDs 163 5.3 Estimating BTVs Based upon Left-Censored Data Sets 164 5.3.1 Computing Upper Prediction Limits (UPLs) for Left-Censored Data Sets 164 5.3.1.1 UPLs Based upon Normal Distribution of Detected Observations and KM Estimates 164 5.3.1.2 UPL Based upon the Chebyshev Inequality 165 5.3.1.3 UPLs Based upon ROS Methods 165 5.3.1.4 UPLs when Detected Data are Gamma Distributed. 165 5.3.1.5 UPLs when Detected Data are Lognormally Distributed. 166 5.3.2 Computing Upper p* 100% Percentiles for Left-Censored Data Sets 166 5.3.2.1 Upper Percentiles Based upon Standard Normal Z-Scores 166 5.3.2.2 Upper Percentiles when Detected Data are Lognormally Distributed 167 XXIX ------- 5.3.2.3 Upper Percentiles when Detected Data are Gamma Distributed. 167 5.3.2.4 Upper Percentiles Based upon ROSMethods 167 5.3.3 Computing Upper Tolerance Limits (UTLs) for Left-Censored Data Sets 168 5.3.3.1 UTLs Based on KM Estimates when Detected Data are Normally Distributed 168 5.3.3.2 UTLs Based on KM Estimates when Detected Data are Lognormally Distributed 168 5.3.3.3 UTLs Based on KM Estimates when Detected Data are Gamma Distributed 168 5.3.3.4 UTLs Based upon ROS Methods 169 5.3.4 Computing Upper Simultaneous Limits (USLs) for Left-Censored Data Sets.. 169 5.3.4.1 USLs Based upon Normal Distribution of Detected Observations and KM Estimates 169 5.3.4.2 USLs Based upon Lognormal Distribution of Detected Observations and KM Estimates 169 5.3.4.3 USLs Based upon Gamma Distribution of Detected Observations and KM Estimates 169 5.3.4.4 USLs Based upon ROS Methods 770 5.4 Computing Nonparametric Upper Limits Based upon Higher Order Statistics 181 CHAPTER 6 Single and Two-sample Hypotheses Testing Approaches 182 6.1 When to Use Single Sample Hypotheses Approaches 182 6.2 When to Use Two-Sample Hypotheses Testing Approaches 183 6.3 Statistical Terminology Used in Hypotheses Testing Approaches 184 6.3.1 Test Form 1 184 6.3.2 Test Form 2 185 6.3.3 Selecting a Test Form 185 6.3.4 Errors Rates and Confidence Levels 185 6.4 Parametric Hypotheses Tests 187 6.5 Nonparametric Hypotheses Tests 187 6.6 Single Sample Hypotheses Testing Approaches 188 6.6.1 The One-Sample t-Test for Mean 188 6.6.1.1 Limitations and Robustness of One-Sample t-Test 188 6.6.1.2 Directions for the One-Sample t-Test 189 6.6.1.3 P-values 189 6.6.1.4 Relation between One-Sample Tests and Confidence Limits of the Mean or Median 190 6.6.2 The One-Sample Test for Proportions 191 6.6.2.1 Limitations and Robustness 797 6.6.2.2 Directions for the One-Sample Test for Proportions 797 6.6.2.3 Use of the Exact Binomial Distribution for Smaller Samples 7 93 6.6.3 The Sign Test 194 6.6.3.1 Limitations and Robustness 194 6.6.3.2 Sign Test in the Presence ofNondetects 194 6.6.3.3 Directions for the Sign Test 194 6.6.4 The Wilcoxon Signed Rank Test 196 6.6.4.1 Limitations and Robustness 196 6.6.4.2 Wilcoxon Signed Rank (WSR) Test in the Presence ofNondetects 797 6.6.4.3 Directions for the Wilcoxon Signed Rank Test 797 6.7 Two-sample Hypotheses Testing Approaches 202 XXX ------- 6.7.1 Student's Two-sample t-Test (Equal Variances) 202 6.7.1.1 Assumptions and their Verification 202 6.7.1.2 Limitations and Robustness 203 6.7.1.3 Guidance on Implementing the Student's Two-sample t-Test 203 6.7.1.4 Directions for the Student's Two-sample t-Test. 203 6.7.2 The Satterthwaite Two-sample t-Test (Unequal Variances) 204 6.7.2.1 Limitations and Robustness 205 6.7.2.2 Directions for the Satterthwaite Two-sample t-Test 205 6.8 Tests for Equality of Dispersions 206 6.8.1 The F-Test forthe Equality of Two-Variances 206 6.8.1.1 Directions for the F-Test. 206 6.9 Nonparametric Tests 209 6.9.1 The Wilcoxon-Mann-Whitney (WMW) Test 209 6.9.1.1 Advantages and Disadvantages 209 6.9.1.2 WMW Test in the Presence of Nondetects 209 6.9.1.3 WMW Test Assumptions and Their Verification 270 6.9.1.4 Directions for the WMWTestwhen the Number of Site and Background Measurements is small (n <20 or m <20) 270 6.9.1.5 Directions for the WMW Test when the Number of Site and Background Measurements is Large (n > 20 and m > 20) 272 6.9.2 GehanTest 216 6.9.2.1 Limitations and Robustness 216 6.9.2.2 Directions for the Gehan Testwhen m>10 andn > 70 216 6.9.3 Tarone-Ware (T-W) Test 218 6.9.3.1 Limitations and Robustness 218 6.9.3.2 Directions for the Tarone-Ware Testwhen m > 70 andn >70 218 CHAPTER 7 Outlier Tests for Data Sets with and without Nondetect Values 223 7.1 Outliers in Environmental Data Sets 224 7.2 Outliers and Normality 225 7.3 Outlier Tests for Data Sets without Nondetect Observations 225 7.3.1 Dixon'sTest 225 7.3.1.1 Directions for the Dixon's Test 226 7.3.2 Rosner's Test 227 7.3.2.1 Directions for the Rosner's Test 227 7.4 Outlier Tests for Data Sets with Nondetect Observations 228 CHAPTER 8 Determining Minimum Sample Sizes for User Specified Decision Parameters and Power Assessment 233 8.1.1 Sample Size Formula to Estimate Mean without Considering Type II ((3) Error Rate 235 8.1.2 Sample Size Formula to Estimate Mean with Consideration to Both Type I (a) and Type II ((3) Error Rates 236 8.2 Sample Sizes for Single-Sample Tests 237 8.2.1 Sample Size for Single-Sample t-test (Assuming Normality) 237 8.2.1.1 Case I (Right-Sided Alternative Hypothesis, Form 1) 238 8.2.1.2 Case II (Left-Sided Alternative Hypothesis, Form 2) 238 8.2.1.3 Case III (Two-Sided Alternative Hypothesis) 239 8.2.2 Single Sample Proportion Test 240 8.2.2.1 Case I (Right-Sided Alternative Hypothesis, Form 1) 241 8.2.2.2 Case II (Left-Sided Alternative Hypothesis, Form 2) 241 XXXI ------- 8.2.2.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 241 8.2.3 Nonparametric Single-sample Sign Test (does not require normality) .............. 243 8.2.3.1 Case I (Right-Sided Alternative Hypothesis) ....................................... 243 8.2.3.2 Case II (Left-Sided Alternative Hypothesis) ........................................ 243 8.2.3.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 243 8.2.4 Nonparametric Single Sample Wilcoxon Sign Rank (WSR) Test ..................... 244 8.2.4.1 Case I (Right-Sided Alternative Hypothesis) ....................................... 244 8.2.4.2 Case II (Left-Sided Alternative Hypothesis) ........................................ 245 8.2.4.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 245 8.3 Sample Sizes for Two-Sample Tests for Independent Sample ....................................... 246 8.3.1 Parametric Two-sample t-test (Assuming Normality) ....................................... 246 8.3.1.1 Case I (Right-Sided Alternative Hypothesis) ....................................... 246 8.3. 1.2 Case II (Left-Sided Alternative Hypothesis) ................................... 247 8.3.1.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 247 8.3.2 Wilcoxon-Mann-Whitney (WMW) Test (Nonparametric Test) ........................ 248 8.3.2.1 Case I (Right-Sided Alternative Hypothesis) ....................................... 248 8.3.2.2 Case II (Left-Sided Alternative Hypothesis) ........................................ 248 8.3.2.3 Case III (Two-Sided Alternative Hypothesis) ...................................... 249 8.3.3 Sample Size forWMW Test Suggested by Noether(1987) ................................ 250 8.4 Acceptance Sampling for Discrete Objects [[[ 251 8.4.1 Acceptance Sampling Based upon Chi-square Distribution .............................. 252 8.4.2 Acceptance Sampling Based upon Binomial/Beta Distribution ........................ 252 CHAPTER 9 Oneway Analysis of Variance Module [[[ 254 9.1 Oneway Analysis of Variance (ANOVA) [[[ 254 9.1.1 General Oneway ANOVA Terminology [[[ 254 9.2 Classical Oneway ANOVA Model [[[ 255 9.3 Nonparametric Oneway ANOVA (Kruskal-Wallis Test) ............................................... 256 CHAPTER 10 Ordinary Least Squares Regression and Trend Analysis .......................... 260 10.1 Ordinary Least Squares Regression [[[ 260 70.7.7 Regression ANOVA Table [[[ 263 10.1.2 Confidence Interval and Prediction Interval around the Regression Line ......... 264 10.2 Trend Analysis [[[ 266 10.2.1 Mann-Kendall Test [[[ 267 10.2.1.1Large Sample Approximation for M-K Test. ......................................... 268 10.2.1. 2Step-by-Step Procedure to perform the Mann-Kendall Test. ................ 269 10.2.2 Theil - Sen Line Test [[[ 272 10. 2. 2. IStep-by-Step Procedure to Compute Theil-Sen Slope ........................... 273 273 1 0. 2. 2. 2Large Sample Inference for Theil - Sen Test Based upon Normal Approximation [[[ 273 ------- REFERENCES 300 XXXlll ------- ------- INTRODUCTION OVERVIEW OF ProUCL VERSION 5.1 SOFTWARE The main objective of the ProUCL software funded by the U.S.EPA is to compute rigorous decision statistics to help the decision makers in making reliable decisions which are cost-effective, and protective of human health and the environment. The development of ProUCL software is based upon the philosophy that rigorous statistical methods can be used to compute representative estimates of population parameters (e.g., site mean, background percentiles) and accurate decision making statistics (including the upper confidence limit [UCL] of the mean, upper tolerance limit [UTL], and upper prediction limit [UPL]) which will assist decision makers and project teams in making sound decisions. The use and applicability of a statistical method (e.g., student's t-UCL, Central Limit Theorem (CLT)-UCL, adjusted gamma-UCL, Chebyshev UCL, bootstrap-t UCL) depend upon data size, data variability, data skewness, and data distribution. ProUCL computes decision statistics using several parametric and nonparametric methods covering a wide-range of data variability, skewness, and sample size. A couple of text book methods described in most of the statistical text books (e.g., Hogg and Craig, 1995) based upon the Student's t-statistic and the CLT alone cannot address all scenarios and situations commonly occurring in environmental studies. It is incorrect to assume that Student's t-statistic and/or CLT based UCLs of mean will provide the desired coverage (e.g., 0.95) to the population mean irrespective of the skewness of the data set/population under consideration. These issues have been discussed in detail in Chapters 2 and 4 of this Technical Guide. Several examples have been discussed throughout this guidance document and also in the accompanying ProUCL 5.1 User Guide to elaborate on these issues. The use of a parametric lognormal distribution on a lognormally distributed data set tends to yield unstable impractically large UCL values, especially when the standard deviation of the log-transformed data is greater than 1.0 and the data set is of small size such as less than 30-50 (Hardin and Gilbert 1993; Singh, Singh, and Engelhardt 1997). Many environmental data sets can be modeled by a gamma as well as a lognormal distribution. Generally, the use of a gamma distribution on gamma distributed data sets yields UCL values of practical merit (Singh, Singh, and laci 2002). Therefore, the use of gamma distribution based decision statistics such as UCLs, UPL, and UTLs cannot be dismissed just because it is easier to use a lognormal model to compute these upper limits. The two distributions do not behave in a similar manner. The advantages of computing the gamma distribution-based decision statistics are discussed in Chapters 2 through 5 of this guidance document. Since many environmental decisions are made based upon a 95% UCL of the population mean, it is important to compute reliable UCLs and other decision making statistics of practical merit. In an effort to compute stable UCLs of the population mean and other decision making statistics, in addition to computing the Student's t statistic and the CLT based statistics (e.g., UCLs, UPLs), significant effort has been made to incorporate rigorous statistical methods for computing UCLs (and other limits) in the ProUCL software, covering a wide-range of data skewness and sample sizes (e.g., Singh, Singh, and Engelhardt, 1997; Singh, Singh, and laci, 2002; and Singh, Singh, 2003). It is anticipated that the availability of the statistical methods in the ProUCL software, which can be applied to a wide range of environmental data sets, will help decision makers in making more informative, practical and sound decisions. ------- It is noted that even for skewed data sets, practitioners tend to use the CLT or Student's t-statistic based UCLs of mean for "large" sample sizes of 25-30 (rule-of-thumb to use CLT). However, this rule-of- thumb does not apply for moderately to highly skewed data sets, specifically when a (standard deviation of the log-transformed data) starts exceeding 1. The large sample size requirement associated with the use of the CLT depends upon the skewness of the data distribution under consideration. The large sample requirement associated with CLT for the sample mean to follow an approximate normal distribution increases with the data skewness; and for highly skewed data sets, even samples of size greater than (>)100 may not be large enough for the sample mean to follow an approximate normal distribution. For moderately skewed to highly skewed environmental data sets, as expected, UCLs based on the CLT and the Student's t-statistic fail to provide the desired coverage of the population mean even when the sample sizes are as large as 100 or more. These facts have been verified in the published simulation experiments conducted on positively skewed data sets (e.g., Singh, Singh, and Engelhardt, 1997; Singh, Singh, and laci, 2002); some graphs showing the simulation results are provided in Appendix B. The initial development and all subsequent upgrades and enhancements of the ProUCL software have been funded by the U.S. EPA through its Office of Research and Development (ORD). Initially ProUCL was developed as a research tool for scientists and researchers of the Technical Support Center and ORD- NERL, Las Vegas. During 1999-2001, the initial intent and objectives of developing the ProUCL software (Version 1.0 and Version 2.0) were to provide a statistical research tool for EPA scientists which can be used to compute theoretically sound 95% upper confidence limits (UCL95s) of the mean routinely used in exposure assessment, risk management and cleanup decisions made at various CERCLA and RCRA sites (EPA 1992a, 2002a). During 2002, the peer-reviewed ProUCL version 2.1 (with Chebyshev inequality based UCLs) was released for public use. Several researchers have developed rigorous parametric and nonparametric statistical methods (e.g., Johnson 1978; Grice and Bain 1980; Efron [1981, 1982]; Efron and Tibshirani 1993; Hall [1988, 1992]; Sutton 1993; Chen 1995; Singh, Singh, and Engelhardt 1997; Singh, Singh, and laci 2002] to compute upper limits (e.g., UCLs) which adjust for data skewness. Since Student's t-UCL, CLT-UCL, and percentile bootstrap UCL fail to provide the desired coverage to the population mean of skewed distributions, several parametric (e.g., gamma distribution based) and nonparametric (e.g., bias-corrected accelerated [BCA] bootstrap and bootstrap-t, Chebyshev UCL) UCL computation methods which adjust for data skewness were incorporated in ProUCL versions 3.0 and 3.00.02 during 2003-2004. ProUCL version 3.00.02 also had graphical Q-Q plots and GOF tests for normal, lognormal, and gamma distributions; capabilities to statistically analyze multiple variables simultaneously were also incorporated in ProUCL 3.00.02 (EPA 2004). It is important to compute decision statistics (e.g., UCLs, UTLs) which are cost-effective and protective of human health and the environment (balancing between Type I and Type II errors), therefore, one cannot dismiss the use of the better [better than t-UCL, CLT-UCL, ROS and KM percentile bootstrap UCL, KM-UCL (t)] performing UCL computation methods including gamma UCLs and the various bootstrap UCLs which adjust for data skewness. During 2004-2007, ProUCL was upgraded to versions 4.00.02, and 4.00.04. These upgrades included exploratory graphical (e.g., Q-Q plots, box plots) and statistical (e.g., maximum likelihood estimation [MLE], KM, and ROS) methods for left-censored data sets consisting of nondetect (NDs) observations with multiple DLs or RLs. For uncensored and left- censored data sets, these upgrades provide statistical methods to compute upper limits: percentiles, UPLs and UTLs needed to estimate site-specific background level constituent concentrations or background threshold values (BTVs). To address statistical needs of background evaluation projects (e.g., MARSSIM/EPA 2000, EPA 2002b), several single-sample and two-sample hypotheses testing approaches were also included in these ProUCL upgrades. ------- During 2008-2010, ProUCL was upgraded to ProUCL 4.00.05. The upgraded ProUCL was enhanced by including methods to compute gamma distribution based UPLs and UTLs (Krishnamoorthy, Mathew, and Mukherjee 2008). The Sample Size module to compute DQOs-based minimum sample sizes, needed to address statistical issues associated with environmental projects (e.g., EPA 2000, 2002c, 2006a, 2006b), was also incorporated in ProUCL 4.00.05. During 2009-2011, ProUCL 4.00.05 was upgraded to ProUCL 4.1 and 4.1.01. ProUCL 4.1 (2010) and 4.1.01 (2011) retain all capabilities of the previous versions of ProUCL software. Two new modules: Oneway ANOVA and Trend Analysis were included in ProUCL 4.1. The Oneway ANOVA module has both parametric and nonparametric ANOVA tests to perform inter-well comparisons. The Trend Analysis module can be used to determine potential upward or downward trends present in constituent concentrations identified in GW monitoring wells (MWs). The Trend Analysis module can compute Mann-Kendall (MK) and Theil-Sen (T-S) trend statistics to determine upward or downward trends potentially present in analyte concentrations. ProUCL 4.1 also has the OLS Regression module. In ProUCL 4.1, some modifications were made in decision tables which are used to make suggestions regarding the use of UCL95 for estimating EPCs. Specifically, based upon experience, developers of ProUCL re-iterated that the use of a lognormal distribution for estimating EPCs and BTVs should be avoided, as the use of the lognormal distribution tends to yield unrealistic and unstable values of decision making statistics including UCLs, UPLs, and UTLs. This is especially true when the sample size is <20- 30 and the data set is moderately to highly skewed. During March 2011, webinars were presented describing the capabilities and use of the methods available in ProUCL 4.1, which can be downloaded from the EPA ProUCL website. ProUCL version 5.0.00 (EPA 2013, 2014) represents an upgrade of ProUCL 4.1.01 (EPA June 2011) which represents an upgrade of ProUCL 4.1.00 (EPA 2010). For uncensored and left-censored data sets, ProUCL 5.0.00 (ProUCL 5.0) contains all statistical and graphical methods that were available in the previous versions of the ProUCL software package except for some poor performing and restricted (e.g., can be used only when a single detection limit is present) estimation methods such as the MLE and winsorization methods for left-censored data sets. ProUCL has GOF tests for normal, lognormal, and gamma distributions for uncensored and left-censored data sets with NDs. ProUCL 5.0 has the extended version of the Shapiro-Wilk (S-W) test to perform normal and lognormal GOF tests for data sets of sizes up to 2000 (Royston [1982, 1982a]). In addition to normal and lognormal distribution- based decision statistics, ProUCL software computes UCLs, UPLs, and UTLs based upon the gamma distribution. Several enhancements were made in the UCLs/EPCs and Upper Limits/BTVs modules of the ProUCL 5.0 software. A new statistic, an upper simultaneous limit (USL) (Singh and Nocerino 2002; Wilks 1963) has been incorporated in the Upper Limits/BTVs module of ProUCL 5.0 for data sets consisting of NDs with multiple DLs. A two-sample hypothesis test, the Tarone-Ware (T-W; Tarone and Ware, 1978) test has also been incorporated in ProUCL 5.0. Nonparametric tolerance limits have been enhanced, and for specific values of confidence coefficients, coverage probability, and sample size, ProUCL 5.0 outputs the confidence coefficient (CC) actually achieved by a UTL. The Trend Analysis and OLS Regression modules can handle missing events when computing trend test statistics and generating trend graphs. Some new methods using KM estimates in gamma (and lognormal) distribution-based UCL, UPL, and UTL equations have been incorporated to compute the decision statistics for data sets consisting of nondetect observations. To facilitate the computation of UCLs from ISM based samples (ITRC 2012); the minimum sample size requirement has been lowered to 3, so that one can compute the UCL95 based upon ISM data sets of sizes >3. ------- All known bugs, typographical errors, and discrepancies found by the developers and users of the ProUCL software package were addressed in ProUCL version 5.0.00. Specifically, a discrepancy found in the estimate of mean based upon the KM method was fixed in ProUCL 5.0. Some changes were made in the decision logic used in the Goodness of Fit and UCLs/EPCs modules. In practice, based upon a given data set, it is well known that the two statistical tests (e.g., T-S and OLS trend tests) can lead to different conclusions. To streamline the decision logic associated with the computation of the various UCLs, the decision tables in ProUCL 5.0 were updated. Specifically, for each distribution if at least one of the two GOF tests (e.g., Shapiro-Wilk or Lilliefors test for normality) determines that the hypothesized distribution holds, then ProUCL concludes that the data set follows the hypothesized distribution, and decision statistics are computed accordingly. Additionally, for gamma distributed data sets, ProUCL 5.0 suggests the use of the: adjusted gamma UCL for samples of sizes < 50 (instead of 40 suggested in previous versions); and approximate gamma UCL for samples of sizes >50. Also, for samples of larger sizes (e.g., with n > 100) and small values of the gamma shape parameter, k (e.g., k < 0.1), significant discrepancies were found in the critical values of the two gamma GOF test statistics (Anderson-Darling [A-D] and Kolmogorov Smirnov [K-S] tests) obtained using the two gamma deviate generation algorithms: Whitaker (1974) and Marsaglia and Tsang (2000). For values of k < 0.2, the critical values of the two gamma GOF tests: A-D and K-S tests have been updated using the currently available more accurate gamma deviate generation algorithm due to Marsaglia and Tsang's (2000); more details about the implementation of their algorithm can be found in Kroese, Taimre, and Botev (2011). For values of the shape parameter, k=0.025, 0.05, 0.1, and 0.2, the critical value tables for these two tests were updated by incorporating the newly generated critical values for the three significance levels: 0.05, 0.1, and 0.01. The updated tables are provided in Appendix A of the ProUCL 5.0/ProUCL 5.1 Technical Guide. It should be noted that for k=0.2, the older and the newly generated critical values are in general agreement; therefore, critical values for k=0.2 were not replaced in tables summarized in Appendix A. ProUCL 5.0 also has a new Background Incremental Sample Simulator (BISS) module (blocked for general public use) which can be used on a large existing discrete background data set to simulate background incremental samples. The availability of a large discrete data set collected from areas with geological formations and conditions comparable to the DUs (background or onsite) of interest is a requirement for successful application of this module. The simulated BISS data can be compared with the actual field ISM (ITRC 2012) data collected from the various DUs using other modules of ProUCL 5.0. The values of the BISS data are not directly available to users; however, the simulated BISS data can be accessed by the various modules of ProUCL 5.0 to perform desired statistical evaluations. For example, the simulated background BISS data can be merged with the actual field ISM data after comparing the two data sets using a two-sample t-test; the simulated BISS or the merged data can be used to compute a UCL of the mean or a UTL. Note: The ISM methodology used to develop the BISS module is a relatively new approach; methods incorporated in this BISS module requires further investigation. For now, the BISS module has been blocked for use in ProUCL 5.0/ProUCL 5.1 as this module is awaiting adequate guidance and instructions for its intended use on discrete background data sets. ProUCL 5.0 is a user-friendly freeware package providing statistical and graphical tools needed to address statistical issues described in several EPA guidance documents. Considerable effort was made to provide a detailed technical guide to help practitioners understand the statistical methods needed to address the statistical needs of their environmental projects. ProUCL generates detailed output sheets and graphical displays for each method which can be used to educate students learning environmental statistical methods. Like previous versions, ProUCL 5.0 can process many variables simultaneously to ------- compute various tests (e.g., ANOVA and trend test statistics) and decision statistics including UCL of the mean, UPLs, and UTLs, a capability not available in other software packages such as Minitab 16 and NADA for R (Helsel 2013). Without the availability of this option, the user has to compute decision and test statistics for one variable at a time which becomes cumbersome when dealing with a large number of variables. ProUCL 5.0 also has the capability of processing data by groups. ProUCL 5.0 is easy to use; it does not require any programming skills as needed when using programs written in R Script. Deficiencies Identified in ProUCL 5.0: For ProUCL to be compatible with Microsoft Office 8 and provide Excel-compatible Spreadsheet functionality (e.g., ability to input/output *.xlsx files), ProUCL 5.0 used FarPoint Spread 5 for .NET; and for graphics, ProUCL 5.0 used the development software package, ChartFx 7. The look and feel of ProUCL 5.0 is quite different from its previous versions; all main menu options were re-arranged. However, the use of upgraded development softwares resulted in some problems. Specifically, it takes an unacceptably long time to save large ProUCL 5.0 generated output files using FarPoint Spread 5. Also the use of ChartFx 7 caused some problems in properly labeling axes for histograms. Additionally, some unhandled exceptions and crashes were noted by users. The unhandled exceptions were mainly noted for "bad" data sets including data sets not following ProUCL input format; data sets with not enough observations; and data sets with not enough detects. ProUCL 5.1: ProUCL 5.1 represents an upgrade of ProUCL 5.0 to address deficiencies identified in ProUCL 5.0. ProUCL 5.1 retains all capabilities of ProUCL 5.0 as described above. All modules in ProUCL 5.1, and their look and feel is the same as in ProUCL 5.0. In this document, any statement made about the capabilities of ProUCL 5.0 also apply to ProUCL version 5.1; and to save time, not all screen shots used in ProUCL 5.0 manuals have been replaced in the ProUCL 5.1 User Guide and Technical Guide. Upgrades in ProUCL 5.1 (not available in earlier versions) have been labeled as New in ProUCL 5.1 in this document. All known bugs, crashes, and unhandled exceptions (e.g., on bad data sets) found in ProUCL 5.0 have been addressed in ProUCL 5.1. In ProUCL 5.1, some enhancements have been made in the Trend Analysis option of the Statistical Test module of ProUCL 5.1. ProUCL 5.1 computes and outputs residuals for the non-parametric T-S trend line which may be helpful to compute a prediction band around the T-S trend line. In addition to generating Q-Q plots based upon detected observations, the Goodness of Fit Tests option of the Statistical Tests module of ProUCL 5.1 generates censored probability plots for data sets with NDs. Some changes have been made in the decision table used to make suggestions for UCL selection based upon a gamma distribution. New licensing agreements were obtained for the development softwares: FarPoint and ChartFx. Due to deficiencies present in the development software, ProUCL 5.1 generated large output files still take a long time to be saved. However, there is a quick work around to this problem, instead of saving the output sheet using ProUCL, one can copy the output spreadsheet and save the copied output sheet using Excel. This operation can be carried out instantly. Note about Histograms: ChartFx 7.0 has some inherent deficiencies, as a result labeling of bins along the x-axis on a histogram is still not as desirable as one would like it to be. The x-axis display will start from zero instead of the proper lowest histogram value. Occurrences are rare but they can occur. Some tools have been added in ProUCL 5.1, and relevant statistics (e.g., start point, midpoint, and end point) of a histogram bar can be displayed by hovering the cursor on that bar. Software ProUCL version 5.1, its earlier versions: ProUCL version 3.00.02, 4.00.02, 4.00.04, 4.1.00, 4.1.01 and ProUCL 5.0, associated Facts Sheet, User Guides and Technical Guides (e.g., EPA [2004, 2007, 2009a, 2009b, 2010a, 2010b, 2013a, 2013b]) can be downloaded from the EPA website: ------- http://www.epa.gov/osp/hstl/tsc/software.htm http://www.epa.gov/osp/hstl/tsc/softwaredocs.htm The Need for ProUCL Software EPA guidance documents (e.g., EPA [1989a, 1989b, 1992a, 1992b, 1994, 1996, 2000, 2002a, 2002b, 2002c, 2006a, 2006b, 2009a, and 2009b]) describe statistical methods including: DQOs-based sample size determination procedures, methods to compute decision statistics: UCL95, UPL, and UTLs, parametric and nonparametric hypotheses testing approaches, Oneway ANOVA, OLS regression, and trend determination approaches. Specifically, EPA guidance documents (2000, 2002c, 2006a, 2006b) describe DQOs-based parametric and nonparametric minimum sample size determination procedures needed: to compute decision statistics (e.g., UCL95); to perform site versus background comparisons (e.g., t-test, proportion test, WMW test); and to determine the number of discrete items (e.g., drums filled with hazardous material) that need to be sampled to meet the DQOs (e.g., specified proportion, po of defective items, allowable error margin in an estimate of mean). Statistical methods are used to compute test statistics (e.g., S-W test, t-test, WMW test, T-S trend statistic) and decision statistics (e.g., 95% UCL, 95% UPL, UTL95-95) needed to address statistical issues associated with CERCLA and RCRA site projects. For example, exposure and risk management and cleanup decisions in support of EPA projects are often made based upon the mean concentrations of the contaminants/constituents of potential concern (COPCs). Site-specific BTVs are used in site versus background evaluation studies. A UCL95 is used to estimate the EPC terms (EPA 1992a, 2002a); and upper limits such as upper percentiles, UPLs, or UTLs are used to estimate BTVs or not-to-exceed values (EPA 1992b, 2002b, and 2009). The estimated BTVs are used to address several objectives: to identify the COPCs; to identify the site areas of concern (AOCs); to perform intra-well comparisons to identify MWs not meeting specified standards; and to compare onsite constituent concentrations with site-specific background level constituent concentrations. Oneway ANOVA is used to perform inter-well comparisons and OLS regression and trend tests are often used to determine potential trends present in constituent concentrations identified in GW monitoring wells (MWs). Most of the methods described in this paragraph are available in the ProUCL 5.1 (ProUCL 5.0) software package. It is noted that not much guidance is available in the guidance documents cited above to compute rigorous UCLs, UPLs, and UTLs for moderately to highly skewed uncensored and left-censored data sets containing NDs with multiple DLs, a common occurrence in environmental data sets. Several parametric and nonparametric methods are available in the statistical literature (Singh, Singh, and Engelhardt 1997; Singh, Singh, and laci 2002; Krishnamoorthy et al. 2008; Singh, Maichle, and Lee, 2006) to compute UCLs and other upper limits which adjust for data skewness. During the years, as new methods became available to address statistical issues related to environmental projects, those methods were incorporated in ProUCL software so that environmental scientists and decision makers can make more accurate and informed decisions. Until 2006, not much guidance was provided on how to compute UCL95s of the mean and other upper limits (e.g., UPLs and UTLs) based upon data sets containing NDs with multiple DLs. For data sets with NDs, Singh, Maichle, and Lee (2006) conducted an extensive simulation study to compare the performances of the various estimation methods (in terms of bias in the mean estimate) and UCL computation methods (in terms of coverage provided by a UCL). They demonstrated that the nonparametric KM method performs well in terms of bias in estimates of mean. They also concluded that UCLs computed using the Student's t-statistic and percentile bootstrap method using the KM estimates do not provide the desired coverage to the population mean of skewed data sets. They demonstrated that depending upon sample size and data skewness, UCLs computed using KM estimates, the BCA bootstrap method (mildly skewed data sets), the bootstrap-t method, and the Chebyshev inequality (moderately to highly skewed data sets) provide better coverage (closer to the specified 95% coverage) to the population ------- mean than other UCL computation methods. Based upon their findings, during 2006-2007, several UCL and other upper limits computation methods based upon KM and ROS estimates were incorporated in the ProUCL 4.0 software. It is noted that since the inclusion of the KM method in ProUCL 4.0 (2007), the use of the KM method based upper limits has become popular in many environmental applications to estimate EPC terms and BTVs. The KM method is also described in the latest version of the unified RCRA guidance document (U.S. EPA 2009). It is not easy to justify distributional assumptions of data sets consisting of both detects and NDs with multiple DLs. Therefore, based upon the published literature and experience, parametric UCL (and other upper limits) computation methods such as the MLE method (Cohen 1991) and the expectation maximization (EM) method (Gleit 1985) for normal and lognormal distributions were not included ProUCL 5.0 (and ProUCL 5.1) even though these methods were available in earlier versions of ProUCL. Additionally, the winsorization method (Gilbert 1987) available in an earlier version of ProUCL has also been excluded from ProUCL 5.0 (ProUCL 5.1) due to its poor performance. During 2015, some researchers (e.g., from New Mexico State University, Las Cruces, NM) suggested that the EM method performs better than some of the methods available in ProUCL 5.0, especially the gamma ROS (GROS) method; a method which can be used on left-censored data sets with multiple DLs. The literature has articles dealing with MLE and EM methods for data sets with a single censoring point (DL). Further research needs to be conducted on methods for computing reliable estimates of the mean, sd, and upper limits based upon parametric MLE and EM methods for data sets with NDs and multiple DLs. As always, it is the desire of the developers of ProUCL to incorporate the best available methods in ProUCL. The developers of ProUCL welcome/encourage other researchers to share their findings about the EM method showing that EM method performs better than methods already available in ProUCL 5.0/ProUCL 5.1 for data sets with single/multiple censoring points. The developers of ProUCL have been enhancing the ProUCL software with better performing methods as those methods become available. Efforts will be made to incorporate contributed code (with acknowledgement) for superior methods in future versions of ProUCL. ProUCL software is also used for teaching environmental statistics courses therefore, in addition to statistical and graphical methods routinely used to address statistical needs of environmental projects, some poor performing methods such as the substitution DL/2 method and Land's (1975) H-statistic based UCL computation method have been retained in ProUCL version 5.1 for research and comparison purposes. Methods incorporated in ProUCL 5.1 and in its earlier versions have been tested and verified extensively by the developers, researchers, scientists, and users. Specifically, the results obtained by ProUCL 5.1 are in agreement with the results obtained by using other software packages including Minitab, SAS®, and programs available in R-Script (not all methods are available in these software packages). Additionally, like ProUCL 5.0, ProUCL 5.1 outputs several intermediate results (e.g., khat and biased corrected kstar estimates of the gamma shape parameter, k, and critical values (e.g., tolerance factor, K, used to compute UTLs; critical value, d2max, used to compute USL) needed to compute decision statistics of interest, which may help interested users to verify statistical results computed by the ProUCL software. Whenever applicable, ProUCL provides warning messages and based upon professional experience and findings of simulation studies, makes suggestions to help a typical user in selecting the most appropriate decision statistic (e.g., UCL). Note: The availability of intermediate results and critical values can be used to compute lower limits and two-sided intervals which are not as yet available in the ProUCL software. For left-censored data sets, ProUCL 5.1 computes decision statistics (e.g., UCL, UPL, and UTL) based upon KM estimates computed in a straight forward manner without flipping the data and re-flipping the decision statistics; these operations are not easy for a typical user to understand and perform and can ------- become quite tedious when multiple analytes need to be processed. Moreover, in environmental applications it is important to compute accurate estimates of sd which are needed to compute decision making statistics including UPLs and UTLs. Decision statistics (UPL, UTL) based upon a KM estimate of the of sd and computed using indirect methods can be different from the statistics computed using an estimate of sd obtained using the KM method directly, especially when one is dealing with a skewed data set or when using a log-transformation. These issues are elaborated by examples discussed in the accompanying ProUCL 5.1 Technical Guide. For uncensored data sets, researchers (e.g., Johnson 1978; Chen 1995; Efron and Tibshirani 1993; Hall [1988, 1992], and additional references found in Chapters 2 and 3) developed parametric (e.g., gamma distribution based) and nonparametric (bootstrap-t and Hall's bootstrap method, modified-t) methods for computation of decision statistics which adjust for data skewness. For uncensored positively skewed data sets, Singh, Singh, and laci (2002) performed simulation experiments to compare the performances (in terms of coverage probabilities) of the various UCL computation methods described in the literature. They demonstrated that for skewed data sets, UCLs based upon Student's t statistic, central limit theorem (CLT), and percentile bootstrap method tend to underestimate the population mean (EPC). It is reasonable to state that the findings of the simulation studies performed on uncensored skewed data sets comparing the performances of the various UCL computation methods can be extended to skewed left-censored data sets. Based upon the findings of those studies performed on uncensored data sets and also using the findings summarized in Singh, Maichle, and Lee (2006), it was concluded that t-statistic, CLT, and the percentile bootstrap method based UCLs computed using KM estimates (and also ROS estimates) underestimate the population mean of moderately skewed to highly skewed data sets. Interested users may want to verify these statements by performing simulation experiments or other forms of rigorous testing. Like uncensored skewed data sets, for left-censored data sets, ProUCL 5.1 offers several parametric and nonparametric methods for computing UCLs and other limits which adjust for data skewness. Due to the lack of research and methods, in earlier versions of the ProUCL software (e.g., ProUCL 4.00.02, ProUCL 4.0), KM estimates were used in the normal distribution based equations for computing the various upper limits for left-censored data sets. However, normal distribution based upper limits (e.g., t-UCL) using KM estimates (or any other estimates such as ROS estimates) fail to provide the specified coverage (e.g., 0.95) of the parameters (e.g., mean, percentiles) of populations with skewed distributions (Singh, Singh, laci 2002; Johnson 1978; Chen 1995). For skewed data sets, ProUCL 5.0/ProUCL 5.1 computes UCLs applying KM estimates in UCL equations for skewed data sets (e.g., gamma and lognormal); therefore, some changes have been made in the decision tables of ProUCL 5.0/ProUCL 5.1 for computing UCL95s. Also, the nonparametric UCL computation methods (e.g., percentile bootstrap) do not provide the desired coverage to the population means of skewed distributions (e.g., Hall [1988, 1992], Efron and Tibshirani, 1993). For example, the use of t-UCL or the percentile bootstrap UCL method on robust ROS estimates or on KM estimates underestimates the population mean for moderately skewed to highly skewed data sets. Chapters 3 and 5 of the ProUCL Technical Guide describe parametric and nonparametric KM methods for computing upper limits (and available in ProUCL 5.0/ ProUCL 5.1) which adjust for data skewness. The KM method yields good estimates of the population mean and std (Singh, Maichle, and Lee2006); however upper limits computed using the KM or ROS estimates in normal equations or in the percentile bootstrap method do not account for skewness present in the data set. Appropriate UCL computation methods which account for data skewness should be used on KM or ROS estimates. For left-censored data sets, ProUCL 5.0/ProUCL 5.1 compute upper limits using KM estimates in gamma (lognormal) UCL, UPL, and UTL equations (e.g., also suggested in U.S. EPA 2009) provided the detected observations in the left-censored data set follow a gamma (lognormal) distribution. ------- Recently, the use of the ISM methodology has been recommended (ITRC 2012) for collecting soil samples with the purpose of estimating mean concentrations of DUs requiring analysis of human and ecological risk and exposure. ProUCL can be used to compute UCLs based upon ISM data as described and recommended in the ITRC ISM Technical and Regulatory Guide (2012). At many sites, large amounts of discrete background data are already available which are not directly comparable to the actual field ISM data (onsite or background). To compare the existing discrete background data with field ISM data, the BISS module (blocked for general use in ProUCL version 5.1 awaiting guidance and instructions for its intended use) of ProUCL 5.1 can be used on a large (e.g., consisting of at least 30 observations) existing discrete background data set. The BISS module simulates the incremental sampling methodology based equivalent incremental background samples; and each simulated BISS sample represents an estimate of the mean of the population represented by the discrete background data set. The availability of a large discrete background data set collected from areas with geological conditions comparable to the DU(s) of interest (onsite DUs) is a requirement for successful application of this module. The user cannot see the simulated BISS data; however, the simulated BISS data can be accessed by other modules of ProUCL 5.0 (ProUCL 5.1) for performing desired statistical evaluations. For example, the simulated BISS data can be merged with the actual field ISM background data after comparing the two data sets using a two-sample t-test. The actual field ISM or the merged ISM and BISS data can be accessed by modules of ProUCL to compute a UCL of the mean or a UTL. ProUCL 5.1 Capabilities Assumptions: Like most statistical methods, statistical methods for computing upper limits (e.g., UCLs, UPLs, UTLs) are also based upon certain assumptions including the availability of a randomly collected data set consisting of independently and identically distributed (i.i.d) observations representing the population (e.g., site area, reference area) under investigation. A UCL of the mean (of a population) and BTV estimates (UPL, UTL) should be computed using a randomly collected (simple random or systematic random) data set representing a single statistical population (e.g., site population or background population). When multiple populations (e.g., background and site data mixed together) are present in a data set, the recommendation is to separate them first by using the population partitioning techniques (e.g., Singh, Singh, and Flatman 1994) prior to computing the appropriate decision statistics (e.g., 95% UCLs). Regardless of how the populations are separated, decision statistics should be computed separately for each identified population. The topic of population partitioning and the extraction of a valid site-specific background data set from a broader mixture data set potentially consisting of both onsite and offsite data are beyond the scope of ProUCL 5.0/ProUCL 5.1. Parametric estimation and hypotheses testing methods (e.g., t-test, UCLs, UTLs) are based upon distributional (e.g., normal distribution, gamma) assumptions. ProUCL includes GOF tests for determining if a data set follows a normal, a gamma, or a lognormal distribution. Multiple Constituents/Variables: Environmental scientists need to evaluate many constituents in their decision making processes including exposure and risk assessment, background evaluations, and site versus background comparisons. ProUCL can process multiple constituents/variables simultaneously in a user-friendly manner; an option not available in other freeware or commercial software packages such as NADA for R (Helsel 2013). This option is very useful when one has to process many variables/analytes and compute decision statistics (e.g., UCLs, UPLs, and UTLs) and/or test statistics (e.g., ANOVA test, trend test) for those variables/analytes. Analysis by a Group Variable: ProUCL also has the capability of processing data by groups. A valid group column should be included in the data file. The analyses of data categorized by a group ID variable such as: 1) Surface versus (vs.) Subsurface; 2) AOC1 vs. AOC2; 3) Site vs. Background; and 4) ------- Upgradient vs. Downgradient MWs are common in many environmental applications. ProUCL offers this option for data sets with and without nondetects. The Group option provides a way to perform statistical tests and methods including graphical displays separately for each of the group (samples from different populations) that may be present in a data set. For example, the same data set may consist of analytical data from multiple groups or populations representing site, background, two or more AOCs, surface soil, subsurface soil, and GW. By using this option, the graphical displays (e.g., box plots, Q-Q plots, histograms) and statistics (including computation of background statistics, UCLs, ANOVA test, trend test and OLS regression statistics) can be easily computed separately for each group in the data set. Exploratory Graphical Displays for Uncensored and Left-Censored Data Sets: Graphical methods included in the Graphs module of ProUCL include: Q-Q plots (data in same column), multiple Q-Q plots (data in different columns), box plots, multiple box plots (data in different columns), and histograms. These graphs can also be generated for data sets containing ND observations. Additionally, the OLS Regression and Trend Analysis module can be used to generate graphs displaying parametric OLS regression lines with confidence and prediction intervals around the regression and nonparametric Theil- Sen trend lines. The Trend Analysis module can generate trend graphs for data sets without a sampling event variable, and also generates time series graphs for data sets with a sampling event (time) variable. Like ProUCL 5.0, ProUCL 5.1 accepts only numerical values for the event variable. Graphical displays of a data set are useful for gaining added insight regarding a data set that may not otherwise be clear by looking at test statistics such as T-S test or MK statistics. Unlike test statistics (e.g., t-test, MK test, AD test) and decision statistics (e.g., UCL, UTL), graphical displays do not get influenced by outliers and ND observations. It is suggested that the final decisions be made based upon statistical results as well as graphical displays. Side-by-side box plots or multiple Q-Q plots are useful to graphically compare concentrations of two or more groups (e.g., several monitoring wells). The GOF module of ProUCL generates Q-Q plots for normal, gamma, and lognormal distributions based upon uncensored as well as left-censored data sets with NDs. All relevant information such as the test statistics, critical values and probability-values (p- values), when available are also displayed on the GOF Q-Q plots. In addition to providing information about the data distribution, a normal Q-Q plot in the original raw scale also helps to identify outliers and multiple populations that may be present in a data set. On a Q-Q plot, observations well-separated from the majority of the data may represent potential outliers coming from a population different from the main dominant population (e.g., background population). In a Q-Q plot, jumps and breaks of significant magnitude suggest the presence of observations coming from multiple populations (onsite and offsite areas). ProUCL can also be used to display box plots with horizontal lines displayed/superimposed at pre-specified compliance limits (CLs) or computed upper limits (e.g., UPL, UTL). This kind of graph provides a visual comparison of site data with compliance limits and/or BTV estimates. Outlier Tests: ProUCL also provides a couple of classical outlier test procedures (EPA 2006b, 2009), the Dixon test and the Rosner test. The details of these outlier tests are described in Chapter 7. These outlier tests often suffer from "masking effects" in the presence of multiple outliers. It is suggested that the classical outlier procedures should always be accompanied by graphical displays including box plots and Q-Q plots. Description and use of the robust and resistant outlier procedures (Rousseeuw and Leroy 1987; Singh and Nocerino 1995) are beyond the scope of ProUCL 5.1. Interested users are encouraged to try the Scout 2008 software package (EPA 2009d) for robust outlier identification methods especially when dealing with multivariate data sets consisting of observations for several variables/analytes/constituents. Outliers represent observations coming from populations different from the main dominant population represented by the majority of the data set. Outliers distort most statistics (e.g., mean, UCLs, UPLs, test 10 ------- statistics) of interest. Therefore, it is desirable to compute decisions statistics based upon data sets representing the main population and not to compute distorted statistics by accommodating a few low probability outliers (e.g., by using a lognormal distribution). Moreover, it should be noted that even though outliers might have minimal influence on hypotheses testing statistics based upon ranks (e.g., WMW test), outliers do distort several nonparametric statistics including bootstrap methods such as bootstrap-t and Hall's bootstrap UCLs and other nonparametric UPLs and UTLs computed using higher order statistics. Goodness-of-Fit Tests: In addition to computing simple summary statistics for data sets with and without NDs, ProUCL 5.1 includes GOF tests for normal, lognormal and gamma distributions. To test for normality (lognormality) of a data set, ProUCL includes the Lilliefors test and the extended S-W test for samples of sizes up to 2000 (Royston 1982, 1982a). For the gamma distribution, two GOF tests: the A-D test (Anderson and Darling 1954) and K-S test (Schneider 1976, 1978) are available in ProUCL. For samples of larger sizes (e.g., with n > 100) and small values of the gamma shape parameter, k (e.g., k < 0.1), significant discrepancies were found in the critical values of the two gamma GOF test statistics (A-D and K-S tests) obtained using the two gamma deviate generation algorithms: Whitaker (1974) and Marsaglia and Tsang (2000). In ProUCL 5.0 (and ProUCL 5.1), for values of k < 0.2, the critical values of the two gamma GOF tests: A-D and K-S tests have been updated using the currently available more efficient gamma deviate generation algorithm due to Marsaglia and Tsang's (2000); more details about the implementation of their algorithm can be found in Kroese, Taimre, and Botev (2011). For these two GOF and values of the shape parameter, k=0.025, 0.05, 0.1, and 0.2, critical value tables have been updated by incorporating the newly generated critical values for three levels of significance: 0.05, 0.1, and 0.01. The updated tables are provided in Appendix A of the ProUCL Technical Guide. It was noted that for k=0.2, the older (generated in 2002) and the newly generated critical values are in general agreement; therefore, critical values for k=0.2 were not replaced in tables summarized in Appendix A. ProUCL also generates GOF Q-Q plots for normal, lognormal, and gamma distributions displaying all relevant statistics including GOF test statistics. GOF tests for data sets with and without NDs are described in Chapters 2 and 3 of the ProUCL Technical Guide. For data sets containing NDs, it is not easy to verify the distributional assumptions correctly, especially when the data set consists of a large percentage of NDs with multiple DLs and NDs exceeding some detected values. Historically, decisions about distributions of data sets with NDs are based upon GOF test statistics computed using the data obtained: without NDs; replacing NDs by 0, DL, or DL/2; using imputed NDs based upon a ROS (e.g., lognormal ROS) method. For data sets with NDs, ProUCL 5.1 can perform GOF tests using the methods listed above. ProUCL 5.1 can also generate censored probability plots (Q-Q plots) which are very similar to Q-Q plots generated using detected data. Using the Imputed NDs using ROS Methods option of the Stats/Sample Sizes module of ProUCL 5.0, additional columns can be generated for storing imputed (estimated) values for NDs based upon normal ROS, gamma ROS, and lognormal ROS (also known as robust ROS) methods. Sample Size Determination and Power Evaluation: The Sample Sizes module in ProUCL can be used to develop DQO-based sampling designs needed to address statistical issues associated with environmental projects. ProUCL 5.1 provides user-friendly options for entering the desired/pre-specified values for decision parameters (e.g., Type I and Type II error rates) and other DQOs used to determine minimum sample sizes for statistical applications including: estimation of the mean, single and two-sample hypothesis testing approaches, and acceptance sampling for discrete items (e.g., drums containing hazardous waste). Both parametric (e.g., t-test) and nonparametric (e.g., Sign test, WRS test) sample size determination methods as described in EPA (2000, 2002c, 2006a, 2006b) guidance documents are available in ProUCL 5.1. ProUCL also has the sample size determination option for acceptance sampling 11 ------- of lots of discrete objects such as a lot (batch, set) of drums containing hazardous waste (e.g., RCRA applications, EPA 2002c). When the sample size for an application (e.g., verification of cleanup level) is not computed using the DQOs-based sampling design process, the Sample Size module can be used to assess the power of the test statistic used in retrospect. The mathematical details of the Sample Sizes module are given in Chapter 8 of the ProUCL Technical Guide. Bootstrap Methods: Bootstrap methods are computer intensive nonparametric methods which can be used to compute decision statistics of interest when a data set does not follow a known distribution, or when it is difficult to analytically derive the distributions of statistics of interest. It is well-known that for moderately skewed to highly skewed data sets, UCLs based upon standard bootstrap and the percentile bootstrap methods do not perform well (e.g., Efron [1981, 1982]; Efron and Tibshirani 1993; Hall [1988,1992]; Singh, Singh, and laci 2002; Singh, Maichle and Lee 2006) as the interval estimates based upon these bootstrap methods fail to provide the specified coverage to the population mean (e.g., UCL95 does not provide adequate 95% coverage of population mean). For skewed data sets, Efron and Tibshirani (1993) and Hall (1988, 1992) considered other bootstrap methods such as the BCA, bootstrap-t and Hall's bootstrap methods. For skewed data sets, bootstrap-t and Hall's bootstrap (meant to adjust for skewness) methods perform better (e.g., in terms of coverage for the population mean) than the other bootstrap methods. However, it has been noted (e.g., Efron and Tibshirani 1993, Singh, Singh, and laci 2002) that these two bootstrap methods tend to yield erratic and inflated UCL values (orders of magnitude higher than other UCLs) in the presence of outliers. Similar behavior of the bootstrap-t UCL and Hall's bootstrap UCL methods is observed for data sets consisting of NDs and outliers. For nonparametric uncensored and left-censored data sets with NDs, depending upon data variability and skewness, ProUCL recommends the use of BCA bootstrap, bootstrap-t, or Chebyshev inequality based methods for computing decision statistics. Due to the reasons described above, whenever applicable, ProUCL 5.0/ProUCL 5.1 provides cautionary notes and warning messages regarding the use of bootstrap-t and Halls bootstrap UCL methods. Hypotheses Testing Approaches: ProUCL software has both single-sample (e.g., Student's t-test, sign test, proportion test, WSR test) and two-sample (Student's t-test, WMW test, Gehan test, and T-W test) parametric and nonparametric hypotheses testing approaches. Hypotheses testing approaches in ProUCL can handle both full-uncensored data sets and left-censored data sets with NDs. Most of the hypotheses tests also report associated /"-values. For some hypotheses tests (e.g., WMW test, WSR test, proportion test), large sample /"-values based upon the normal approximation are computed using continuity correction factors. The mathematical details of the various single-sample and two-sample hypotheses testing approaches are described in Chapter 6 the ProUCL Technical Guide. • Single-Sample Tests: Parametric (Student's t-test) and nonparametric (Sign test, WSR test, tests for proportions and percentiles) hypotheses testing approaches are available in ProUCL. Single-sample hypotheses tests are used when environmental parameters such as the cleanup standard, action level, or compliance limits are known, and the objective is to compare site concentrations with those known threshold values. A t-test (or a sign test) may be used to verify the attainment of cleanup levels in an AOC after a remediation activity has taken place or a test for proportion may be used to verify if the proportion of exceedances of an action level (Ao or a CL) by sample observations collected from an AOC (or a MW) exceeds a certain specified proportion (e.g., 1%, 5%, 10%). The differences between these tests should be noted and understood. A t-test or a Wilcoxon Signed Rank (WSR) test are used to compare the measures of location and central tendencies (e.g., mean, median) of a site area (e.g., AOC) to a cleanup standard, Cs, or action level also representing a measure of central tendency (e.g., mean, median); whereas, a proportion test determines if the 12 ------- proportion of site observations from an AOC exceeding a compliance limit (CL) exceeds a specified proportion, P0 (e.g., 5%, 10%). The percentile test compares a specified percentile (e.g., 95th) of the site data to a pre-specified upper threshold (e.g., action level). • Two-Sample Tests: Hypotheses tests (Student's t-test, WMW test, Gehan test, T-W test) are used to perform site versus background comparisons, compare concentrations of two or more AOCs, or to compare concentrations of GW collected from MWs. As cited in the literature, some of the hypotheses testing approaches (e.g., nonparametric two-sample WMW) deal with a single detection limit scenario. When using the WMW test on a data set with multiple detection limits, all observations (detects and NDs) below the largest detection limit need to be considered as NDs (Gilbert 1987). This in turn tends to reduce the power and increase uncertainty associated with test. As mentioned before, it is always desirable to supplement the test statistics and conclusions with graphical displays such as multiple Q-Q plots and side-by-side box plots. The Gehan test or T-W test (new in ProUCL 5.1) should be used in cases where multiple detection limits are present. Note about Ouantile Test: For smaller data sets, the Quantile test as described in U.S. EPA documents (U.S. EPA [1994, 2006b]; Hollander and Wolfe, 1999) is available in ProUCL 4.1(see ProUCL 4.1 Technical Guide). In the past, some users incorrectly used this test for larger data sets. Due to lack of resources, this test has not been expanded for data sets of all sizes. Therefore, to avoid confusion and its misuse for larger data sets, the Quantile test was not included in ProUCL 5.0 and ProUCL 5.1. Computation of Upper Limits including UCLs. UPLs. UTLs. and USLs: ProUCL software has parametric and nonparametric methods including bootstrap and Chebyshev inequality based methods to compute decision making statistics such as UCLs of the mean (EPA 2002a), percentiles, UPLs for future k (>1) observations, UTLs (U.S. EPA [1992b and 2009]) and upper simultaneous limits (USLs) (Singh and Nocerino [1995, 2002]) based upon uncensored full data sets and left-censored data sets containing NDs with multiple DLs. Methods incorporated in ProUCL cover a wide range of skewed data distributions with and without NDs. In addition to normal and lognormal distributions based upper limits, ProUCL 5.0 can compute parametric UCLs, percentiles, UPLs for future k (>1) observations, UTLs, and USLs based upon gamma distributed data sets. For data sets with NDs, ProUCL has several estimation methods including the Kaplan-Meier (KM) method (1958), ROS methods (Helsel 2005) and substitution methods such as replacing NDs with the DL or DL/2 (Gilbert 1987; U.S. EPA 2006b). Substitution method and other poor performing methods (e.g., H-UCL for lognormal distribution) have been retained, as requested by U.S. EPA scientists, in ProUCL 5.0/ProUCL 5.1 for research and comparison purposes. One may not interpret the availability of these poor performing methods in ProUCL as recommended methods by ProUCL or by the U.SEP A for computing decision statistics. Computation of UCLs Based upon Uncensored Data Sets without NDs: Parametric UCL computation methods in ProUCL for uncensored data sets include: Student's t-UCL, Approximate gamma UCL (using chi-square approximation), Adjusted gamma UCL (adjusted for level significance), Land's H-UCL, and Chebyshev inequality-based UCL (using minimum variance unbiased estimates (MVUEs) of parameters of a lognormal distribution). Nonparametric UCL computation methods for data sets without NDs include: CLT-based UCL, Modified-t-statistic-based UCL (adjusted for skewness), Adjusted-CLT-based UCL (adjusted for skewness), Chebyshev inequality-based UCL (using sample mean and standard deviation), Jackknife method-based UCL, UCL based upon standard bootstrap, UCL based upon percentile bootstrap, UCL based upon BCA bootstrap, UCL based upon bootstrap-t, and UCL based upon Hall's bootstrap method. The details of UCL computation methods for uncensored data sets are summarized in Chapter 2 of the ProUCL Technical Guide. 13 ------- Computations of UPLs. UTLs. and USLs Based upon Uncensored Data Sets without NDs: For uncensored data sets without NDs, ProUCL can compute parametric percentiles, UPLs for k (k>l) future observations, UPLs for mean of k (>1) future observations, UTLs, and USLs based upon the normal, gamma, and lognormal distributions. Nonparametric upper limits are typically based upon order statistics of a data set. Depending upon the size of the data set, the higher order statistics (maximum, second largest, third largest, and so on) are used to compute these upper limits (e.g., UTLs). Depending upon the sample size, specified CC and coverage probability, ProUCL 5.1 outputs the actual CC achieved by a nonparametric UTL. The details of the parametric and nonparametric computation methods for UPLs, UTLs, and USLs are described in Chapter 3 of the ProUCL Technical Guide. Computation of UCLs. UPLs. UTLs. and USLs Based upon Left-Censored Data Sets with NDs: For data sets with NDs, ProUCL computes UCLs, UPLs, UTLs, and USLs based upon the mean and sd computed using lognormal ROS (LROS, robust ROS), Gamma ROS (GROS), KM, and DL/2 substitution methods. To adjust for skewness in non-normally distributed data sets, ProUCL uses bootstrap methods and Chebyshev inequality when computing UCLs and other limits using estimates of the mean and sd obtained using the methods (details in Chapters 4 and 5) listed above. ProUCL 5.1 (new in ProUCL 5.0) uses parametric methods on KM (and ROS) estimates, provided detected observations in the left-censored data set follow a parametric distribution. For example, if the detected data follow a gamma distribution, ProUCL uses KM estimates in gamma distribution-based equations when computing UCLs, UTLs, and other upper limits. When detected data do not follow a discernible distribution, depending upon size and skewness of detected data, ProUCL recommends the use of Kaplan-Meier (1958) estimates in bootstrap methods and the Chebyshev inequality for computing nonparametric decision statistics (e.g., UCL95, UPL, UTL) of interest. ProUCL computes KM estimates directly using left-censored data sets without flipping data and requiring re-flipping of decision statistics. The KM method incorporated in ProUCL computes both sd and standard error (SE) of the mean. As mentioned earlier, for historical reasons and for comparison and research purposes, the DL/2 substitution method and H-UCL based upon LROS method have been retained in ProUCL 5.0/ProUCL 5.1. The inclusion of the substitution and LROS methods in ProUCL should not be inferred as an endorsement of those methods by ProUCL software and its developers. The details of the UCL computation methods for data sets with NDs are given in Chapter 4 and the detail description of the various other upper limits: UPLs, UTLs, and USLs for data sets with NDs are given in Chapter 5 of the ProUCL Technical Guide. Oneway ANOVA. OLS Regression and Trend Analysis: The Oneway ANOVA module has both classical and nonparametric K-W ANOVA tests as described in EPA guidance documents (e.g., EPA [2006b, 2009]). Oneway ANOVA is used to compare means (or medians) of multiple groups such as comparing mean concentrations of several areas of concern or performing inter-well comparisons of COPC concentrations at several MWs. The OLS Regression option computes the classical OLS regression line and generates graphs displaying the OLS line, confidence bands and prediction bands around the regression line. All statistics of interest including slope, intercept, and correlation coefficient are displayed on the OLS line graph. The Trend Analysis module has two nonparametric trend tests: the M-K trend test and T-S trend test. Using this option, one can generate trend graphs and time-series graphs displaying a T-S trend line and all other statistics of interest with associated />-values. In addition to slope and intercept, the T-S test in ProUCL 5.1 computes and outputs residuals based upon the computed nonparametric T-S line. In GW monitoring applications, OLS regression, trend tests, and time series plots are often used to identify trends (e.g., upwards, downwards) in constituent concentrations of GW monitoring wells over a 14 ------- certain period of time (U.S. EPA 2009). The details of Oneway ANOVA are given in Chapter 9 and OLS regression line and Trend tests methods are described in Chapter 10 of the ProUCL Technical Guide. BISS Module: At many sites, a large amount of discrete onsite and background data are already available which are not directly comparable to actual field ISM data. In order to provide a tool to compare the existing discrete data with ISM data, the BISS module of ProUCL 5.0 may be used on a large existing discrete data set. The ISM methodology used to develop the BISS module is a relatively new approach; methods incorporated in this BISS module require further investigation. For now, the BISS module has been blocked for use in ProUCL 5.0/ProUCL 5.1 as this module is awaiting adequate guidance for its intended use on discrete background data sets. Note: It is pointed out that in this document, all statements made about the capabilities of ProUCL 5.0 also apply to ProUCL version 5.1; and to save time, many screen shots used in ProUCL 5.0 manuals have been used in ProUCL 5.1 manuals (User Guide and Technical Guide). All upgrades in ProUCL 5.1 (not available in earlier versions) have been identified as new in ProUCL 5.1 in this document. Recommendations and Suggestions in ProUCL: Until 2006, not much guidance was available on how to compute a UCL95 of the mean and other upper limits (e.g., UPLs and UTLs) for skewed left-censored data sets containing NDs with multiple DLs, a common occurrence in environmental data sets. For uncensored positively skewed data sets, Singh, Singh, and laci (2002) summarize some simulation results comparing the performances (in terms of coverage probabilities) of several UCL computation methods described in the statistical and environmental literature. They noted that the optimal choice of a decision statistic (e.g., UCL95) depends upon the sample size, data distribution and data skewness. They incorporated the results of their findings in ProUCL 3.1 and higher versions to select the most appropriate UCL to estimate the EPC term. For data sets with NDs, Singh, Maichle, and Lee (2006) conducted a similar simulation study to compare the performances of the various estimation methods (in terms of bias in the mean estimate); and some UCL computation methods (in terms of coverage provided by a UCL). They demonstrated that the KM estimation method performs well in terms of bias in estimates of the mean; and for skewed data sets, the t- statistic, CLT, and the percentile bootstrap method based UCLs computed using KM estimates (and ROS estimates) underestimate the population mean. From these findings summarized in Singh, Singh, and laci (2002) and Singh, Maichle, and Lee (2006), it is natural to state and assume the findings of the simulation studies performed on uncensored skewed data sets comparing performances of the various UCL computation methods can be extended to skewed left-censored data sets. Like uncensored data sets without NDs, for data sets with NDs, there is no one single best UCL (and other upper limits such as UTL, UPL) which can be used to estimate an EPC (and background threshold values) for all data sets of varying sizes, distribution, and skewness. The optimal choice of a decision statistic depends upon the size, distribution, and skewness of detected observations. For data sets with and without NDs, ProUCL computes decision statistics including UCLs, UPLs, and UTLs using several parametric and nonparametric methods covering a wide-range of sample size, data variability and skewness. Using the results and findings summarized in the literature cited above, and based upon the sample size, data distribution, and data skewness, modules of ProUCL make suggestions about using the most appropriate decision statistic(s) to estimate population parameter(s) of interest (e.g., EPC). The suggestions made in ProUCL are based upon the extensive professional applied and theoretical experience of the developers in environmental statistical methods, published literature, results of simulation studies conducted by the developers of ProUCL and procedures described in many U.S. EPA 15 ------- guidance documents. These suggestions are made to help the users in selecting the most appropriate UCL to estimate an EPC which is routinely used in exposure assessment and risk management studies of the U.S. EPA. It should be pointed out that a typical simulation study cannot cover all data sets of various sizes and skewness from all types of distributions. For an analyte (data set) with skewness (sd of logged data) near the end points of the skewness intervals described in decision tables of Chapter 2 (e.g., Tables 2-9 through 2-11) of the ProUCL Technical Guide, the user/project team may select the most appropriate UCL based upon the site CSM, expert site knowledge, toxicity of the analyte, and exposure risks associated with that analyte. The project team should make the final decision regarding using or not using the suggestions/recommendations made by ProUCL. If deemed necessary, the project team may want to consult a statistician. Even though, ProUCL software has been developed using limited government funding, ProUCL 5.1 provides many statistical and graphical methods described in U.S. EPA documents for data sets with and without NDs. However, one may not compare the availability of methods in ProUCL 5.1 with methods available in the commercial software packages such as SAS® and Minitab 16. For example, trend tests correcting for seasonal/spatial variations and geostatistical methods are not available in the ProUCL software. For those methods, the user is referred to commercial software packages such as SAS®. As mentioned earlier, is the developers of ProUCL recommended supplementing test results (e.g., two- sample test) with graphical displays (e.g., Q-Q plots, side-by-side box plots) especially when data sets contain NDs and outliers. With the inclusion of the BISS, Oneway ANOVA, OLS Regression Trend and the user-friendly DQOs based Sample Size modules, ProUCL represents a comprehensive software package equipped with statistical methods and graphical tools needed to address many environmental sampling and statistical needs as described in the various CERCLA (U.S. EPA 1989a, 1992a, 2002a, 2002b, 2006a, 2006b), MARSSIM (U.S. EPA 2000), and RCRA (U.S. EPA 1989b, 1992b, 2002c, 2009) guidance documents. Finally, the users of ProUCL are cautioned about the use of methods and suggestions described in some recent environmental literature. For example, many decision statistics (e.g., UCLs, UPLs, UTLs,) computed using the methods (e.g., percentile bootstrap, statistics using KM estimates and t-critical values) described in Helsel (2005, 2012) will fail to provide the desired coverage for environmental parameters of interest (mean, upper percentile) of moderately skewed to highly skewed populations and conclusions derived based upon those decisions statistics may lead to incorrect conclusions which may not be cost-effective or protective of human health and the environment. Note: The look and feel of ProUCL 5.1 is similar to that of ProUCL 5.0; and they share the same names for the various modules and drop-down menus. For modules where no changes have been made in ProUCL since 2010 (e.g., Sample Sizes), screen shots as used in ProUCL 5.0 documents have been used in ProUCL 5.1 documents. Some of the screen shots generated using ProUCL 5.1 might have ProUCL 5.0 in their titles as those screen shots have not been re-generated and replaced. ProUCL 5.1 User Guide In addition to this Technical Guide, a User Guide also accompanies the ProUCL 5.1 software, providing details of using the statistical and graphical methods incorporated in ProUCL 5.1. The User Guide provides details about the input and output operations that can be performed using ProUCL 5.1. The User guide also provides details about saving edited input files, output Excel-type spreadsheets and graphical displays generated by ProUCL 5.1. 16 ------- CHAPTER 1 Guidance on the Use of Statistical Methods in ProUCL Software Decisions based upon statistics computed using discrete data sets of small sizes (e.g., < 6) cannot be considered reliable enough to make decisions that affect human health and the environment. For example, a background data set of size < 6 is not large enough to characterize a background population, compute BTV estimates, or to perform background versus site comparisons. Several U.S. EPA guidance documents (e.g., EPA 2000, 2006a, 2006b) detail DQOs and minimum sample size requirements needed to address statistical issues associated with different environmental applications. In order to obtain reliable statistical results, an adequate amount of data should be collected using project-specified DQOs (i.e., CC, decision error rates). The Sample Sizes module of ProUCL computes minimum sample sizes based on DQOs specified by the user and described in many guidance documents. In some cases, it may not be possible (e.g., due to resource constraints) to collect the calculated number of samples needed to meet the project-specific DQOs. Under these circumstances one can use the Sample Sizes module to assess the power of the test statistic resulting from the reduced number of samples which were collected. Based upon professional experience, the developers of ProUCL 4 software and its later versions have been making some rule-of-thumb suggestions regarding minimum sample size requirements needed to perform statistical evaluations such as: estimation of environmental parameters of interest (i.e., EPCs and BTVs), comparing site data with background data or with some pre-established screening levels (e.g., action levels [ALs], compliance limits [CLs]). Those rule-of thumb suggestions are described later in Section 1.7 of this chapter. It is noted that those minimum sample requirements have been adopted by some other guidance documents including the RCRA Guidance Document (EPA 2009). This chapter also describes the differences between the various statistical upper limits including upper confidence limits (UCLs) of the mean, upper prediction limits (UPLs) for future observations, and upper tolerance intervals (UTLs) often used to estimate the environmental parameters of interest including EPC terms and BTVs. The use of a statistical method depends upon the environmental parameter(s) being estimated or compared. The measures of central tendency (e.g., means, medians, or their UCLs) are used to compare site mean concentrations with a cleanup standard, Cs, also representing some central tendency measure of a reference area or some other known threshold representing a measure of central tendency. The upper threshold values, such as the CLs, alternative concentration limits (ACL), or not-to-exceed values, are used when individual point-by-point observations are compared with those threshold values. Depending upon whether the environmental parameters (e.g., BTVs, not-to-exceed value, or EPC term) are known or unknown, different statistical methods with different data requirements are needed to compare site concentrations with pre-established (known) or estimated (unknown) standards and BTVs. Several upper limits, and single and two sample hypotheses testing approaches, for both full-uncensored and left-censored data sets are available in the ProUCL software package for performing the comparisons described above. 1.1 Background Data Sets Based upon the CSM and regional and expert knowledge about the site, the project team selects background or reference areas. Depending upon the site activities and the pollutants, the background area can be site-specific or a general reference area with conditions comparable to the site before contamination due to site related activities. An appropriate random sample of independent observations 17 ------- (i.i.d) should be collected from the background area. A defensible background data set represents a "single" environmental population possibly without any outliers. In a background data set, in addition to reporting and/or laboratory errors, statistical outliers may also be present. A few elevated statistical outliers present in a background data set may actually represent potentially contaminated locations belonging to an impacted site area and/or possibly from other sources; those elevated outliers may not be coming from the background population under evaluation. Since the presence of outliers in a data set tends to yield distorted (poor and misleading) values of the decision making statistics (e.g., UCLs, UPLs and UTLs), elevated outliers should not be included in background data sets and estimation of BTVs. The objective here is to compute background statistics based upon a data set which represents the main background population, and does not accommodate the few low probability high outliers (e.g., coming from extreme tails of the data distribution) that may also be present in the sampled data. The occurrence of elevated outliers is common when background samples are collected from various onsite areas (e.g., large Federal Facilities). The proper disposition of outliers, to include or not include them in statistical computations, should be decided by the project team. The project team may want to compute decision statistics with and without the outliers to evaluate the influence of outliers on the decision making statistics. A couple of classical outlier tests (Dixon and Rosner tests) are available in ProUCL. Since both of these classical tests suffer from masking effects (e.g., some extreme outliers may mask the occurrence of other intermediate outliers), it is suggested that these classical outlier tests be supplemented with graphical displays such as a box plot and a Q-Q plot on a raw scale. The use of exploratory graphical displays helps in determining the number of outliers potentially present in a data set. The use of graphical displays also helps in identifying extreme high outliers as well as intermediate and mild outliers. The use of robust and masking-resistant outlier identification procedures (Singh and Nocerino, 1995, Rousseeuw and Leroy, 1987) is recommended when multiple outliers are present in a data set. Those methods are beyond the scope of ProUCL 5.1. However, several robust outlier identification methods are available in the Scout 2008 version 1.0 software package (EPA 2009d) available at An appropriate background data set of a reasonable size (preferably computed using the DQOs processes) is needed for the data set to be representative of background conditions and to compute upper limits (e.g., estimates of BTVs) and compare site and background data sets using hypotheses testing approaches. A background data set should have a minimum of 10 observations, however more observations is preferable. 1.2 Site Data Sets A data set collected from a site population (e.g., AOC, exposure area [EA], DU, group of MWs) should be representative of the population under investigation. Depending upon the areas under investigation, different soil depths and soil types may be considered as representing different statistical populations. In such cases, background versus site comparisons may have to be conducted separately for each of those sub-populations (e.g., surface and sub-surface layers of an AOC, clay and sandy site areas). These issues, such as comparing depths and soil types, should also be considered in the planning stages when developing sampling designs. Specifically, the availability of an adequate amount of representative data is required from each of those site sub-populations/strata defined by sample depths, soil types, and other characteristics. Site data collection requirements depend upon the objective(s) of the study. Specifically, in background versus site comparisons, site data are needed to perform: 18 ------- point-by-point onsite comparisons with pre-established ALs or estimated BTVs. Typically, this approach is used when only a small number (e.g., < 6) of onsite observations are compared with a BTV or some other not-to-exceed value. If many onsite values need to be compared with a BTV, the recommended upper limit to use is the UTL or upper simultaneous limit (USL) to control the false positive error rate (Type I Error Rate). More details can be found in Chapter 3 of this guidance document. Alternatively, one can use hypothesis testing approaches (Chapter 6) provided enough observations (at least 10, more are preferred) are available. single-sample hypotheses tests to compare site data with a pre-established cleanup standards, Cs (e.g., representing a measure of central tendency); proportion test to compare site proportion of exceedances of an AL with a pre-specified allowable proportion, Po. These hypotheses testing approaches are used on site data when enough site observations are available. Specifically, when at least 10 (more are desirable) site observations are available; it is preferable to use hypotheses testing approaches to compare site observations with specified threshold values. The use of hypotheses testing approaches can control both types of error rates (Type 1 and Type 2) more efficiently than the point-by-point individual observation comparisons. This is especially true as the number of point-by-point comparisons increases. This issue is illustrated by the following table summarizing the probabilities of exceedances (false positive error rate) of a BTV (e.g., 95th percentile) by onsite observations, even when the site and background populations have comparable distributions. The probabilities of these chance exceedances increase as the site sample size increases. Sample Size 1 2 5 8 10 12 64 Probability of Exceedance 0.05 0.10 0.23 0.34 0.40 0.46 0.96 • two-sample hypotheses tests to compare site data distribution with background data distribution to determine if the site concentrations are comparable to background concentrations. An adequate amount of data needs to be made available from the site as well as the background populations. It is preferable to collect at least 10 observations from each population under comparison. Notes: From a mathematical point of view, one can perform hypothesis tests on data sets consisting of only 3-4 data values; however, the reliability of the test statistics (and the conclusions derived) thus obtained is questionable. In these situations it is suggested to supplement the test statistics decisions with graphical displays. 1.3 Discrete Samples or Composite Samples? ProUCL can be used for discrete sample data sets, as well as on composite sample data sets. However, in a data set (background or site), samples should be either all discrete or all composite. In general, both discrete and composite site samples may be used for individual point-by-point site comparisons with a threshold value, and for single and two-sample hypotheses testing applications. 19 ------- • When using a single-sample hypothesis testing approach, site data can be obtained by collecting all discrete or all composite samples. The hypothesis testing approach is used when many (> 10) site observations are available. Details of the single-sample hypothesis approaches are widely available in EPA guidance documents (MARSSIM 2000, EPA 1989a, 2006b). Several single- sample hypotheses testing procedures available in ProUCL are described in Chapter 6 of this document. • If a two-sample hypothesis testing approach is used to perform site versus background comparisons, then samples from both of the populations should be either all discrete samples, or all composite samples. The two-sample hypothesis testing approaches are used when many (e.g., at least 10) site, as well as background, observations are available. For better results with higher statistical power, the availability of more observations perhaps based upon an appropriate DQOs process (EPA 2006a) is desirable. Several two-sample hypotheses tests available in ProUCL 5.1 are described in Chapter 6 of this document. 1.4 Upper Limits and Their Use The computation and use of statistical limits depend upon their applications and the parameters (e.g., EPC term, BTVs) they are supposed to be estimating. Depending upon the objective of the study, a pre- specified cleanup standard, Cs, can be viewed as representing: 1) an average (or median) constituent concentration, //o; or 2) a not-to-exceed upper threshold concentration value, Ao. These two threshold values, /no, and Ao, represent two significantly different parameters, and different statistical methods and limits are used to compare the site data with these two very different threshold values. Statistical limits, such as a UCL of the population mean, a UPL for an independently obtained "single" observation, or independently obtained "&" observations (also called future k observations, next k observations, or k different observations), upper percentiles, and UTLs are often used to estimate the environmental parameters: EPC (/no) and a BTV (Ao). A new upper limit, USL was included in ProUCL 5.0 which may be used to estimate a BTV based upon a well-established background data set representing a single statistical population without any outliers. It is important to understand and note the differences between the uses and numerical values of these statistical limits so that they can be properly used. The differences between UCLs and UPLs (or upper percentiles), and UCLs and UTLs should be clearly understood. A UCL with a 95% confidence limit (UCL95) of the mean represents an estimate of the population mean (measure of the central tendency), whereas a UPL95, a UTL95%-95% (UTL95-95), and an upper 95th percentile represent estimates of a threshold from the upper tail of the population distribution such as the 95th percentile. Here, UPL95 represents a 95% upper prediction limit, and UTL95-95 represents a 95% confidence limit of the 95th percentile. For mildly skewed to moderately skewed data sets, the numerical values of these limits tend to follow the order given as follows. Sample Mean < UCL95 of Mean < Upper 95th Percentile < UPL95 of a Single Observation < UTL95-95 Example 1-1. Consider a real data set collected from a Superfund site. The data set has several inorganic COPCs, including aluminum (Al), arsenic (As), chromium (Cr), iron (Fe), lead (Pb), manganese (Mn), thallium (Tl) and vanadium (V). Iron concentrations follow a normal distribution. This data set has been used in several examples throughout the two ProUCL guidance documents (Technical Guide and User Guide), therefore it is provided as follows. 20 ------- Aluminum 6280 3830 3900 5130 9310 15300 9730 7840 10400 16200 6350 10700 15400 12500 2850 9040 2700 1710 3430 6790 11600 4110 7230 4610 Arsenic 1.3 1.2 2 1.2 3.2 5.9 2.3 1.9 2.9 3.7 1.8 2.3 2.4 2.2 1.1 3.7 1.1 1 1.5 2.6 2.4 1.1 2.1 0.66 Chromium 8.7 8.1 11 5.1 12 20 12 11 13 20 9.8 14 17 15 8.4 14 4.5 3 4 11 16.4 7.6 35.5 6.1 Iron 4600 4330 13000 4300 11300 18700 10000 8900 12400 18200 7340 10900 14400 11800 4090 15300 6030 3060 4470 9230 Lead 16 6.4 4.9 8.3 18 14 12 8.7 11 12 14 14 19 21 16 25 20 11 6.3 13 98.5 53.3 109 8.3 Manganese 39 30 10 92 530 140 440 130 120 70 60 110 340 85 41 66 21 8.6 19 140 72.5 27.2 118 22.5 Thallium 0.0835 0.068 0.155 0.0665 0.071 0.427 0.352 0.228 0.068 0.456 0.067 0.0695 0.07 0.214 0.0665 0.4355 0.0675 0.066 0.067 0.068 0.13 0.068 0.095 0.07 Vanadium 12 8.4 11 9 22 32 19 17 21 32 15 21 28 25 8 24 11 7.2 8.1 16 Several upper limits for iron are summarized as follows, and it be seen that they follow the order (in magnitude) as described above. Table 1-1. Computation of Upper Limits for Iron (Normally Distributed) Mean 9618 Median 9615 Min 3060 Max 18700 UCL95 11478 UPL95 for a Single Observation 18145 UPL95 for 4 Observations 21618 UTL95-95 21149 95% Upper Percentile 17534 For highly skewed data sets, these limits may not follow the order described above. This is especially true when the upper limits are computed based upon a lognormal distribution (Singh, Singh, and Engelhardt 1997). It is well known that a lognormal distribution based H-UCL95 (Land's UCL95) often yields unstable and impractically large UCL values. An H-UCL95 often becomes larger than UPL95 and even larger than a UTL 95%-95% and the largest sample value. This is especially true when dealing with skewed data sets of smaller sizes. Moreover, it should also be noted that in some cases, a H-UCL95 21 ------- becomes smaller than the sample mean, especially when the data are mildly skewed and the sample size is large (e.g., > 50, 100). There is a great deal of confusion about the appropriate use of these upper limits. A brief discussion about the differences between the applications and uses of the statistical limits described above is provided as follows. • A UCL represents an average value that is compared with a threshold value also representing an average value (pre-established or estimated), such as a mean Cs. For example, a site 95% UCL exceeding a Cs, may lead to the conclusion that the cleanup standard, Cs has not been attained by the average site area concentration. It should also be noted that UCLs of means are typically computed from the site data set. • A UCL represents a "collective" measure of central tendency, and it is not appropriate to compare individual site observations with a UCL. Depending upon data availability, single or two-sample hypotheses testing approaches are used to compare a site average or a site median with a specified or pre-established cleanup standard (single-sample hypothesis), or with the background population average or median (two-sample hypothesis). • A UPL, an upper percentile, or a UTL represents an upper limit to be used for point-by-point individual site observation comparisons. UPLs and UTLs are computed based upon background data sets, and point-by-point onsite observations are compared with those limits. A site observation exceeding a background UTL may lead to the conclusion that the constituent is present at the site at levels greater than the background concentrations level. • When enough (e.g., at least 10) site observations are available, it is preferable to use hypotheses testing approaches. Specifically, single-sample hypotheses testing (comparing site to a specified threshold) approaches should be used to perform site versus a known threshold comparison; and two- sample hypotheses testing (provided enough background data are also available) approaches should be used to perform site versus background comparison. Several parametric and nonparametric single and two-sample hypotheses testing approaches are available in ProUCL 5.0/ProUCL 5.1. It is re-emphasized that only averages should be compared with averages or UCLs, and individual site observations should be compared with UPLs, upper percentiles, UTLs, or USLs. For example, the comparison of a 95% UCL of one population (e.g., site) with a 90% or 95% upper percentile of another population (e.g., background) cannot be considered fair and reasonable as these limits (e.g., UCL and UPL) estimate and represent different parameters. 1.5 Point-by-Point Comparison of Site Observations with BTVs, Compliance Limits and Other Threshold Values The point-by-point observation comparison method is used when a small number (e.g., < 6) of site observations are compared with pre-established or estimated BTVs, screening levels, or preliminary remediation goals (PRGs). Typically, a single exceedance of the BTV by an onsite (or a monitoring well) observation may be considered an indication of the presence of contamination at the site area under investigation. The conclusion of an exceedance by a site value is sometimes confirmed by re-sampling (taking a few more collocated samples) at the site location (or a monitoring well) exhibiting constituent concentrations in excess of the BTV. If all collocated sample observations (or all sample observations collected during the same time period) from the same site location (or well) exceed the BTV or PRO, then 22 ------- it may be concluded that the location (well) requires further investigation (e.g., continuing treatment and monitoring) and possibly cleanup. When BTV constituent concentrations are not known or pre-established, one has to collect or extract a background data set of an appropriate size that can be considered representative of the site background. Statistical upper limits are computed using the background data set thus obtained, which are used as estimates of BTVs. To compute reasonably reliable estimates of BTVs, a minimum of 10 background observations should be collected, perhaps using an appropriate DQOs process as described in EPA (2000, 2006a). Several statistical limits listed above are used to estimate BTVs based upon a defensible (free of outliers, representing the background population) background data set of an adequate size. The point-by-point comparison method is also useful when quick turnaround comparisons are required in real time. Specifically, when decisions have to be made in real time by a sampling/screening crew, or when only a few site samples are available, then individual point-by-point site concentrations are compared either with pre-established cleanup goals or with estimated BTVs. The sampling crew can use these comparisons to: 1) screen and identify the COPCs, 2) identify the potentially polluted site AOCs, or 3) continue or stop remediation or excavation at an onsite area of concern. If a larger number of samples (e.g., >10) are available from the AOC, then the use of hypotheses testing approaches (both single-sample and a two-sample) is preferred. The use of hypothesis testing approaches tends to control the error rates more tightly and efficiently than the individual point-by-point site comparisons. 1.6 Hypothesis Testing Approaches and Their Use Both single-sample and two-sample hypotheses testing approaches are used to make cleanup decisions at polluted sites, and also to compare constituent concentrations of two (e.g., site versus background) or more populations (e.g., MWs). 1.6.1 Single Sample Hypotheses (Pre-established BTVs and Not-to-Exceed Values are Known) When pre-established BTVs are used such as the U.S. Geological Survey (USGS) background values (Shacklette and Boerngen 1984), or thresholds obtained from similar sites, there is no need to extract, establish, or collect a background data set. When the BTVs and cleanup standards are known, one-sample hypotheses are used to compare site data (provided enough site data are available) with known and pre- established threshold values. It is suggested that the project team determine (e.g., using DQOs) or decide (depending upon resources) the number of site observations that should be collected and compared with the "pre-established" standards before coming to a conclusion about the status (clean or polluted) of the site AOCs. As mentioned earlier, when the number of available site samples is < 6, one might perform point-by-point site observation comparisons with a BTV; and when enough site observations (at least 10) are available, it is desirable to use single-sample hypothesis testing approaches. Depending upon the parameter (fio, Ao), represented by the known threshold value, one can use single-sample hypotheses tests for population mean or median (t-test, sign test), or use single-sample tests for proportions and percentiles. The details of the single-sample hypotheses testing approaches can be found in EPA (2006b) guidance document and in Chapter 6 of this document. One-Sample t-Test: This test is used to compare the site mean, fi, with some specified cleanup standard, Cs, where the Cs represents an average threshold value, /no. The Student's t-test (or a UCL of the mean) is 23 ------- used (assuming normality of site data set or when sample size is large, such as larger than 30, 50) to verify the attainment of cleanup levels at a polluted site after some remediation activities. One-Sample Sign Test or Wilcoxon Signed Rank (WSR) Test: These tests are nonparametric tests and can also handle ND observations, provided the detection limits of all NDs fall below the specified threshold value, Cs. These tests are used to compare the site location (e.g., median, mean) with some specified Cs representing a similar location measure. One-Sample Proportion Test or Percentile Test: When a specified cleanup standard, Ao, such as a PRG or a BTV represents an upper threshold value of a constituent concentration distribution rather than the mean threshold value, juo, then a test for proportion or a test for percentile (equivalently UTL 95-95 UTL 95-90) may be used to compare site proportion (or site percentile) with the specified threshold or action level, Ao. 1.6.2 Two-Sample Hypotheses (BTVs and Not-to-Exceed Values are Unknown) When BTVs, not-to-exceed values, and other cleanup standards are not available, then site data are compared directly with the background data. In such cases, two-sample hypothesis testing approaches are used to perform site versus background comparisons. Note that this approach can be used to compare concentrations of any two populations including two different site areas or two different monitoring wells (MWs). In order to use and perform a two-sample hypothesis testing approach, enough data should be available from each of the two populations. Site and background data requirements (e.g., based upon DQOs) for performing two-sample hypothesis test approaches are described in EPA (2000, 2002b, 2006a, 2006b) and also in Chapter 6 of this Technical Guide. While collecting site and background data, for better representation of populations under investigation, one may also want to account for the size of the background area (and site area for site samples) in sample size determination. That is, a larger number (>15-20) of representative background (and site) samples should be collected from larger background (and site) areas; every effort should be made to collect as many samples as determined by the DQOs- based sample sizes. The two-sample (or more) hypotheses approaches are used when the site parameters (e.g., mean, shape, distribution) are being compared with the background parameters (e.g., mean, shape, distribution). The two-sample hypotheses testing approach is also used when the cleanup standards or screening levels are not known a priori. Specifically, in environmental applications, two-sample hypotheses testing approaches are used to compare average or median constituent concentrations of two or more populations. To derive reliable conclusions with higher statistical power based upon hypothesis testing approaches, an adequate amount of data (e.g., minimum of 10 samples) should be collected from all of the populations under investigation. The two-sample hypotheses testing approaches incorporated in ProUCL 5.1 are listed as follows: • Student t-test (with equal and unequal variances) - Parametric test assumes normality • Wilcoxon-Mann-Whitney (WMW) test - Nonparametric test handles data with NDs with one DL - assumes two populations have comparable shapes and variability • Gehan test - Nonparametric test handles data sets with NDs and multiple DLs - assumes comparable shapes and variability • Tarone-Ware (T-W) test - Nonparametric test handles data sets with NDs and multiple DLs - assumes comparable shapes and variability 24 ------- The Gehan and T-W tests are meant to be used on left-censored data sets with multiple DLs. For best results, the samples collected from the two (or more) populations should all be of the same type obtained using similar analytical methods and apparatus; the collected site and background samples should all be discrete or all composite (obtained using the same design and pattern), and be collected from the same medium (soil) at similar depths (e.g., all surface samples or all subsurface samples) and time (e.g., during the same quarter in groundwater applications) using comparable (preferably same) analytical methods. Good sample collection methods and sampling strategies are given in EPA (1996, 2003) guidance documents. Note: ProUCL 5.1 (and previous versions) has been developed using limited government funding. ProUCL 5.1 is equipped with statistical and graphical methods needed to address many environmental sampling and statistical issues as described in the various CERCLA, MARSSIM, and RCRA documents cited earlier. However, one may not compare the availability of methods in ProUCL 5.1 with methods incorporated in commercial software packages such as SAS® and Minitab 16. Not all methods available in the statistical literature are available in ProUCL. 1.7 Minimum Sample Size Requirements and Power Evaluations Due to resource limitations, it is not be possible (nor needed) to sample the entire population (e.g., background area, site area, AOCs, EAs) under study. Statistics is used to draw inference(s) about the populations (clean, dirty) and their known or unknown statistical parameters (e.g., mean, variance, upper threshold values) based upon much smaller data sets (samples) collected from those populations. To determine and establish BTVs and site specific screening levels, defensible data set(s) of appropriate size(s) representing the background population (e.g., site-specific, general reference area, or historical data) need to be collected. The project team and site experts should decide what represents a site population and what represents a background population. The project team should determine the population area and boundaries based upon all current and intended future uses, and the objectives of data collection. Using the collected site and background data sets, statistical methods supplemented with graphical displays are used to perform site versus background comparisons. The test results and statistics obtained by performing such site versus background comparisons are used to determine if the site and background level constituent concentrations are comparable; or if the site concentrations exceed the background threshold concentration level; or if an adequate amount of remediation approaching the BTV or some cleanup level has been performed at polluted site AOCs. To perform statistical tests and compute upper limits, determine the number of samples that need to be collected from the populations (e.g., site and background) under investigation using appropriate DQOs processes (EPA 2000, 2006a, 2006b). ProUCL has the Sample Sizes module which can be used to develop DQOs based sampling designs needed to address statistical issues associated with polluted sites projects. ProUCL provides user-friendly options to enter the desired/pre-specified values of decision parameters (e.g., Type I and Type II error rates) to determine minimum sample sizes for the selected statistical applications including: estimation of mean, single and two-sample hypothesis testing approaches, and acceptance sampling. Sample size determination methods are available for the sampling of continuous characteristics (e.g., lead or Radium 226), as well as for attributes (e.g., proportion of occurrences exceeding a specified threshold). Both parametric (e.g., t-tests) and nonparametric (e.g., Sign test, test for proportions, WRS test) sample size determination methods are available in ProUCL 5.1 and in its earlier versions (e.g., ProUCL 4.1). ProUCL also has sample size determination methods for acceptance sampling of lots of discrete objects such as a batch of drums containing hazardous waste (e.g., RCRA applications, U.S. EPA 2002c). 25 ------- However, due to budgetary or logistical constraints, it may not be possible to collect the same number of samples as determined by applying a DQO process. For example, the data might have already been collected (as often is the case) without using a DQO process, or due to resource constraints, it may not have been possible to collect as many samples as determined by using a DQO-based sample size formula. In practice, the project team and the decision makers tend not to collect enough background samples. It is suggested to collect at least 10 background observations before using statistical methods to perform background evaluations based upon data collected using discrete samples. The minimum sample size recommendations described here are useful when resources are limited, and it may not be possible to collect as many background and site samples as computed using DQOs based sample size determination formulae. In case data are collected without using a DQO process, the Sample Sizes module can be used to assess the power of the test statistic in retrospect. Specifically, one can use the standard deviation of the computed test statistic (EPA 2006b) and compute the sample size needed to meet the desired DQOs. If the computed sample size is greater than the size of the data set used, the project team may want to collect additional samples to meet the desired DQOs. Note: From a mathematical point of view, the statistical methods incorporated in ProUCL and described in this guidance document for estimating EPC terms and BTVs, and comparing site versus background concentrations can be performed on small site and background data sets (e.g., of sizes as small as 3). However, those statistics may not be considered representative and reliable enough to make important cleanup and remediation decisions which will potentially impact human health and the environment. ProUCL provides messages when the number of detects is <4-5, and suggests collecting at least 8-10 observations. Based upon professional judgment, as a rule-of-thumb, ProUCL guidance documents recommend collecting a minimum of 10 observations when data sets of a size determined by a DQOs process (EPA 2006) cannot be collected. This however, should not be interpreted as the general recommendation and every effort should be made to collect DQOs based number of samples. Some recent guidance documents (e.g., EPA 2009) have also adopted this rule-of-thumb and suggest collecting a minimum of about 8-10 samples in the circumstance that data cannot be collected using a DQO-based process. However, the project team needs to make these determinations based upon their comfort level and knowledge of site conditions. • To allow users to compute decision statistics using data from ISM (ITRC, 2012) samples, ProUCL 5.1 will compute decision statistics (e.g., UCLs, UPLs, UTLs) based upon samples of sizes as small as 3. The user is referred to the ITRC ISM Technical Regulatory Guide (2012) to determine which UCL (e.g., Student's t-UCL or Chebyshev UCL) should be used to estimate the EPC term. 1.7.1 Why a data set of minimum size, n = 8 through 10? Typically, the computation of parametric upper limits (UPL, UTL, UCL) depends upon three values: the sample mean, sample variability (standard deviation) and a critical value. A critical value depends upon sample size, data distribution, and confidence level. For samples of small size (< 8-10), the critical values are large and unstable, and upper limits (e.g., UTLs, UCLs) based upon a data set with fewer than 8-10 observations are mainly driven by those critical values. The differences in the corresponding critical values tend to stabilize when the sample size becomes larger than 8-10 (see tables below, where degrees of freedom \df\ = sample size - 1). This is one of the reasons ProUCL guidance documents suggest a minimum data set size of 10 when the number of observations determined from sample-size calculations based upon EPA DQO process exceed the logistical/financial/temporal/constraints of a project. For samples of sizes 2-11, 95% critical values used to compute upper limits (UCLs, UPLs, UTLs, and USLs) 26 ------- based upon a normal distribution are summarized in the subsequent tables. In general, a similar pattern is followed for critical values used in the computation of upper limits based upon other distributions. For the normal distribution, Student's t-critical values are used to compute UCLs and UPLs which are summarized as follows. Table of Critical Values of t-Statistic df= sample size-l= (n-1) Upper-tail probability p df 1 2 3 4 5 t, 7 & 9 10 .10 3.07 S .886 .633 .533 .476 .440 .415 .397 .383 .372 .05 6.314 2.920 2.353 2.132 2.015 .943 .895 .860 .833 .812 .025 12.71 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 .112 15.89 4.849 3.482 2.999 2.757 2.612 2.517 2.449 2.398 2.359 .(.11 31.S2 6.965 4.541 3.747 3.365 3.143 2.99S 2.896 1.821 2.764 One can see that once the sample size starts exceeding 9-10 (df= 8, 9), the difference between the critical values starts stabilizing. For example, for upper tail probability (= level of significance) of 0.05, the difference between critical values for df= 9 and df=\0 is only 0.021, where as the difference between critical values for df= 4 and 5 is 0.117; similar patterns are noted for other levels of significance. For the normal distribution, critical values used to compute UTL90-95, UTL95-95, USL90, and USL95 are described as follows. One can see that once the sample size starts exceeding 9-10, the difference between the critical values starts decreasing significantly. UTL90-95 UTL95-95 USL90 USL95 3 4 5 6 7 8 9 10 11 6.155 4.162 3.407 3.006 2.755 2.582 2.454 2.355 2.275 7.656 5.144 4.203 3.708 3.399 3.187 3.031 2.911 2.815 1.148 1.425 1.602 1.729 1.828 1.909 1.977 2.036 2.088 1.153 1.462 1.671 1.822 1.938 2.032 2.11 2.176 2.234 Note: Nonparametric upper limits (UPLs, UTLs, and USLs) are computed using higher order statistics of a data set. To achieve the desired confidence coefficient, samples of sizes much greater than 10 are required. For details, refer to Chapter 3. It should be noted that critical values of USLs are significantly lower than critical values for UTLs. Critical values associated with UTLs decrease as the sample size increases. Since, as the sample size increases the maximum of the data set also increases, and critical values associated with USLs increase with the sample size. 1.7.2 Sample Sizes for Bootstrap Methods Several nonparametric methods including bootstrap methods for computing UCL, UTL, and other limits for both full-uncensored data sets and left-censored data sets with NDs are available in ProUCL 5.1. Bootstrap resampling methods are useful when not too few (e.g., < 15-20) and not too many (e.g., > 500- 27 ------- 1000) observations are available. For bootstrap methods (e.g., percentile method, BCA bootstrap method, bootstrap-t method), a large number (e.g., 1000, 2000) of bootstrap resamples are drawn with replacement from the same data set. Therefore, to obtain bootstrap resamples with at least some distinct values (so that statistics can be computed from each resample), it is suggested that a bootstrap method should not be used when dealing with small data sets of sizes less than 15-20. Also, it is not necessary to bootstrap a large data set of size greater than 500 or 1000; that is when a data set of a large size (e.g., > 500) is available, there is no need to obtain bootstrap resamples to compute statistics of interest (e.g., UCLs). One can simply use a statistical method on the original large data set. Note: Rules-of-thumb about minimum sample size requirements described in this section are based upon professional experience of the developers. ProUCL software is not a policy software. It is recommended that the users/project teams/agencies make determinations about the minimum number of observations and minimum number of detects that should be present in a data set before using a statistical method. 1.8 Statistical Analyses by a Group ID The analyses of data categorized by a group ID variable such as: 1) Surface vs. Subsurface; 2) AOC1 vs. AOC2; 3) Site vs. Background; and 4) Upgradient vs. Downgradient monitoring wells are common in environmental applications. ProUCL 5.1 offers this option for data sets with and without NDs. The Group Option provides a tool for performing separate statistical tests and for generating separate graphical displays for each member/category of the group (samples from different populations) that may be present in a data set. The graphical displays (e.g., box plots, quantile-quantile plots) and statistics (e.g., background statistics, UCLs, hypotheses tests) of interest can be computed separately for each group by using this option. Moreover, using the Group Option, graphical methods can display multiple graphs (e.g., Q-Q plots) on the same graph providing graphical comparison of multiple groups. It should be pointed out that it is the user's responsibility to provide an adequate amount of data to perform the group operations. For example, if the user desires to produce a graphical Q-Q plot (e.g., using only detected data) with regression lines displayed, then there should be at least two detected data values (to compute slope, intercept, sd) in the data set. Similarly if the graphs are desired for each group specified by the group ID variable, there should be at least two observations in each group specified by the group variable. When ProUCL data requirements are not met, ProUCL does not perform any computations, and generates a warning message (colored orange) in the lower Log Panel of the output screen of ProUCL 5.1. 1.9 Statistical Analyses for Many Constituents/Variables ProUCL software can process multiple analytes/variables simultaneously in a user-friendly manner This option is useful when one has to process multiple variables and compute decision statistics (e.g., UCLs, UPLs, and UTLs) and test statistics (e.g., ANOVA test, trend test) for multiple variables. It is the user's responsibility to make sure that each selected variable has an adequate amount of data so that ProUCL can perform the selected statistical method correctly. ProUCL displays warning messages when a selected variable does not have enough data needed to perform the selected statistical method. 1.10 Use of Maximum Detected Value as Estimates of Upper Limits Some practitioners use the maximum detected value as an estimate of the EPC term. This is especially true when the sample size is small such as < 5, or when a UCL95 exceeds the maximum detected values 28 ------- (EPA 1992a). Also, many times in practice, the BTVs and not-to-exceed values are estimated by the maximum detected value (e.g., nonparametric UTLs, USLs). 1.10.1 Use of Maximum Detected Value to Estimate BTVs and Not-to-Exceed Values BTVs and not-to-exceed values represent upper threshold values from the upper tail of a data distribution; therefore, depending upon the data distribution and sample size, the BTVs and other not-to-exceed values may be estimated by the largest or the second largest detected value. A nonparametric UPL, UTL, and USL are often estimated by higher order statistics such as the maximum value or the second largest value (EPA 1992b, 2009, Hahn and Meeker 1991). The use of higher order statistics to estimate the UTLs depends upon the sample size. For data sets of size: 1) 59 to 92 observations, a nonparametric UTL95-95 is given by the maximum detected value; 2) 93 to 123 observations, a nonparametric UTL95-95 is given by the second largest maximum detected value; and 3) 124 to 152 observations, a UTL95-95 is given by the third largest detected value in the sample, and so on. 1.10.2 Use of Maximum Detected Value to Estimate EPC Terms Some practitioners tend to use the maximum detected value as an estimate of the EPC term. This is especially true when the sample size is small such as < 5, or when a UCL95 exceeds the maximum detected value. Specifically, the EPA (1992a) document suggests the use of the maximum detected value as a default value to estimate the EPC term when a 95% UCL (e.g., the H-UCL) exceeds the maximum value in a data set. ProUCL computes 95% UCLs of the mean using several methods based upon normal, gamma, lognormal, and non-discernible distributions. In the past, a lognormal distribution was used as the default distribution to model positively skewed environmental data sets. Additionally, only two methods were used to estimate the EPC term based upon: 1) normal distribution and Student's t-statistic, and 2) lognormal distribution and Land's H-statistic (Land 1971, 1975). The use of the H-statistic often yields unstable and impractically large UCL95 of the mean (Singh, Singh, and Engelhardt 1997; Singh, Singh, and laci 2002). For highly skewed data sets of smaller sizes (< 30, < 50), H-UCL often exceeds the maximum detected value. Since the use of a lognormal distribution has been quite common (suggested as a default model in the risk assessment guidance for Superfund [RAGS] document [EPA 1992a]), the exceedance of the maximum value by an H-UCL95 is frequent for many skewed data sets of smaller sizes (e.g., < 30, < 50). These occurrences result in the possibility of using the maximum detected value as an estimate of the EPC term. It should be pointed out that in some cases, the maximum observed value actually might represent an impacted location. Obviously, it is not desirable to use an observation potentially representing an impacted location to estimate the EPC for an AOC. The EPC term represents the average exposure contracted by an individual over an EA during a long period of time; the EPC term should be estimated by using an average value (such as an appropriate 95% UCL of the mean) and not by the maximum observed concentration. One needs to compute an average exposure and not the maximum exposure. As can be seen in figures described in Appendix B, for data sets of small sizes (e.g., < 10-20), the Max Test (U.S. EPA 1996)does not provide the specified 95% coverage to the population mean, and for larger data sets it overestimates the EPC term, which may lead to unnecessary further remediation. Several methods, some of which are described in EPA (2002a) and other EPA documents, are available in versions of ProUCL (i.e., ProUCL 3.00.02 [EPA 2004], ProUCL 4.0 [U.S. EPA 2007], ProUCL 4.00.05 [EPA 2009, 2010], ProUCL 4.1 [EPA 2011]) for estimating the EPC terms. For data sets with NDs, ProUCL 5.0 (and ProUCL 5.1) has some new UCL (and other limits) computation methods which were 29 ------- not available in earlier versions of ProUCL. It is unlikely that the UCLs based upon those methods will exceed the maximum detected value, unless some outliers are present in the data set. 1.10.2.1 Chebyshev Inequality Based UCL95 ProUCL 5.1 (and its earlier versions) displays a warning message when the suggested 95% UCL (e.g., Hall's or bootstrap-t UCL with outliers) of the mean exceeds the detected maximum concentration. When a 95% UCL does exceed the maximum observed value, ProUCL suggests the use of an alternative UCL computation method based upon the Chebyshev inequality. One may use a 97.5% or 99% Chebyshev UCL to estimate the mean of a highly skewed population. The use of the Chebyshev inequality to compute UCLs tends to yield more conservative (but stable) UCLs than other methods available in ProUCL software. In such cases, when the sample size is large (and other UCL methods such as the bootstrap-t method yield unrealistically high values due to presence of outliers), one may want to use a 95% Chebyshev UCL or a Chebyshev UCL with a lower confidence coefficient such as 90% as an estimate of the population mean, especially when the sample size is large (e.g., >100, 150). The details (as functions of sample size and skewness) for the use of those UCLs are summarized in various versions of ProUCL Technical Guides (EPA 2004, 2007, 2009, 2010d, 2011, 2013a). Notes: Using the maximum observed value to estimate the EPC term representing the average exposure contracted by an individual over an EA is not recommended. For the sake of interested users, ProUCL displays a warning message when the recommended 95% UCL (e.g., Hall's bootstrap UCL) of the mean exceeds the observed maximum concentration. For such scenarios (when a 95% UCL does exceed the maximum observed value), an alternative UCL computation method based upon Chebyshev inequality is suggested by the ProUCL software. 1.11 Samples with Nondetect Observations ND observations are inevitable in most environmental data sets. Singh, Maichle, and Lee (2006) studied the performances (in terms of coverages) of the various UCL95 computation methods including the simple substitution methods (such as the DL/2 and DL methods) for data sets with ND observations. They concluded that the UCLs obtained using the substitution methods, including the replacement of NDs by DL/2; do not perform well even when the percentage of ND observations is low, such as less than 5% to 10%. They recommended avoiding the use of substitution methods for computing UCL95 based upon data sets with ND observations. 7.77.7 Avoid the Use of the DL/2 Substitution Method to Compute UCL95 Based upon the results of the report by Singh, Maichle, and Lee (2006), it is recommended to avoid the use of the DL/2 substitution method when performing a GOF test, and when computing the summary statistics and various other limits (e.g., UCL, UPL, UTLs) often used to estimate the EPC terms and BTVs. Until recently, the substitution method has been the most commonly used method for computing various statistics of interest for data sets which include NDs. The main reason for this has been the lack of the availability of the other rigorous methods and associated software programs that can be used to estimate the various environmental parameters of interest. Today, several methods (e.g., using KM estimates) with better performance, including the Chebyshev inequality and bootstrap methods, are available for computing the upper limits of interest. Several of those parametric and nonparametric methods are available in ProUCL 4.0 and higher versions. The DL/2 method is included in ProUCL for historical reasons as it had been the most commonly used and recommended method until recently (EPA 30 ------- 2006b). EPA scientists and several reviewers of the ProUCL software had suggested and requested the inclusion of the DL/2 substitution method in ProUCL for comparison and research purposes. Notes: Even though the DL/2 substitution method has been incorporated in ProUCL, its use is not recommended due to its poor performance. The DL/2 substitution method has been retained in ProUCL 5.1 for historical and comparison purposes. NERL-EPA, Las Vegas strongly recommends avoiding the use of this method even when the percentage of NDs is as low as 5% to 10%. 1.11.2 ProUCL Does Not Distinguish between Detection Limits, Reporting limits, or Method Detection Limits ProUCL 5.1 (and all previous versions) does not make distinctions between method detection limits (MDLs), adjusted MDLs, sample quantitation limits (SQLs), reporting limits (RLs), or DLs. Multiple DLs (or RLs) in ProUCL mean different values of the detection limits. It is user's responsibility to understand the differences between these limits and use appropriate values (e.g., DLs) for nondetect values below which the laboratory cannot reliably detect/measure the presence of the analyte in collected samples (e.g., soil samples). A data set consisting of values less than the DLs (or MDLs, RLs) is considered a left-censored data set. ProUCL uses statistical methods available in the statistical literature for left-censored data sets for computing statistics of interest including mean, sd, UCL, and estimates of BTVs. The user determines which qualifiers (e.g., J, U, UJ) will be considered as nondetects. Typically, all values with U or UJ qualifiers are considered as nondetect values. It is the user's responsibility to enter a value which can be used to represent a ND value. For NDs, the user enters the associated DLs or RLs (and not zeros or half of the detection limits). An indicator column/variable, D_x taking a value, 0, for all nondetects and a value, 1, for all detects is assigned to each variable, x, with NDs. It is the user's responsibility to supply the numerical values for NDs (should be entered as reported DLs) not qualifiers (e.g., J, U, B, UJ). For example, for thallium with nondetect values, the user creates an associated column labeled as D_thallium to tell the software that the data set will have nondetect values. This column, D_thallium consists of only zeros (0) and ones (1); zeros are used for all values reported as NDs and ones are used for all values reported as detects. 1.12 Samples with Low Frequency of Detection When all of the sampled values are reported as NDs, the EPC term and other statistical limits should also be reported as a ND value, perhaps by the maximum RL or the maximum RL/2. The project team will need to make this determination. Statistics (e.g., UCL95) based upon only a few detected values (e.g., < 4) cannot be considered reliable enough to estimate EPCs which can have a potential impact on human health and the environment. When the number of detected values is small, it is preferable to use ad hoc methods rather than using statistical methods to compute EPCs and other upper limits. Specifically, for data sets consisting of < 4 detects and for small data sets (e.g., size < 10) with low detection frequency (e.g., < 10%), the project team and the decision makers should decide, on a site-specific basis, how to estimate the average exposure (EPC) for the constituent and area under consideration. For data sets with low detection frequencies, other measures such as the median or mode represent better estimates (with lesser uncertainty) of the population measure of central tendency. Additionally, when most (e.g., > 95%) of the observations for a constituent lie below the DLs, the sample median or the sample mode (rather than the sample average) may be used as an estimate of the EPC. Note that when the majority of the data are NDs, the median and the mode may also be represented by a ND ------- value. The uncertainty associated with such estimates will be high. The statistical properties, such as the bias, accuracy, and precision of such estimates, would remain unknown. In order to be able to compute defensible estimates, it is always desirable to collect more samples. 1.13 Some Other Applications of Methods in ProUCL 5.1 In addition to performing background versus site comparisons for CERCLA and RCRA sites, performing trend evaluations based upon time-series data sets, and estimating EPCs in exposure and risk evaluation studies, the statistical methods in ProUCL can be used to address other issues dealing with environmental investigations that are conducted at Superfund or RCRA sites. 1.13.1 Identification of COPCs Risk assessors and remedial project managers (RPMs) often use screening levels or BTVs to identify COPCs during the screening phase of a cleanup project at a contaminated site. The screening for COPCs is performed prior to any characterization and remediation activities that are conducted at the site. This comparison is performed to screen out those constituents that may be present in the site medium of interest at low levels (e.g., at or below the background levels or some pre-established screening levels) and may not pose any threat and concern to human health and the environment. Those constituents may be eliminated from all future site investigations, and risk assessment and risk management studies. To identify the COPCs, point-by-point site observations are compared with some pre-established soil screening levels (SSL) or estimated BTVs. This is especially true when the comparisons of site concentrations with screening levels or BTVs are conducted in real time by the sampling or cleanup crew onsite. The project team should decide the type of site samples (discrete or composite) and the number of site observations that should be collected and compared with the screening levels or the BTVs. In case BTVs or screening levels are not known, the availability of a defensible site-specific background or reference data set of reasonable size (e.g., at least 10) is required for computing reliable and representative estimates of BTVs and screening levels. The constituents with concentrations exceeding the respective screening values or BTVs may be considered COPCs, whereas constituents with concentrations (e.g., in all collected samples) lower than the screening values or BTVs may be omitted from all future evaluations. 1.13.2 Identification of Non-Compliance Monitoring Wells In MW compliance assessment applications, individual (often discrete) constituent concentrations from a MW are compared with some pre-established limits such as an ACL or a maximum concentration limit (MCL). An exceedance of the MCL or the BTV (e.g., estimated by a UTL95-95 or a UPL95) by a MW concentration may be considered an indication of contamination in that MW. For individual concentration comparisons, the presence of contamination (determined by an exceedance) may have to be confirmed by re-sampling from that MW. If concentrations of constituents in the original sample and re-sample(s) exceed the MCL or BTV, then that MW may require further scrutiny, perhaps triggering remediation activates. If the concentration data from a MW for 4 to 5 continuous quarters (or some other designated time period determined by the project team) are below the MCL or BTV level, then that MW may be considered as complying with (achieving) the pre-established or estimated standards. 32 ------- 1.13.3 Verification of the Attainment of Cleanup Standards, Cs Hypothesis testing approaches are used to verify the attainment of the cleanup standard, Cs, at site AOCs after conducting remediation and cleanup at those site AOCs (EPA 1989a, 1994). In order to assess the attainment of cleanup levels, a representative data set of adequate size perhaps obtained using the DQO process (or a minimum of 10 observations should be collected) needs to be made available from the remediated/excavated areas of the site under investigation. The sample size should also account for the size of the remediated site areas: meaning that larger site areas should be sampled more (with more observations) to obtain a representative sample of the remediated areas under investigation. Typically, the null hypothesis of interest is H0: Site Mean, /& > Cs versus the alternative hypothesis, Hi: Site Mean, /4 < Cs, where the cleanup standard, Cs, is known a priori. 1.13.4 Using BTVs (Upper Limits) to Identify Hot Spots The use of upper limits (e.g., UTLs) to identify hot spot(s) has also been mentioned in the Guidance for Comparing Background and Chemical Concentrations in Soil for CERCLA Sites (EPA 2002b). Point-by- point site observations are compared with a pre-established or estimated BTV. Exceedances of the BTV by site observations may represent impacted locations with elevated concentrations (hot spots). 1.14 Some General Issues, Suggestions and Recommendations made by ProUCL Some general issues regarding the handling of multiple DLs by ProUCL and recommendations made about various substitution and ROS methods for data sets with NDs are described in the following sections. 1.14.1 Handling of Field Duplicates ProUCL does not pre-process field duplicates. The project team determines how field duplicates will be handled and pre-processes the data accordingly. For an example, if the project team decides to use average values for field duplicates, then averages need to be computed and field duplicates need to be replaced by their respective average values. It is the user's responsibility to feed in appropriate values (e.g., averages, maximum) for field duplicates. The user is advised to refer to the appropriate EPA guidance documents related to collection and use of field duplicates for more information. 1.14.2 ProUCL Recommendation about ROS Method and Substitution (DL/2) Method For data sets with NDs, ProUCL can compute point estimates of population mean and standard deviation using the KM and ROS methods (and also using the DL/2 substitution method). The substitution method has been retained in ProUCL for historical and research purposes. ProUCL uses Chebyshev inequality, bootstrap methods, and normal, gamma, and lognormal distribution based equations on KM (or ROS) estimates to compute upper limits (e.g., UCLs, UTLs). The simulation study conducted by Singh, Maichle and Lee (2006) demonstrated that the KM method yields accurate estimates of the population mean. They also demonstrated that for moderately skewed to highly skewed data sets, UCLs based upon KM estimates and BCA bootstrap (mild skewness), KM estimates and Chebyshev inequality (moderate to high skewness), and KM estimates and bootstrap-t method (moderate to high skewness) yield better (in terms of coverage probability) estimates of EPCs than other UCL methods based upon the Student's t- statistic on KM estimates, percentile bootstrap method on KM or ROS estimates. 33 ------- 1.14.3 UnhandledExceptions and Crashes in ProUCL A typical statistical software, especially developed under limited resources may not be able to accommodate data sets with all kinds of deficiencies such as all missing values for a variable, or all nondetect values for a variable. An inappropriate/insufficient data set can occur in various forms and not all of them can be addressed in a scientific program like ProUCL. Specifically, from a programming point of view, it can be quite burdensome on the programmer to address all potential deficiencies that can occur in a data set. ProUCL 5.1 addresses many data deficiencies and produces warming messages. All data deficiencies causing unhandled exceptions which were identified by users have been addressed in ProUCL 5.1. However, when ProUCL yields an unhandled exception or crashes, it is highly likely that there is something wrong with the data set; the user is advised to review the input data set to make sure that the data set follows ProUCL data and formatting requirements. 1.15 The Unofficial User Guide to ProUCL4 (Helsel and Gilroy 2012) Several ProUCL 4.1 users sent inquiries about the validity of the comments made about the ProUCL software in the Unofficial User Guide to ProUCL4 (Helsel and Gilroy, 2012) and in the Practical Stats webinar, "ProUCL v4: The Unofficial User Guide," presented by Dr. Helsel on October 15, 2012 (Helsel 2012a). Their inquiries led us to review comments made about the ProUCL4 software and its associated guidance documents (EPA 2007, 2009a, 2009b, 2010c, 2010d, and 2011) in the "The Unofficial Users Guide to ProUCL4" and in the webinar, "ProUCL v4: The Unofficial User Guide". These two documents collectively are referred to as the Unofficial ProUCLv4 User Guide in this ProUCL document. The pdf document describing the material presented in the Practical Stats Webinar (Helsel 2012a) was downloaded from the http://www.practicalstats.com website. In the "ProUCL v4: The Unofficial User Guide", comments have been made about the software and its guidance documents, therefore, it is appropriate to address those comments in the present ProUCL guidance document. It is necessary to provide the detailed response to assure that: 1) rigorous statistical methods are used to compute decision making statistics; and 2) the methods incorporated in ProUCL software are not misrepresented and misinterpreted. Some general responses and comments about the material presented in the webinar and in the Unofficial User Guide to ProUCLv4 are described as follows. Specific comments and responses are also considered in the respective chapters of ProUCL 5.1 (and ProUCL 5.0) guidance documents. Note: It is noted that the Kindle version of "ProUCL v4: Unofficial User Guide" is no longer available on Amazon. Several incorrect theoretical statements and statements misrepresenting ProUCL 4 were made in that Unofficial User Guide; therefore, a brief response to some of those statements has been retained in ProUCL 5.1 guidance documents. ProUCL is a freeware software package which has been developed under limited government funding to address statistical issues associated with various environmental site projects. Not all statistical methods (e.g., Levene test) described in the statistical literature have been incorporated in ProUCL. One should not compare ProUCL with commercial software packages which are expensive and not as user-friendly as the ProUCL software when addressing environmental statistical issues. The existing and some new statistical methods based upon the research conducted by ORD-NERL, EPA Las Vegas during the last couple of decades have been incorporated in ProUCL to address the statistical needs of various environmental site projects and research studies. Some of those new methods may not be available in text books, in the 34 ------- library of programs written in R-script, and in commercial software packages. However, those methods are described in detail in the cited published literature and also in the ProUCL Technical Guides (e.g., EPA 2007, 2009a, 2009b, 2010c, 2010d, and 2011). Even though for uncensored data sets, programs which compute gamma distribution based UCLs and UPLs are available in R Script, programs which compute a 95% UCL of mean based upon a gamma distribution on KM estimates are not as easily available. • In the Unofficial ProUCL v4 User Guide, several statements have been made about percentiles. There are several ways to compute percentiles. Percentiles computed by ProUCL may or may not be identical (don't have to be) to percentiles computed by NADA for R (Helsel 2013) or described in Helsel and Gilroy (2012). To address users' requests, ProUCL 4.1 (2011) and its higher versions compute percentiles that are comparable to the percentiles computed by Excel 2003 and higher versions. • The literature search suggests that there are a total of nine (9) known types of percentiles, i.e., 9 different methods of calculating percentiles in statistics literature (Hyndman and Fan, 1996). The R programming language (R Core Team 2012) computes percentiles using those 9 methods using the following statement in R Quantile (x, p, type=k) where p = percentile, k = integer between 1-9 ProUCL computes percentiles using Type 7; Minitab 16 and SPSS compute percentiles using Type 6. It is simply a matter of choice, as there is no 'best' type to use. Many software packages use one type for calculating a percentile, and another for generating a box plot (Hyndman and Fan 1996). • An incorrect statement "By definition, the sample mean has a 50% chance of being below the true population mean" has been made in Helsel and Gilroy (2012) and also in Helsel (2012a). The above statement is not correct for means of skewed distributions (e.g., lognormal or gamma) commonly occurring in environmental applications. Since Helsel (2012) prefers to use a lognormal distribution, the incorrectness of the above statement has been illustrated using a lognormal distribution. The mean and median of a lognormal distribution (details in Section 2.3.2 of Chapter 2 of this Technical Guide) are given by: mean =jul = exp(ju + 0.5a2); and median =M = exp(//) From the above equations, it is clear that the mean of a lognormal distribution is always greater than the median for all positive values of a (sd of log-transformed variable). Actually the mean is greater than the pth percentile when a >2zp. For example, when p = 0.80, zp = 0.845, and mean of a lognormal distribution, ju\ exceeds xo.so, the 80th percentile when a > 1.69. In other words, when a > 1.69 the lognormal mean will exceed the 80th percentile of a lognormal distribution. Here zp represents thepth percentile of the standard normal distribution (SND) with mean 0 and variance 1. To demonstrate the incorrectness of the above statement, a small simulation study was conducted. The distribution of sample means based upon samples of size 100 were generated from lognormal distributions with ju =4, and varying skewness. The experiment was performed 10,000 times to generate the distributions of sample means. The probabilities of sample means less than the population means were computed. The following results are noted. 35 ------- Table 1-2. Probabilities p(~x < ^) Computed for Lognormal Distributions with u=4 and Varying Values of a Results are based upon 10000 Simulation Runs for Each Lognormal Distribution Considered Parameter />(*------- impute NDs) as the objective is not to impute NDs. To impute NDs, ProUCL uses ROS methods (Gamma ROS and log ROS) requiring place holders; and ProUCL computes plotting positions for all detects and NDs to generate a proper regression model which is used to impute NDs. Also for comparison purposes, ProUCL can be used to generate Q-Q plots on data sets obtained by replacing NDs by their respective DLs or DL/2s. In these cases, no NDs are imputed, and there is no need to retain placeholders for NDs. On these Q-Q plots, ProUCL displays some relevant statistics which are computed based upon the data displayed on those graphs. Helsel (2012a) states that the Summary Statistics module does not display KM estimates and that statistics based upon logged data are useless. Typically, estimates computed after processing the data do not represent summary statistics. Therefore, KM and ROS estimates are not displayed in the Summary Statistics module. These statistics are available in several other modules including the UCL and BTV modules. At the request of several users, summary statistics are computed based upon logged data. It is believed that the mean, median, or standard deviation of logged data do provide useful information about data skewness and data variability. To test for the equality of variances, the F-test, as incorporated in ProUCL, performs fairly well and the inclusion of the Levene's (1960) test will not add any new capability to the ProUCL software. Therefore, taking budget constraints into consideration, Levene's test has not been incorporated in the ProUCL software. o Although it makes sense to first determine if the two variances are equal or unequal, this is not a requirement to perform a t-test. The t-distribution based confidence interval or test for fii - fi2 based on the pooled sample variance does not perform better than the approximate confidence intervals based upon Satterthwaite's test. Hence testing for the equality of variances is not required to perform a two-sample t-test. The use of Welch-Satterthwaite's or Cochran's method is recommended in all situations (see Hayes 2005). Helsel (2012a) suggests that imputed NDs should not be made available to the users. The developers of ProUCL and other researchers like to have access to imputed NDs. As a researcher, for exploratory purposes only, one may want to have access to imputed NDs to be used in exploratory advanced methods such as multivariate methods including data mining, cluster and principal component analyses. It is noted that one cannot easily perform exploratory methods on multivariate data sets with NDs. The availability of imputed NDs makes it possible for researchers and scientists to identify potential patterns present in complex multivariate data by using data mining exploratory methods on those multivariate data sets with NDs. Additional discussion on this topic is considered in Chapter 4 of this Technical Guide. o The statements summarized above should not be misinterpreted. One may not use parametric hypothesis tests such as a t-test or a classical ANOVA on data sets consisting of imputed NDs. These methods require further investigation as the decision errors associated with such methods remain unquantified. There are other methods such as the Gehan and T-W tests in ProUCL 5.0/ProUCL 5.1 which are better suited to perform two-sample hypothesis tests using data sets with multiple detection limits. Outliers: Helsel (2012a) and Helsel and Gilroy (2012) make several comments about outliers. The philosophy (with input from EPA scientists) of the developers of ProUCL about the outliers in environmental applications is that those outliers (unless they represent typographical errors) may potentially represent impacted (site related or otherwise) locations or monitoring wells; and therefore 37 ------- may require further investigation. Moreover, decision statistics such as a UCL95 based upon a data set with outliers gets inflated and tends to represent those outliers instead of representing the population average. Therefore, a few low probability outliers coming from the tails of the data distribution may not be included in the computation of the decision making upper limits (UCLs, UTLs), as those upper limits get distorted by outliers and tend not to represent the parameters they are supposed to estimate. o The presence of outliers in a data set tends to destroy the normality of the data set. In other words, a data set with outliers can seldom (may be when outliers are mild, lying around the border of the central and tail parts of a normal distribution) follow a normal distribution. There are modern robust and resistant outlier identification methods (e.g., Rousseeuw and Leroy 1987; Singh and Nocerino 1995) which are better suited to identify outliers present in a data set; several of those robust outlier identification methods are available in the Scout 2008 version 1.0 (EPA 2009) software package. o For both Rosner and Dixon tests, it is the data set (also called the main body of the data set) obtained after removing the outliers (and not the data set with outliers) that needs to follow a normal distribution (Barnett and Lewis 1994). Outliers are not known in advance. ProUCL has normal Q-Q plots which can be used to get an idea about potential outliers (or mixture populations) present in a data set. However, since a lognormal model tends to accommodate outliers, a data set with outliers can follow a lognormal distribution; this does not imply that the outlier which may actually represent an impacted/unusual location does not exist! In environmental applications, outlier tests should be performed on raw data sets, as the cleanup decisions need to be made based upon values in the raw scale and not in log-scale or some other transformed space. More discussion about outliers can be found in Chapter 7 of this Technical Guide. In Helsel (2012a), it is stated, "Mathematically, the lognormal is simpler and easier to interpret than the gamma (opinion)." We do agree with the opinion that the lognormal is simpler and easier to use but the log-transformation is often misunderstood and hence incorrectly used and interpreted. Numerous examples (e.g., Example 2-1 and 2-2, Chapter 2) are provided in the ProUCL guidance documents illustrating the advantages of the using a gamma distribution. It is further stated in Helsel (2012a) that ProUCL prefers the gamma distribution because it downplays outliers as compared to the lognormal. This argument can be turned around - in other words, one can say that the lognormal is preferred by practitioners who want to inflate the effect of the outlier. Setting this argument aside, we prefer the gamma distribution as it does not transform the variable so the results are in the same scale as the collected data set. As mentioned earlier, log- transformation does appear to be simpler but problems arise when practitioners are not aware of the pitfalls (e.g., Singh and Ananda 2002; Singh, Singh, and laci 2002) associated with the use of lognormal distribution. Helsel (2012a) and Helsel and Gilroy (2012) state that "lognormal and gamma are similar, so usually if one is considered possible, so is the other." This is another incorrect and misleading statement; there are significant differences in the two distributions and in their mathematical properties. Based upon the extensive experience in environmental statistics and published literature, for skewed data sets that follow both lognormal and gamma distributions, the developers favor the use of the gamma distribution over the lognormal distribution. The use of the gamma distribution based decision statistics is preferred to estimate the environmental parameters (mean, upper percentile). A lognormal 38 ------- model tends to hide contamination by accommodating outliers and multiple populations whereas a gamma distribution adjusts for skewness but tends not to accommodate contamination (elevated values) as can be seen in Examples 2-1 and 2-2 of Chapter 2 of this Technical Guide. The use of the lognormal distribution on a data set with outliers tends to yield inflated and distorted estimates which may not be protective of human health and the environment; this is especially true for skewed data sets of small of sizes <20-30; the sample size requirement increases with skewness. o In the context of computing a UCL95 of mean, Helsel and Gilroy (2012) and Helsel (2012a) state that GROS and LROS methods are probably never better than the KM method. It should be noted that these three estimation methods compute estimates of mean and standard deviation and not the upper limits used to estimate EPCs and BTVs. The use of the KM method does yield good estimates of the mean and standard deviation as noted by Singh, Maichle, and Lee (2006). The problem of estimating mean and standard deviation for data sets with nondetects has been studied by many researchers as described in Chapter 4 of this document. Computing good estimates of mean and sd based upon left-censored data sets addresses only half of the problem. The main issue is to compute decision statistics (UCL, UPL, UTL) which account for uncertainty and data skewness inherently present in environmental data sets. o Realizing that for skewed data sets, Student's t-UCL, CLT-UCL, and standard and percentile bootstrap UCLs do not provide the specified coverage to the population mean for uncensored data sets, many researchers (e.g., Johnson 1978; Chen 1995; Efron and Tibshirani 1993; Hall [1988, 1992]; Grice and Bain 1980; Singh, Singh, and Engelhardt 1997; Singh, Singh, and laci 2002) developed parametric (e.g., gamma) and nonparametric (e.g., bootstrap-t and Hall's bootstrap method, modified-t, and Chebyshev inequality) methods for computing confidence intervals and upper limits which adjust for data skewness. One cannot ignore the work and findings of the researchers cited above, and assume that Student's t-statistic based upper limits or percentile bootstrap method based upper limits can be used for all data sets with varying skewness and sample sizes. o Analytically, it is not feasible to compare the various estimation and UCL computation methods for skewed data sets containing ND observations. Instead, researchers use simulation experiments to learn about the distributions and performances of the various statistics (e.g., KM-t- UCL, KM-percentile bootstrap UCL, KM-bootstrap-t UCL, KM-Gamma UCL). Based upon the suggestions made in published literature and findings summarized in Singh, Maichle, and Lee (2006), it is reasonable to state and assume that the findings of the simulation studies performed on uncensored skewed data sets comparing the performances of the various UCL computation methods can be extended to skewed left-censored data sets. o Like uncensored skewed data sets, for left-censored data sets, ProUCL 5.0/ProUCL 5.1 has several parametric and nonparametric methods to compute UCLs and other limits which adjust for data skewness. Specifically, ProUCL uses KM estimates in gamma equations; in the bootstrap-t method, and in the Chebyshev inequality to compute upper limits for left-censored skewed data sets. Helsel (2012a) states that ProUCL 4 is based upon presuppositions. It is emphasized that ProUCL does not make any suppositions in advance. Due to the poor performance of a lognormal model, as demonstrated in the literature and illustrated via examples throughout ProUCL guidance documents, the use of a gamma distribution is preferred when a data set can be modeled by a lognormal model and a gamma model. To provide the desired coverage (as close as possible) for the population mean, 39 ------- in earlier versions of ProUCL (version 3.0), in lieu of H-UCL, the use of Chebyshev UCL was suggested for moderately and highly skewed data sets. In later (3.00.02 and higher) versions of ProUCL, depending upon skewness and sample size, for gamma distributed data sets, the use of the gamma distribution was suggested for computing the UCL of the mean. Upper limits (e.g., UCLs, UPLs, UTLs) computed using the Student's t statistic andpercentile bootstrap method (Helsel 2012, NADAfor R, 2013) often fail to provide the desired coverage (e.g., 95% confidence coefficient) to the parameters (mean, percentile) of most of the skewed environmental populations. It is suggested that the practitioners compute the decision making statistics (e.g., UCLs, UTLs) by taking: data distribution; data set size; and data skewness into consideration. For uncensored and left-censored data sets, several such upper limits computation methods are available in ProUCL 5.1 and its earlier versions. Contrary to the statements made in Helsel and Gilroy (2012), ProUCL software does not favor statistics which yield higher (e.g., nonparametric Chebyshev UCL) or lower (e.g., preferring the use of a gamma distribution to using a lognormal distribution) estimates of the environmental parameters (e.g., EPC and BTVs). The main objectives of the ProUCL software funded by the U.S. EPA is to compute rigorous decision statistics to help the decision makers and project teams in making sound decisions which are cost-effective and protective of human health and the environment. Cautionary Note: Practitioners and scientists are cautioned about: 1) the suggestions made about the computations of upper limits described in some recent environmental literature such as the NADA books (Helsel [2005, 2012]); and 2) the misleading comments made about the ProUCL software in the training courses offered by Practical Stats during 2012 and 2013. Unfortunately, comments about ProUCL made by Practical Stats during their training courses lack professionalism and theoretical accuracy. It is noted that NADA packages in R and Minitab (2013) developed by Practical Stats do not offer methods which can be used to compute reliable or accurate decision statistics for skewed data sets. Decision statistics (e.g., UCLs, UTLs, UPLs) computed using the methods (e.g., UCLs computed using percentile bootstrap, and KM and LROS estimates and t-critical values) described in the NADA books and incorporated in NADA packages do not take data distribution and data skewness into consideration. The use of statistics suggested in NADA books and in Practical Stats training sessions often fail to provide the desired specified coverage to environmental parameters of interest for moderately skewed to highly skewed populations. Conclusions derived based upon those statistics may lead to incorrect conclusions which may not be cost-effective or protective of human health and the environment. Page 75 (Helsel [2012]): One of the reviewers of the ProUCL 5.0 software drew our attention to the following incorrect statement made on page 75 of Helsel (2012): "If there is only 1 reporting limit, the result is that the mean is identical to a substitution of the reporting limit for censored observations." An example of a left-censored data set containing ND observations with one reporting limit of 20 which illustrates this issue is described as follows. Y D_y 20 0 20 0 20 0 7 1 58 1 40 ------- 92 1 100 1 72 1 11 1 27 1 The mean and standard deviation based upon the KM and two substitution methods: DL/2 and DL are summarized as follows: Kaplan-Meier (KM) Statistics Mean 39.4 SD 35.56 DL Substitution method (replacing censored values by the reporting limit) Mean 42.7 SD 34.77 DL/2 Substitution method (replacing NDs by the reporting limit) Mean 39.7 SD 37.19 The above example illustrates that the KM mean (when only 1 detection limit is present) is not actually identical to the mean estimate obtained using the substitution, DL (RL) method. The statement made in Helsel's text (and also incorrectly made in his presentations such as the one made at the U.S. EPA 2007 National Association of Regional Project Managers (NARPM) Annual Conference: http ://www.ttemidev. com/narpm2007 Admin/conference/) holds only when all observations reported as detects are greater than the single reporting limit, which is not always true for environmental data sets consisting of analytical concentrations. 1.16 Box and Whisker Plots At the request of ProUCL users, a brief description of box plots (also known as box and whisker plots) as developed by Tukey (Hoaglin, Mosteller and Tukey 1991) is provided in this section. A box and whiskers plot represents a useful and convenient exploratory tool and provides a quick five point summary of a data set. In statistical literature, one can find several ways to generate box plots. The practitioners may have their own preferences to use one method over the other. Box plots are well documented in the statistical literature and description of box plots can be easily obtained by surfing the net. Therefore, the detailed description about the generation of box plots is not provided in ProUCL guidance documents. ProUCL also generates box plots for data set with NDs. Since box plots are used for exploratory purposes to identify outliers and also to compare concentrations of two or more groups, it does not really matter how NDs are displayed on those box plots. ProUCL generates box plots using detection limits and draws a horizontal line at the highest detection limit. Users can draw up to four horizontal lines at other levels (e.g., a screening level, a BTV, or an average) of their choice. All box plot methods, including the one in ProUCL, represent five-point summary graphs including: the lowest and the highest data values, median (50th percentile=second quartile, Q2), 25th percentile (lower quartile, Ql), and 75th percentile (upper quartile, Q3). A box and whisker plot also provides information 41 ------- about the degree of dispersion (interquartile range (IQR) = Q3-Ql=length/height of the box in a box plot), the degree of skewness (suggested by the length of the whiskers) and unusual data values known as outliers. Specifically, ProUCL (and other software packages) use the following to generate a box and whisker plot. • Q1= 25th percentile, Q2= 50th (median), and Q3 = 75th percentile • Interquartile range= IQR = Q3-Q1 (the length/height of the box in a box plot) • Lower whisker starts at Q1 and the upper whisker starts at Q3. • Lower whisker extends up to the lowest observation or (Ql - 1.5 * IQR) whichever is higher • Upper whisker extends up to the highest observation or (Q3 + 1.5* IQR) whichever is lower • Horizontal bars (also known as fences) are drawn at the end of whiskers • Guidance in statistical literature suggests that observations lying outside the fences (above the upper bar and below the lower bar) are considered potential outliers An example box plot generated by ProUCL is shown in the following graph. Box Plot for Lead Box Plot with Fences and Outlier It should be pointed out that the use of box plots in different scales (e.g., raw-scale and log-scale) may lead to different conclusions about outliers. Below is an example illustrating this issue. Example 1-2. Consider an actual data set consisting of copper concentrations collected a Superfund Site. The data set is: 0.83, 0.87, 0.9, 1, 1, 2, 2, 2.18, 2.73, 5, 7, 15, 22, 46, 87.6, 92.2, 740, and 2960. Box plots using data in the raw-scale and log-scale are shown in Figures 1-1 and 1-2. 42 ------- Box Plot for Copper Figure 1-1. Box Plot of Raw Data in Original Scale Based upon the last bullet point of the description of box plots described above, from Figure 1-1, it is concluded that two observations 740 and 2960 in the raw scale represent outliers. Box Plot for In(copper) Figure 1-2. Box Plot of Data in Log-Scale However, based upon the last bullet point about box plots, from Figure 1-2, it is concluded that two observations 740 and 2960 in the log-scale do not represent outliers. The log-transformation has accommodated the two outliers. This is one of the reasons ProUCL guidance suggests avoiding the use of log-transformed data. The use of a log-transformation tends to hide/accommodate outliers/contamination. 43 ------- Note: ProUCL uses a couple of development tools such as FarPoint spread (for Excel type input and output operations) and ChartFx (for graphical displays). ProUCL generates box plots using the built-in box plot feature in ChartFx. The programmer has no control over computing various statistics (e.g., Ql, Q2, Q3, IQR) using ChartFx. So box plots generated by ProUCL can differ slightly from box plots generated by other programs (e.g., Excel). However, for all practical and exploratory purposes, box plots in ProUCL are equally good (if not better) as available in the various commercial software packages for investigating data distribution (skewed or symmetric), identifying outliers, and comparing multiple groups (main objectives of box plots). Precision in Box Plots: Box plots generated using ChartFx round values to the nearest integer. For increased precision of graphical displays (all graphical displays generated by ProUCL), the user can use the process described as follows. Position your cursor on the graph and right-click, a popup menu will appear. Position the cursor on Properties and right-click; a windows form labeled Properties will appear. There are three choices at the top: General, Series and Y-Axis. Position the e cursor over the Y-Axis choice and left-click. You can change the number of decimals to increase the precision, change the step to increase or decrease the number Y-Axis values displayed and/or change the direction of the label. To show values on the plot itself, position your cursor on the graph and right-click; a pop-up menu will appear. Position the cursor on Point Labels and right-click. There are other options available in this pop-up menu including changing font sizes. 44 ------- CHAPTER 2 Goodness-of-Fit Tests and Methods to Compute Upper Confidence Limit of Mean for Uncensored Data Sets without Nondetect Observations 2.1 Introduction Many environmental decisions including exposure and risk assessment and management, and cleanup decisions are made based upon the mean concentrations of the contaminants/constituents of potential COPCs. To address the uncertainty associated with the sample mean, a UCL95 is used to estimate the unknown population mean, fj,\. A UCL95 is routinely used to estimate the EPC) term (EPA 1992a; EPA 2002a). A UCL95 of the mean represents that limit such that one can be 95% confident that the population mean, JJL\, will be less than that limit. From a risk point of view, a UCL95 of the mean represents a number that is considered health protective when used to compute risk and health hazards. Since, many environmental decisions are made based upon a UCL95, it is important to compute a reliable, defensible (from human health point of view) and cost-effective estimate of the EPC. To compute reliable estimates of practical merit, ProUCL software provides several parametric and nonparametric UCL computation methods covering a wide-range of skewed distributions (e.g., symmetric, mildly skewed to highly skewed) for data sets of various sizes. Based upon simulation results summarized in the literature (Singh, Singh, and Engelhardt [1997], Singh, Singh and laci [2002]), data distribution, data set size, and skewness, ProUCL makes suggestions on how to select an appropriate UCL95 of the mean to estimate the EPC. It should be noted that a simulation study cannot cover all possible real world data sets of various sizes and skewness following different probability distributions. This ProUCL Technical Guide provides sufficient guidance to help a user select the most appropriate UCL as an estimate of the EPC. The ProUCL software makes suggestions to help a typical user select an appropriate UCL from all the UCLs incorporated in ProUCL and those available in the statistical literature. UCL values, other than those suggested by ProUCL, may be selected based upon project personnel's experiences and project needs. The user may want to consult a statistician before selecting an appropriate UCL95. The ITRC (2012) regulatory document recommends the use of a Student's t-UCL95 and Chebyshev inequality based UCL95 to estimate EPCs for ISM based soil samples collected from DUs. In order to facilitate the computation of ISM data based estimates of the EPC, ProUCLS.l (and ProUCL 5.0) can compute a UCL95 of the mean based upon data sets of sizes as small as 3. Additionally, the UCL module of ProUCL can be used on ISM-based data sets with NDs. However, it is advised that the users do not compute decision making statistics (e.g., UCLs, upper prediction limits [UPLs], upper tolerance limits [UTLs])from discrete data sets consisting of less than 8- 10 observations. 45 ------- For uncensored data sets without ND observations, theoretical details of the Student's t- and percentile bootstrap UCL computation methods, as well as the more complicated bootstrap-t and gamma distribution methods, are described in this Chapter. One should not ignore the use of gamma distribution based UCLs (and other upper limits) just because it is easier to use a lognormal distribution. Typically, environmental data sets are positively skewed, and a default lognormal distribution (EPA 1992a) is used to model such data distributions. Additionally, an H-statistic based Land's (Land, 1971, 1975) H-UCL is then typically used to estimate the EPC. Hardin and Gilbert (1993), Singh, Singh, and Engelhardt (1997, 1999), Schultz and Griffin (1999), and Singh, Singh, and laci (2002) pointed out several problems associated with the use of the lognormal distribution and the H-statistic to compute UCL of the mean. For lognormal data sets with high standard deviation (sd), a, of the natural log-transformed data (e.g., a exceeding 1.0 to 1.5), the H-UCL becomes unacceptably large, exceeding the 95% and 99% data quantiles, and even the maximum observed concentration, by orders of magnitude (Singh, Singh, and Engelhardt 1997). The H-UCL is also very sensitive to a few low or a few high values. For example, the addition of a single low measurement can cause the H-UCL to increase by a large amount (Singh, Singh, and laci, 2002) by increasing variability. Realizing that the use of the H-statistic can result in an unreasonably large UCL, it has been recommended (EPA 1992a) that the maximum value be used as an estimate of the EPC in cases when the H-UCL exceeds the largest value in the data set. For uncensored data sets without any NDs, ProUCL makes suggestions/recommendations on how to compute an appropriate UCL95 based upon data set size, data skewness and distribution. In practice, many skewed data sets follow a lognormal as well as a gamma distribution. Singh, Singh, and laci (2002) observed that UCLs based upon a gamma distribution yield reliable and stable values of practical merit. It is, therefore, desirable to test whether an environmental data set follows a gamma distribution. A gamma distribution based UCL95 of the mean provides approximately 95% coverage to the population mean, ju\ = kO of a gamma distribution, G (k, 9) with k and 9 respectively representing the shape and scale parameters. For data sets following a gamma distribution with shape parameter, k > 1, the EPC should be estimated using an adjusted gamma (when «<50) or approximate gamma (when «>50) UCL95 of the mean. For highly skewed gamma distributed data sets with values of the shape parameter, k < 1.0, a 95% UCL may be computed using the bootstrap-t-method or Hall's bootstrap method when the sample size, n, is smaller, such as <15 to 20. For larger sample sizes with n> 20, a UCL of the mean may be computed using the adjusted or approximate gamma UCL (Singh, Singh, and laci 2002) computation method. Based upon professional judgment and practical experience of the authors, some of these suggestions have been modified. Specifically, in earlier versions ProUCL, the cutoff value for the shape parameter, k was 0.1 which has been changed to 1.0 in this version. Unlike the percentile bootstrap and bias-corrected accelerated bootstrap (BCA) methods, bootstrap-t and Hall's bootstrap methods (Efron and Tibshirani, 1993) account for data skewness and their use is recommended on skewed data sets when computing UCLs of the mean. However, the bootstrap-t and Hall's bootstrap methods sometimes result in erratic, inflated, and unstable UCL values, especially in the presence of outliers (Efron and Tibshirani 1993). Therefore, these two methods should be used with caution. The user should examine the various UCL results and determine if the UCLs based upon the bootstrap-t and Hall's bootstrap methods represent reasonable and reliable UCL values. If the results of these two methods are much higher than the rest of the UCL computation methods, it could be an indication of erratic behavior of these two bootstrap UCL computation methods. ProUCL prints out a warning message whenever the use of these two bootstrap methods is recommended. In cases where these two bootstrap methods yield erratic and inflated UCLs, the UCL of the mean may be computed using the Chebyshev inequality. 46 ------- ProUCL has graphical (e.g., quantile-quantile [Q-Q] plots) and formal goodness-of-fit (GOF) tests for normal, lognormal, and gamma distributions. These GOF tests are available for data sets with and without NDs. The critical values of the Anderson-Darling (A-D) test statistic and the Kolmogorov-Smirnov (K-S) test statistic to test for gamma distributions were generated using Monte Carlo simulation experiments (Singh, Singh, and laci 2002). Those critical values have been incorporated in ProUCL software and are tabulated in Appendix A for various levels of significance. ProUCL computes summary statistics for raw, as well as, log-transformed data sets with and without ND observations. In this Technical Guide and in ProUCL software, log -transformation (log) stands for the natural logarithm (In, LN) or log to the base e. For uncensored data sets, mathematical algorithms and formulae used in ProUCL to compute the various UCLs are summarized in this chapter. ProUCL also computes the maximum likelihood estimates (MLEsj and the minimum variance unbiased estimates (MVUEs) of the population parameters of normal, lognormal, and gamma distributions. Nonparametric UCL computation methods in ProUCL include: Jackknife, central limit theorem (CLT), adjusted-CLT, modified Student's t (adjusts for skewness) Chebyshev inequality, and bootstrap methods. It is well known that the Jackknife method (with sample mean as an estimator) and Student's t-method yield identical UCL values. Moreover, it is noted that UCLs based upon the standard bootstrap and the percentile bootstrap methods do not perform well by not providing the specified coverage of the mean for skewed data sets. Note on Computing Lower Confidence Limits (LCLs) of Mean: For some environmental projects an LCL of the unknown population mean is needed to achieve the project DQOs. At present, ProUCL does not directly compute LCLs of mean. However, for data sets with and without nondetects, excluding the bootstrap methods, gamma distribution, and H-statistic based LCLs of mean, the same critical value (e.g., normal z value, Chebyshev critical value, or t-critical value) can be used to compute a LCL of mean as used in the computation of the UCL of the mean. Specifically, to compute a LCL, the '+' sign used in the computation of the corresponding UCL needs to be replaced by the '-' sign in the equation used to compute that UCL (excluding gamma, lognormal H-statistic, and bootstrap methods). For specific details, the user may want to consult a statistician. For data sets without nondetect observations, the user may want to use the Scout 2008 software package (EPA 2009d, 2010) to directly compute the various parametric and nonparametric LCLs of mean. 2.2 Goodness-of-Fit (GOF) Tests Let x\, X2, ..., xn be a representative random sample (e.g., representing lead concentrations) from the underlying population (e.g., site areas under investigation) with unknown mean, [j.\, and variance, oi1. Let fj. and a represent the population mean and the population standard deviation (sd) of the log -transformed (natural log to the base e) data. Let y and sy (a ) be the sample mean and sample sd, respectively, of the log -transformed data, yt = log (xi); 1= 1,2, ... ,n. Specifically, let n (2-2) Similarly, let x and sx be the sample mean and sd of the raw data, xi , X2 , .. , xn, obtained by replacing y by x in equations (2-1) and (2-2), respectively. In this chapter, irrespective of the underlying distribution, 47 ------- ju\, and a\2 represent the mean and variance of the random variable X (in original units), whereas // and a1 represent the mean and variance of Y = loge(X). Three data distributions have been considered in ProUCL 5.1 (and in older versions). These include the normal, lognormal, and the gamma distributions. Shapiro-Wilk, for n <2000, and Lilliefors (1967) test statistics are used to test for normality or lognormality of a data set. Lilliefors test (along with graphical Q-Q plot) seems to perform fairly well for samples of size 50 and higher. In ProUCL 5.1, updated critical values of Lilliefors test developed by Moling and Abdi (2007) and provided in the Encyclopedia of Measurement and Statistics have been used. The empirical distribution function (EDF) based methods, the K-S and A-D tests, are used to test for a gamma distribution. Extensive critical values for these two test statistics have been obtained via Monte Carlo simulation experiments (Singh, Singh, and laci 2002). For interested users, those critical values are given in Appendix A for various levels of significance. In addition to these formal tests, the informal histogram and Q-Q plots (also called probability plots) are also available for visual inspection of the data distributions (Looney and Gulledge 1985). Q-Q plots also provide useful information about the presence of potential outliers and multiple populations in a data set. A brief description of the GOF tests follows. No matter which normality test is used, it may fail to detect the actual non-normality of the population distribution if the sample size is small, «<20 and with large sample sizes, «>50 or so, a small deviation from normality will lead to rejection of the normality hypothesis. The modified K-S test known as Lilliefors test is suggested for checking the normality assumption when the mean and sd of population distribution is not known. 2.2.1 Test Normality and Lognormality of a Data Set ProUCL tests for normality and lognormality of a data set using three different methods described below. The program tests normality or lognormality at three different levels of significance, 0.01, 0.05, and 0.1 (or confidence levels: 0.99, 0.95, and 0.90). For normal distributions, ProUCL outputs approximate probability values (p-values) for the S-W GOF test. The details of those methods can be found in the cited references. 2.2.1.1 Normal Quantile-quantile (Q-Q) Plot A normal Q-Q represents a graphical method to test for approximate normality or lognormality of a data set (Hoaglin, Mosteller, and Tukey 1983; Singh 1993; Looney and Gulledge, 1985). A linear pattern displayed by the majority of the data suggests approximate normality or lognormality (when performed on log-transformed data) of the data set. For example, a high value, 0.95 or greater, of the correlation coefficient of the linear pattern may suggest approximate normality (or lognormality) of the data set under study. However, on this graphical display, observations well-separated from the linear pattern displayed by the majority of data may represent outlying observations not belonging to the dominant population, whose distribution one is assessing based upon a data set. Apparent jumps and breaks in the Q-Q plot suggest the presence of multiple populations. The correlation of the Q-Q plot for such a data set may still be high but that does not signify that the data set follows a normal distribution. Notes: Graphical displays provide added insight into a data set which might not be apparent based upon statistics such as S-W statistic or a correlation coefficient. The correlation coefficient of a Q-Q plot with curves, jumps and breaks can be high, which does not necessarily imply that the data follow a normal or lognormal distribution. AGOF test of a data set should always be judged based upon a formal (e.g., S-W statistic) as well as informal graphical displays. The normal Q-Q plot is used as an exploratory tool to 48 ------- identify outliers or to identify multiple populations. On all Q-Q plots, ProUCL displays relevant statistics including: mean, sd, GOF test statistic, associated critical value, /"-value (when available), and the correlation coefficient. There is no substitute for graphical displays of data sets. Graphical displays provide added insight about data sets and do not get distorted by outliers and/or mixture populations. The final conclusion regarding the data distribution should be based upon the formal GOF tests as wells as Q-Q plots. This statement is true for all GOF tests: normal, lognormal, and gamma distributions. 2.2.1.2 Shapiro-Wilk (S-W) Test The S-W test is a powerful test used to test the normality or lognormality of a data set. ProUCL performs this test for samples of size up to 2000 (Royston 1982, 1982a). For sample sizes < 50, in addition to a test statistic and critical value, an approximate /"-value associated with S-W test is also displayed. For samples of size >50, only approximate />-values are displayed. Based upon the selected level of significance and the computed test statistic, ProUCL informs the user if the data set is normally (or lognormally) distributed. This information should be used to compute an appropriate UCL of the mean. 2.2.1.3 Lilliefors Test This test is useful for data sets of larger size (Lilliefors 1967; Dudewicz and Misra 1988; Conover 1999). This test is a slight modification of the Kolmogorov-Smirnov (K-S) test and is more powerful than a one- sample K-S (with the estimated population mean and sd). In version 5.1 of ProUCL, critical values of Lilliefors test developed by Moling and Abdi and provided in the Encyclopedia of Measurement and Statistics (Salkind, N. Editor 2007) have been used and incorporated in the program. The critical values as described in Salkind (2007) are used for n up to 50, and for values of «>50 approximate critical values are computed using the following formula: Critical Values = Factor If (n); where /(«) = —= 0.01. V« The Factor used in the above equation depends upon the level of significance, a; Factor values are 0.741, 0.819, 0.895, and 1.035 for a = 0.20, 0.1, 0.05, and 0.01 respectively. Based upon the selected level of significance and the computed test statistic, ProUCL informs the user if the data set is normally or lognormally distributed. This information should be used to compute an appropriate UCL of the mean. The program outputs the relevant statistics on the Q-Q plot of data. • For convenience, normality, lognormality, or gamma distribution test results for a built-in level of significance of 0.05 are displayed on the UCL and background statistics output sheets. This helps the user in selecting the most appropriate UCL to estimate the EPC. It should be pointed out that sometimes, the two GOF tests may lead to different conclusions. In such situations, ProUCL displays a message that data are approximately normally or lognormally distributed. It is suggested that the user makes a decision based upon the information provided by the associated Q-Q plot and the values of the GOF test statistics. New in ProUCL 5.1: Based upon the author's professional experience and in an effort to streamline the decision process for computing upper limits (e.g., UCL95), some changes were made in the decision logic applied in ProUCL for suggesting/recommending UCL values. Specifically, ProUCL 5.1 makes decisions about the data distribution based upon both the Lilliefors and S-W GOF test statistics for 49 ------- normal and lognormal distributions and both the A-D and K-S GOF test statistics for the gamma distribution. When a data set passes one of the two GOF tests for a distribution, ProUCL outputs a statement that the data set follows that approximate distribution and suggests using appropriate decision statistic(s). Specifically, when only one of the two GOF statistic leads to the conclusion that data are normal, lognormal or gamma, ProUCL outputs the conclusion that the data set follows that approximate distribution and all suggestions provided by ProUCL regarding the use of parametric or nonparametric decision statistics are made based upon this conclusion. As a result, UCLs suggested by ProUCL 5.1 can differ from the UCLs suggested by earlier versions of ProUCL. Note: When dealing with a small data set, n <50, and Lilliefors test suggests that data are normal and the S-W test suggests that data are not normal, ProUCL will suggest that the data set follows an approximate normal distribution. However, for smaller data sets, Lilliefors test results may not be reliable, therefore the user is advised to review GOF tests for other distributions and proceed accordingly. It is emphasized, when a data set follows a distribution (e.g., distribution A) using all GOF tests, and also follows an approximate distribution (e.g., distribution B) using one of the available GOF tests, it is preferable to use distribution A over distribution B. However, when distribution A is a highly skewed (e.g., sd of logged data >1.0) lognormal distribution, use the guidance provided on the ProUCL generated output. In practice, depending upon the power associated with statistical tests, two tests (e.g., two sample t-test vs. WMW test; S-W test vs. Lilliefors test) used to address the same statistical issue (comparing two groups, assessing data distribution) can lead to different conclusions (e.g., GOF tests for normality in Example 2-4); this is especially true when dealing with data sets of smaller sizes. The power of a test can be increased by collecting more data. If this is not feasible due to resource constraints, the collective project team should determine which conclusion to use in the decision making process. It may, in these cases, be appropriate to consult a statistician. 2.2.2 Gamma Distribution A continuous random variable, X (e.g., concentration of an analyte), is said to follow a gamma distribution, G(k, 6) with parameters k > 0 (shape parameter) and 6 > 0 (scale parameter), if its probability density function is given by the following equation: f(x; k, 9} = —i— x^e-*16; x> 0 •s \ ? ? / /)/tT—< /- 7 \ ? /^->\ (2-3) = 0; otherwise Many positively skewed data sets follow a lognormal as well as a gamma distribution. The use of a gamma distribution tends to yield reliable and stable 95% UCL values of practical merit. It is therefore desirable to test if an environmental data set follows a gamma distribution. If a skewed data set does follow a gamma model, then a 95% UCL of the population mean should be computed using a gamma distribution. For data sets which follow a gamma distribution, the adjusted 95% UCL of the mean based upon a gamma distribution is optimal (Bain and Engelhardt 1991) and approximately provides the specified 95% coverage of the population mean, jj.\ = W (Singh, Singh, and laci 2002). The GOF test statistics for a gamma distribution are based upon the EDF. The two EDF tests incorporated in ProUCL are the K-S test and the A-D test, which are described in D'Agostino and Stephens (1986) and Stephens (1970). The graphical Q-Q plot for a gamma distribution has also been incorporated in ProUCL. The critical values for the two EDF tests are not available, especially when the shape parameter, k, is 50 ------- small (k < 1). Therefore, the associated critical values have been computed via extensive Monte Carlo simulation experiments (Singh, Singh, and laci 2002). The critical values for the two test statistics are given in Appendix A. The 1%, 5%, and 10% critical values of these two test statistics have been incorporated in ProUCL. The GOF tests for a gamma distribution depend upon the MLEs of the gamma parameters, k and 6, which should be computed before performing the GOF tests. Information about estimation of gamma parameters, gamma GOF tests, and construction of gamma Q-Q plots is not readily available in statistical textbooks. Therefore, a detailed description of the methods for a gamma distribution is provided as follows. 2. 2. 2. 1 Quantile-Quantile (Q-Q) Plot for a Gamma Distribution Let xi, Jt2, ... , xn be a random sample from the gamma distribution, G(k,9); and let X(i) < X(2) < ... < X(n) represent the ordered sample. Let k and 9 represent the maximum likelihood estimates (MLEs) of k and 9, respectively; details of the computation of the MLEs of k and 9 can be found in Singh, Singh, and laci (2002). The Q-Q plot for a gamma distribution is obtained by plotting the scatter plot of pairs, (x0i,x(i)) i := 1, 2, ..., n. The gamma quantiles, Jtoi, are given by the equation, x0i = z0i0/2; i := 1, 2, ..., n, where the quantiles zoi (already ordered) are obtained by using the inverse chi-square distribution and are given as follows: ; /:=1,2,...,« (2-4) 9 * In (2-4), x2k represents a chi-square random variable with 2k degrees of freedom (df). The program, PPCHI2 (Algorithm AS91) described in Best and Roberts (1975) has been used to compute the inverse chi-square percentage points given by equation (2-4). All relevant statistics including the MLE of k are also displayed on a gamma Q-Q plot. Like a normal Q-Q plot, a linear pattern displayed by the majority of the data on a gamma Q-Q plot suggests that the data set follows an approximate gamma distribution. For example, a high value (e.g., 0.95 or greater) of the correlation coefficient of the linear pattern may suggest an approximate gamma distribution of the data set under study. However, on this Q-Q plot, points well-separated from the bulk of data may represent outliers. Apparent breaks and jumps in the gamma Q-Q plot suggest the presence of multiple populations. The correlation coefficient of a Q-Q plot with outliers and jumps can still be high which does not signify that the data follow a gamma distribution. Therefore, a graphical Q-Q plot and other formal GOF tests, the A-D test or K-S test, should be used on the same data set to determine the distribution of a data set. 2.2.2.2 Empirical Distribution Function (EDF) -Based Goodness-of Fit Tests Let F(x) be the cumulative distribution function (CDF) of a gamma distributed random variable, X. Let Z = F(X), then Z represents a uniform U(0,l) random variable (Hogg and Craig 1995). For each x,, compute z; by using the incomplete gamma function given by the equation z; = F (x;); /':=!, 2, ..., n. The algorithm (Algorithm AS 239, Shea 1988) as given in the book Numerical Recipes in C, the Art of Scientific Computing (Press et al. 1990) has been used to compute the incomplete gamma function. 51 ------- Arrange the re suiting z; in ascending order as Z(i) < zp) <...< Z(n). Le t z = zt / « be the mean of the \t=i ) n, zi; i := 1, 2, ..., n. Compute the following two statistics: D+ = max . {1 / n - z(i) } , and D = max . {z(i) - (i -!)/«} (2-5) The K-S test statistic is given byD = max(D+,D~) ; and the A-D test statistic is given as follows: A2 =-n- (l/«) {(2/ - l)[log z(!) + log(l - z(B+1_0)]} (2-6) i=l As mentioned before, the critical values for these two statistics, D and A2, are not readily available. For the A-D test, only the asymptotic critical values are available in the statistical literature (D'Agostino and Stephens 1986). Some raw critical values for the K-S test are given in Schneider (1978), and Schneider and Clickner (1976). Critical values of these test statistics are computed via Monte Carlo experiments (Singh, Singh, and laci 2002). It is noted that the distributions of the K-S test statistic, D, and the A-D test statistic, A2, do not depend upon the scale parameter, 6; therefore, the scale parameter, 6, has been set equal to 1 in all simulation experiments. In order to generate critical values, random samples from gamma distributions were generated using the algorithm as given in Whittaker (1974). It is observed that the simulated critical values are in close agreement with all available published critical values. The critical values simulated by Singh, Singh, and laci (2002) for the two test statistics have been incorporated in the ProUCL software for three levels of significance, 0.1, 0.05, and 0.01. For each of the two tests, if the test statistic exceeds the corresponding critical value, then the hypothesis that the data set follows a gamma distribution is rejected. ProUCL computes the GOF test statistics and displays them on the gamma Q-Q plot and also on the UCL and background statistics output sheets generated by ProUCL. Like all other tests, in practice these two GOF test may lead to different conclusions. In such situations, ProUCL outputs a message that the data follow an approximate gamma distribution. The user should make a decision based upon the information provided by the associated gamma Q-Q plot and the values of the GOF test statistics. Computation of the Gamma Distribution Based Decision Statistics and Critical Values: When computing the various decision statistics (e.g., UCL and BTVs), ProUCL uses biased corrected estimates, kstar, k and theta star, 9* (described in Section 2.3.3) of the shape, k, and scale, 9 , parameters of the gamma distribution. It is noted that the critical values for the two gamma GOF tests reported in the literature (D'Agostino and Stephens 1986; Schneider and Clickner 1976; and Schneider 1978) are computed using the MLE estimates, k and 9 of the two gamma parameters, k and$. Therefore, the critical values of A- D and K-S tests incorporated in ProUCL have also been computed using the MLE estimates: khat, k and theta hat, 9 of the two gamma parameters, k and 9 . 52 ------- Updated Critical Values of Gamma GOF Test Statistics (New in ProUCL 5.0): For values of the gamma distribution shape parameter, k < 0.1, critical values of the two gamma GOF tests, A-D and K-S tests, have been updated in ProUCL 5.0 and higher versions. Critical values incorporated in earlier versions were simulated using the gamma deviate generation algorithm (Whittaker 1974) available at the time and with the source code described in the book Numerical Recipes in C, the Art of Scientific Computing (Press et al. 1990). Th gamma deviate generation algorithm available at the time was not very efficient especially for smaller values of the shape parameter, k < 0.1. For values of the shape parameter, £< 0.1, significant discrepancies were found in the critical values of the two gamma GOF test statistics obtained using the two gamma deviate generation algorithms: Whitaker (1974) and Marsaglia and Tsang (2000). Therefore, for values of k < 0.2, critical values for the two gamma GOF tests have been re-generated and tables of critical values of the two gamma GOF tests have been updated in Appendix A. Specifically, for values of the shape parameter, k < 0.1, critical values of the two gamma GOF tests have been generated using the more efficient gamma deviate generation algorithm as described in Marsaglia and Tsang's (2000) and Best (1983). Detailed description about the implementation of Marsaglia and Tsang's algorithm to generate gamma deviates can be found in Kroese, Taimre, and Botev (2011). From a practical point of view, for values of k greater than 0.1, the simulated critical values obtained using Marsaglia and Tsang's algorithm (2000) are in general agreement with the critical values of the two GOF test statistics incorporated in ProUCL 4.1 for the various values of the sample size considered. Therefore, those critical values for values of k > 0.1 do not have to be updated. Note: In March 2015 minor discrepancies were identified in critical values of the gamma GOF A-D tests, as summarized in Tables A1-A6 of ProUCL 5.0 Technical Guide. For example, for a specified sample size and level of significance, a, the critical values for GOF tests are expected to decrease as k increases. Due to inherent random variability in the simulated gamma data sets, critical values do not follow (deviations are minor occurring in 2nd or 3rd decimal places) this trend in a few cases. However, from a practical and decision making point of view those differences are minor (see below). These discrepancies can be eliminated by performing simulation experiments using more iterations. In ProUCL 5.1, these discrepancies in the critical values of gamma GOF tests have been fixed via interpolation. For example, in Table A-3, for the A-D test, with significance level a= 0.05 and «=7, critical values for £=10, 20, and 50 are 0.708, 0.707, and 0.708. Also, in Table A-4 for «=200 and £=0.025, the critical value is 0.070489, and for «=200, £=0.05, the critical value is 0.07466. Due to a lack of resources and time, the critical values have not been re-simulated; however, this value has been replaced by an interpolated value using simulated values for £=0.025 and £=0.1. 2.3 Estimation of Parameters of the Three Distributions Incorporated in ProUCL Let jui and a\- represent the mean and variance of the random variable, X, and ju and o2 represent the mean and variance of the random variable Y = log(X). Also, a represents the standard deviation of the log- transformed data. For both lognormal and gamma distributions, the associated random variable can take only positive values. It is typical of environmental data sets to consist of only positive concentrations. 2.3.7 Normal Distribution Let X be a continuous random variable (e.g., lead concentrations in surface soils of a site), which follows a normal distribution, N (MI, a\2) with mean, ju\, and variance, a\2. The probability density function of a normal distribution is given by the following equation: 53 ------- -GO------- For positively skewed data sets, the various levels of skewness can be defined in terms a or its MLE estimate, sy. These levels are described as follows in Table 2-1. ProUCL software uses the sample sizes and skewness levels defined below to make suggestions/recommendations to select an appropriate UCL as an estimate of the EPC. 55 ------- Table 2-1. Skewness as a Function of a (or its MLE, sy = 6), sd of log(X) Standard Deviation of Logged Data Skewness a < 0.5 Symmetric to mild skewness 0.5 < a < 1.0 Mild skewness to moderate skewness 1.0 < a < 1.5 Moderate skewness to high skewness 1.5 < a < 2.0 High skewness 9 0 < < 1 0 Very high skewness (moderate probability of ~ ' outliers and/or multiple populations) - „ Extremely high skewness (high probability of ~ ' outliers and/or multiple populations) Note: When data are mildly skewed with a < 0.5, the three distributions considered in ProUCL tend to yield comparable upper limits irrespective of the data distribution. 2.3.2.3 MLEs of the Quantiles of a Lognormal Distribution For highly skewed (a > 1.5) lognormally distributed populations, the population mean, jj.\, often exceeds the higher quantiles (80%, 90%, 95%) of the distribution. Therefore, the estimation of these quantiles is also of interest. This is especially true when one may want to use MLEs of the higher order quantiles such as 95%, 97.5%, etc. as estimates of the EPC. The formulae to compute these quantiles are described here. The />th quantile (or 100 />th percentile), xp, of the distribution of a random variable, X, is defined by the probability statement, P(X < xp) = p. If zp is the pth quantile of the standard normal random variable, Z, with P(Z < zp) = p, then the pth quantile of a lognormal distribution is given by xp = exp(« + zpa). Thus the MLE of the pih quantile is given by: xp = exp(/} + zpa) (2-13) It is expected that 95% of the observations coming from a lognormal LN(w, o2) distribution would lie at or below exp(« + 1.650). The 0.5th quantile of the standard normal distribution is ZQ.S = 0, and the 0.5th quantile (or median) of a lognormal distribution is M= exp(«), which is obviously smaller than the mean, [j.\, as given by equation (2-8). Notes: The mean, ju\, is greater than xp if and only if a > 2zp. For example, when/? = 0.80, zp = 0.845, ju\ exceeds xo.so, the 80th percentile if and only if a > 1.69, and, similarly, the mean, jj.\, will exceed the 95th percentile if and only if a > 3.29 (extremely highly skewed). ProUCL computes the MLEs of the 50% (median), 90%, 95%, and 99% percentiles of lognormally distributed data sets. 56 ------- 2.3.2.4 MVUEs of Parameters of a Lognormal Distribution Even though the sample mean x is an unbiased estimator of the population mean, /j.\, it does not possess the minimum variance (MV). The MVUEs of ju\ and a\2 of a lognormal distribution are given as follows: (2-14) (2-15) The series expansion of the function g»(x) is given in Bradu and Mundlak (1970), and Aitchison and Brown (1969). Tabulations of this function are also provided by Gilbert (1987). Bradu and Mundlak (1970) computed the MVUE of the variance of the estimate, fa, -gn((n-2)s2yl(n-m (2-16) The square root of the variance given by equation (2-16) is called the standard error (SE) of the estimate, fa, given by equation (2-14). The MVUE of the median of a lognormal distribution is given by M = exp(y)gn[-s2y l(2(n -1))] (2-17) For a lognormally distributed data set, ProUCL also computes these MVUEs given by equations (2-14) through (2-17). 2.3.3 Estimation of the Parameters of a Gamma Distribution The population mean and variance of a two-parameter gamma distribution, G(k, 0), are functions of both parameters, k and 6. In order to estimate the mean, one has to obtain estimates of k and 9. The computation of the MLE of k is quite complex and requires the computation of Digamma and Trigamma functions. Several researchers (Choi and Wette 1969; Bowman and Shenton 1988; Johnson, Kotz, and Balakrishnan 1994) have studied the estimation of the shape and scale parameters of a gamma distribution. The MLE method to estimate the shape and scale parameters of a gamma distribution is described below. As before, let x\, X2, ..., xn be a random sample (e.g., representing constituent concentrations) of size n from a gamma distribution, G(k, 0), with unknown shape and scale parameters, k and 0, respectively. The log- likelihood function (obtained using equation (2-3)) is given as follows: ~~ ' (2-18) To find the MLEs of k and 0, one differentiates the log-likelihood function as given in (2-18) with respect to k and 0, and sets the derivatives to zero. This results in the following two equations: 1 ~ " ' \and (2-19) 57 ------- 1 k9 = -^_jxi=x (2-20) Solving equation (2-20) for 9 , and substituting the result in (2-19), we get following equation: ^ (2-21) There does not exist a closed form solution of equation (2-21). This equation needs to be solved numerically fork , which requires the use of digamma and trigamma functions. An estimate of k can be computed iterative ly by using the Newton-Raphson method (Press et al. 1990), leading to the following iterative equation: , ,, , k,=k,,- *~y - - (2-22) The iterative process stops when k starts to converge. In practice, convergence is typically achieved in fewer than 10 iterations. In equation (2-22), Here *V(k) is the digamma function and ~*V'(k) is the trigamma function. Good approximate values for these two functions (Choi and Wette 1969) can be obtained using the following two approximations. For k > 8, these functions are approximated by: log( £) - {l + [l - (1 / 1 0 - 1 1(2 Ik2 ))/ £2 }(6k)]l(2k) , and (2-23) - (1/5-1 /(Ik2 ))/k2 ^(3k)}/(2k)}/k (2-24) For k < 8, one can use the following recurrence relations to compute these functions: + 1) - 1 / k , and (2-25) (2-26) In ProUCL, equations (2-23) through (2-26) have been used to estimate k. The iterative process requires an initial estimate of k. A good starting value for k in this iterative process is given by ko = 1 / (2M). Thorn (1968) suggested the following approximation as an initial estimate of k: 4M V 3 1 1 + ,1 + -M (2-27) 58 ------- Bowman and Shenton (1988) suggest using k , given by (2-27) as a starting value of k for the iterative procedure, calculating kl at the 7th iteration using the following formula: ( } M Both equations (2-22) and (2-28) have been used to compute the MLE of k. It is observed that the estimate, k , based upon the Newton-Raphson method, as given by equation (2-22), is in close agreement with the one obtained using equation (2-28) with Thorn's approximation as an initial estimate. Choi and Wette (1969) further concluded that the MLE of k, k , is biased high. A bias-corrected (Johnson, Kotz, and Balakrishnan 1994) estimate of k is given by: k* = (n - 3)k I n + 2 /(3n) (2-29) In (2-29), k is the MLE of k obtained using either (2-22) or (2-28). Substitution of equation (2-29) in equation (2-20) yields an estimate of the scale parameter, 6, given as follows: 0* =xlk* (2-30) ProUCL computes simple MLEs of £ and 0, and also bias-corrected estimates given by (2-29) and (2-30) of k and 6. The bias -corrected estimate (called k star and theta star in ProUCL graphs and output sheets) of k as given by (2-29) has been used in the computation of the UCLs (as given by equations (2-34) and (2-35) below) of the mean of a gamma distribution. Note on Bias Corrected Estimates, k and 6 : As mentioned above, Choi and Wette (1969) concluded that the MLE, k , of k is biased high. They suggested the use of the bias-corrected (Johnson, Kotz, and Balakrishnan 1994) estimate of £ given by (2-29) above. However, recently the developers performed a simulation study to evaluate the bias in the MLE of the mean of a gamma distribution for various values of the shape parameter, k and sample size, n. For smaller values of £ (e.g., <0.2), the bias in the mean estimate (in absolute value) and mean square error (MSE) based upon the biased corrected MLE, k are higher than those computed using the MLE estimate, k ; and for higher values of £ (e.g., >0.2), the bias in the mean estimate and MSE computed using the biased corrected MLE, k are lower than those computed using the MLE, k . For values of k around 0.2, the use of k and k yields comparable results for all values of the sample size. The bias in the mean estimate obtained using the MLE, k , increases as k increases, and as expected, bias and MSE decrease as the sample size increases. The results of this study will be published elsewhere. At present for uncensored and left-censored data sets, ProUCL computes all gamma UCLs and other upper limits (Chapters 3, 4 and 5) using bias corrected estimates, k and 6 of k and 6. ProUCL generated output sheets display many intermediate results including k and k* ; 6 and 9* . Interested users may want to compute UCLs and other upper limits using MLE estimates, k and 6 ofk and 6 for values of k described in the above paragraph. 59 ------- 2.4 Methods for Computing a UCL of the Unknown Population Mean ProUCL computes a (1 - a)*100 UCL of the population mean, ju\, using several parametric and nonparametric methods. ProUCL can compute a (1 - a)*100 UCL (except for adjusted gamma UCL and Land's H-UCL) of the mean for any user selected confidence coefficient, (1 - a), lying in the interval [0.5, 1.0]. For the computation of the adjusted gamma UCL, three confidence levels, namely: 0.90, 0.95, and 0.99 are supported by the ProUCL software. An approximate gamma UCL can be computed for any level of significance in the interval [0.5, 1.0]. Parametric UCL Computation Methods in ProUCL include: • Student's t-statistic (assumes normality or approximate normality) based UCL, • Approximate gamma UCL (assumes approximate gamma distribution), • Adjusted gamma UCL (assumes approximate gamma distribution), • Land's H-Statistic UCL (assumes lognormality), and • Chebyshev inequality based UCL: Chebyshev (MVUE) UCL obtained using MVUE of the parameters (assumes lognormality). Nonparametric UCL Computation Methods in ProUCL include: • Modified-t-statistic (modified for skewness) UCL, • Central Limit Theorem (CLT) UCL to be used for large samples, • Adjusted Central Limit Theorem UCL: adjusted-CLT UCL (adjusted for skewness), • Chebyshev UCL: Chebyshev (Mean, sd) obtained using classical sample mean and standard deviation, • Jackknife UCL (yields the same result as Student's t-statistic UCL), • Standard bootstrap UCL, • Percentile bootstrap UCL, • BCA bootstrap UCL, • Bootstrap-t UCL, and • Hall's bootstrap UCL. For skewed data sets, Modified-t and adjusted CLT methods adjust for skewness. However, this adjustment is not adequate (Singh, Singh, and laci, 2002) for moderately skewed to highly skewed data sets (levels of skewness described in Table 2-1). Even though some UCL methods (e.g., CLT, UCL based upon Jackknife method, standard bootstrap, and percentile bootstrap methods) do not perform well enough to provide the specified coverage to the population mean of skewed distributions. These methods have been included in ProUCL for comparison, academic, and research purposes. These comparisons are also necessary to demonstrate why the use of a Student's t-based UCL and Kaplan-Meier (KM) method based UCLs using t-critical values as suggested in some environmental books should be avoided. Additionally, the inclusion of these methods also helps the user to make better decisions. Based upon the sample size, n, data skewness, a, and data distribution, ProUCL makes suggestions regarding the use of one or more 95% UCL methods to estimate the EPC. For additional gudidance, the users may want to consult a statistician to select the most appropriate UCL95 to estimate an EPC. It is noted that in the environmental literature, recommendations about the use of UCLs have been made without accounting for the skewness and sample size of the data set. Specifically, Helsel (2005, 2012) suggests the use t-statistic and percentile bootstrap method on robust regression on order statistics (ROS) and KM estimates to compute UCL95s without considering data skewness and sample size. For 60 ------- moderately skewed to highly skewed data sets, the use of such UCLs underestimates the population mean. These issues are illustrated by examples discussed in the following sections and also in Chapters 4 and 5. 2. 4.1 (1 - a) *100 UCL of the Mean Based upon Student 's t-Statistic The widely used Student's t-statistic is given by: - (2-31) sxn Where x and sx are, respectively, the sample mean and sample standard deviation obtained using the raw data. For normally distributed data sets, the test statistic given by equation (2-31) follows the Student's t- distribution with (n -1) df. Let ta,n-\ be the upper a* quantile of the Student's t-distribution with (n -1) df. A (1 - a)*100 UCL of the population mean, ju\, is given by: (2-32) For a normally (when the skewness is approximately 0) distributed data sets, equation (2-32) provides the best (optimal) way of computing a UCL of the mean. Equation (2-32) may also be used to compute a UCL of the mean based upon symmetric or mildly skewed (|skewness|<0.5) data sets, where the skewness is defined in Table 2-1. For moderately skewed data sets (e.g., whencr, the sd of log-transformed data, starts approaching and exceeding 0.5), the UCL given by (2-32) fails to provide the desired coverage of the population mean. This is especially true when the sample size is smaller than 20-25 (graphs summarized in Appendix B). The situation gets worse (coverage much smaller) for higher values of the sd, tj , or its MLE, sy. Notes: ProUCL 5.0 and later versions make a decision about the data distribution based upon both of the GOF test statistics: Lilliefors and Shapiro-Wilk GOF statistics for normal and lognormal distributions; and A-D and K-S GOF test statistics for gamma distribution. Specifically, when only one of the two GOF statistic lead to the conclusion that data are normal (lognormal or gamma), ProUCL outputs the conclusion that the data set follows an approximate normal (lognormal, gamma) distribution; all decision statistics (parametric or nonparametric) are computed based upon this conclusion. Due to these changes, UCL(s) suggested by ProUCL 5.1 can differ from the UCL(s) suggested by ProUCL 4.1. Some examples illustrating these differences have been considered later in this chapter and also in Chapter 4.0. 2. 4.2 Computation of the UCL of the Mean of a Gamma, G (k, 6), Distribution It is well-known that the use of a lognormal distribution often yields unstable and unrealistic values of the decision statistics including UCLs and UTLs for moderately skewed to highly skewed lognormally distributed data sets; especially when the data set is of a small size (e.g., <30, 50, ...). Even though methods exist to compute 95% UCLs of the mean, UPLs and UTLs based upon gamma distributed data sets (Grice and Bain 1980; Wong 1993; Krishnamoorthy et al. 2008), those methods have not become popular due to their computational complexity and/or the lack of their availability in commercial software packages (e.g., Minitab 16). Despite the better performance (in terms of coverage and stability) of the decision making statistics based upon a gamma distribution, some practitioners tend to dismiss the use of gamma distribution based decision statistics by not acknowledging them (EPA 2009; Helsel 2012) and/or 61 ------- stating that the use of a lognormal distribution is easier to compute the various upper limits. Throughout this document, several examples have been used to illustrate these issues. For gamma distributions, ProUCL software has both approximate (used for n>50) and adjusted (when «<50) UCL computation methods. Critical values of the chi-square distribution and an estimate of the gamma shape parameter, k along with the sample mean are used to compute gamma UCLs. As seen above, computation of an MLE of k is quite involved, and this works as a deterrent to the use of a gamma distribution-based UCL of the mean. However, the computation of a gamma UCL currently should not be a problem due to the easy availability of statistical software to compute these estimates. It is noted that some of the gamma distribution based methods incorporated in ProUCL (e.g., prediction limits, tolerance limits) are also available in the R Script library. Update in ProUCL 5.0 and Higher Versions: For gamma distributed data sets, all versions of ProUCL compute both adjusted and approximate gamma UCLs. However, in earlier versions of ProUCL, an adjusted gamma UCL was recommended for data sets of sizes <40 (instead of 50 as in ProUCL 5.1), and an approximate gamma UCL was recommended for data sets of sizes>40, whereas ProUCL 5.1 suggests using approximate gamma UCL for sample sizes >50. Given a random sample, x\, x2, ... , xn, of size n from a gamma, G(k, 6), distribution, it can be shown that 1m, 19 follows a chi-square distribution, j^nk with v = Ink degrees of freedom (df). When the shape parameter, k, is known, a uniformly most powerful test of size of a of the null hypothesis, H0: [j.\ > Cs, against the alternative hypothesis, HA: n\ < Cs, is to reject H0 if x/Cs < x^nlc(a)/2nk . The corresponding (1 - a) 100% uniformly most accurate UCL for the mean, ju\, is then given by the probability statement. l-a (2-33) Where, %2u(a) denotes the cumulative percentage point of the chi-square distribution (e.g., a is the area in the left tail) with v (=2nk) df. That is, if Y follows xl, thenP(7 < xl(.aJ) = a • m practice, k is not known and needs to be estimated from data. A reasonable method is to replace k by its bias-corrected estimate,A: , as given by equation (2-29). This yields the following approximate (1 - a)*100 UCL of the mean,//i. Approximate - UCL = 2nk*x/xlnp («) (2-34) It should be pointed out that the UCL given by equation (2-34) is an approximate UCL without guarantee that the confidence level of (1 - a) will be achieved by this UCL. Simulation results summarized in Singh, Singh, and laci (2002) suggest that an approximate gamma UCL given by (2-34) does provide the specified coverage (95%) for values of k > 0.5. Therefore, for values of k> 0.5, one should use the approximate gamma UCL given by equation (2-34) to estimate the EPC. For smaller sample sizes, Grice and Bain (1980) computed an adjusted probability level, ft (adjusted level of significance), which can be used in (2-34) to achieve the specified confidence level of (7 - a). For a = 0.05 (confidence coefficient of 0.95), a = 0.1, and a = 0.01, these probability levels are given below in Table 2-2 for some values of the sample size n. One can use interpolation to obtain an adjusted ft for values of n not covered in Table 2-2. The adjusted (1 - a)*100 UCL of the gamma mean, ju\ = kO, is given by the following equation: 62 ------- Adjusted- UCL = 2nk*x/x^- Off) (2-35) Where ft is given in equation (2-2) for a = 0.05, 0.1, and 0.01. Note that as the sample size, n, becomes large, the adjusted probability level, /?, approaches the specified level of significance, a. Except for the computation of the MLE of k, equations (2-34) and (2-35) provide simple chi-square-distribution-based UCLs of the mean of a gamma distribution. It should also be noted that the UCLs given by (2-34) and (2- 35) only depend upon the estimate of the shape parameter, k, and are independent of the scale parameter, 6, and its ML estimate. Consequently, coverage probabilities for the mean associated with these UCLs do not depend upon the values of the scale parameter, 6. Table 2-2. Adjusted Level of Significance, /? « = 0.05 « = 0.1 « = 0.01 N probability level, /? probability level, /? probability level, /? 5 10 20 40 — 0.0086 0.0267 0.0380 0.0440 0.0500 0.0432 0.0724 0.0866 0.0934 0.1000 0.0000 0.0015 0.0046 0.0070 0.0100 For gamma distributed data sets, Singh, Singh, and laci (2002) noted that the coverage probabilities provided by the 95% UCLs based upon bootstrap-t and Hall's bootstrap methods (discussed below) are in close agreement. For larger samples, these two bootstrap methods approximately provide the specified 95% coverage and for smaller data sets (from a gamma distribution), the coverage provided by these two methods is slightly lower than the specified level of 0.95. Notes Note 1: Gamma UCLs do not depend upon the standard deviation of the data set which gets distorted by the presence of outliers. Thus, unlike the lognormal distribution, outliers have reduced influence on the computation of the gamma distribution based upon decision statistics including the UCL of the mean - a fact generally not known to a typical user. Note 2: For all gamma distributed data sets for all values of k and n, all modules and all versions of ProUCL compute the various upper limits based upon the mean and standard deviation obtained using the bias-corrected estimate, k*. As noted earlier, the estimate k* does yield better estimates (reduced bias) for all values of k >0.2. For values of k <0.2, the differences between the various limits obtained using k and A: are not that significant. However from a theoretical point of view, when k <0.2, it is desirable to compute the mean, standard deviation, and the various upper limits using the MLE estimate, k . ProUCL generated output sheets display many intermediate results including k andk ; 9 and 9 . Interested users may want to compute UCLs and other upper limits using MLE estimates, k and 0,ofk and 6 for values of £ described in the above paragraph. 63 ------- 2.4.3 (1- a) *100 UCL of the Mean Based Upon H-Statistic (H-UCL) The one-sided (1 - aj*100 UCL for the mean, ju\, of a lognormal distribution as derived by Land (1971, 1975) is given as follows: UCL = exp(y + 0.5s* + SyH^/Jn-l) (2-36) Tables of H-statistic critical values can be found in Land (1975). When the population is lognormal, Land (1971) showed that theoretically the UCL given by equation (2-36) possesses optimal properties and is the uniformly most accurate unbiased confidence limit. However, in practice, the H-statistic based UCL can be quite disappointing and misleading, especially when the data set is skewed and/or consists of outliers, or represents a mixture data set coming from two or more populations (Singh, Singh, and Engelhardt 1997, 1999; Singh, Singh, and laci 2002). Even a minor increase in the sd, sy, drastically inflates the MVUE of pi and the associated H-UCL. The presence of low as well as high data values increases sy, which in turn inflates the H-UCL. Furthermore, it has been observed (Singh, Singh, Engelhardt 1997, 1999) that for samples of sizes smaller than 20-30 (sample size requirement also depends upon skewness), and for values of a approaching and exceeding 1.0 (moderately skewed to highly skewed data), the use of the H-statistic results in impractical and unacceptably large UCL values. Notes: In practice, many skewed data sets can be modeled by both gamma and lognormal distributions; however, there are differences in the properties and behavior of these two distributions. Decision statistics computed using the two distributions can differ significantly (see Example 2-2 below). It is noted that some recent documents (Helsel and Gilroy, 2012) incorrectly state that the two distributions are similar. Helsel (2012, 2012a) suggests the use a lognormal distribution due its computational ease. However, one should not compromise the accuracy and defensibility of estimates and decision statistics by using easier methods which may underestimate (e.g., using a percentile bootstrap UCL based upon a skewed data set) or overestimate (e.g., H-UCL) the population mean. Computation of defensible estimates and decision statistics taking the sample size and data skewness into consideration is always recommended. For complicated and skewed data sets, several UCL computation methods (e.g., bootstrap-t, Chebyshev inequality, and Gamma UCL) are available in ProUCL to compute appropriate decision statistics (UCLs, UTLs) covering a wide-range of data skewness and sample sizes. For lognormally distributed data sets, the coverage provided by the bootstrap-t 95% UCL is a little lower than the coverage provided by the 95% UCL based upon Hall's bootstrap method (Appendix B). However, it is noted that for lognormally distributed data sets, the coverage provided by these two bootstrap methods is significantly lower than the specified 0.95 coverage for samples of all sizes. This is especially true for moderately skewed to highly skewed (a >1.0) lognormally distributed data sets. For such data sets, a Chebyshev inequality based UCL can be used to estimate the population mean. The H- statistic often results in unstable values of the UCL95, especially when the sample size is small, «<20, as shown in Examples 2-1 through 2-3. Example 2-1. Consider the silver data set with n=56 (from NADA for R package [Helsel, 2013]). The normal GOF test graph is shown in Figure 2-1. It can be seen that the data set has an extreme outlier (an observation significantly different from the main body of the data set). The data set contains NDs, and therefore is considered in Chapter 4 and 5 again. Here this data set is considered assuming that all observations represent detected values. The data set does not follow a gamma distribution (Figure 2-3) but can be modeled by a lognormal distribution as shown in Figure 2-2, accommodating the outlier 560. The histogram shown in Figure 2-4 suggests that data are highly skewed. The sd of the logged data = 1.74. 64 ------- The various UCLs computed using ProUCL 5.0 are displayed in Table 2-3 (with outlier) and Table 2-4 (without outlier) following the Q-Q plots. Normal Q-Q Plot for Silver Silva (i-BG Mean = 1545 Sd = 75.19 Slope = 31.03 Intercept = 15.45 Conelalion. R = 0.406 Shapiro-WifcTesI Appiox.Tesl Value = 11206 p Value = 0.000 • Best Fit Line Theoretical Ouantiles (Standard Normal) Figure 2-1. Normal Q-Q Plot of Raw Data in Original Scale Lognormal Q-Q Plot for Silver Mean = 0.6 Sd = 1.746 Slope-1.732 Intercept = 16 Correlation, R =0.975 Lilhflurj I eii Test Statistic -0.117 Critical Value(0.05l = 0.11 Data Appeal Lognoimal • Best Fit Line Theoretical Ouantiles (Standard Normal) Figure 2-2. Lognormal Q-Q plot with GOF Test Statistics 65 ------- Gamma Q-Q Plot for Silver N=56 Mean = 15. 4482 k star = 0 3141 tbetasl------- Table 2-3. Lognormal and Nonparametric UCLs for Silver Data including the outlier 560 Silver Total Number of Observations General Statistics 56 Minimum 0.1 Maximum 560 SD 75.19 Coefficient of Variation 4.868 Number of Distinct Observations 22 Number of Missing Observations 0 Mean 15.45 Median 1.3 Std, Error of Mean 10.05 Skewness 7.174 Lognormal GOF Test Shapiro Wilk Test Statistic 0.951 Shapiro Wilk Lognormal GOF Test 5% Shapiro Wilk P Value 0.0464 Data Not Lognormal at 5% Significance Level Ulliefore Test Statistic 0117 LJIIiefors Lognormal GOF Test 5% Ljlliefore Critical Value 0.118 Data appear Lognormal at 5% Significance Level Data appear Approximate Lognormal at 5% Significance Level Lognormal Statistics Minimum of Logged Data -2.303 Maximum of Logged Data 6.328 Mean of logged Data SD of logged Data 0.6 1.746 Assuming Lognonnal Distribution 95%H-UCL 1S.54 95% Chebyshev (MVUE) UCL 19.12 99% Chebyshev (MVUE) UCL 33.59 Nonparametric Distribution Free UCLs 95%CLTUCL 31.98 95% Standard Bootstrap UCL 32.23 95% Hall's Bootstrap UCL 94.1 95% BCA Bootstrap UCL 52.45 90% Chebyshev(Mean. Sd} UCL 45.59 97.5% Chebyshev{Mean, Sd) UCL 78.2 90% Chebyshev (MVUE) UCL 15.61 97.5% Chebyshev (MVUE) UCL 24 95% Jackknife UCL 32.26 95% Bootstrap-t UCL 180.4 95% Percentile Bootstrap UCL 35.5 95% Chebyshev{Mean. Sd) UCL 59.25 99% Chebyshev(Mean, Sd) UCL 115.4 Suggested UCL to Use 95% H-UCL 13.54 The histogram without the outlier is shown in Figure 2-5. The data is positively skewed with skewness = 5.45. UCLs based upon the data set without the outlier are summarized in Table 2-4 as follows. A quick comparison of the results presented in Tables 2-3 and 2-4 reveals how the presence of an outlier distorts the various decision making statistics. 67 ------- Histogram for Silver without Outlier 560 Silva Minimum Maximum SD Ekewneii Kurtosis 0 Medlar 0 Normal Distiibutior Figure 2-5. Histogram of Silver Data Set Excluding Outlier 560 Table 2-4. Lognormal and Nonparametric UCLs Not Including the Outlier Observation 560 Silver General Statistics Total Number of Observations 55 Number of Distinct Observations 21 Number of Missing Observations 0 Minimum 0.1 Maximum 90 SD 12.95 Coefficient of Variation 2.334 Mean 5.547 Median 1.2 Std. Error of Mean 1.746 Skewness 5.45 Lognormal GOF Test Shapiro Wilk Test Statistic 0.959 5% Shapiro Wilk P Value 0.114 Ulliefors Test Statistic 0.122 5% LJlliefors Critical Value 0.119 Shapiro Wilk Lognormal GOF Test Data appear Lognormal at 5% Significance Level I lilliefors Lognonnal GOF Test Data Not Lognormal at 5% Significance Level Data appear Approximate Lognormal at 5% Significance Level Lognonnal Statistics Minimum of Logged Data -2.303 Maximum of Logged Data 4.5 Mean of logged Data 0.496 SD of logged Data 1.577 Assuming Lognormal Distribution 95%H-UCL 11.11 90% Chebyshev (MVUE) UCL 10.13 95% Chebyshev (MVUE) UCL 12.26 97.5% Chebyshev (M VU E) UC L 1 5.22 99'. Chebyshev (MVUE) UCL 21 .04 Nonparametric Distribution Free UCLs 95SCLTUCL S.419 95% Jackknife UCL 8.469 95% Standard Bootstrap UCL S.371 95% Bootstrap UCL 12.12 95't Halls Bootstrap UCL 19.2 95% BCA Bootstrap UCL 10.47 90% Chebyshev(Hean, Sd) UCL 10.78 97.5*4 Chebyshev(Mean.Sd) UCL 16.45 95% Percentile Bootstrap UCL 8.642 95% Chebyshev(Mean, Sd) UCL 1 3.16 99% Chebyshev(Mean. Sd) UCL 22.92 Suggested UCL to Use 95%H-UCL 11.11 68 ------- Example 2-2: The positively skewed data set consisting of 25 observations, with values ranging from 0.35 to 170, follows a lognormal as well as a gamma distribution. The data set is: 0.3489, 0.8526, 2.5445, 2.5602, 3.3706, 4.8911, 5.0930, 5.6408, 7.0407, 14.1715, 15.2608, 17.6214, 18.7690, 23.6804, 25.0461, 31.7720, 60.7066, 67.0926, 72.6243, 78.8357, 80.0867, 113.0230, 117.0360, 164.3302, and 169.8303. The mean of the data set is 44.09. The data set is positively skewed with sd of log-transformed data = 1.68. The normal GOF results are shown in the Q-Q plot of Figure 2-6, it is noted that the data do not follow a normal distribution. The data set follows a lognormal as well as a gamma distribution as shown in Figures 2-6a and 2-6b and also in Tables 2-5 and 2-6. The various lognormal and nonparametric UCL95s (Table 2-5) and Gamma UCL95s (Table 2-6) are summarized in the following. • The lognormal distribution based UCL95 is 229.2 which is unacceptably higher than all other UCLs and an order of magnitude higher than the sample mean of 44.09. A more reasonable Gamma distribution based UCL95 of the mean is 74.27 (recommended by ProUCL). • The data set is highly skewed (Figure 2-6) with sd of the log-transformed data = 1.68; a Student's t- UCL of 61.66 and a nonparametric percentile bootstrap UCL95 of 60.32 may represent underestimates of the population mean. • The intent of the ProUCL software is to provide users with methods which can be used to compute reliable decision statistics required to make decisions which are cost-effective and protective of human health and the environment. Q-Q Plot for X Theoretical Quantiles (Standard Normal) Figure 2-6. Normal Q-Q Plot of X 69 ------- Gamma Q-Q Plot for X Mean-44 0892 kstai-0.5924 thetastar = 74.4199 Slope = 0.9291 Intercept = 4.0114 Correlation, R =0.9685 Anderson-Darling Test Test Statistic-0.374 Critical Value[0 05) = 0.794 Data appear Gamma Distributed • Best Fit Line Theoretical Quantiles of Gamma Distribution Figure 2-6a. Gamma Q-Q Plot of X Lognormal Q-Q Plot for X Slope = 1696 Intercept-^. 335 Correlation, R = 0.97S Shapiio-Wilk Test Exact Test Statistic = 0948 DiticalValue(0.05] = 0.918 Data Appear Lognormal Appiox. Test Value-0.949 p-Vabe = 0.247 • Best Fit Line Theoretical Quantiles (Standard Normal) Figure 2-6b. Lognormal Q-Q Plot of X 70 ------- Table 2-5. Nonparametric and Lognormal UCL95 General Statistics Total Number of Observations 25 Minimum 0-349 Maximum 163.3 SD 5134 Coefficient of Variation 1-1 64 Number of Distinct Observations 25 Number of Missing Observations 0 Mean 44.09 Median 18.77 Std. Eirorof Mean 10.27 Skewness 1.234 Lognormal GOF Test Shapiro Wilk Test Statistic 0.94B Shapiro Wilk Lognonrial GOF Test 5% Shapiro Wilk Critical Value 0.318 Data appear Lognormal at 5% Signtficance Level Uliefors Test Statistic 0.135 Lilliefors Lognonnal GOF Test 5% Lilliefors Critical Value (3.177 Data appear Lognormai at 5% Signtficance Level Data appear Lognormal at 5% Significance Level Lognonnal Statistics Minimum of Logged Data -1.053 Maximum of Logged Data 5.135 Assigning Lognonnal Distribution 35% H-LICL 223.2 95% Chebyshev (MVUE) UCL 176.3 99% Chebyshev (MVUE) UCL 323 Nonparametric Distribution Free UCLs 35% CLT UCL 60.93 95% Standard Bootstrap UCL 60.57 95% Halls Bootstrap UCL 62.55 95% BCA Bootstrap UCL 64.8 30%Chebyshev{Mean,Sd)UCL 74.83 97.5% ChebyshevfMean. Sd) UCL 108.2 Mean of logged Data SD of logged Data 2.S35 1.68 90% Chebyshev (M VU E) DC L 140.6 97.5% Chebyshev (MVUE) UCL 225.8 95% Jackfcnife UCL 61.66 95% Bootstrap^ UCL 65.58 95% Percentile Bootstrap UCL 60.32 35% ChebyshevfMean, Sd} UCL 83.85 39% ChebyshevfMean. Sd} UCL 146.3 Notes: The use of H-UCL is not recommended for moderately skewed to highly skewed data sets of smaller sizes (e.g., 30, 50, 70, etc.). ProUCL computes and outputs H-statistic based UCLs for historical and academic reasons. This example further illustrates that there are significant differences between a lognormal and a gamma model; for positively skewed data sets, it is recommended to test for a gamma model first. If data follow a gamma distribution, then the UCL of the mean should be computed using a gamma distribution. The use of nonparametric methods is preferred when computing a UCL95 for skewed data sets which do not follow a gamma distribution. 71 ------- Table 2-6. Gamma UCL95 Total Number of Observations General Statistics 25 Minimum 0.349 Maximum 163.8 SD 51.34 Coefficient of Variation 1.164 Number of Distinct Observations 25 Number of Missing Observations 0 Mean 44.09 Median 18.77 SD of logged Data 1.68 Skewness 1.294 Gamma GOF Test A-D Test Statistic 0.374 Anderson-Darling Gamma GOF Test 5% A-D Critical Value 0.794 Data appear Gamma Distributed at 5% Significance Level K-S Test Statistic 0113 Kolmogrov Smirnoff Gamma GOF Test 5% K-S Critical Value 0.1S3 Data appear Gamma Distributed at 5% Significance Level Data appear Gamma Distributed at 5% Significance Level Gamma Statistics k hat (MLE) 0.643 Theta hat (MLE) 68.58 nu hat (MLE) 32.15 MLE Mean (bias corrected) 44.09 Adjusted Level of Significance 8.0395 Assuming Gamma Distribution 95% .Approximate Gamma UCL 71.77 Suggested UCL to Use 35% Adjusted Gamma UCL 7^.27 k star (bias corrected MLE) 0.592 Theta star {bias corrected MLE) 74.42 nu star {bias corrected) 29.62 MLE Sd (bias corrected) 57.28 Approximate Chi Square Value {0.05} 18.2 Adjusted Chi Square Value 17.59 35% .Adjusted Gamma UCL 2.4.4 (1 - a)* 100 UCL of the Mean Based upon Modified-t-Statistic for Asymmetrical Populations It is well known that percentile bootstrap, standard bootstrap, and Student's t-statistic based UCL of the mean do not provide the desired coverage of a population mean (Johnson 1978, Sutton 1993, Chen 1995, Efron and Tibshirani 1993) of skewed data distributions. Several researchers including: Chen (1995), Johnson (1978), Kleijnen, Kloppenburg, and Meeuwsen (1986), and Sutton (1993) suggested the use of the modified-t-statistic and skewness adjusted CLT for testing the mean of a positively skewed distribution. The UCLs based upon the modified t-statistic and adjusted CLT methods were included in earlier versions of ProUCL (e.g., versions 1.0 and 2.0) for research and comparison purposes prior to the availability of Gamma distribution based UCLs in ProUCL 3.0 (2004). Singh, Singh, and laci (2002) noted that these two skewness adjusted UCL computation methods work only for mildly skewed distributions. These methods have been retained in later versions of ProUCL for academic reasons. The (1 - a)*100 UCL of the mean based upon a modified t-statistic is given by: (2-37) 72 ------- Where ju3, an unbiased moment estimate (Kleijnen, Kloppenburg, and Meeuwsen 1986) of the third central moment is given as follows: This modification for a skewed distribution does not perform well even for mildly to moderately skewed data sets. Specifically, the UCZ given by equation (2-37) may not provide the desired coverage of the population mean, ju\, when a starts approaching and exceeding 0.75 (Singh, Singh, and laci 2002). This is especially true when the sample size is smaller than 20-25. This small sample size requirement increases as a increases. For example, when a starts approaching and exceeding 1 to 1.5, the UCL given by equation (2-37) does not provide the specified coverage (e.g., 95%), even for samples as large as 100. 2. 4.5 (1 - a) *100 UCL of the Mean Based upon the Central Limit Theorem The CLT states that the asymptotic distribution, as n approaches infinity, of the sample mean, xn , is normally distributed with mean, [j.\, and variance, o\2ln irrespective of the distribution of the population. More precisely, the sequence of random variables given by: a/ -\in has a standard normal limiting distribution. For large sample sizes, n, the sample mean, X , has an approximate normal distribution irrespective of the underlying distribution function (Hogg and Craig 1995). The large sample requirement depends upon the skewness of the underlying distribution function of individual observations. The large sample requirement for the sample mean to follow a normal distribution increases with skewness. Specifically, for highly skewed data sets, even samples of size 100 may not be large enough for the sample mean to follow a normal distribution. This issue is illustrated in Appendix B. Since the CLT method requires no distributional assumptions, this is a nonparametric method. As noted by Hogg and Craig (1995), if a\ is replaced by the sample standard deviation, sx, the normal approximation for large n is still valid. This leads to the following approximate large sample (1 - a)* 100 UCL of the mean: UCL =x + zasjn (2-40) An often cited and used rule of thumb for a sample size associated with a CLT based method is n > 30. However, this may not be adequate if the population is skewed, specifically when a (sd of log- transformed variable) starts exceeding 0.5 to 0.75 (Singh, Singh, laci 2002). In practice, for skewed data sets, even a sample as large as 100 is not large enough to provide adequate coverage to the mean of skewed populations. Noting these observations, Chen (1995) proposed a refinement of the CLT approach, which makes a slight adjustment for skewness. 73 ------- 2. 4.6 (1 - a)* 100 UCL of the Mean Based upon the Adjusted Central Limit Theorem (Adjusted-CLT) The "adjusted-CLT" UCL is obtained if the standard normal quantile, za, in the upper limit of equation (2-40) is replaced by the following adjusted critical value (Chen 1995): Thus, the adjusted- CLT (1 - a)* 100 UCL for the mean, ju\, is given by UCL = x + [za + 4 (1 + 2zza )/(6 Vw) jyx /Vw (2-42) Here k3 , the coefficient of skewness (raw data), is given by Skewness (raw data) k3 = Ju3/s3x (2-43) where, fi3, an unbiased estimate of the third moment, is given by equation (2-38). This is another large sample approximation for the UCL of the mean of skewed distributions. This is a nonparametric method, as it does not depend upon any of the distributional assumptions. Just like the modified-t-UCL, it is observed that the adjusted-CLT UCL also does not provide the specified coverage to the population mean when the population is moderately skewed, specifically when a becomes larger than 0.75. This is especially true when the sample size is smaller than 20 to25. This large sample size requirement increases as the skewness (or a) increases. For example, when a starts approaching and exceeding 1.5, the UCL given by equation (2-42) does not provide the specified coverage (e.g., 95%), even for samples as large as 100. It is noted that UCL given by (2-42) does not provide adequate coverage to the mean of a gamma distribution, especially when the shape parameter (or its estimate) k< 1.0 and the sample size is small. Notes: UCLs based upon these skewness adjusted methods, such as the Johnson's modified-t and Chen's adjusted-CLT, do not provide the specified coverage to the population mean even for mildly to moderately skewed (e.g., a in [0.5, 1 .0]) data sets. The coverage of the population mean provided by these UCLs becomes worse (much smaller than the specified coverage) for highly skewed data sets. These methods have been retained in ProUCL 5.1 for academic and research purposes. 2. 4. 7 Chebyshev (1 - a) *100 UCL of the Mean Using Sample Mean and Sample sd Several commonly used UCL95 computation methods (e.g., Student's t-UCL, percentile and BCA bootstrap UCLs) fail to provide the specified coverage (e.g., 95%) to the population mean of skewed data sets. The use of a lognormal distribution based H-UCL (EPA 2006a, EPA 2009) is still commonly used to estimate EPCs based upon lognormally distributed skewed data sets. However, the use of Land's H- statistic yields unrealistically large UCL95 values for moderately skewed to highly skewed data sets. On the other hand, when the mean of a logged data set is negative, the H-statistic tends to yield an 74 ------- impractically low value of H-UCL (See Example 2-1 above) especially when the sample size is large (e.g., > 30-50). To address some of these issues associated with lognormal H-UCLs, Singh, Singh, and Engelhardt (1997) proposed the use of the Chebyshev inequality to compute a UCL of the mean of skewed distributions. They noted that a Chebyshev UCL tends to yield stable, realistic, and conservative estimates of the EPCs. The use of the Chebyshev UCL has been recently adopted by the ITRC (2012) to compute UCLs of the mean based upon data sets obtained using the incremental sampling methodology (ISM) approach. For moderately skewed data sets, the Chebyshev inequality yields conservative but realistic UCL95 . For highly skewed data sets, even a Chebyshev inequality fails to yield a UCL95 providing 95% coverage for the population mean (Singh, Singh, and laci 2002; Appendix B). To address these issues, ProUCL computes and displays 97.5% or 99% Chebyshev UCLs. The user may want to consult a statistician to select the most appropriate UCL (e.g., 95% or 97.5% UCL) for highly skewed nonparametric data sets. Since the use of the Chebyshev inequality tends to yield conservative UCL95s, especially for moderately skewed data sets of large sizes (e.g., >50), ProUCL 5.1 also outputs a UCL90 based upon the Chebyshev inequality. The two-sided Chebyshev theorem (Hogg and Craig 1995) states that given a random variable, X, with finite mean and standard deviation, /j.\ and o\, we have P(-kal < x - ft < kxr1 ) > 1 - 1 / k2 (2-44) This result can be applied to the sample mean, x (with mean, ju\ and variance, a\ /«), to compute a conservative UCL for the population mean, /j.\. For example, if the right side of equation (2-44) is equated to 0.95, then k = 4.47, and UCL = x + 4.47^ / ~Jn represents a conservative 95% upper confidence limit for the population mean, ju\. Of course, this would require the user to know the value of a\. The obvious modification would be to replace a\ with the sample standard deviation, sx, but since this is estimated from data, the result is not guaranteed to be conservative. However, in practice, the use of the sample sd does yield conservative values of the UCL95 unless the data set is highly skewed with sd of the log- transformed data exceeding 2 to 2.5, and so forth. In general, the following equation can be used to obtain a (1 - a)* 100 UCL of the population mean, ju\: UCL =x + V(1/«K /Jn (2-45) A slight refinement of equation (2-45) is given as follows: UCL = x + ((l/a)-V)sx n (2-46) All versions of ProUCL compute the Chebyshev (1 - a)* 100 UCL of the population mean using equation (2-46). This UCL is labeled as Chebyshev (Mean, Sd) on the output sheets generated by ProUCL. Since this Chebyshev method requires no distributional assumptions, it is a nonparametric method. This UCL may be used to estimate the population mean, ju\, when the data are not normal, lognormal, or gamma distributed, especially when sd, a (or its estimate, sy) becomes large such as > 1.5. From simulation results summarized in Singh, Singh, and laci (2002) and graphical results presented in Appendix B, it is observed that for highly skewed gamma distributed data sets (with shape parameter k < 0.5), the coverage provided by the Chebyshev 95% UCL (given by equation (2-46)) is smaller than the 75 ------- specified coverage of 0.95. This is especially true when the sample size is smaller than 10-20. As expected, for larger samples sizes, the coverage provided by the 95% Chebyshev UCL is at least 95%. For larger samples, the Chebyshev 95% UCL tends to result in a higher (but stable) UCL of the mean of positively skewed gamma distributions. Based upon the number of observations and data skewness, ProUCL suggests using a 95%, 97.5%, or a 99% Chebyshev UCL. If these limits appear to be higher than expected, collectively the project team should make the decision regarding using an appropriate confidence coefficient (CC) to compute a Chebyshev inequality based upper limit. ProUCL can compute upper limits (e.g., UCLs, UTLs) for any user-specified level of confidence coefficient in the interval [0.5, 1]. For convenience, ProUCL 5.0 also displays Chebyshev inequality based 90% UCL of the mean. Note about Chebyshev Inequality based UCLs: The developers of ProUCL have made significant efforts to make suggestions that allows the user to choose the most appropriate UCL95 to estimate the EPC. However, suggestions made in ProUCL may not cover all real world data sets, especially smaller data sets with higher variability. Based upon the results of the simulation studies and graphical displays presented in Appendix B, the developers noted that for smaller data sets with high variability (e.g., sd of logged data >1, 1.5, etc.) even a conservative Chebyshev UCL95 tends not to provide the desired 95% coverage to the population mean. In these scenarios, ProUCL suggests the use of a Chebyshev UCL97.5 or a Chebyshev UCL99 to provide the desired coverage (0.95) for the population mean. It is suggested that when data are highly skewed and ProUCL is recommending the use of a Chebyshev inequality based UCL, the project team collectively determines which UCL will be the most appropriate to address the project needs. ProUCL can calculate UCLs for many levels including non-typical levels such as 98%, 96%, 92%. 2.4.8 Chebyshev (1 - a) *100 UCL of the Mean of a Lognormal Population Using the MVUE of the Mean and its Standard Error Earlier versions of ProUCL (when gamma UCLs were not available in ProUCL) used equation (2-44) on the MVUEs of the lognormal mean and sd to compute a UCL (denoted by (1 - a)* 100 Chebyshev (MVUE)) of the population mean of a lognormal population. In general, if ^i is an unknown mean, /^ is an estimate, and al (/}j) is an estimate of the standard error of//j, then the following equation: UCL = & +V((l/«)-l)------- For a confidence coefficient of 0.95, ProUCL UCLs/EPCs module makes suggestions which are based upon the extensive experience of the developers of ProUCL with environmental statistical methods, published literature (Singh, Singh, and Engelhardt 1997, Singh and Nocerino 2002, Singh, Singh, and laci 2002, and Singh, Maichle, and Lee 2006) and procedures described in the various guidance documents. However, the project team is responsible for determining whether to use the suggestions made by ProUCL. This determination should be based upon the conceptual site model (CSM), expert site and regional knowledge. The project team may want to consult a statistician. 2.4.9 (1 - a) *100 UCL of the Mean Using the Jackknife and Bootstrap Methods Bootstrap and jackknife methods (Efron 1981, 1982; Efron and Tibshirani 1993) are nonparametric statistical resampling techniques which can be used to reduce the bias in point estimates and construct approximate confidence intervals for parameters, such as the population mean, population percentiles. These methods do not require any distributional assumptions and can be applied to a variety of situations. The bootstrap methods incorporated in ProUCL for computing upper limits include: the standard bootstrap method, percentile bootstrap method, BCA percentile bootstrap method, bootstrap-t method (Efron, 1981, 1982; Hall 1988), and Hall's bootstrap method (Hall 1992; Manly 1997). As before, let x\, x2, ... , x» represent a random sample of size n from a population with an unknown parameter, 6, and let 6 be an estimate of 9, which is a function of all n observations. Here, the parameter, #, could be the population mean and a reasonable choice for the estimate, 6, might be the sample mean, x . Another choice for 6 is the MVUE of the mean of a lognormal population, especially when dealing with lognormally distributed data sets. 2.4.9.1 (l-o.) *100 UCL of the Mean Based upon the Jackknife Method For the jackknife method, n estimates of 9 are computed by deleting one observation at a time (Dudewicz and Misra 1988). For each index, / (1=1,2,...n), denote by #(!), the estimate of 9(computed similarly as 6) omit the /'th observation from the original sample of size n and compute the arithmetic mean of these n jackknifed estimates using: 0=-£0(0 (2-48) n i=\ A quantity known as the /'th "pseudo-value" is given by: J,=nO-(n-1)0(0 (2-49) Using equations (2-48) and (2-49), compute the jackknife estimator of #as follows: J(0) = - Y J. = n6 - (n -1)0 (2-50) 11 ------- If the original estimate 6 is biased, then under certain conditions, part of the bias is removed by the /*, jackknife method, and an estimate of the standard error (SE) of the jackknife estimate, J(0), is given by (2-51) // Next, using the jackknife estimate, compute a t-type statistic given by (2-52) The t-type statistic given above follows an approximate Student's t- distribution with (n - 1) df, which can be used to derive the following approximate (1-a)* 100% UCL for 9, If the sample size, n, is large, then the upper a* Hpantile in the above equation can be replaced with the corresponding upper a* standard normal quantile, za. Observe, also, that when 6 is the sample mean, x , then the jackknife estimate is the same as the sample mean, J(x) = x, the estimate of the standard error given by equation (2-51) simplifies to sjn112, and the UCL in equation (2-53) reduces to the familiar t- statistic based UCL given by equation (2-32). ProUCL uses the jackknife estimate as the sample mean, that yields J(x) = X , which in turn translates equation (2-53) to Student's t- UCL given by equation (2- 32). This method has been included in ProUCL to satisfy the curiosity of those users who are unaware that the jackknife method (with sample mean as the estimator) yields a UCL of the population mean identical to the UCL based upon the Student's t- statistic as given by equation (2-32). Notes: It is well known that the Jackknife method (with sample mean as an estimator) and Student's t- method yield identical UCLs. However, some users may be unaware of this fact, and some researchers may want to see these issues described and discussed in one place. Also, it has been suggested that a 95% UCL based upon the Jackknife method on the full data set obtained using the robust ROS (LROS) method may provide adequate coverage (Shumway, Kayhanian, and Azari 2002) to the population mean of skewed distributions, which of course is not true since like Student's t-UCL, the Jackknife UCL does not account for data skewness. Finally, users are cautioned to note that for large data sets («>10,000), the Jackknife method may take a long time (several hours) to compute a UCL of the mean. 2.4.9.2 (1 - a)*100 UCL of the Mean Based upon the Standard Bootstrap Method In bootstrap resampling methods, repeated samples of size n each are drawn with replacement from a given data set of size n. The process is repeated a large number of times (e.g., 2000 times), and each time an estimate,$., of 6 is computed. The estimates are used to compute an estimate of the SE of 9. A description of the bootstrap methods, illustrated by application to the population mean, /j.\, and the sample mean, x , is given as follows. 78 ------- Step 1. Let (Jen, Xi2, ... , xm) represent the /'th bootstrap sample of size n with replacement from the original data set, (x\, x2, ..., x»); denote the sample mean using this bootstrap sample by xt. Step 2. Repeat Step 1 independently TV times (e.g., 1000-2000), each time calculating a new estimate. Denote these estimates (KM means, ROS means) by xl,x2,..., XN. The bootstrap estimate of the population mean is the arithmetic mean, XB, of the TV estimates xf: /' := 1, 2, ..., N. The bootstrap estimate of the SE of the estimate, x , is given by: (2-54) If some parameter, 6 (e.g., the population median), other than the mean is of concern with an associated estimate (e.g., the sample median), then same steps described above are applied with the parameter and its estimates used in place of ^i and x . Specifically, the estimate, Qi , would be computed, instead of xt , for each of the TVbootstrap samples. The general bootstrap estimate, denoted by<9B , is the arithmetic mean of those TV estimates. The difference,^ -0, provides an estimate of the bias in the estimate, 6 , and an estimate of the SE of 9 is given by: (2-55) A (l-a)*100 standard bootstrap UCL for 6>is given by UCL=9 + zaaB (2-56) ProUCL computes the standard bootstrap UCL by using the population mean and sample mean, given by //i and x . The UCL obtained using the standard bootstrap method is quite similar to the UCL obtained using the Student's t-statistic given by equation (2-32), and, as such, does not adequately adjust for skewness. For skewed data sets, the coverage provided by the standard bootstrap UCL is much lower than the specified coverage (e.g., 0.95). Notes: Typically, bootstrap methods are not recommended for small data sets consisting of less than 10- 15 distinct values. Also, it is not desirable to use bootstrap methods on larger (« > 500) data sets. For small data sets, several bootstrap re-samples could be identical and/or all values in a bootstrap re-sample could be identical; no statistical computations can be performed on data sets with all identical observations. For larger data sets, there is no need to perform and use bootstrap methods as a large data set is already representative of the population itself. Methods based upon normal approximations, applied to data sets of larger sizes (n > 500), yield good estimates and results. Also, for larger data, bootstrap methods and especially the Jackknife method can take a long time to compute statistics of interest. 79 ------- 2.4.9.3 (1 - a) *100 UCL of the Mean Based upon the Simple Percentile Bootstrap Method Bootstrap resampling of the original data set of size n is used to generate the bootstrap distribution of the unknown population mean. In this method, the TV bootstrapped means, xt, /:=1,2,...,7V, are arranged in ascending order as^(1) < x(2} <•••< x(N). The (1 - a)* 100 UCL of the population mean, [j,\, is given by the value that exceeds the (1 - a)* 100 of the generated mean values. The 95% UCL of the mean is the 95th percentile of the generated means and is given by: 95%Percentile UCL = 95th% jf.;;: = 1, 2, ..., N (2-57) For example, when N= 1000, the bootstrap 95% percentile UCL is given by the 950th ordered mean value given by jc . It is well-known that for skewed data sets, the UCL95 of the mean based upon the percentile bootstrap method does not provide the desired coverage (95%) for the population mean. The users of ProUCL and other software packages are cautioned about the suggested use of the percentile bootstrap method for computing UCL95s of the mean based upon skewed data sets. Noting the deficiencies associated with the upper limits (UCLs) computed using the percentile bootstrap method, researchers (Efron 1981; Hall 1988, 1992; Efron and Tibshirani 1993) have developed and proposed the use of skewness adjusted bootstrap methods. Simulations results and graphs presented in Appendix A verify that for skewed data sets, the coverage provided by the percentile bootstrap UCL95 and standard bootstrap UCL is much lower than the coverages provided by the UCL95s based upon the bootstrap-t and the Hall's bootstrap methods. It is observed that for skewed (lognormal and gamma) data sets, the BCA bootstrap method performs slightly better (in terms of coverage probability) than the percentile method. 2.4.9.4 (1 - a)*100 UCL of the Mean Basedupon the Bias-Corrected Accelerated (BCA) Percentile Bootstrap Method The BCA bootstrap method adjusts for bias in the estimate (Efron and Tibshirani 1993; and Manly 1997). Results and graphs summarized in Appendix B suggest that the BCA method does provide a slight improvement over the simple percentile and standard bootstrap methods. However, for skewed data sets (parametric or nonparametric), the improvement is not adequate enough and yields UCLs with a coverage probability much lower than the coverage provided by bootstrap-t and Hall's bootstrap methods. This is especially true when the sample size is small. For skewed data sets, the BCA method also performs better than the modified-t-UCL. Based upon gamma distributed data sets, the coverage provided by the BCA 95%UCL approaches 0.95 as the sample size increases. For lognormally distributed data sets, the coverage provided by the BCA 95%UCL is much lower than the specified coverage of 0.95. The BCA upper confidence limit of intended (1 - a)*100 coverage is given by the following equation: BCA - UCL = Jc("2) (2-58) 80 ------- Here x("2) is the a2*100*percentile computed using TVbootstrap means xt; /': = 1, 2, ..., N. For example, when N = 2000, x(a^ = (a2N)th ordered statistic of the TV bootstrapped means, xt; /': = 1, 2, ..., TV denoted by x(a JV) represents a BCA-UCL; 0.2 is given by the following probability statement: O z0+z^ (2-59) ------- From the simulation results summarized in Singh, Singh, and laci (2002) and in Appendix B, it is observed that for skewed data sets, the bootstrap-t method tends to yield more conservative (higher) UCL values than the other UCLs obtained using the Student's t, modified-t, adjusted-CZr, and other bootstrap methods described above. It is noted that for highly skewed (k < 0.1 or a > 2) data sets of small sizes (n < 10 to 15), the bootstrap-t method performs better (in terms of coverage) than other (adjusted gamma UCL, or Chebyshev inequality UCL) UCL computation methods. 2.4.9.6 (l-o.) *100 UCL of the Mean Based upon Hall's Bootstrap Method Hall (1992) proposed a bootstrap method that adjusts for bias as well as skewness. This method has been included in UCL guidance document for CERCLA sites (EPA 2002a). In this method, xt, sX:i , and&3., the sample mean, the sample standard deviation, and the sample skewness, respectively, are computed from the /'th bootstrap re-sample (/' = 1, 2,..., N) of the original data. Let x be the sample mean, sxbe the /*, sample standard deviation, and k3 be the sample skewness (as given by equation (2-43)) computed using the original data set of size n. The quantities, Wt and Qi, given below are computed for the TV bootstrap samples: W, = (3c;. - 3c)/sx, , and Q, (W, } = W,+ k3,W,213 + kffi 127 + k3i l(6n) The quantities, Qi (Wi.) are arranged in ascending order. For a specified (1 - a) confidence coefficient, compute the (aN)ih ordered value, qa, of the quantities, <2,(J^). Next, compute W(qa~) using the inverse function, which is given as follows: W(qa) = 3^[\ + k3(qa-k3/(6n))) -Ij/k3 (2-63) In equation (2-63), k3 is computed using equation (2-43). Finally, the (1 - a)*100 UCL of the population mean based upon Hall's bootstrap method is given as follows: UCL=x-W(qJsx (2-64) For both lognormal and gamma distributions, bootstrap-t and Hall's bootstrap methods perform better than the other bootstrap methods, namely, the standard bootstrap method, simple percentile, and bootstrap BCA percentile methods. For highly skewed lognormal data sets, the coverages based upon Hall's method and bootstrap-t method are significantly lower than the specified coverage, 0.95. This is true even for samples of larger sizes (n > 100). For lognormal data sets, the coverages provided by Hall's bootstrap and bootstrap-t methods do not increase much with the sample size, n. For highly skewed (sd > 1.5, 2.0) data sets of small sizes (n < 15), Hall's bootstrap method and the bootstrap-t method perform better than the Chebyshev UCL, and for larger samples, the Chebyshev UCL performs better than Hall's and bootstrap-t methods. Notes: The bootstrap-t and Hall's bootstrap methods sometimes yield inflated and erratic values, especially in the presence of outliers (Efron and Tibshirani 1993). Therefore, these two methods should 82 ------- be used with caution. If outliers are present in a data set and the project team decides to use them in UCL computations, the use of alternative UCL computation methods (e.g., based upon the Chebyshev inequality) is suggested. These issues are examined in Example 2-3. Also, when a data set follows a normal distribution without outliers, these nonparametric bootstrap methods, percentile bootstrap method, BCA bootstrap method and bootstrap-t method, will yield comparable results to the Student's t-UCL and modified-t UCL. Moreover, when a data set is mildly skewed sd of logged data <0.5), parametric methods and bootstrap methods discussed in this chapter tend to yield comparable UCL values. Example 2-3: Consider the pyrene data set with n = 56 selected from the literature (She 1997; Helsel 2005). The pyrene data set has been used in several chapters of this technical guide to illustrate the various statistical methods incorporated in ProUCL. The pyrene data set contains several NDs and will be considered again in Chapter 4. However, here, the data set is considered as an uncensored data set to discuss the issues associated with skewed data sets containing outliers; and how outliers can distort UCLs based upon bootstrap-t and Hall's bootstrap UCL computation methods. The Rosner outlier test (see Chapter 7) and normal Q-Q plot displayed in Figure 2-7 below confirm that the observation, 2982.45, is an extreme outlier. However, the lognormal distribution accommodated this outlier and the data set with this outlier follows a lognormal distribution (Figure 2-8). Note that the data set including the outlier does not follow a gamma distribution. Q-Q Plot for Pyrene Mean-1732 Sd - 391 4 Sbpe-195.9 lnte(ceDt = 173.2 Cotrelation, R ^0.432 Theoretical Quantiles (Standard Normal) Figure 2-7. Normal Q-Q Plot of She's Pyrene Data Set 83 ------- Lognormal Q-Q Plot for Pyrene Pyrene Mean - 4 66 Srj, 0.787 Slope-0.762 lni«cep,-4.66 Ctmeiation,R= 0.951 Tesi Slaiiste = 0 099 Difc^V^iie[005)-011E Data Appear Lggrormal • Be;! Fit Line Theoretical Quantiles (Standard Normal) Figure 2-8. Lognormal Q-Q Plot of She's Pyrene Data Set Several lognormal and nonparametric UCLs (with outlier) are summarized in Table 2-7 below. Table 2-7. Nonparametric and Lognormal UCLs on Pyrene Data Set with Outlier 2982 Pyrene General Statistics Total Number of Observations 56 Minimum 28 Maximum 2982 SD 391.4 Coefficient of Variation 2.26 Number of Distinct Observations 44 Number of Missing Observations 0 Mean 173.2 Median 104 Std. Error of Mean 52.3 Skewness 6.967 Lognormal GOF Test Shapiro Wilk Test Statistic 0.924 Shapiro Wilk Lognormal GOF Test 5% Shapiro Wilk P Value 0.00174 Data Not Lognormal at 5% Significance Level LJIIiefors Test Statistic 0.0992 Lilliefors Lognormal GOF Test 5% Lilliefors Critical Value 0.118 Data appear Lognormal at 5% Significance Level Data appear Approximate Lc Locator Minimum of Logged Data 3.3j Maximum of Logged Data 8 Assuming Loc 95%H-IJCL 130.2 95% Chebyshev (MVUE) UCL 21 6.3 99% Chebyshev (MVUE) UCL 310.4 Nonparametnc I 95%CLTUCL 259.2 95'i Standard Bootstrap UCL 254.5 m Hall's Bootstrap UCL 588.5 95% BCA Bootstrap UCL 336.7 90% ChebysneviMean. Sd) UCL 330.1 97.5% Chebyshev(Mean. Sd) UCL 499.8 ignormal at 5% Significance Level Tial Statistics 2 Mean of logged Data SD of logged Data normal Distribution 90% Chebyshev (MVUE) UCL 97.5% Chebyshev (MVUE) UCL distribution Free UCLs 95%JackknifeUCL 95% Bootstrap^ UCL 95% Percentile Bootstrap UCL 95% ChebyshevfMean. Sd) UCL 99% ChebyshevfMean. Sd) UCL 4.66 0.787 193.5 248.1 26C.7 525.2 276.5 401.1 693.6 84 ------- Looking at the mean (173.2), standard deviation (391.4), and SE (52.3) in the original scale, the H-UCL (180.2) appears to represent an underestimate of the population mean; a nonparametric UCL such as a 90% Chebyshev or a 95% Chebyshev UCL may be used to estimate the population mean. Since there is an outlier present in the data set, both bootstrap-t (UCL=525.2) and Hall's bootstrap (UCL=588.5) methods yield elevated values for the UCL95. A similar pattern was noted in Example 2-1 where the data set included an extreme outlier. Computations of UCLs without the Outlier 2982 The data set without the outlier follows both a gamma and lognormal distribution with sd of the log- transformed data = 0.649 suggesting that the data are moderately skewed. The gamma GOF test results are shown in Figure 2-9. The UCL output results for the pyrene data set without the outlier are summarized in Table 2-8. Since the data set is moderately skewed and the sample size of 55 is fairly large, all UCL methods (including bootstrap-t and Hall's bootstrap methods) yield comparable results. ProUCL suggested the use of a gamma UCL95. This example illustrates how the inclusion of even a single outlier distorts all statistics of interest. The decision statistics should be computed based upon a data set representing the main dominant population. Gamma Q-Q Plot for Pyrene-1 N-55 kstai- 24544 tlleta star - 49.7506 Steps-1.0899 llllelcepl-.1D.5e72 Cpmelatini. R - 0.9852 Anderson-Darling Test Test Statistic = D 461 Critical Vakle(0 051-0.760 Data appear Gamma Distributed Theoretical Quantiles of Gamma Distribution Figure 2-9. Gamma GOF Test on Pyrene Data Set without the Outlier 85 ------- Table 2-8. Gamma, Nonparametric and Lognormal UCLs on Pyrene Data Set without Outlier=2982 Gamma GOF Test A-D Test Statistic 0.461 Andenson-Darting Gamma GOF Test 0.76 Detected data appear Gamma Distributed at 5% Significance Level 0.0316 Kolmogrov-Smirnoff Gamma GOF Test 0.121 Detected data appear Gamma Distributed at 55, Significance Level 5% A-D Critical Value K-S Test Statistic 5% K-S Critical Value Detected data appear Gamma Distributed at 5% Significance Level k hat (MLE) ThetahatfMLE) nu hat (MLE) MLE Mean (bias coirectedj Gamma Statistics 2.5S3 47.27 234.2 1221 Adjusted Level of Significance 0.0456 k star (bias corrected MLE} Theta star (bias corrected MLE) nu star [bias corrected) MLE Sd (bias corrected) Approximate Chi Square Value (D.05) Adjusted Chi Square Value 2454 49.75 270 77.94 232.9 232 Assuming Gamma Distribution 55% Approximate Gamma UCL (use when n;=5D) 141.5 95% Adjusted Gamma UCL (use when n;50) 142.1 Lognomal GOF Test Shapiro Wilk Test Statistic D.976 Shapiro Wilk Lognomal GOF Test 5'i Shapiro Wilk P Value 0.552 Data appear Lognormal at 5% Significance Level Lilliefore Test Statistic 0.0553 LJIIiefors Lognomal GOF Test 5% Ulliefors Critical Value 0.115 Data appear Lognormal at 5'i Significance Level Data appear Lognormal at 5% Significance Level Lognormal Statistics Minimum of Logged Data 3.332 Maximum of Logged Data €.129 Assuming LognormaJ Distribution 95% H-UCL 146.2 95% Chebyshev (MVU E) UCL 172.6 99% Chebyshev (MVU E) UCL 237.3 Mean of logged Data SD of logged Data 4.599 0.649 90% Chebyshev (MVUE) UCL 156.8 97.5% Chebyshev (MVU E) UCL 194.4 Table 2-8 (continued). Gamma, Nonparametric and Lognormal UCLs on Pyrene Data Set without Outlier=2982 Nonparametric Distribution Free UCL Statistics Data appear to follow a Discernible Distribution at 5% Significance Level Nonparametric Distribution Free UCLs 95% CLT UCL 95% Standard Bootstrap UCL 95% Halls Bootstrap UCL 95% BCA Bootstrap UCL 90%Chebyshev(Mean, Sd) UCL 97.5%Chebyshev(Mean. Sd) UCL 141 141 145 145.1 156.6 193.8 95%JackknifeUCL 141.3 95% Bootstrap-t UCL 146.2 95% Percentile Bootstrap UCL 141.5 95% ChebyshevfMean, Sd) UCL 172.2 99% ChebyshevfMean, Sd) UCL 236.4 Suggested UCL to Use 95% Approximate Gamma UCL 141.5 86 ------- Example 2-4: Consider the chromium concentration data set of size 24 from a real polluted site to illustrate the differences in UCL95 suggested by ProUCL 4.1 and ProUCL 5.0/ProUCL 5.1. The data set is provided here in full as it has been also used in several examples in Chapter 3. Aluminum Arsenic Chromium 6280 3830 3900 5130 9310 15300 9730 7840 10400 16200 6350 10700 15400 12500 2850 9040 2700 1710 3430 6790 11600 4110 7230 4610 1.3 1.2 2 1.2 3.2 5.9 2.3 1.9 2.9 3.7 1.8 2.3 2.4 2.2 1.1 3.7 1.1 1 1.5 2.6 2.4 1.1 2.1 0.66 8.7 8.1 11 5.1 12 20 12 11 13 20 9.8 14 17 15 8.4 14 4.5 3 4 11 16.4 7.6 35.5 6.1 Iron 4600 4330 13000 4300 11300 18700 10000 8900 12400 18200 7340 10900 14400 11800 4090 15300 6030 3060 4470 9230 Lead 16 6.4 4.9 8.3 18 14 12 8.7 11 12 14 14 19 21 16 25 20 11 6.3 13 98.5 53.3 109 8.3 Mn 39 30 10 92 530 140 440 130 120 70 60 110 340 85 41 66 21 8.6 19 140 72.5 27.2 118 22.5 Thallium Vanadium 0.0835 0.068 0.155 0.0665 0.071 0.427 0.352 0.228 0.068 0.456 0.067 0.0695 0.07 0.214 0.0665 0.4355 0.0675 0.066 0.067 0.068 0.13 0.068 0.095 0.07 12 8.4 11 9 22 32 19 17 21 32 15 21 28 25 8 24 11 7.2 8.1 16 The chromium concentrations follow an approximate normal distribution (determined using the two normality tests) and also a gamma distribution. ProUCL 5.1 uses the conclusion based upon both (Shapiro-Wilk and Lilliefors) normality tests and ProUCL 4.1 uses the conclusion based only upon the Shapiro-Wilk test leading to the conclusion that the data set does not follow a normal distribution and suggested the use of gamma UCLs. UCL results computed and suggested by ProUCL 5.1 and ProUCL 4.1 are summarized as follows. Data are mildly skewed (with sd of logged data = 0.57), therefore, UCL95s obtained using normal and gamma distributions are comparable. 87 ------- UCLs Suggested by ProUCL 5.0/ProUCL 5.1 Chromium Total Number of Observations General Satisfies 24 Minimum 3 Maximum 35.5 SD 6.832 Coefficient of Variation G.576 Number of Distinct Observations 19 Number of Missing Observations 0 Mean 11.37 Median 11 3d. Error of Mean 1.407 Skewness 1.728 Shapiro Wilk Test Statistic 5% Shapiro Wilk Critical Value Ljlliefors Test Statistic 5% Ljlliefors Critical Value Normal GOF Test 8.87 0.316 0.134 8.181 Shapiro Wilk GOF Test Data Not Normal at 5% Significance Level Ljlliefors GOF Test Data appear Normal at 5% Significance Level Data appear Approximate Normal at 5% Significance Level Assuming Normal Distribution 95!'. Normal UCL 35% Student s-t UCL 14.3-8 35% UCLs (Adjusted for Skewness) 35% Adjusted-C LT UCL fChen-1395} 14.81 95% Modified^ UCL (Johnson-1378) 14.46 Suggested UCL to Use 95% Students* UCL 14.38 UCLs Suggested by ProUCL 4.1 Gamma Distribution Test k star (bias corrected) 3.128 Theta Star 3.825 MLEofMean 11.97 MLE of Standard Deviation 6.766 nustar 150.2 Approximate Chi Square Value (.05) 122.8 Adjusted Level of Significance 0.0332 Adjusted Chi Square Value 121.1 Anderson-Darling Test Statistic 0.208 Anderson-Darling 5% Critical Value 0.75 Kdmogorov-Smirnov Test Statistic 0.0325 Kolmogorov-Smirnov 5% Critical Value 0.179 Data appear Gamma Distribided at 5% Significance Levd Assuming Gamma Distribution 95% Approximate Gamma UCL (Use when n >=40) 14.63 95% .Adjusted Garrrra UCL (Use when n < iO) Kgi Potential UCLto Use Data Distribution Data appear Gamma Distributed at 5% Significance Levd Nonparametric Statistics 95% CLT UCL 14.28 95% Jackknife UCL 14.38 95% Standard Bootstrap UCL 14.28 95% Bootstrap-t UCL 15.19 35% Hall's Bootstrap UCL 16.77 95% Percentile Bootstrap UCL 14.37 95% BCA Bootstrap UCL 14.95 95% Chebyshev(Mean. Sd) UCL 18.1 37.5% Cnebyshev(Mean. Sd) UCL 20.75 93% Chebyshev(Mean. Sd) UCL 25.96 Use 95% Adjusted Garrrra UCL 14.84 88 ------- Example 2-5: Consider another mildly skewed real-world data set consisting of lead (Pb) concentrations from a polluted site Questions were raised regarding ProUCL suggesting that the data are approximate normal and suggesting the use of the Student's t-UCL This example is included to illustrate that when data are mildly skewed (sd of logged data <0.5), the differences between UCLs computed using different distributions are not substantial from a practical point of view. The mildly skewed (with sd of logged data =0.47), zinc (Zn) data set of size 11 is given by: 38.9, 45.4, 40.1, 101.4, 166.7, 53.9, 57. 35.7, 43.2, 72.9, and 72.1. The Zn data set follows an approximate normal (using the Lilliefors test). As we know, the Lilliefors test works well for data sets of size >50; so it is valid to question why ProUCL suggests the use of a normal Student's t-UCL. This data set also follows a gamma (using both tests) and lognormal distribution (using both tests). Student's t-UCL95 suggested by ProUCL (using approximate normality) = 87.26, Gamma UCL95 (adjusted) = 93.23, Gamma UCL95 (approximate) = 88.75, and a lognormal UCL95 = 90.51. So all UCLs are comparable for this mildly skewed data set. Note: When a data set follows all three distributions (when this happens, it is highly likely that data set is mildly skewed), one may want to use a UCL for the distribution with the highest p-value. Also when skewness in terms of sd of logged data is <0.5, all three distributions yield comparable UCLs. New in ProUCL 5.0 and ProUCL 5.1: Some changes have been made in the decision tables which are used to make suggestions for selecting a UCL to estimate EPCs. In earlier versions, data distribution conclusions (internally) in the UCL and BTV modules were based upon only one GOF test statistic (e.g., Shapiro Wilk test for normality or lognormality). In ProUCL 5.0 and ProUCL 5.1, data distribution conclusions are based upon both GOF statistics (e.g., both Shapiro -Wilk and Lilliefors tests for normality) available in ProUCL. When only one of the GOF test passes, it is determined that the data set follows an approximate distribution and ProUCL makes suggestions accordingly. However, when a data set follows more than one distribution, the use of the distribution passing both GOF tests is preferred. For data sets with NDs, ProUCL 5.0/5.1 offers more UCL computation methods than ProUCL 4.1. These updates and additions have been incorporated in the decision tables of ProUCL 5.1. Due to these upgrades and additions, suggestions regarding the use of a UCL made by ProUCL 4.1 and ProUCL 5.1 can differ for some data sets. Suggestions made by ProUCL are based upon simulations performed by the developers. A typical simulation study does not (cannot) cover all data sets of various sizes and skewness from the various distributions. The ProUCL Technical Guide provides sufficient guidance which can help a user select the most appropriate UCL as an estimate of the EPC. ProUCL makes these UCL suggestions to help a typical user select the appropriate UCL from the various available UCLs. Non-statisticians may want to seek help from a qualified statistician. 2.5 Suggestions and Summary The suggestions provided by ProUCL for selecting an appropriate UCL of the mean are summarized in this section. These suggestions are made to help the users in selecting the most appropriate UCL to estimate the EPC which is routinely used in exposure assessment and risk management studies of the USEPA. The suggestions are based upon the findings of the simulation studies described in Singh, Singh, and Engelhardt (1997, 1999); Singh, Singh, and laci (2002); Singh et al. (2006); and Appendix B. A typical simulation study does not (cannot) cover all data sets of all sizes and skewness from all distributions. For an analyte (data set) with skewness (sd of logged data) near the end points of the skewness intervals described in decision tables, Table 2-9 through Table 2-11, the user may select the most appropriate UCL based upon expert site knowledge, toxicity of the analyte, and exposure risk associated with that analyte. ProUCL makes these UCL suggestions to help a typical user in selecting the 89 ------- appropriate UCL from the many available UCLs. Non-statisticians may want to seek help from a qualified statistician. UCL suggestions have been summarized for: 1) normally distributed data sets, 2) gamma distributed data sets, 3) lognormally distributed data sets, and 4) nonparametric data sets (data sets not following any of the three distributions available in ProUCL). For a given data set, an appropriate UCL can be computed by using more than one method. Therefore, depending upon the data size, distribution, and skewness, sometimes ProUCL may suggest more than one UCL. In such situations, the user may choose any of the suggested UCLs. If needed, the user may consult a statistician for additional insight. When the use of a Chebyshev inequality based UCL (e.g., UCL95) is suggested, the user may want to compare that UCL95 with other UCLs including the Chebyshev UCL90 (as Chebyshev inequality tends to yield conservative UCLs), before deciding upon the use of an appropriate UCL to estimate the population (site) average. 2.5.7 Suggestions for Computing a 95% UCL of the Unknown Population Mean, pi, Using Symmetric and Positively Skewed Data Sets For mildly skewed data sets with a or a < 0.5, most of the parametric and nonparametric methods (excluding Chebyshev inequality which is used on skewed data sets) tend to yield comparable UCL values. Any UCL computation method may be used to estimate the EPC. However, for highly skewed (<7>2.0) parametric and nonparametric data sets, there is no simple solution to compute a reliable 95% UCL of the population mean, ju\. As mentioned earlier, the UCL95 based upon skewness adjusted methods, such as Johnson's modified-t and Chen's adjusted-CLT, do not provide the specified coverage to the population mean even for moderately skewed (a in the interval [0.5, 1.0]) data sets for samples of sizes as large as 100. The coverage of the population mean by these skewness-adjusted UCLs gets poorer (much smaller than the specified level) for highly skewed data sets, where skewness levels have been defined in Table 2-1 as functions of (7 (standard deviation of logged data). Interested users may also want to consult graphs provided in Appendix B for a better understanding of the summary and suggestions described in this section. 2.5.1.1 Normally or Approximately Normally Distributed Data Sets For normally distributed data sets, several methods such as: the Student's t-statistic, modified-t-statistic, and bootstrap-t computation methods yield comparable UCL95s providing coverage probabilities close to the nominal level, 0.95. • For normally distributed data sets, a UCL based upon the Student's t-statistic, as given by equation (2-32), provides the optimal UCL of the population mean. Therefore, for normally distributed data sets, one should always use a 95% UCL based upon the Student's t-statistic. • The 95% UCL of the mean given by equation (2-32) based upon the Student's t-statistic (preferably modified-t) may also be used on non-normal data sets with sd, sy of the log- transformed data less than 0.5, or when the data set follows an approximate normal distribution. A data set is approximately normal when: 1) the normal Q-Q plot displays a linear pattern (without outliers, breaks and jumps) and the resulting correlation coefficient is high (0.95 or higher); and/or 2) one of the two GOF tests for a normal distribution incorporated in ProUCL suggests that data are normally distributed. 90 ------- • Student's t-UCL may also be used to estimate the EPC when the data set is symmetric (but possibly not normally distributed). A measure of symmetry (or skewness) is k3, which is given /*, by equation (2-43). A value of k3 close to zero (absolute value of skewness is roughly less than 0.2 or 0.3) suggests approximate symmetry. The approximate symmetry of a data distribution can also be judged by looking at a box plot and/or a histogram. Note: Use Student's t-UCL for normally distributed data sets. For approximately normally distributed data sets, non-normal symmetric data sets (when skewness is less than 0.2-0.3), and mildly skewed data sets with logged sd<0.5, one may use the modified t-UCL. 2.5.1.2 Gamma or Approximately Gamma Distributed Data Sets In practice, many skewed data sets can be modeled both by a lognormal distribution and a gamma distribution. Estimates of the unknown population mean based upon the two distributions can differ significantly (see Example 2- 2 above). For data sets of small size (<20 and even <50) the 95% H-UCL of the mean based upon a lognormal model often results in unjustifiably large and impractical 95% UCL values. In such cases, a gamma model, G (k, 0), may be used to compute a 95% UCL provided the data set follows a gamma distribution. • One should always first check if a given skewed data set follows a gamma distribution. If a data set does follow a gamma distribution or an approximate gamma distribution (suggested by gamma Q-Q plots and gamma GOF tests), one should use a 95% UCL based upon a gamma distribution to estimate the EPC. For gamma distributed data sets of sizes > 50 with shape parameter, k>\, the use of the approximate gamma UCL95 is recommended to estimate the EPC. • For gamma distributed data sets of sizes <50, with shape parameter, k >1, the use of the adjusted gamma UCL95 is recommended. • For highly skewed gamma distributed data sets of small sizes (e.g., <15 or <20) and small values of the shape parameter, k (e.g., k < =1.0), a gamma UCL95 may fail to provide the specified 0.95 coverage for the population mean (Singh, Singh, and laci 2002); the use of a bootstrap-t UCL95 or Hall's bootstrap UCL95 is suggested for small highly skewed gamma distributed data sets to estimate the EPC. The small sample size requirement increases as skewness increases. That is as k decreases, the required sample size, n, increases. In the case Hall's bootstrap and bootstrap-t methods yield inflated and erratic UCL results (e.g., when outliers are present), the 95% UCL of the mean may be computed based upon the adjusted gamma 95% UCL. • For highly skewed gamma distributed data sets of sizes > 15 and small values of the shape parameter, k (k < 1.0), the adjusted gamma UCL95 (when available) may be used to estimate the EPC, otherwise one may want to use the approximate gamma UCL. • For highly skewed gamma distributed data sets of sizes > 50 and small values of the shape parameter, k (k < 1.0), the approximate gamma UCL95 may be used to estimate the EPC. • The use of an H-UCL should be avoided for highly skewed (a > 2.0) lognormally distributed data sets. For such highly skewed lognormally distributed data sets that cannot be modeled by a gamma or an approximate gamma distribution, the use of nonparametric UCL computation 91 ------- methods based upon the Chebyshev inequality (larger samples) or bootstrap-t and Hall's bootstrap methods (smaller samples) is preferred. Notes: Bootstrap-t and Hall's bootstrap methods should be used with caution as sometimes these methods yield erratic, unreasonably inflated, and unstable UCL values, especially in the presence of outliers (Efron and Tibshirani 1993). In the case Hall's bootstrap and bootstrap-t methods yield inflated and erratic UCL results, the 95% UCL of the mean may be computed based upon the adjusted gamma 95% UCL. ProUCL prints out a warning message associated with the recommended use of the UCLs based upon the bootstrap-t method or Hall's bootstrap method. Table 2-9. Summary Table for the Computation of a 95% UCL of the Unknown Mean, fii, of a Gamma Distribution; Suggestions are made Based upon Biased Adjusted Estimates k"(Skewness „ , „. „ Sample Size, n Suggestion Bias Adjusted) Approximate gamma 95% UCL (Gamma KM or k* > 1.0 n>=50 GROS) ,% <-„ Adjusted gamma 95% UCL (Gamma KM or GROS) K ^ l.U n^-ju £* ,, 95% UCL based upon bootstrap-t ~ ' or Hall's bootstrap method* Adjusted gamma 95% UCL (Gamma KM) if k* <1.0 n>\5,n<50 available, otherwise use approximate gamma 95% UCL(Gamma KM) k* <1.0 n > 50 Approximate gamma 95% UCL (Gamma KM) *In case the bootstrap-t or Hall's bootstrap method yields an erratic, inflated, and unstable UCL value, the UCL of the mean should be computed using the adjusted gamma UCL method. Note: Suggestions made in Table 2-9 are used for uncensored as well as left-censored data sets. This table is not repeated in Chapter 4. All suggestions have been made based upon bias adjusted estimates, K of k. When the data set is uncensored, use upper limits based upon the sample size and bias adjusted MLE estimates; and when the data set is left-censored, use upper limits based upon the sample size and biased adjusted estimates obtained using the KM method or GROS method provided k*>\. When k*>\, UCLs based upon the GROS method and gamma UCLs computed using KM estimates tend to yield comparable UCLs from a practical point of view. 2.5.1.3 Lognormally or Approximately Lognormally Distributed Skewed Data Sets For lognormally, LN (ju, a1), distributed data sets, the H-statistic-based UCL provides the specified 0.95 coverage for the population mean for all values of a; however, the H-statistic often results in unjustifiably large UCL values that do not occur in practice. This is especially true when skewness is high (a > 1.5-2.0) and the data set is small («<20-50). For skewed (a or a > 0.5) lognormally distributed data sets, the Student's t-UCL95, modified-t-UCL95, adjusted-CLT UCL95, standard bootstrap and percentile 92 ------- bootstrap UCL95 methods fail to provide the specified 0.95 coverage for the population mean for samples of all sizes. Based upon the results of the research conducted to evaluate the appropriateness of the applicability of a lognormal distribution based estimates of the EPC (Singh, Singh, and Engelhardt 1997; Singh, Singh, and laci 2002), the developers of ProUCL suggest avoiding the use of the lognormal distribution to estimate the EPC. Additionally, the use of the lognormal distribution based Chebyshev (MVUE) UCL should also be avoided unless skewness is mild with the sd of log-transformed data <0.5- 0.75. The Chebyshev (MVUE) UCL has been retained in ProUCL software for historical and information purposes. ProUCL 5.0 and higher versions do not suggest its use. • ProUCL5.0 computes and outputs H-statistic based UCLs and Chebyshev (MVUE) UCLs for historical, research, and comparison purposes as it is noted that some recent guidance documents (EPA 2009) are recommending the use of lognormal distribution based decision statistics. ProUCL can compute an H-UCL of the mean for samples of sizes up to 1000. • It is suggested that all skewed data sets be first tested for a gamma distribution. For gamma distributed data sets, decisions statistics should be computed using gamma distribution based exact or approximate statistical methods as summarized in Section 2.5.1.2. • For lognormally distributed data sets that cannot be modeled by a gamma distribution, methods as summarized in Table 2-10 may be used to compute a UCL of the mean to estimate the EPC. For highly skewed (e.g., sd >1.5) lognormally distributed data sets which do not follow a gamma distribution, one may want to compute a UCL using nonparametric bootstrap methods (Efron and Tibshirani 1993) and the Chebyshev (Mean, Sd) UCL. Table 2-10. Summary Table for the Computation of a UCL of the Unknown Mean, [it, of a Lognormal Population to Estimate the EPC 25 «<20 20<«<50 «>50 «<20 20<«<50 50<«<70 «>70 «<30 30<«<70 70<«< 100 Suggestions Student's t, modified-t, or H-UCL H-UCL 95% Chebyshev (Mean, Sd) UCL H-UCL 97.5% or 99% Chebyshev (Mean, Sd) UCL 95% Chebyshev (Mean, Sd) UCL H-UCL 99% Chebyshev (Mean, Sd) UCL 97.5% Chebyshev (Mean, Sd) UCL 95% Chebyshev (Mean, Sd) UCL H-UCL 99% Chebyshev (Mean, Sd) 97.5% Chebyshev (Mean, Sd) UCL 95% Chebyshev (Mean, Sd) UCL 93 ------- 3.5« Sample Size, n n> 100 n< 15 15<«<50 50<«< 100 100 150 For all n Suggestions H-UCL Bootstrap-t or Hall's bootstrap method* 99% Chebyshev(Mea«, Sd) 97.5% Chebyshev (Mean, Sd) UCL 95% Chebyshev (Mean, Sd) UCL H-UCL Use nonparametric methods* *In the case that the Hall's bootstrap or bootstrap-t methods yield an erratic unrealistically large UCL95 value, a UCL of the mean may be computed based upon the Chebyshev inequality: Chebyshev (Mean, Sd) UCL ** For highly skewed data sets with a exceeding 3.0, 3.5, pre-process the data. It is very likely that the data includes outliers and/or come from multiple populations. The population partitioning methods may be used to identify mixture populations present in the data set. 2.5.1.4 Nonparametric Skewed Data Sets without a Discernible Distribution For moderately and highly skewed data sets which are neither gamma nor lognormal, one can use a nonparametric Chebyshev UCL, bootstrap-t, or Hall's bootstrap UCL (for small samples) of the mean to estimate the EPC. For skewed nonparametric data sets with negative and zero values, use a 95% Chebyshev (Mean, Sd) UCL for the population mean, ju\. For all other nonparametric data sets with only positive values, the following procedure may be used to estimate the EPC. The suggestions described here are based upon simulation experiments and may not cover all skewed data sets or various sizes originating from the real world practical studies and applications. • As noted earlier, for mildly skewed data sets with a (or 30) with a< 0.5 one can use the BCA bootstrap method or the adjusted CLT to compute a 95% UCL of the mean, ju\. • For nonparametric moderately skewed data sets (e.g., a or its estimate, a in the interval [0.5, 1]), one may use a 95% Chebyshev (Mean, Sd) UCL of the population mean, jj.\. In practice, for values of a closer to 0.5, a 95% Chebyshev (Mean, Sd) UCL may represent an over estimate of the EPC. The user is advised to compare 95% and 90% Chebyshev (Mean, Sd) UCLs. • For nonparametric moderately and highly skewed data sets (e.g., a in the interval [1.0, 2.0]), depending upon the sample size, one may use a 97.5% Chebyshev (Mean, Sd) UCL or a 95% Chebyshev (Mean, Sd) UCL to estimate the EPC. 94 ------- • For highly and extremely highly skewed data sets with a in the interval [2.0, 3.0], depending upon the sample size, one may use Hall's UCL95 or the 99% Chebyshev (Mean, Sd) UCL or the 97.5% Chebyshev (Mean, Sd) UCL or the 95% Chebyshev (Mean, Sd) UCL to estimate the EPC. For skewed data sets with(7>3, none of the methods considered in this chapter provide the specified 95% coverage for the population mean, ju\. The coverages provided by the various methods decrease as a (a) increases. For such data sets of sizes less than 30, a 95% UCL can be computed based upon Hall's bootstrap method or bootstrap-t method. Hall's bootstrap method provides the highest coverage (but < 0.95) when the sample size is small; and the coverage for the population mean provided by Hall's method (and the bootstrap-t method) does not increase much as the sample size, n, increases. However, as the sample size increases, the coverage provided by the Chebyshev (Mean, Sd) UCL increases. Therefore, for larger skewed data sets wither>3, the EPC may be estimated by the 99% Chebyshev (Mean, Sd) UCL. The large sample size requirement increases as a increases. Suggestions are summarized in Table 2-11. Table 2-11. Summary Table for the Computation of a 95% UCL of the Unknown Mean, [ti, Based upon a Skewed Data Set (with All Positive Values) without a Discernible Distribution, Where 3.5** Sample Size, n For all n For all n For all n n<20 20 ------- *If Hall's bootstrap method yields an erratic and unstable UCL value (e.g., happens when outliers are present), a UCL of the population mean may be computed based upon the 99% Chebyshev (Mean, Sd) method. ** For highly skewed data sets with a exceeding 3.0 to 3.5, pre-process the data. Data sets with such high skewness are complex and it is very likely that the data includes outliers and/or come from multiple populations. The population partitioning methods may be used to identify mixture populations present in the data set. 2.5.2 Summary of the Procedure to Compute a 95% UCL of the Unknown Population Mean, pi, Based upon Full UncensoredData Sets without Nondetect Observations A summary of the process used to compute an appropriate UCL95 of the mean is summarized as follows. • Formal GOF tests are performed first so that based on the determined data distribution, an appropriate parametric or nonparametric UCL of the mean can be computed to estimate the EPC. ProUCL generates formal GOF Q-Q plots to graphically evaluate the distribution (normal, lognormal, or gamma) of the data set. • For a normally or approximately normally distributed data set, the user is advised to use a Student's t-distribution-based UCL of the mean. Student's t-UCL or modified-t-statistic based UCL can be used to compute the EPC when the data set is symmetric (e.g., to 0.3) or mildly skewed, that is, when a or a is less than 0.5. is smaller than 0.2 For mildly skewed data sets with a (sd of logged data) less than 0.5, all distributions available in ProUCL tend to yield comparable UCLs. Also, when a data set follows all three distributions in ProUCL, compute the upper limits based upon the distribution with highest /"-value. For gamma or approximately gamma distributed data sets, the user is advised to: 1) use the approximate gamma UCL when biased adjusted MLE, k* of k >1 and n > 50; 2) use the adjusted gamma UCL when biased MLE, k* of k > 1 and n < 50; 3) use the bootstrap-t method or Hall's bootstrap method when k < 1 and the sample size, n < 15 (or <20, sample size requirement depends upon k); 4) use the adjusted gamma UCL (if available) for k* < 1 and sample size, 15 < n < 50; and 5) use approximate gamma UCL when k* <\ but n >50. If the adjusted gamma UCL is not available, then use the approximate gamma UCL as an estimate of the EPC. When the bootstrap-t method or Hall's bootstrap method yields an erratic inflated UCL (when outliers are present) result, the UCL may be computed using the adjusted gamma UCL (if available) or the approximate gamma UCL. For lognormally or approximately lognormally distributed data sets, ProUCL recommends a UCL computation method based upon the sample size, n, and standard deviation of the log-transformed data, a. These suggestions are summarized in Table 2-10. 96 ------- • For nonparametric data sets, which are not normally, lognormally, or gamma distributed, a nonparametric UCL is used to estimate the EPC. Methods used to estimate EPCs based upon nonparametric data sets are summarized in Table 2-11. For example, for mildly skewed nonparametric data sets (sd of logged data <0.5) of smaller sizes (n <30), one may use a modified-t UCL or BCA bootstrap UCL; and for larger mildly skewed data sets, one may use a CLT-UCL, adjusted-CLT UCL, or BCA bootstrap UCL. • For moderately skewed to highly skewed nonparametric data sets, the use of a Chebyshev (Mean, Sd) UCL is suggested. For extremely skewed data sets (a > 3.0), even a Chebyshev inequality- based 99% UCL of the mean fails to provide the desired coverage (e.g., 0.95) of the population mean. It is likely that such high skewed data sets do not occur with high probability representing a single statistical population. • For highly skewed data sets with a exceeding 3.0, 3.5, it is suggested the user pre-processes the data. It is very likely that the data contains outliers and/or come from multiple populations. Population partitioning methods (available in Scout; EPA 2009d) may be used to identify mixture populations present in the data set; and decision statistics, such as EPCs, may be computed separately for each of the identified sub-population. Notes: It should be pointed out that when dealing with a small data set (e.g., <50), and the Lilliefors test suggests that data are normal and S-W test suggests that data are not normal, ProUCL will suggest that the data set follows an approximate normal distribution. However, for smaller data sets, Lilliefors test results may not be reliable, therefore the user is advised to review GOF tests for other distributions and proceed accordingly. It is emphasized, when a data set follows a distribution (e.g., distribution A) using all GOF tests, and also follows an approximate distribution (e.g., Distribution B) using one of the available GOF tests, it is preferable to use distribution A over distribution B. However, when distribution A is a highly skewed (sd of logged data >1.0) lognormal distribution, use the guidance provided on the ProUCL generated output. Once again, contrary to the common belief and practice, for moderately skewed to highly skewed data sets, the CLT and t-statistic based UCLs of the mean cannot provide defensible estimates of EPCs. Depending upon data skewness of a nonparametric data set, sample size as large as 50, 70, or 100 is not large enough to apply the CLT and conclude that the sample mean approximately follows a normal distribution. The sample size requirement increases with skewness. The use of nonparametric methods such as bootstrap-t and Chebyshev inequality based upper limits is suggested for skewed data sets. Finally, ProUCL makes suggestions about the use of one or more UCLs based upon the data distribution, sample size, and data skewness. Most of the suggestions made in ProUCL are based upon the simulation studies performed by the developers and their professional experience. However, simulations performed do not cover all real world scenarios and data sets. The users may use UCLs values other than those suggested by ProUCL based upon their own experiences and project needs. 97 ------- CHAPTER 3 Computing Upper Limits to Estimate Background Threshold Values Based Upon Uncensored Data Sets without Nondetect Observations 3.1 Introduction In background evaluation studies, site-specific (e.g., soils, groundwater) background level constituent concentrations are needed to compare site concentrations with background level concentrations also known as background threshold values (BTVs). The BTVs are estimated, based upon sampled data collected from reference areas and/or unimpacted site-specific background areas (e.g., upgradient wells) as determined by the project team. The first step in establishing site-specific background level constituent concentrations is to collect an appropriate number of samples from the designated background or reference areas. The Stats/Sample Sizes module of ProUCL software can be used to compute DQOs- based sample sizes. Once an adequate amount of data has been collected, the next step is to determine the data distribution. This is typically done using exploratory graphical tools (e.g., Q-Q plots) and formal GOF tests. Depending upon the data distribution, one will use a parametric or nonparametric methods to estimate BTVs. In this chapter and also in Chapter 5 of this document, a BTV is a parameter of the background population representing an upper threshold (e.g., 95th upper percentile) of the background population. When one is interested in comparing averages, a BTV may represent an average value of a background population which can be estimated by a UCL95 (e.g., Chapter 21 of EPA 2009 RCRA Guidance). However, in ProUCL guidance and in ProUCL software, a BTV represents an upper threshold of the background population. The Upper Limits/BTVs module of ProUCL software computes upper limits which are often used to estimate a BTV representing an upper threshold of the background population. With this definition of a BTV, an onsite observation in exceedance of a BTV estimate may be considered as not coming from the background population; such a site observation may be considered as exhibiting some evidence of contamination due to site-related activities. Sometimes, locations exhibiting concentrations higher than a BTV estimate are re-sampled to verify the possibility of contamination. Onsite values less than BTVs represent unimpacted locations and can be considered part of the background (or comparable to the background) population. This approach, comparing individual site or groundwater (GW) monitoring well (MW) observations with BTVs, is particularly helpful to: 1) identify and screen constituents/contaminants of concern (COCs); and 2) use after some remediation activities (e.g., installation of a GW treatment plant) have already taken place and the objective is to determine if the remediated areas have been remediated close enough to the background level constituent concentrations. Background versus site comparisons can also be performed using two-sample hypothesis tests (see Chapter 6). However, BTV estimation methods described in this chapter are useful when not enough site data are available to perform hypotheses tests such as the two-sample t-test or the nonparametric Wilcoxon Rank Sum (WRS) test. When enough (more than 8 to 10 observations) site data are available, 98 ------- hypotheses testing approaches can be used to compare onsite and background data or onsite data with some pre-established threshold or screening values. The single-sample hypothesis tests (e.g., t-test, WRS test, proportion test) are used when screening levels or BTVs are known or pre-established. The two- sample hypotheses tests are used when enough data (at least 8-10 observations from each population) are available from background (e.g., upgradient wells) as well as site (e.g., monitoring wells) areas. This chapter describes statistical limits that may be used to estimate the BTVs for full uncensored data sets without any ND observations. Statistical limits for data sets consisting of NDs are discussed in Chapter 5. It is implicitly assumed that the background data set used to estimate BTVs represents a single statistical population. However, since outliers (well-separated from the main dominant data) are inevitable in most environmental applications, some outliers such as the observations coming from populations other than the background population may also be present in a background data set. Outliers, when present, distort decision statistics of interest (e.g., upper prediction limits [UPLs], upper tolerance limits [UTLs]), which in turn may lead to incorrect remediation decisions that may not be cost-effective or protective of human health and the environment. The BTVs should be estimated by statistics representing the dominant background population represented by the majority of the data set. Upper limits computed by including a few low probability high outliers (e.g., coming from the far tails of data distribution) tend to represent locations with those elevated concentrations rather than representing the main dominant background population. It is suggested that all relevant statistics be computed using the data sets with and without low probability occasional outliers. This extra step often helps the project team to see the potential influence of outlier(s) on the decision making statistics (UCLs, UPLs, UTLs) and to make informative decisions about the disposition of outliers. That is, the project team and experts familiar with the site should decide which of the computed statistics (with outliers or without outliers) represent more accurate estimate(s) of the population parameters (e.g., mean, EPC, BTV) under consideration. Since the treatment and handling of outliers in environmental applications is a subjective and controversial topic, the project team (including decision makers, site experts) may decide to treat outliers on a site-specific basis using all existing knowledge about the site and reference areas under investigation. A couple of classical outlier tests, incorporated in ProUCL, are discussed in Chapter 7. Extracting a Site-Specific Background Data Set from a Broader Mixture Data Set: Typically, not many background samples are collected due to resource constraints and difficulties in identifying suitable background areas with anthropogenic activities and natural geological characteristics comparable to onsite areas (e.g., at large Federal Facilities, mining sites). Under these conditions, due to confounding of site related chemical releases with anthropogenic influences and natural geological variability, it becomes challenging to:l) identify background/reference areas with comparable anthropogenic activities and geological conditions/formations; and 2) collect an adequate amount of data needed to perform meaningful and defensible site versus background comparisons for each geological stratum to determine chemical releases only due to the site related operations and releases. Moreover, a large number of background samples (not impacted by site related chemical releases) may need to be collected representing the various soil types and anthropogenic activities present at the site; which may not be feasible due to resource constraints and difficulties in identifying background areas with anthropogenic activities and natural geological characteristics comparable to onsite areas. The lack of sufficient background data makes it difficult to perform defensible background versus site comparisons and compute reliable estimates of BTVs. A small background data set may not adequately represent the background population; and due to uncertainty and larger variability, the use of a small data set tends to yield non-representative estimates of BTVs. Knowing the complexity of site conditions and that within all environmental site samples (data sets) exist both background level concentrations and concentrations indicative of site-related releases, sometimes it 99 ------- is desirable to extract a site-specific background data set from a mixture data set consisting of all available onsite and offsite concentrations. This is especially true for larger sites including Federal Facilities and Mining Sites. Several researchers (Sinclair 1976; Holgresson and Jorner 1978; Fleischhauer and Korte 1990) have used normal Q-Q/probability plots methods to delineate multiple populations which can be present in a mixture data set collected from environmental, geological and mineral exploration studies. Therefore, when not enough observations are available from reference areas with geological and anthropogenic influences comparable to onsite areas, the project team may want to use an iterative population partitioning methods (Singh, Singh, and Flatman 1994; Fleischhauer and Korte 1990) on a broader mixture data set to extract a site-specific background data set with geological conditions and anthropogenic influences comparable to those of the various onsite areas. Using the information provided by iteratively generated Q-Q plots, the project team then determines a background breakpoint (BP) distinguishing between background level concentrations and onsite concentrations potentially representing locations impacted by onsite releases. The background BP is determined based upon the information provided by iterative Q-Q plots, site CSM, expert site knowledge, and toxicity of the contaminant. The extracted background data set is used to compute upper limits (BTVs) which take data (contaminant) variability into consideration. If all parties of a project team do not come to a consensus on a background BP, then the best approach is to: identify comparable background areas and collect a sufficient amount of background data representing all formations and potential anthropogenic influences present at the site. The topics of population partitioning and the extraction of a site-specific background data set from a mixture data set are beyond the scope of ProUCL software and this technical guidance document. It requires the development of a separate chapter describing the iterative population partitioning method including the identification and extraction of a defensible background data set from a mixture data set consisting of all available data collected from background areas (if available), and unimpacted and impacted onsite locations. A review of the environmental literature reveals that one or more of the following statistical upper limits are used to estimate BTVs: • Upper percentiles • Upper prediction limits (UPLs) • Upper tolerance limits (UTLs) • Upper Simultaneous Limits (USLs) - New in ProUCL 5.0/ProUCL 5.1 Note: The upper limits which are selected to estimate the BTV are dependent on the project objective (e.g., comparing a single future observation, or comparing an unknown number of observations with a BTV estimate). . ProUCL does not provide suggestions as to which estimate of a BTV is appropriate for a project; the appropriate upper limit is determined by the project team. Once the project team has decided on an upper limit (e.g., UTL95-95), a similar process used to select a UCL95 can be used for selecting a UTL95-95 from among the UTLs computed by ProUCL. The differences between the various limits used to estimate BTVs are not clear to many practitioners. Therefore, a detailed discussion about the use of the different limits with their interpretation is provided in the following sections. Since 0.95 is the most commonly used confidence coefficient (CC), these limits are described for a CC of 0.95 and coverage probability of 0.95 associated with a UTL. ProUCL can compute these limits for any valid combination of CC and coverage probabilities including some commonly used values of CC levels (0.80, 0.90, 0.95, 0.99) and coverage probabilities (0.80, 0.90, 0.95, 0.975). 100 ------- Caution: To provide a proper balance between false positives and false negatives, the upper limits described above, especially a 95% USL (USL95), should be used only when the background data set represents a single environmental population without outliers (observations not belonging to background). Inclusion of multiple populations and/or outliers tends to yield elevated values of USLs (and also of UPLs and UTLs) which can result in a high number (and not necessarily high percentage) of undesirable false negatives, especially for data sets of larger sizes (n > 30). Note on Computing Lower Limits: In many environmental applications (e.g., in GW monitoring), one needs to compute lower limits including: lower confidence limits (LCL) of the mean, lower prediction limits (LPLs), lower tolerance limits (LTLs), or lower simultaneous limit (LSLs). At present, ProUCL does not directly compute a LCL, LPL, LTL, or a LSL. For data sets with and without NDs, ProUCL outputs several intermediate results and critical values (e.g., khat, nuhat, tolerance factor K for UTLs, d2max for USLs) needed to compute the interval estimates and lower limits. For data sets with and without NDs, except for the bootstrap methods, the same critical value (e.g., normal z value, Chebyshev critical value, or t-critical value) can be used to compute a parametric LPL, LSL, or a LTL (for samples of size >30 to be able to use Natrella's approximation in LTL) as used in the computation of a UPL, USL, or a UTL (for samples of size >30). Specifically, to compute a LPL, LSL, and LTL («>30) the '+' sign used in the computation of the corresponding UPL, USL, and UTL («>30) needs to be replaced by the '-' sign in the equations used to compute UPL, USL, and UTL («>30). For specific details, the user may want to consult a statistician. For data sets without ND observations, the Scout 2008 software package (EPA 2009d) can compute the various parametric and nonparametric LPLs, LTLs (all sample sizes), and LSLs. 3.1.1 Description and Interpretation of Upper Limits used to Estimate BTVs Based upon a background data set, upper limits such as a 95% upper confidence limit of the 95th percentile (UTL95-95) are used to estimate upper threshold value(s) of the background population. It is expected that observations coming from the background population will lie below that BTV estimate with a specified CC. BTVs should be estimated based upon an "established" data set representing the background population under consideration. Established Background Data Set: This data set represents background conditions free of outliers which potentially represent locations impacted by the site and/or other activities. An established background data set should be representative of the environmental background population. This can be determined by using a normal Q-Q plot on a background data set. If there are no jumps and breaks in the normal Q-Q plot, the data set may be considered representative of a single environmental population. A single environmental background population here means that the background (and also the site) can be represented by a single geological formation, or by single soil type, or by a single GW aquifer etc. Outliers, when present in a data set, result in inflated values of many decision statistics including UPLs, UTLs, and USLs. The use of inflated statistics as BTV estimates tends to result in a higher number of false negatives. However, when a site consists of various formations or soil types, separate background data sets may need to be established for each formation or soil type, therefore the project team may want to establish separate BTVs for different formations. When it is not feasible (e.g., due to implementation complexities) or desirable to establish separate background data sets for different geological formations present at a site (e.g., large mining sites), the project team may decide to use the same BTV for all formations.. In this case, a Q-Q plot of background data set collected from unimpacted areas may display discontinuities as concentrations in different formations may vary naturally. In these scenarios, use a Q-Q plot and outlier 101 ------- test only to identify outliers (well separated from the rest of the data) which may be excluded from the computation of BTV estimates. Notes: The user specifies the allowable false positive error rate, a (=1-CC The false negative error rate (declaring a location clean when in fact it is contaminated) is controlled by making sure that one is dealing with a defensible/established background data set representing a background population and the data set is free of outliers. Let x\, X2, xn represent sampled concentrations of an established background data set collected from some site-specific or general background reference area. Upper Percentile. XQQS: Based upon an established background data set, a 95th percentile represents that statistic such that 95% of the sampled data will be less than or equal to (<) x0.gs . It is expected that an observation coming from the background population (or comparable to the background population) will be < xo.gs with probability 0.95. A parametric percentile takes data variability into account. Upper Prediction Limit (UPL): Based upon an established background data set, a 95% UPL (UPL95) represents that statistic such that an independently collected observation (e.g., new/future) from the target population (e.g., background, comparable to background) will be less than or equal to the UPL95 with CC of 0.95. We are 95% sure that a single future value from the background population will be less than the UPL95 with CC= 0.95. A parametric UPL takes data variability into account. In practice, many onsite observations are compared with a BTV estimate. The use of a UPL95 to compare many observations may result in a higher number of false positives; that is the use of a UPL95 to compare many observations just by chance tends to incorrectly classify observations coming from the background or comparable to background population as coming from the impacted site locations. For example, if many (e.g., 30) independent onsite comparisons (e.g., Ra-226 activity from 30 onsite locations) are made with the same UPL95, each onsite value may exceed that UPL95 with a probability of 0.05 just by chance. The overall probability, aactuaiof at least one of those 30 comparisons being significant (exceeding BTV) just by chance is given by: aactuai = l-(l-a)k =1 - 0.9530 -1-0.21 = 0.79 (false positive rate). This means that the probability (overall false positive rate) is 0.79 (and is not equal to 0.05) that at least one of the 30 onsite locations will be considered contaminated even when they are comparable to background. The use of a UPL95 is not recommended when multiple comparisons are to be made. Upper Tolerance Limit (UTL): Based upon an established background data set, a UTL95-95 represents that statistic such that 95% of observations (current and future) from the target population (background, comparable to background) will be less than or equal to the UTL95-95 with CC of 0.95. A UTL95-95 represents a 95% UCL of the 95th percentile of the data distribution (population). A UTL95-95 is designed to simultaneously provide coverage for 95% of all potential observations (current and future) from the background population (or comparable to background) with a CC of 0.95. A UTL95-95 can be used when many (unknown) current or future onsite observations need to be compared with a BTV. A parametric UTL95-95 takes the data variability into account. By definition a UTL95-95 computed based upon a background data set just by chance can classify 5% of background observations as not coming from the background population with CC 0.95. This percentage (false positive error rate) stays the same irrespective of the number of comparisons that will be made with 102 ------- a UTL95-95. However, when a large number of observations coming from the target population (background, comparable to background) are compared with a UTL95-95, the number of exceedances (not the percentage of exceedances) of UTL95-95 by background observations can be quite large. This implies that a larger number (but not greater than 5%) of onsite locations comparable to background may be falsely declared as requiring additional investigation which may not be cost-effective. To avoid this situation, ProUCL provides a limit called USL which can be used to estimate the BTV provided the background data set represents a single population free of outliers. The use of a USL is not advised when the background data set may represent several geological formations/soil types. Upper Simultaneous Limit (USL): Based upon an established background data set free of outiers and representing a single statistical population (representing a single formation, representing the same soil type, same aquifer), a USL95 represents that statistic such that all observations from the "established" background data set are less than or equal to the USL95 with a CC of 0.95. Outliers should be removed before computing a USL as outliers in a background data set tend to represent observations coming from a population other than the background population represented by the majority of observations in the data set. Since USL represents an upper limit on the largest value in the sample, that largest value should come from the same background population. A parametric USL takes the data variability into account. It is expected that all current or future observations coming from the background population (comparable to background population, unimpacted site locations) will be less than or equal to the USL95 with CC, 0.95 (Singh and Nocerino 2002). The use of a USL as a BTV estimate is suggested when a large number of onsite observations (current or future) need to be compared with a BTV. The false positive error rate does not change with the number of comparisons, as the USL95 is designed to perform many comparisons simultaneously. Furthermore, the USL95 also has a built in outlier test (Wilks 1963), and if an observation (current or future) exceeds the USL95, then that value definitely represents an outlier and does not come from the background population. The false negative error rate is controlled by making sure that the background data set represents a single background population free of outliers. Typically, the use of a USL95 tends to result in a smaller number of false positives than a UTL95-95, especially when the size of the background data set is greater than 15. 3.7.2 Confidence Coefficient (CC) and Sample Size This section briefly discusses sample sizes and the selection of CCs associated with the various upper limits used to estimate BTVs. • Higher statistical limits are associated with higher levels of CCs. For example, a 95% UPL is higher than a 90% UPL. • Higher values of a CC (e.g., 99%) tend to decrease the power of a test, resulting in a higher number of false negatives - dismissing contamination when present. Therefore, the CC should not be set higher than necessary. • Smaller values of the CC (e.g., 0.80) tend to result in a higher number of false positives (e.g., declaring contamination when it is not present). • In most practical applications, choice of a 95% CC provides a good compromise between confidence and power. 103 ------- • Higher level of uncertainty in a background data set (e.g., due to a smaller background data set) and higher values of critical values associated with smaller (n < 15-20) samples tend to dismiss contamination as representing background conditions (results in higher number of false negatives; identifying a location that may be dirty as background). This is especially true when one uses UTLs and UPLs to estimate BTVs. • Nonparametric upper limits based upon order statistics (e.g., the largest, the second largest, etc.) may not provide the desired coverage as they do not take data variability into account. Nonparametric methods are less powerful than the parametric methods; and they require larger data sets to achieve power comparable to parametric methods. 3.2 Treatment of Outliers The inclusion of outliers in a background data set tends to yield distorted and inflated estimates of BTVs. Outlying observations which are significanly higher than the majority of the background data may not be used in establishing background data sets and in the computation of BTV estimates. A couple of classical outlier tests cited in environmental literature (Gilbert 1987; EPA 2006b, 2009; Navy 2002a, 2002b) are available in the ProUCL software. The classical outlier procedures suffer from masking effects as they get distorted by the same outlying observations that they are supposed to find! It is therefore recommended to supplement outlier tests with graphical displays such as box plots, Q-Q plots. On a Q-Q plot, elevated observations which are well-separated from the majority of data represent potential outliers. It is noted that nonparametric upper percentiles, UPLs and UTLs, are often represented by higher order statistics such as the largest value or the second largest value. When high outlying observations are present in a background data set, the higher order statistics may represent observations coming from the contaminated onsite/offsite areas. Decisions made based upon outlying observations or distorted parametric upper limits can be incorrect and misleading. Therefore, special attention should be given to outlying observations. The project team and the decision makers involved should decide about the proper disposition of outliers, to include or not include them, in the computation of the decision making statistics such as the UCL95 and the UTL95-95. Sometimes, performing statistical analyses twice on the same data set, once using the data set with outliers and once using the data set without outliers, can help the project team in determining the proper disposition of high outliers. Examples elaborating on these issues are discussed in several chapters (Chapters 2, 4, 7) this document. Notes: It should be pointed out that methods incorporated in ProUCL can be used on any data set with or without NDs and with or without outliers. Do not misinterpret that ProUCL is restricted and can only be used on data sets without outliers. It is not a requirement to exclude outliers before using any of the statistical methods incorporated in ProUCL. The intent of the developers of the ProUCL software is to inform the users on how the inclusion of occasional outliers coming from the low probability tails of the data distribution can yield distorted values of UCL95, UPLs, UTLs, and various other statistics. The decision limits and test statistics should be computed based upon the majority of data representing the main dominant population and not by accommodating a few low probability outliers resulting in distorted and inflated values of the decision statistics. Statistics computed based upon a data set with outliers tend to represent those outliers rather than the population represented by the majority of the data set. The inflated decision statistics tend to represent the locations with those elevated observations rather than representing the main dominant population. The outlying observations may be separately investigated to determine the reasons for their occurrences (e.g., errors or contaminated locations). It is suggested to compute the statistics with and without the outliers, and compare the potential impact of outliers on the decision making processes. 104 ------- Let Jti, X2, ..., xn represent concentrations of a contaminant/constituent of concern (COC) collected from some site-specific or general background reference area. The data are arranged in ascending order and the ordered sample (called ordered statistics) is denoted by X(i) < xp) < ... < X(n). The ordered statistics are used as nonparametric estimates of upper percentiles, UPLs, UTLs and USLs. Also, let ji = In (x;); i = 1,2, ... , n, and y and sy represent the mean and standard deviation (sd) of the log-transformed data. Statistical details of some parametric and nonparametric upper limits used to estimate BTVs are described in the following sections. 3.3 Upper p*100% Percentiles as Estimates of BTVs In most statistical textbooks (e.g., Hogg and Craig 1995), the/?* (e.g.,/? = 0.95) sample percentile of the measured sample values is defined as that value, x , such that/?*100% of the sampled data set lies at or below it. The carat sign over xp, indicates that it represents a statistic/estimate computed using the sampled data. The same use of the carat sign is found throughout this guidance document. The statistic x represents an estimate of the /?th population percentile. It is expected that about/?* 100% of the population values will lie below the pth percentile. Specifically, x095 represents an estimate of the 95th percentile of the background population. 3.3.1 Nonparametric p *100% Percentile Nonparametric 95% percentiles are used when the background data (raw or transformed) do not follow a discernible distribution at some specified (e.g., a = 0.05, 0.1) level of significance. Different software packages (e.g., SAS, Minitab, and Microsoft Excel) use different formulae to compute nonparametric percentiles, and therefore yield slightly different estimates of population percentiles, especially when the sample size is small, such as less than 20-30. Specifically, some software packages estimate the /?th percentile by using the p*nth order statistic, which may be a whole number between 1 and n or a fraction lying between 1 and n, while other software packages compute the /?th percentile by the p*(n+\)ih order statistic (e.g., used in ProUCL versions 4.00.02 and 4.00.04) or by the (pn+0.5) th order statistic. For example, if n = 20, and p = 0.95, then 20*0.95 = 19, thus the 19th ordered statistic represents the 95th percentile. If n = 17, and/? = 0.95, then 17*0.95= 16.15, thus the 16.15th ordered value represents the 95th percentile. The 16.15th ordered value lies between the 16th and the 17th order statistics and can be computed by using a simple linear interpolation given by: X(16.15)=*(16)+0.15 (^(17)-X(16)). (3-1) Earlier versions of ProUCL (e.g., ProUCL 4.00.02, 4.00.04) used thep*(n+\)th order statistic to estimate the nonparametric/?* percentile. However, since most users are familiar with Excel and some consultants have developed statistical software packages using Excel, and at the request of some users, it was decided to use the same algorithm as incorporated in Excel to compute nonparametric percentiles. ProUCL 4.1 and higher versions compute nonparametric percentiles using the same algorithm as used in Excel 2007. This algorithm is used on data sets with and without ND observations. Notes: From a practical point of view, nonparametric percentiles computed using the various percentile computation methods described in the literature are comparable unless the data set is small (e.g., n <20- 30) and/or comes from a mixed population consisting of some extreme high values. No single percentile computation method should be considered superior to other percentile computation methods available in 105 ------- the statistical literature. In addition to nonparametric percentiles, ProUCL also computes several parametric percentiles described as follows. 3.3.2 Normal p*l 00% Percentile The sample mean, x . and sd, s, are computed first. For normally distributed data sets, the p* 100th sample percentile is given by the following statement: xp=x + Szp (3-2) Here zp is the p* 100th percentile of a standard normal, N(0, 1), distribution, which means that the area (under the standard normal curve) to the left of zp is p. If the distributions of the site and background data are comparable, then it is expected that an observation coming from a population (e.g., site) comparable to the background population would lie at or below the/?* 100% upper percentile, x , with probability p. p- 3.3.3 Lognormalp *100% Percentile To compute the pth percentile, xp , of a lognormally distributed data set, the sample mean, y , and sd, sy, of log-transformed data, y are computed first. For lognormally distributed data sets, the p* 100th percentile is given by the following statement: (3-3) zp is thep* 100th percentile of a standard normal, N(0,l), distribution. 3.3.4 Gamma p* 100% Percentile Since the introduction of a gamma distribution, G (k, 9), is relatively new in environmental applications, a brief description of the gamma distribution is given first; more details can be found in Section 2.3.3. The maximum likelihood estimator (MLE) equations to estimate gamma parameters, k (shape parameter) and 9 (scale parameter), can be found in Singh, Singh, and laci (2002). A random variable (RV), X (arsenic concentrations), follows a gamma distribution, G(k, 9), with parameters k > 0 and 9 > 0, if its probability density function is given by the following equation: f(x; k, 9) = —i— x^e-*16; x>0 "*— (3.4) = 0; otherwise The mean, variance, and skewness of a gamma distribution are: // = kQ, variance = o2 = kff, and skewness = 2/v&- Note that as k increases, the skewness decreases, and, consequently, a gamma distribution starts approaching a normal distribution for larger values of k (e.g., k > 10). In practice, k is not known and a normal approximation may be used even when the MLE estimate of k is as small as 6. Let k and 6 represent the MLEs of k and 9 respectively. The relationship between a gamma RV, X = G (k, 9), and a chi-square RV, Y, is given by X = Y * 912, where Y follows a chi-square distribution with 106 ------- 2k degrees of freedom (df). Thus, the percentiles of a chi-square distribution (as programmed in ProUCL) can be used to determine the percentiles of a gamma distribution. In practice, k is replaced by its MLE. Once an a* 100% percentile, y(a) 2k, of a chi-square distribution with 2k df is obtained, the a* 100% percentile for a gamma distribution is computed using the following equation: (3-5) 3.4 Upper Tolerance Limits A UTL (l-a)-p (e.g., UTL95-95) based upon an established background data set represents that limit such that/?* 100% of the observations (current and future) from the target population (background, comparable to background) will be less than or equal to UTL with a CC, (1-a). It is expected that/?* 100% of the observations belonging to the background population will be less than or equal to a UTL with a CC, (1-a). A UTL (l-a)-p represents a (1-a) 700% UCL for the unknown pih percentile of the underlying background population. A UTL95-95 is designed to provide coverage for 95% of all observations potentially coming from the background or comparable to background population(s) with a CC of 0.95. A UTL95-95 will be exceeded by all (current and future) values coming from the background population less than 5% of the time with a CC of 0.95, that is 5 exceedances per 100 comparisons (of background values) can result just by chance for an overall CC of 0.95. Unlike a UPL95, a UTL95-95 can be used when many, or an unknown number of current or future onsite observations need to be compared with a BTV. A parametric UTL95-95 takes the data variability into account. When a large number of comparisons are made with a UTL95-95, the number of exceedances (not the percentage of exceedances) of the UTL95-95 by those observations can also be large just by chance. This implies that just by chance, a larger number (but not larger than 5%) of onsite locations comparable to background can be greater than a UTL95-95 potentially requiring unnecessary investigation which may not be cost-effective. In order to avoid this situation, it is suggested to use a USL95 to estimate a BTV, provided the background data set represents a single statistical population, free of outliers. 3.4.1 Normal Upper Tolerance Limits First, compute the sample mean, x , and sd, s, using a defensible data set representing a single background population. For normally distributed data sets, an upper (1 - a)* 100% UTL with coverage coefficient, p, is given by the following statement. UTL=x + K*s (3-6) Here, K = K (n, a, p) is the tolerance factor and depends upon the sample size, n, CC = (1 - a), and the coverage proportion = p. For selected values of n, p, and (1-a), values of the tolerance factor, K, have been tabulated extensively in the various statistical books (e.g., Hahn and Meeker 1991). Those K values are based upon the non-central t-distribution. Also, some large sample approximations (Natrella 1963) are available to compute the K values for one-sided tolerance intervals (same for both UTLs and lower tolerance limits). The approximate value of K is also a function of the sample size, n, coverage coefficient, p, and the CC, (1 - a). For samples of small sizes, n< 30, ProUCL uses the tabulated (Hahn and Meeker 1991) AT values. Tabulated K values are available only for some selected combinations ofp (0.90, 0.95, 0.975) and (1-a) values (0.90, 0.95, 0.99). For sample sizes larger than 30, ProUCL computes the K values using the large sample approximations, as given in Natrella (1963). The Natrella's 107 ------- approximation seems to work well for samples of sizes larger than 30. ProUCL computes these K values for all valid values ofp and (1-a) and samples of sizes as large as 5000. 3.4.2 Lognormal Upper Tolerance Limits The procedure to compute UTLs for lognormally distributed data sets is similar to that for normally distributed data sets. First, the sample mean, y , and sd, sy, of the log-transformed data are computed. An upper (1 - a)* 100% tolerance limit with tolerance or coverage coefficient, p is given by the following statement: UTL = exp(>> + K * s ) (3-7) The K factor in (3-7) is the same as the one used to compute the normal UTL. Notes: Even though there in no back-transformation bias present in the computation of a lognormal UTL, a lognormal distribution based UTL is typically higher (sometimes unrealistically higher as shown in the following example) than other parametric and nonparametric UTLs; especially when the sample size is less than 20. Therefore, the use of lognormal UTLs to estimate BTVs should be avoided when skewness is high (sd of logged data > 1 or 1.5) and sample size is small (e.g., n < 20-30). 3.4.3 Gamma Distribution Upper Tolerance Limits Positively skewed environmental data can often be modeled by a gamma distribution. ProUCL software has two goodness-of-fit tests: the Anderson-Darling (A-D) and Kolmogorov-Smirnov (K-S) tests for a gamma distribution. These GOF tests are described in Chapter 2. UTLs based upon normal approximation to the gamma distribution (Krishnamoorthy et al. 2008) have been incorporated in ProUCL 4.00.05 (EPA 2010d) and higher versions. Those approximations are based upon Wilson-Hilferty (WH)(Wilson and Hilferty 1931) and Hawkins-Wixley (HW) (Hawkins and Wixley 1986) approximations. Note: It should be pointed out that the performance of gamma UTLs and gamma UPLs based upon these HW and WH approximations is not well-studied and documented. Interested researchers may want to evaluate the performance of these gamma upper limits based upon HW and WH approximations. A description of method to compute gamma UTLs is given as follows. Let xi, x2, ..., xn represent a data set of size n from a gamma distribution, G(k, 6) with shape parameter, k and scale parameter 6. • According to the WH approximation, the transformation, Y = X1/3 follows an approximate normal distribution. The mean, n and variance, a2 of the transformed normally distributed variable, Y are given as follows: ju = [6>1/3r(£ +1 / 3)] / r(£); and a2 = [02/3r(k + 2/ 3)] / r(£) - //2 • According to the HW approximation, the transformation, Y = X1/4 follows an approximate normal distribution. Let y and sy represent the mean and sd of the observations in the transformed scale (Y). 108 ------- Using the WH approximation, the gamma UTL (in original scale, X), is given by: UTL = max o,y + K*sy (3-8) Similarly, using the HW approximation, the gamma UTL in original scale is given by: UTL= (y+K*sy)4 (3-9) The tolerance factor, K is defined earlier in (3-6) while computing a UTL based upon normal distribution. Note: For mildly skewed to moderately skewed gamma distributed data sets, HW and WH approximations yield fairly comparable UTLs. However, for highly skewed data sets (k ------- 3.4.4.1 Determining the Order, r, of the Statistic, X(r), to Compute UTLp,(l-a) Using the cumulative binomial probabilities, a number, r. 1 < r < n, is chosen such that the cumulative i=r fn\ binomial probability: "V />'(! - p}("~'} becomes as close as possible to (1 - a). The binomial %(*) distribution (BD) based algorithm has been incorporated in ProUCL for data sets of sizes up to 2000. For data sets of size, n >2000, ProUCL computes the rth (r. 1 < r < n) order statistic by using the normal approximation (Conover, 1999) given by the equation (3-10). p)+0.5 (3-10) Depending upon the sample size, p, and (1 - a) the largest, the second largest, the third largest, and so forth order statistic is used to estimate the UTL. As mentioned earlier for a given data set of size n, the rth order statistic, X(r) may or may not achieve the specified CC, (1 - a). ProUCL uses the F-distribution based probability statement to compute the CC achieved by the UTL determined by the rth order statistic. 3.4.4.2 Determining the Achieved Confidence Coefficient, CC achieve, Associated with X(r) For a given data set of size, n, once the rth order statistic, X(r), has been determined, ProUCL can be used to determine if a UTL computed using X(r) achieves the specified CC, (1 - a). ProUCL computes the achieved CC by using the following approximate probability statement based upon the F-distribution with vi and V2 degrees of freedom. CCAcUm=(l-a.)= Probability (F(v^ ------- when the second largest value, X(n-i) is used to compute a UTL, and m=n-r+l when the rth order statistic, X(r), is used to compute a UTL. For example, if the largest sample value, X(»), is used to compute a UTL95- 95, then a minimum sample size of 59 (see equation (3-12)) will be needed to achieve a confidence level of 0.95 providing coverage to 95% of the observations coming from the target population. A UTL95-95 estimated by the largest value and computed based upon a data set of size less than 59 may not achieve the desired confidence of 0.95 for the 95th percentile of the target population. Note: The minimum sample size requirement of 59 cited in the literature is valid when the largest value, X(n) (with m=l) in the data set is used to compute a compute a UTL95-95. For example, when the largest order statistic, X(») (with m=l) is used to compute a nonparametric UTL95-95, the approximate minimum sample size needed 0.25*5.99*1.95/0.05 ~ 58.4 (using equation (3-12)) which is rounded upward to 59; and when the second largest value, X(n-i) (with m=2) is used to compute a UTL95-95, the approximate minimum sample size needed = [(0.25*9.488*1.95)70.05] + 0.5 ~ 93. Similarly, to compute a UTL90-95 by the largest sample value, about 29 observations will be needed to provide coverage for 90% of the observations from the target population with CC = 0.95. Other sample sizes for various values of p and (I-a.) can be computed using equation, (3-12). In environmental applications, the number of available observations from the target population is much smaller than 29, 59 or 93 and a UTL computed based upon those data sets may not provide specified coverage with the desired CC. For specified values of CC, (I-a.) and coverage, p, ProUCL 5.1 outputs the achieved CC by a computed UTL and the minimum sample size needed to achieve the pre-specified CC. 3.4.4.4 Nonparametric UTL Based upon the Percentile Bootstrap Method A couple of bootstrap methods to compute nonparametric UTLs are also available in ProUCL 5.1. Like the percentile bootstrap UCL computation method, for data sets without a discernible distribution, one can use percentile bootstrap resampling method to compute UTL^,^_a; =UTL/>, (1 - a). The TV bootstrapped nonparametric pth percentiles, p, (i:=l,2,...,N), are arranged in ascending order: pl------- 3.5 Upper Prediction Limits Based upon a background data set, UPLs are computed for a single (UPLi) and k (UPLk) future observations. Additionally, in groundwater monitoring applications, an upper prediction limit of the mean of the k future observations, UPLk (mean) is also used. A brief description of parametric and nonparametric upper prediction limits is provided in this section. for a Single Future Observation: A UPLi computed based upon an established background data set represents that statistic such that a single future observation from the target population (e.g., background, comparable to background) will be less than or equal to the UPLi 95 with a CC of 0.95. A parametric UPL takes the data variability into account. A UPLi is designed for a single future observation comparison; however in practice users tend to use UPLi95 to perform many future comparisons which results in a high number of false postives (observations declared contaminated when in fact they are clean). When k>7 future comparisons are made with a UPLi, some of those future observations will exceed the UPLi just by chance, each with probability 0.05. For proper comparison, a UPL needs to be computed accounting for the number of comaprisons that will be performed. For example, if 30 independent onsite comparisons (e.g., Pu-238 activity from 30 onsite locations) are made with the same background UPLi95, each onsite value comparable to background may exceed that UPLi 95 with probability 0.05. The overall probability of at least one of those 30 comparisons being significant (exceeding the BTV) just by chance is given by: aactuai = l-(l-a)k =1 - 0.9530 -1-0.21 = 0. 79 (false positive rate). This means that the probability (overall false positive rate) is 0.79 (and not 0.05) that at least one of the 30 onsite observations will be considered contaminated even when they are comparable to background. Similar arguments hold when multiple (=j, a positive integer) constituents are analyzed, and status (clean or impacted) of an onsite location is determined based upon j comparisons (one for each analyte). The use of a UPLi is not recommended when multiple comparisons are to be made. 3. 5. 1 Normal Upper Prediction Limit The sample mean, x , and the sd, s, are computed first based upon a defensible background data set. For normally distributed data sets, an upper (1 - a)* 100% prediction limit is given by the following well known equation: UPL = x + t((1_aun_1)} * s * VC + l/H) (3-13) Here t(na) („_!« is a critical value from the Student's t-distribution with (n-\) df. 3.5.2 Lognormal Upper Prediction Limit An upper (1 - a)*100% lognormal UPL is similarly given by the following equation: UPL = exp(y +1((1_al(n_1}} *sy * J(\ + l/n)} (3-14) Here t(na) („_!« is a critical value from the Student's t-distribution with (n-\) df. 112 ------- 3.5.3 Gamma Upper Prediction Limit Given a sample, xi, x2, ..., xn of size n from a gamma distribution G(k, 9), approximate (based upon WH and HW approximations described earlier in Section 3.4.3, Gamma Distribution Upper Tolerance Limits), (1 - a)*100% upper prediction limits for a future observation from the same gamma distributed population are given by: Wilson-Hilferty(WH)UPL= max\OAy + t^ w ,**s*Jl+y] (3-15) ^ \ U1 «M» !)) ^ Af /«/ J (/ \4 y + f,,, *s *./!+ I/ (3-16) ^ ((l-a),(H-l)t 7 V / ft I ^ ' Here t^_a) (n_^ is a critical value from the Student's t-distribution with (n-\)df. Note: As noted earlier, the performance of gamma UTLs and gamma UPLs based upon these WH and HW approximations is not well-studied. Interested researchers may want to evaluate their performances via simulation experiments. These approximations are also available in R script. 3.5.4 Nonparametric Upper Prediction Limit A one-sided nonparametric UPL is simple to compute and is given by the following mth order statistic. One can use linear interpolation if the resulting number, m, given below does not represent a whole number (a positive integer). UPL = X(m), where m = (n + 1) * (1 - a). (3-17) For example, for a nonparametric data set of size n=25, a 90% UPL is desired. Then m = (26*0.90) = 23.4. Thus, a 90% nonparametric UPL can be obtained by using the 23rd and the 24th ordered statistics and is given by the following equation: UPL = X(23) + 0.4 * (X(24) -X(23) ) Similarly, if a nonparametric 95% UPL is desired, then m = 0.95 * (25 + 1) = 24.7, and a 95% UPL can be similarly obtained by using linear interpolation between the 24th and 25th order statistics. However, if a 99% UPL needs to be computed, then m = 0.99 * 26 = 25.74, which exceeds 25, the sample size; for such cases, the highest order statistic is used to compute the 99% UPL of the background data set. The largest value(s) should be used with caution (as they may represent outliers) to estimate the BTVs. Since nonparametric upper limits (e.g., UTLs, UPLs) are based upon higher order statistics, often the CC achieved by these nonparametric upper limits is much lower than the specified CC of 0.95, especially when the sample size is small. In order to address this issue, one may want to compute a UPL based upon the Chebyshev inequality. In addition to various parametric and nonparametric upper limits, ProUCL computes Chebyshev inequality based UPL. 113 ------- 3.5.4.1 Upper Prediction Limit Based upon the Chebyshev Inequality Like a UCL of the mean, the Chebyshev inequality can be used to compute a conservative but stable UPL and is given by the following equation: UPL = Jc+[^((1/00-1)* This is a nonparametric method since the Chebyshev inequality does not require any distributional assumptions. It should be noted that just like the Chebyshev UCL, a UPL based upon the Chebyshev inequality tends to yield higher estimates of BTVs than the various other methods. This is especially true when skewness is mild (sd of log-transformed data is low < 0.75), and the sample size is large (n > 30). The user is advised to apply professional judgment before using this method to compute a UPL. Specifically, for larger skewed data sets, instead of using a 95% UPL based upon the Chebyshev inequality, the user may want to compute a Chebyshev UPL with a lower CC (e.g., 85%, 90%) to estimate a BTV. ProUCL can compute a Chebyshev UPL (and all other UPLs) for any user specified CC in the interval [0.5, 1]. 3.5.5 Normal, Lognormal, and Gamma Distribution based Upper Prediction Limits for k Future Comparisons A UPLk95 computed based upon an established background data set represents that statistic such that k future (next, independent and not belonging to the current data set) observations from the target population (e.g., background, comparable to background) will be less than or equal to the UPLk95 with a CC of 0.95. A UPLk95 for k (>1) future observations is designed to compare k future observations; we are 95% sure that "k" future values from the background population will be less than or equal to UPLk95 with CC of 0.95. In addition to UPLk, ProUCL also computes an upper prediction limit of the mean of k future observations, UPLk (mean). A UPLk (mean) is commonly used in groundwater monitoring applications. A UPLk controls the false positive error rate by using the Bonferroni inequality based critical values to perform k future comparisons. These UPLs statisfy the relationship: UPLi ------- UPL395 = \x A lognormal distribution based UPLk (1 - a) for k future observations, xn+1,xn+2,...,xn+k is given by the following equation: A gamma distribution based UPLk for the next k > 7 (k future observations) are computed similarly using the WH and HW approximations described in Section 3.4.3. 3. 5. 6 Proper Use of Upper Prediction Limits It is noted that some users tend to use UPLs without taking their definition and intended use into consideration; this is an incorrect application of a UPL. Some important points to note about the proper use of UPLi and UPLk for k>7 are described as follows. • When a UPLk is computed to compare k future observations collected from a site area or a group of MW within an operating unit (OU), it is assumed that the project team will make a decision about the status (clean or not clean) of the site (MWs in an OU) based upon those k future observations. • The use of an UPLk implies that a decision about the site-wide status will be made only after k comparisons have been made with the UPLk. It does not matter if those k observations are collected (and compared) simultaneously or successively. The k observations are compared with the UPLk as they become available and a decision (about site status) is made based upon those k observations. • An incorrect use of a UPLi 95 is to compare many (e.g., 5, 10, 20, etc.) future observations. This results in a higher than 0.05 false positive rate. Similarly, an inappropriate use of a UPLioo would be to compare less than 100 (i.e., 10, 20, or 50 observations) future observations. Using a UPLioo to compare 10 or 20 observations can potentially result in a high number of false negatives (a test with reduced power), declaring contaminated areas comparable to background. • The use of other statistical limits such as 95%-95% UTLs (UTL95-95) is preferred to estimate BTVs and not-to-exceed values. The computation of a UTL does not depend upon the number of future comparisons which will be made with the UTL. 3.6 Upper Simultaneous Limits An (1 - a) * 100% upper simultaneous limit (USL) based upon an established background data set is meant to provide coverage for all observations, xt, i = 1, 2, n simultaneously in the background data set. It is implicitly assumed that the data set comes from a single background population and is free of outliers (established background data set). A USL95 represents that statistic such that all observations from the "established" background data set will be less than or equal to the USL95 with a CC of 0.95. It is expected that observations coming from the background population will be less than or equal to the USL95 with a 95% 115 ------- CC. A USL95 can be used to perform any number (unknown) of comparisons of future observations. The false positive error rate does not change with the number of comparisons as the purpose of the USL95 is to perform any number of comparisons simultaneously. Notes: If a background population is established based upon a small data set; as one collects more observations from the background populations, some of the new background observations will exceed the largest value in the existing data set. In order to address these uncertainties, the use of a USL is suggested, provided the data set represents a single population without outliers. 3.6.1 Upper Simultaneous Limits for Normal, Lognormal and Gamma Distributions The normal distribution based two-sided (1 - a) 100% simultaneous interval obtained using the first order Bonferroni inequality (Singh and Nocerino 1995, 1997) is given as follows: P(x-sdba------- Nonparametric USL: For nonparametric data sets, the largest value, X(n) is used to compute a nonparametric USL. Just like a nonparametric UTL, a nonparametric USL may fail to provide the specified coverage, especially when the sample size is small (e.g., <60). The confidence coefficient actually achieved by a USL can be computed using the same process as used for a nonparametric UTL described in Sections 3.4.4.2 and 3.4.4.3. Specifically, by substituting r = n in equation (3-11), the confidence coefficient achieved by a USL can be computed, and by substituting m=l in equation (3-12), one can compute the sample size needed to achieve the desired confidence. Note: Nonparametric USLs, UTLs or UPLs should be used with caution; nonparametric upper limits are based upon order statistics and therefore do not take the variability of the data set into account. Often nonparametric BTVs estimated by order statistics do not achieve the specified CC unless the sample size is fairly large. Dependence of UTLs and USLs on the Sample Size: For smaller samples (n <10), a UTL tends to yield impractically large values, especially when the data set is moderately skewed to highly skewed. For data sets of larger sizes, the critical values associated with UTLs tend to stabilize whereas critical values associated with a USL increase as the sample size increases. Specifically, a USL95 is less than a UTL95- 95 for samples of sizes, n <16, they are equal/comparable for samples of size 17, and a USL95 becomes greater than a UTL95-95 as the sample size becomes greater than 17. Some examples illustrating the computations of the various upper limits described in this chapter are discussed as follows. Example 3-1. Consider the real data set used in Example 2-4 of Chapter 2 consisting of concentrations for several constituents of potential concern, including aluminum, arsenic, chromium (Cr), and lead. The computation of background statistics obtained using ProUCL for some of the metals are summarized as follows. Upper Limits Based upon a Normally Distributed Data Set: The aluminum data set follows a normal distribution as shown in the following GOF Q-Q plot of Figure 3-1. Normal Q-Q Plot for Aluminum n = 24 Mean = 7789 Sd = 4264 Slope = 4293 Intercept = 7789 Correlation, R= 0.976 Shapiro-WilkTest DilicalVal|0.05)-0.916 Dala Appeal Normal Appro*. TestVdlue-D.939 p-Value-0.161 • Best Fil Line Theoretical Quantiles (Standard Normal) Figure 3-1. Normal Q-Q plot of Aluminum with GOF Statistics 117 ------- From the normal Q-Q plot shown in Figure 3-1, it is noted that the 3 largest values are higher (but not extremely high) than the rest of the 21 observations. These observations may or may not come from the same population as the rest of the 21 observations. Table 3-1. BTV Estimated Based upon All 24 Observations Aluminum General Statistics Total Number of Observations 24 Minimum 1710 Second Largest 154QO Maximum 16200 Mean 7789 Coefficient of Variation 0.547 Mean of logged Data 8.79S Number of Distinct Observations 24 First Quartile 405S Median 7010 Third Quartile 10475 SD 4264 Skewness 0.542 SD of logged Data 0.61 Critical Values for Background Threshold Values (BTVs) Tolerance Factor K (For UTL) 2,309 d2max for USL) 2.644 Shapiro Wilk Test Statistic 5% Shapiro Wilk Critical Value Lilliefors Test Statistic 5% Lilliefors Critical Value Normal GOF Test 0.939 0.916 0.109 0.181 Shapiro Wilk GOF Test Data appear Normal at 5% Significance Level Lilliefors GOF Test Data appear Normal at 5% Significance Level Data appear Normal at 5% Significance Level Background Statistics Assuming Normal Distribution 95% UTL with 95% Coverage 17635 95%UPL|t) 15248 95% USL 19063 90% Percentile (z) 132-54 95% Percentile (z) 14803 93% Percentile (2) 17708 The classical outlier tests (Dixon and Rosner tests) did not identify these 3 data points as outliers. Robust outlier tests, MCD (Rousseeuw and Leroy 1987), and PROP influence function (Singh and Nocerino, 1995) based tests identified the 3 high values as statistical outliers. The project team should decide whether or not the 3 higher concentrations represent outliers. A brief discussion about robust outlier methods is given in Chapter 7. The inclusion of the 3 higher values in the data set resulted in higher upper limits. The various upper limits have been computed with and without the 3 high observations and are summarized respectively, in Tables 3-1 and 3-2 as follows. The project team should make a determination of which statistics (with outliers or without outliers) should be used to estimate BTVs. 118 ------- Table 3-2. BTV Estimated Based upon 21 Observations without 3 Higher Values Aluminum General Statistics Total Number of Observations 21 Minimum 1710 Second Largest 11 SOD Maximum 125DD Mean 6669 Coefficient of Variation 0.482 Number of Distinct Observations 21 Number of Missing Observations 3 First Quartile 3900 Median 6350 Third Quartile 9310 SD 3215 Skewness 0.25 Mean of logged Data 8.676 SD of logged Data 0.549 Critical Values for Background Threshold Values (BTVs) Tolerance Factor K (For UTL) 2.371 d2max for USL) 2.58 Normal GOF Test Shapiro Wilk Test Statistic 0.955 5% Shapiro Wilk Critical Value 0.903 Lilliefors Test Statistic 0.12 5% Lilliefors Critical Value 0.193 Data appear Normal at Shapiro Wilk GOF Test Data appear Normal at 5V. Significance Level Lilliefors GOF Test Data appear Normal at 5*'= Significance Level 5% Significance Level Background Statistics Assuming Normal Distribution 95% UTL with 95*4 Coverage 14291 90% Percerrtile (z) 10789 95%UPL------- Table 3-3. Lognormal Distribution Based UPLs, UTLs, and USLs Chromium General Statistics Total Number of Observations 24 Minimum 3 Second Largest 20 Maximum 35.5 Mean 11.97 Coefficient of Variation 0.576 Mean of logged Data 2.334 Number of Distinct Observations First Quartile Median Third Quartile SD Skewness SD of logged Data 19 7.975 11 14.25 6.892 1.728 0.568 Gitical Values for Background Threshold Values (BTVs) Tolerance Factor K (For UTL) 2.309 Lognormal Shapiro Wilk Test Statistic 0.978 5% Shapiro Wilk Critical Value 0.91 6 Lilliefors Test Statistic 0.123 5% Lilliefons Critical Value 0.181 d2max for USL) GOFTest Shapiro Wilk Lognonnal GOF Test Data appear Lognormal at 5% Significance Level Lilliefors Lognonnal GOF Test Data appear Lognormal at 5% Significance Level 2.644 Data appear Lognoimal at 5% Syiificance Level Background Statistics assui 95*4 UTL with 95% Coverage 38.3 95%UPL(t) 27.37 95% U P L f or Next 5 Observations 43.96 95% LIPLfor Mean of 5 Observations 1 6.66 ning Lognormal Distribution 90% Percentile (z) 95% Percentile (z) 99% Percentile ft 95% USL 21.37 26.27 38.68 46.33 Example 3-3. Arsenic concentrations of the data set used in Example 2-4 follow a gamma distribution. The background statistics, obtained using a gamma model, are shown in Table 3-4. Figure 3-3 is the gamma Q-Q plot with GOF statistics. Gamma Q-Q Plot for Arsenic Aisenic = 24 ear, = 2.1*83 tot-3-6616 els slai-0.5887 lope-1.0264 creep! - 0.01 S3 .rcijtion. R - i) 3770 dels dn-D a i ling Test iiSi.jSisiif-0.5f1 CiilicalValue[0 05) - OL748 D ala appeal Gamma Dishibuted • Best Fit Line Theoretical Quantiles of Gamma Distribution Figure 3-3. Gamma Q-Q plot of Arsenic with GOF Statistics 120 ------- Table 3-4. Gamma Distribution Based UPLs, UTLs, and USLs Arsenic General Statistics Total Number of Observations 24 Minimum 8.66 Second Largest 3.7 Maximum 5.9 Mean 2.148 Coefficient of Variation 0.54 Mean of logged Data 0.639 Number of Distinct Observations 18 First Quartile 1.2 Median 2.05 Third Quartile 2.45 SD 1.159 Skewness 1.554 SD of logged Data 0.51 Critical Values for Background Threshold Values (BTVs) Tolerance Factor K (For UTL) 2.309 d2maxforUSL} 2.644 Gamma GOF Test A-D Test Statistic 0.341 Anderson-Darling Gamma GOF Test 5% A-D Critical Value 0.748 Detected data appear Gamma Distributed at 5% Significance Level K-S Test Statistic 0114 Kolmogrov-Smimoff Gamma GOF Test 5% K-S Critical Value 0.179 Detected data appear Gamma Distributed at 5% Significance Level Detected data appear Gamma Distributed at 5% Significance Level khat(MLE) Theta hat (MLE) nu hat (MLE} MLE Mean |bias corrected) Gamma Statistics 4.153 0.517 199.3 2.142 k star (bias corrected MLE} 3.662 Theta star (bias corrected MLE) Q.587 nu star fbias corrected} 175.8 M LE Sd (bias corrected) 1.123 Background Statistics Assuming Gamma Distribution 95% Wilson Hiferty (WH) Approx. Gamma UPL 4.345 95% Hawkins Wixley (HW) Approx. Gamma UPL 4.397 95% WH Approx. Gamma UTL with 95% Coverage 5.382 95% HW .Approx. Gamma UTL with 95% Coverage 5.524 95%WHUSL 6.074 90% Percentile 3.654 95% Percentile 4.264 99% Percentile 5.574 95% HW USL 6.294 121 ------- Example 3-4. Lead concentrations of the data set used in Example 2-4 do not follow a discernible distribution. The various nonparametric background statistics for lead are shown in Table 3-5. Table 3-5. Nonparametric UPLs, UTLs, and USLs for Lead in Soils Lead General Statistics Total Number of Observations 24 Minimum 4.9 Second Largest 38.5 Maximum 109 MCJI| 22.49 Coefficient of Variation 1 .193 Mean of logged Data 2.743 Number of Distinct Observations 1 8 First Quartile 10.43 Median 14 Third Quartile 19.25 SD| 26.83 Skewness 2.665 SD of logged Data 0.771 Critical Values for Background Threshold Values (BTVs) Tolerance Factor K (For UTL) 2.309 d2max (For USL) 2.644 Nonparamelric Distribution Free Background Statistics Data do not follow a Discernible Distribution (0.05) Nonparametric Upper Limits for Background Threshold Values Order of Statistic, r 24 r^praxirnate f 1 .263 95% Percentile Bootstrap UTL with 95% Coverage 109 95% UPL 106.4 mChebyshevUPL 104.6 95%ChebyshevUPL 141.8 95% USL 109 55% UTL with 55% Coverage 109 Confidence Coefficient (CC) achieved by UTL 0.708 95% BCA Bootstrap UTL with 95% Coverage 109 90% Percentile 44.81 95% Percentile 91.72 99% Percentile 106.6 Notes: Note: As mentioned before, nonparametric upper limits are computed by higher order statistics, or by some value in between (based upon linear interpolation) the higher order statistics. In practice, nonparametric upper limits do not provide the desired coverage to the population parameter (upper threshold) unless the sample size is large. From Table 3-5, it is noted that a UTL95-95 is estimated by the maximum value in the data set of size 24. However, the CC actually achieved by UTL95-95 (and also by USL95) is only 0.708. Therefore, one may want to use other upper limits such as 95% Chebyshev UPL = 141.8 to estimate a BTV. Note: As mentioned earlier, for symmetric and mildly skewed nonparametric data sets (when sd of logged data is <=0.5), one can use the normal distribution to compute percentiles, UPLs, UTLs and USLs. Example 3-5: Why Use a Gamma Distribution to Model Positively Skewed Data Sets? The data set considered in Example 2-2 of Chapter 2 is used to illustrate the deficiencies and problems associated with the use of a lognormal distribution to compute upper limits. The data set follows a lognormal as well as a gamma model; the various upper limits, based upon a lognormal and a gamma model, are summarized as follows. The data set is highly skewed with sd of logged data = 1.68. The 122 ------- largest value in the data set is 169.8, the UTL95-95 and UPL95 based upon a lognormal model are 799.7 and 319 both of which are significantly higher than the maximum value of 169.8. UTL95-95s based upon WH and HW approximations to gamma distributions are 245.3 and 285.6; UPLs based upon WH and HW approximations are 163.5 and 178.2 which appear to represent more reasonable estimates of the BTV. These statistics are summarized in Table 3-6 (lognormal) and Table 3-7 (gamma) below. Table 3-6. Background Statistics Based upon a Lognormal Model x General Satisfies Total Number of Observations 25 Minimum 0.349 Second Largest 164.3 Maximum 1S9.8 Mean 44.09 Coefficient of Variation 1.164 Mean of logged Data 2.S35 Number of Distinct Observations 25 First Quartile 5.093 Median 18.77 Third Quartile 72.62 SD 51.34 Skewness 1.294 SD of logged Data 1.68 Critical Values for Background Threshold Values (BTVs) Tolerance Factor K (For UTL) 2.292 d2max for USL) 2.663 Lognonnal GOF Test Shapiro Wilk Test Statistic 0.948 Shapiro Wilk Lognormal GOF Test 5% Shapiro Wilk Critical Value 0.918 Data appear Lognormal at 5% Significance Level Lilliefors Test Statistic 0.135 Ljlliefors Lognormal GOF Test 5% Ulliefors Critical Value 0.177 Data appear Lognormal at 5% Significance Level Data appear Lognonnal at 5"4 Significance Level Background Statistics assuming Lognormal Distribution 95% UTL with 35% Coverage 799.7 35%UPLft) 319 90% Percentile ',z] 146.5 95% Percentile (2] 269.7 123 ------- Table 3-7. Background Statistics Based upon a Gamma Model x General Statistics Total Number of Observations 25 Minimum 0.349 Second Largest 164.3 Maximum 163.8 Mean 44.09 Coefficient of Variation 1.164 Mean of logged Data 2.835 Number of Distinct Observations 25 First Quartile 5.093 Median 18.77 Third Quartile 72.62 SD 51.34 Skewness 1.294 SD of logged Data 1.68 Critical Values for Background Threshold Values (BTVs) Tolerance Factor K (For UTL) 2.292 d2max for USL) 2.663 Gamma GOF Test A-D Test Statistic 0.374 Anderson-Darling Gamma GOF Test 5% A-D Critical Value Q.794 Detected data appear Gamma Distributed at 5% Significance Level K-S Test Statistic 0113 Kolmogrov-Smimoff Gamma GOF Test 5% K-S Critical Value 0.183 Detected data appear Gamma Distributed at 5% Significance Level Detected data appear Gamma Distributed at 5% Significance Level k hat (MLE) Theta hat (MLE) nu hat {MLE} MLE Mean (bias corrected) Gamma Statistics 0.643 68.58 32.15 44.03 k star (bias corrected M LE} 0.532 Theta star (bias corrected MLE) 74.42 nu star {bias corrected} 29.62 MLE Sd {bias corrected} 57.28 Background Statistics Assuming Gamma Distribution 95% Wilson Hilferty (WH) Approx. Gamma UPL 163,5 35% Hawkins Wbdey (HW) Approx. Gamma UPL 178.2 95% WH Approx. Gamma UTL with 95% Coverage 245,3 95% HW Approx. Gamma UTL with 95% Coverage 285.6 90% Percentile 115 95% Percentile 159.4 99% Percentile 266.8 124 ------- CHAPTER 4 Computing Upper Confidence Limit of the Population Mean Based upon Left-Censored Data Sets Containing Nondetect Observations 4.1 Introduction Nondetect (ND) observations are inevitable in most environmental data sets. It should be noted that the estimation of the mean and sd, and the computation of the upper limits (e.g., upper confidence limits [UCLs], upper tolerance intervals [UTLs]) are two different tasks. For left-censored data sets with NDs, in addition to the availability of good estimation methods, the availability of rigorous statistical methods which account for data skewness is needed to compute the decision making statistics such as UCLs, UTLs, and UPLs. For left-censored data sets consisting of multiple detection limits (DLs) or reporting limits (RLs), ProUCL 4.0 (2007) and its higher versions offer methods to: 1) impute NDs using regression on order statistics (ROS) methods; 2) perform GOF tests; 3) estimate the mean, standard deviation (sd), and standard error of the mean; and 4) compute skewness adjusted upper limits (e.g., UCLs, UTLs, UPLs). Based upon KM (Kaplan and Meier 1958) estimates, and the distribution and skewness of detected observations, several upper limit computation methods which adjust for data skewness have also been incorporated in ProUCL 5.1. For left-censored data sets with NDs, Singh and Nocerino (2002) compared the performances of the various estimation methods (in terms of bias and MSB) to estimate the population mean, ^ , and sd, crl including the MLE method (Cohen 1950, 1959), restricted MLE (RMLE) method (Perrson and Rootzen 1977); Expectation Maximization (EM) method (Gleit 1985), EPA Delta lognormal method (EPA 1991; Hinton 1993), Winsorization method (Gilbert 1987), and regression on order statistics (ROS) method (Helsel 1990). Singh, Maichle, and Lee (EPA 2006) performed additional simulation experiments to study and evaluate the performances (in terms of bias and MSE) of KM and ROS methods for estimating the population mean. They concluded that the KM method yields better estimates, in terms of bias, of population mean in comparison with other estimation methods including the LROS (ROS on logged data) method. Singh, Maichle, and Lee (EPA 2006) also studied the performances, in terms of coverage probabilities, of some parametric and nonparametric UCL computation methods based upon ROS, KM, and other estimation methods. They concluded that for skewed data sets, KM estimates based UCLs computed using bootstrap methods (e.g., BCA bootstrap, bootstrap-t) and Chebyshev inequality perform better than the Student's t statistic UCL and percentile bootstrap UCL computed using ROS and KM estimates as described in Helsel (2005, 2012) and incorporated in NADA packages (2013). As mentioned above, computing good estimates of the mean and sd based upon left-censored data sets addresses only half of the problem. The main issue is computing decision statistics (UCL, UPL, UTL) which account for NDs as well as uncertainty and data skewness inherently present in environmental data sets. Until recently (ProUCL 4.0, 4.00.05, 4.1; Singh, Maichle, and Lee 2006), not much guidance was available on how to compute the various upper limits (UCLs, UPLs, UTLs) based upon skewed left- censored data sets with multiple DLs. For left-censored data sets, the existing literature (Helsel 2005, 2012) suggests computing upper limits using a Student's t-type statistic and percentile bootstrap methods on KM and LROS estimates without adjusting for data skewness. Environmental data sets tend to follow skewed distributions, and UCL95s and other upper limits computed using methods described in Helsel 125 ------- (2005, 2012) will under estimate the population parameters of interest including EPCs and background threshold values. In earlier versions of ProUCL (ProUCL versions 4 [2007, 2009, 2010]), all evaluated estimation methods including the poor performing methods (MLE and RMLE, and Winsorization methods) and better performing, in terms of bias in the mean estimate, estimation (KM method) and UCL computation methods (BCA bootstrap, bootstrap-t) were incorporated in ProUCL version 4 (2007, 2009, 2010). Currently, the KM estimation method is widely used in environmental applications to compute parametric (when detected data follow a known distribution) and nonparametric upper limits needed to estimate environmental parameters of interest such as the population mean and upper thresholds of a background population. Note that the KM method is now included in a recent EPA RCRA groundwater monitoring guidance document (2009). Due to the poor performances and/or failure to correctly verify probability distributions for data sets with multiple DLs, the parametric MLE and RMLE methods, the normal ROS and the Winsorization estimation methods for computing upper limits are no longer available in ProUCL version 5.0/5.1. The normal ROS method is available only under the Stats/Sample Sizes module of ProUCL 5.0/5.1 to impute NDs based upon the normal distribution assumption for advanced users who may want to use the imputed data in other graphical and exploratory methods such as scatter plots, box plots, cluster analysis and principal component analysis (PCA). The estimation methods for computing upper limits retained in ProUCL 5.0/5.1 include the two ROS (lognormal, and gamma) methods and the KM method. The KM estimation method can be used on a wide-range of skewed data sets with multiple DLs and NDs exceeding detected observations. Also, the substitution methods such as replacing NDs by half of their respective DLs and the H-UCL method (EPA 2009 recommends its use in Chapter 15) have been retained in ProUCL 5.0/5.1 for historical reasons, and academic and research purposes. Inclusion of the DL/2 method (substitution of 1A the DL for NDs) in ProUCL should not be inferred as a recommended method. The developers of ProUCL are not endorsing the use of the DL/2 estimation method or H-UCL computation method. Note on the use of letter k (k): Not to get confused with the use of letter "k (k)" in this Chapter and in Chapters 2, 3, 4, and 5. Following the standard statistical terminology, "k" is used to denote the shape parameter of a gamma distribution, G(k, 9) as described in Chapter 2; "k" is used to represent future (next) observations (Chapter 3 and 5), and "k" is used to represent the number of ND observations present in a data set (Chapters 4 and 5). Notes on Skewness of Left-Censored Data Sets: Skewness of a data set is measured as a function of sd, a (or its estimate, cr) of log-transformed data. Like uncensored full data sets, a, or its estimate, a, of the log-transformed detected data is used to get an idea about the skewness of a data set consisting of ND observations. This information along with the distribution of detected observations is used to decide which UCL should be used to estimate the EPC and other upper limits for data sets consisting of both detects and NDs. For data sets with NDs, output sheets generated by ProUCL 5.0/5.1 display the sd, ------- 4.2 Pre-processing a Data Set and Handling of Outliers Throughout this chapter (and in other chapters such as Chapters 2, 3, and 5), it has been implicitly assumed that the data set under consideration represents a "single" statistical population as a UCL is computed for the mean of a "single" statistical population. In addition to representing "wrong" values (e.g., typos, lab errors), outliers may also represent observations coming from population(s) significantly different from the main dominant population whose parameters (mean, upper percentiles) we are trying to estimate based upon the available data set. The main objective of using a statistical procedure is to model the majority of data representing the main dominant population and not to accommodate a few low probability (coming from far and extreme tails) outlying observations potentially representing impacted locations (site related or otherwise). Statistics such as a UCL95 of the mean computed using data sets with occasional low probability outliers tend to represent locations exhibiting those elevated low probability outlying observations rather than representing the main dominant population. 4.2.1 Assessing the Influence of Outliers and Disposition of Outliers One can argue against "not using the outliers" while estimating the various environmental parameters such as the EPCs and BTVs. An argument can be made that outlying observations are inevitable and can be naturally occurring (not impacted by site activities) in some environmental media (and therefore in data sets). For example, in groundwater applications, a few elevated values (coming from the far tails of the data distribution with low probabilities) may be considered to be naturally occurring and as such may not represent the impacted MW data values. However, the inclusion of a few outliers (impacted or naturally occurring observations) tends to yield distorted and elevated values of the decision statistics of interest (UCLs, UPLs, and UTLs); and those statistics tend not to represent the main dominant population (MW concentrations). As mentioned earlier, instead of representing the main dominant population, the inflated decision statistics (UCLs, UTLs) computed with outliers included, tend to represent those low probability outliers. This is especially true when one is dealing with smaller data sets (n <20-30) and a lognormal distribution is used to model those data sets. To assess the influence of outliers on the various statistics (upper limits) of interest, it is suggested to compute all relevant statistics using data sets with outliers and without outliers, and then compare the results. This extra step often helps the project team/users to see the direct potential influence of outlier(s) on the various statistics of interest (mean, UPLs, UTLs). This in turn will help the project team to make informative decisions about the disposition of outliers. That is, the project team and experts familiar with the site should decide which of the computed statistics (with outliers or without outliers) represent better and more accurate estimate(s) of the population parameters (mean, EPC, BTV) under consideration. 4.2.2 Avoid Data Transformation Data transformations are performed to achieve symmetry of the data set and be able to use parametric (normal distribution based) methods on transformed data. In most environmental applications, the cleanup decisions are made based on statistics and results computed in the original scale as the cleanup goals need to be attained in the original scale. Therefore, statistics and results need to be back- transformed in the original scale before making any cleanup decisions. Often, the back-transformed statistics (UCL of the mean) in the original scale suffer from an unknown amount of transformation bias; many times the transformation bias can be unacceptably large (for highly skewed data sets) leading to incorrect decisions. The use of a log-transformation on a data set tends to accommodate outliers and hide 127 ------- contaminated locations instead of revealing them. Specifically, an observation that is a potential outlier (representing a contaminated location) in the original raw scale may not appear to be an outlier in the log- scale. This does not imply that the location with elevated concentrations in the original scale does not represent an impacted location. This issue has been considered and illustrated throughout this guidance document. The use of a gamma model does not require any data transformation therefore whenever applicable the use of a gamma distribution is suggested to model skewed data sets. In cases when a data set in the original scale cannot be modeled by a normal or a gamma distribution, it is better to use nonparametric methods rather than testing or estimating parameters in the transformed space. For data sets which do not follow a discernible parametric distribution, nonparametric and computer intensive bootstrap methods can be used to compute the upper limits needed to estimate environmental parameters. Several of those methods are available in ProUCL 5.1(ProUCL 5.0) for data sets consisting of NDs with multiple DLs. 4.2.3 Do Not Use DL/2(t) UCL Method In addition to environmental scientists, ProUCL is also used by students and researchers. Therefore, for historical and comparison purposes, the substitution method of replacing NDs by half of the associated DLs (DL/2) is retained in ProUCL 5.1.; that is the DL/2 GOF tests, UCL, UPL, and UTL computation methods have been retained in ProUCL 5.0/5.1 for historical reasons, and comparison and academic purposes. For data sets with NDs, output sheets generated by ProUCL display a message suggesting that DL/2 is not a recommended method. It is suggested that the use of the DL/2 (t) UCL method (UCL computed using Student's t-statistic) be avoided when estimating a EPC or BTVs, unless the data set consists of only a small fraction of NDs (<5%) and the data are mildly skewed. The DL/2 UCL computation method does not provide adequate coverage (Singh, Maichle, and Lee 2006) for the population mean, even for censoring levels as low as 10% or 15%. This is contrary to statements (EPA 2006b) made that the DL/2 UCL method can be used for lower (< 20%) censoring levels. The coverage provided by the DL/2 (t) UCL method deteriorates fast as the censoring intensity, percentage of NDs, increases and/or data skewness increases. 4.2.4 Minimum Data Requirement Whenever possible, it is suggested that a sufficient number of samples be collected to satisfy the requirements for the data quality objectives (DQOs) for the site. Often, in practice, it is not feasible to collect the number of samples as determined by DQOs-based sample size formulae. Therefore, some rule-of-thumb minimum sample size requirements are described in this section. At the minimum, collect a data set consisting of about 10 observations to compute reasonably reliable and accurate estimates of EPCs (UCLs) and BTVs (UPLs, UTLs). The availability of at least 15 to 20 observations is desirable to compute UCLs and other upper limits based upon re-sampling bootstrap methods. Some of these issues have also been discussed in Chapter 1 of this Technical Guide. However, from a theoretical point of view, ProUCL can compute various statistics (KM UCLs) based upon data sets consisting of at least 3 detected observations. The accuracy of the decisions based upon statistics computed using such small data sets remains questionable. 4.3 Goodness-of-Fit (GOF) Tests and Skewness for Left-Censored Data Sets It is not easy to assess and verify the distribution of data sets with NDs, especially when multiple DLs are present and those DLs exceed the detected values. One can perform GOF tests on detected data and consider/expect that NDs (not the DLs) also follow the same distribution of detected data. For data sets 128 ------- with NDs, ProUCL has GOF tests for normal, lognormal, and gamma distributions which are also supplemented with graphical Q-Q plots. GOF tests in ProUCL include: 1) exclude all NDs; 2) replace NDs by their DL/2s; and 3) ROS methods. In the environmental literature (Helsel 2005, 2012), some other graphs such as censored probability plots have also been described. However, the usefulness of those graphs in the computation of decision making statistics is not clear. Some practitioners have criticized that ProUCL does not offer censored probability plots, therefore, even though those graphs do not provide additional useful information, ProUCL 5.1 now offers those graphs as well. Formally, let x\, X2, ..., xn (including k NDs and (n-k) detected measurements) represent a random sample of n observations obtained from a population under investigation (e.g., background area, or an area of concern [AOC]). Out of the n observations, k: l------- Formally, let x\,x2, ...,xn represent « data values of a left-censored data set. Let p,KM and a^ represent KM estimates of the mean and variance based upon such a data set with NDs. Let x[ denote the smallest xt. Then Fx = \, x x j ] ' wlth Xo= ° t4' Using the PLE (or KM) method, an estimate of the SE of the mean is given by the following equation. Where k = number of ND observations, and 7=1 The KM variance is computed as follows: (4'3) U, \ in f = KM mean of the data, x ' (x)—KM JU, 2\_f't/r = KM mean of the square of the data, x (second raw moment) In addition to the KM mean, ProUCL computes both the SE of the mean given by (4-2) and the variance given by (4-3). The SE is used to estimate EPCs (e.g., UCLs) whereas the variance is used to compute BTV estimates (e.g., UTLs, USLs). The KM method in ProUCL can be used directly on left-censored environmental data sets without requiring any flipping of data and back flipping of the KM estimates and 130 ------- other statistics (e.g., flipping LCL to compute a UCL) which may be burdensome for most users and practitioners. Note: Decision making statistics (e.g., UPLs and UTLs) used in background evaluations projects require good estimates of the population standard deviation, sd. The decision statistics (e.g., UTLs) obtained using the direct estimate of sd (Equation 4-3) and an indirect "back door" estimate of sd (Helsel 2012) can differ significantly, especially for skewed data sets. An example illustrating this issue is described as follows. Example 4-1 (Oahu Data Set): Consider the moderately skewed well-cited Oahu data set (Helsel 2012). A direct KM estimate of the sd obtained using equation (4-3) is o= 0.713; and an indirect KM estimate of sd = sqrt (24)*SE = 4.899 * 0.165 = 0.807 (Helsel 2012, p 87). A UTL95-95 (direct) = 2.595 and a UTL95-95 (based upon indirect estimate ofsd) = 2.812. The discrepancy between the two estimates ofsd and upper limits (e.g., UTL95-95) computed using the two estimates increases with skewness. Cautionary notes for NADA (2013) in R Users: It is well known that the KM method yields a good (in terms of bias) estimate of the population mean (Singh, Maichle, and Lee 2006). However, the use of KM estimates in the Student's t-statistic based UCL equation or percentile bootstrap method as included in NADA packages do not guarantee that those UCLs will provide the desired (e.g., 0.95) coverage for the population mean in all situations. Specifically, it is highly likely that for moderately skewed to highly skewed data sets (determined using detected values) the Student's t-statistic or percentile bootstrap method based UCLs computed using KM estimates will fail to provide the desired coverage to the population mean, as these methods do not account for skewness. Several UCL (and other limits) computation methods based upon KM estimates which adjust for data skewness are available in ProUCL 5.0 and ProUCL 5.1; those methods were not available in ProUCL 4.1. 4.5 Regression on Order Statistics (ROS) Methods In this guidance document and in ProUCL software, LROS represents the ROS (also known as robust ROS) method for a lognormal distribution and GROS represents the ROS method for a gamma distribution. The ROS methods impute NDs based upon a hypothesized distribution such as a gamma or a lognormal distribution. The "Stats/Sample Sizes" menu option of ProUCL 5.1 can be used to impute and store imputed NDs along with the original detected values in additional columns generated by ProUCL. ProUCL assigns self-explanatory titles for those generated columns. It is a good idea to store the imputed values to determine the validity of the imputed NDs and assess the distribution of the complete data set consisting of detects and imputed NDs. As a researcher, one may want to have access to imputed NDs to be used by other methods such as regression analysis and PCA. Moreover, one cannot easily perform multivariate methods on data sets with NDs; and the availability of imputed NDs makes it possible for researchers to use multivariate methods on data sets with NDs. The developers believe that statistical methods to evaluate data sets with NDs require further investigation and research. Providing the imputed values along with the detected values may be helpful to practitioners conducting research in this area. For data sets with NDs, ProUCL 5.0/ProUCL 5.1 also performs GOF tests on data sets obtained using the LROS and GROS methods. The ROS methods yield a data set of size n with (n-k) original detected observations and k imputed NDs. The full data set of size n thus obtained can be used to compute the various summary statistics, and to estimate the EPCs and BTVs using methods described in Chapters 2 and 3 of this technical guidance document. In a ROS method, the distribution (e.g., gamma, lognormal) of the (n-k) detected observations is assessed first; and assuming that the £ND observations, x\, X2, ..., Xk follow the same distribution (e.g., gamma or a 131 ------- lognormal distribution when used on logged data) of the (n-k) detected observations, the NDs are imputed using an OLS regression line obtained using the (n-k) pairs: (ordered detects, hypothesized quantiles). Earlier versions of ProUCL software also included the normal ROS (NROS) method for computing the various upper limits. The use of NROS on environmental data sets (with positive values) tends to yield unfeasible and negative imputed ND values; and the use of negative imputed NDs yields biased and incorrect results (e.g., UCL, UTLs). Therefore, the NROS method is no longer available in the UCLs/EPCs and Upper Limits/BTVs modules of ProUCL version 5.0 and ProUCL 5.1. Instead, when detected data follow a normal distribution, the use of KM estimates in normal equations is suggested for computing the upper limits as described in Chapters 2 and 3. 4.5.1 Computation of the Plotting Positions (Percentiles) and Quantiles Before computing the n hypothesized (lognormal, gamma) quantiles, q®; i:=k+l, k+2,...,n, and q^dif, i: = 1, 2, ..., k, the plotting positions (also known as percentiles) need to be computed for the n observations with £NDs and (n-k) detected values. There are several methods available in the literature (Blom 1958; Barnett, 1976; Singh and Nocerino, 1995, Johnson and Wichern, 2002) to compute the plotting positions (percentiles). Note that plotting positions for the three ROS methods: LROS, GROS, and NROS are the same. For a full data set of size n, the most commonly used plotting position for the ith observation (ordered) is given by (/' - 3/s) / (« + %) or (i - 'Aj/n; i: =1,2,... ,n. These plotting positions are routinely used to generate Q-Q plots based upon full uncensored data sets (Singh 1993; Singh and Nocerino 1995; ProUCL 3.0 and higher versions). For the single DL case (with all observations below the DL reported as NDs), ProUCL uses Blom's percentiles, (/' - %) / (n + %) for normal and lognormal distributions, and uses empirical percentiles given by (i - %)/n for a gamma distribution. Specifically, for normal and lognormal distributions, once the plotting positions have been obtained, the n normal quantiles, g® are computed using the probability statement: P(Z < g®) = (/' - 3/8) /(« + %),/: = 1, 2, ...,«, where Z represents a standard normal variate (SNV). The gamma quantiles are computed using the probability statement: P(X - ------- 4.5.2 Computing OLS Regression Line to Impute NDs An ordinary least squares (OLS) regression model is obtained by fitting a linear straight line to the (n-k) ordered (in ascending order) detected values, x^ (perhaps after a suitable transformation), and the (n-k) hypothesized (e.g., normal, gamma) quantiles, qpy, i:=k+l, k+2,...,n, associated with those (n-k) detected ordered observations. The hypothesized quantiles are obtained for all of the n data values by using the hypothesized distribution for the (n-k) detected observations. The quantiles associated with (n-k) detected values are denoted by q^y, i:=k+l, k+2,...,n, and the k quantiles associated with ND observations are denoted by q^y, i: = 1, 2, ..., k.. An OLS regression line is obtained first by using the (n - k) pairs, (q®, x®); /':= k + 1, k + 2, ...,«, where Xft) are the (n-k) detected values arranged in ascending order. The OLS regression line fitted to the (n - k) pairs (qft), Xft)); /':= k + \,k + 2, ...,n corresponding to the detected values is given by: = a i:= k + 1, k + 2, n. (4-4) Table 4-1. Plotting Positions, Gamma and Lognormal (Normal) Quantiles (Q) Pyrene 28 31 32 34 35 35 40 47 48 58 59 63 64 64 67 67 67 72 73 84 86 86 87 34 98 100 103 103 D_pyrene a 1 1 1 a a 1 1 1 a 1 1 1 1 1 1 1 1 1 1 a 1 1 1 1 1 1 1 Percentiles 0.01818162 0.063635671 0.090908101 §.118180531 §.048484321 0.096968641 Q. 163634582 0.181816202 0.199997822 0.109089721 0.238013937 0.257848432 0.277682927 0.297517422 0.317351916 0.337186411 0.357020906 0.376855401 0.396689895 0.41652439 0.218179443 0.455406297 Q.4744537Q8 0.493.50112 0.512548532 0.531595943 0.550643355 0.569690767 Gamma-Q(Hat) 4.339031664 14.41837445 20.46764835 26.60142257 11.07571144 21.82204786 37.09766155 41.41222876 45.80230241 24.54516765 55.25114882 60.33991965 65.54790077 70.88334133 76.3549565 81.97203137 87.74452837 93.68320235 99.79972782 1Q6.1G68416 50,27369978 119.0785779 125.7564975 132.6689087 139.8343146 147.2733809 155.009301 163.0682381 Normal-Q -2.0928422 -1 .5249509 -1.3351838 -1.1841314 -1.6597307 -1.2990194 -0.9796292 •0.3084654 -0.841629 -1.2313836 •0.7127057 -0.6499928 -0.5897387 -0.5315541 -0.4751166 -04201.542 -0.3664333 •0.3137502 •0.2619243 -0.210793 •0.7783565 -0.1120136 •0.0640789 -0.016291 0.0314597 0.0792823 0.1272869 0.1755869 105 107 110 111 117 113 119 122 122 132 133 133 138 163 163 163 163 174 187 190 222 23S 273 289 306 333 453 29S2 1 1 1 1 Q 1 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0.588738178 0.60778553 0.626833001 0.645880413 0.332463312 8.67836071 0.631793595 0.35261324 0.721551168 0.737875855 0.754200542 0.770525229 0.786849916 0.200733651 0.401587302 0.602380952 0.812301587 0.410714286 0.837662338 0.853896104 0.87012387 0.886363636 0.902537403 0.918831169 0.935064935 0.951238701 0.367532468 0.383766234 171.4798683 180.2780505 189.501663 199.1956546 80.62092045 216.3648821 224.8326032 86.44778239 243.5316694 254.6417323 266.4543419 279.06604.34 292.5350126 45.33627612 101.3388027 177.7401595 315.8763629 104.2387681 342.4104955 361.6446136 383.1239177 407.44B911 435.4987369 468.636129 509.1426864 561.2340357 634.6836503 759.9003157 0.2243003 0.2735521 0.323477 0.374222 -0.4331137 0.4631197 0.5009408 -0.3782748 0.5874557 0.63S8105 0.687768 0.7405777 0.7355388 -0.8387838 -0.2492407 0.2535148 0.8864096 -0.225708 0.9848957 1.0532907 1.1270053 1.2074141 1.2364944 1.3972525 1.5146142 1.6575784 1.8457049 2.1386067 When ROS is used on transformed data (e.g., log-transformed), then ordered values, x© ; /': = k + 1, k + 2, ..., n represent ordered detected data in that transformed scale (e.g., log-scale, Box-Cox (BC)-type 133 ------- transformation). Equation (4-4) is then used to impute or estimate the ND values. Specifically, for quantile, q(ndi) corresponding to the ith ND, the imputed ND is given by X(ndi) = a + bq(ndi) ,• i:=l,2,...k. When there is only a single DL and all values lying below the DL represent ND observations, then the quantiles corresponding to those ND values typically are lower than the quantiles associated with the detected observations. However, when there are multiple DLs, and when some of those DLs exceed detected values, then quantiles, #(ndi) corresponding to some of those ND values might become greater than the quantiles, q® associated with some of the detected values. 4.5.2.1 Influence of Outliers on Regression Estimates and Imputed NDs Like all other statistics, it is well-known (Rousseeuw and Leroy 1987; Singh and Nocerino 1995; Singh and Nocerino 2002) that presence of outliers (detects) also distorts the regression estimates of slope and intercept which are used to impute NDs based upon a ROS method. It is noted that for skewed data sets with outliers, the imputed values computed using the ROS method on raw data in the original scale become negative (e.g., GROS method). Therefore, inclusion of outliers (e.g., impacted locations) can yield distorted statistics and upper limits computed using the ROS method. This issue is also discussed later in this chapter. Note: It is noted that a linear regression line can be obtained even when only two detected observations are available. Therefore, methods (e.g., ROS) discussed here and incorporated in ProUCL can be used on data sets with 2 or more detected observations. However, to obtain a reliable OLS model (slope and intercept) and imputed NDs for computation of defensible upper limits, enough (> 4-6 as a rule of thumb, more are desirable) detected observations should be made available. 4.5.3 ROS Method for Lognormal Distribution Let Org stand for the data in the original unit and Ln stand for the data in the natural logarithmic unit. The LROS method may be used when the log-transformed detected data follow a lognormal distribution. For the LROS method, the OLS model given by (4-4) is obtained using the log-transformed detected data and the corresponding normal quantiles. Using the OLS linear model on log-transformed, detected observations, the NDs in log-transformed scale are imputed corresponding to the k normal quantiles, #(ndi) associated with the ND observations which are back-transformed in original, Org scale by exponentiation. 4.5.3.1 Fully Parametric Log ROS Method Once the k NDs have been imputed, the sample mean and sd can be computed using the back- transformation formula (El Shaarawi, 1989) given by equation (4-5) below. This method is called the fully parametric method (Helsel, 2005). The mean, jj.Ln, and sd, oLn, are computed in log-scale using a full data set obtained by combining the (n - k) detected log-transformed data values and the k imputed ND (in log scale) values. Assuming lognormality, El-Shaarawi (1989) suggested estimating ju and erby back- transformation using the following equations as one of the several ways of computing these estimates. The estimates given by equation (4-5) are neither unbiased nor have minimum variance (Gilbert 1987). Therefore, it is recommended to avoid the use of this version of ROS method on log-transformed data to compute UCL95s and other statistics. This method is not available in the ProUCL software. n 12), and a20rg = ^Org (e^(------- 4.5.3.2 Robust ROS Method on Log-Transformed Data The robust ROS method is performed on log-transformed data as described above. In the robust ROS method, ND observations are first imputed in the log-scale, based upon a linear ROS model fitted to the log-transformed detects and normal quantiles. The imputed NDs are transformed back in the original scale by exponentiation. The process of using the ROS method based upon a lognormal distribution and imputing NDs by exponentiation does not yield negative estimates for ND values; perhaps that is why it got the name robust ROS (or LROS in ProUCL). This process yields a full data set of size n, and methods described in Chapters 2 and 3 can be used to compute the decision statistics of interest including estimates of EPCs and BTVs. If the detected observations follow a lognormal, the data set consisting of detects and imputed NDs also follow a lognormal distribution. As expected, the process of imputing NDs using the LROS method does not reduce the skewness of the data set and therefore, appropriate methods need to be used to compute upper limits (Chapters 2 and 3) which provide specified (e.g., 0.95) coverage by adjusting for skewness. Note: The use of the robust ROS method has become quite popular. Helsel (2012) suggests the use of a classical t-statistic or a percentile bootstrap method to compute a UCL of the mean based upon the full data set obtained using the LROS method. These methods are also available in his NADA packages. However, these methods do not adjust for skewness and for moderately skewed to highly skewed data sets, and UCLs based upon these two methods fail to provide the specified coverage to the population mean. For skewed data sets, methods described in Chapter 2 can be used on LROS data sets to compute UCLs of the mean. Example 4-3 (Oahu Data Set). Consider the Oahu arsenic data set of size 24 with 13 NDs. The detected data set of size 11 follows a lognormal distribution as shown in Figure 4-1; this graph simply represents a Q-Q plot of detects and does not account for NDs when computing quantiles. The censored probability plot (new in ProUCL 5.1) is shown in Figure 4-2; its details can be found in the literature (Chapter 15 of Unified Guidance, EPA 2009). A censored probability plot is also based upon detected observations and it computes quantiles by accounting for NDs. The LROS data set consisting of 11 detects and 13 imputed NDs also follows a lognormal distribution as shown in Figure 4-3. Summary statistics and LROS UCLs are summarized in Table 4-2. 135 ------- Lognormal Q-Q Plot (Statistics using Detected Data) for Arsenic Theoretical Quantiles (Standard Normal) Arsenic Tol.alNumberoIData = 24 Number or NDs=13 Ma«DL=2 N =11 Percent MDs = 542 Mean- -0.0255 Sd = 0.634 Slope = 0.631 Intercept 0.0255 Correlation, Ft = 0.333 Shapiro Wifc Test E xact T esl H talisftr - 0.860 Critical Value(0.05,= 0.850 Data Appear Lognormal Appro* TeslValue = a875 p Value^O 0882 Distribution Test Suspect • Best Fit Line Figure 4-1 Quantiles Lognormal GOF Test on Detected Oahu Data Set - Does not Account for NDs to Compute Lognormal Q-Q Plot for Arsenic Statistics Displayed using Censored Quantiles and Detected Data Theoretical Quantiles (Standard Normal) Only Quantiles for Detects Displayed Total Number of Data = 24 Number or Detects = 11 mum Censored Quamtile = 1.347 :enlNDs = 542 DL = 2 nx of D elects = -0.0255 tdvx of Detects = U634 lope = 0.559 ntercept =• tl 13T> relation. R -- 0.97G hapir o-Wik Ted rilicalValue(0.05)-0.850 ata Appear Log normal Appro* Test Value-0.07T. p-Value-0.0882 Distribution Test Suspect • 8est Fit Line Figure 4-2. Lognormal Censored Probability Plot (Oahu Data) - Uses Only Detects but Accounts for NDs to Compute Quantiles Note: The two graphs displayed in Figures 4-1 and 4-2 provide similar information about data distributions, as GOF tests simply use detected values (and not quantiles). Both graphs are okay without any preference. 136 ------- Lognormal Q-Q Plot for Arsenic Statistics using ROS Lognormal Imputed Estimates NumbeiofNDs = 13 Mean =- -0.209 Set = 0.571 Slope -0.568 Intercept = 4.209 CoFielalion, R -= 0.363 Shapiro Wik Test Ex------- The data set is moderately skewed with sd of logged detects equal to 0.694. All methods tend to yield comparable results. One may want to use a 95% BCA bootstrap UCL or a bootstrap-t UCL to estimate the EPC. However, the detected data follow a gamma distribution, therefore ProUCL recommends gamma UCLs as shown in the following section. 4. 5. 3. 3 Gamma ROS Method Many positively skewed data sets tend to follow a lognormal as well as a gamma distribution. Singh, Singh, and laci (2002) noted that the gamma distribution is better suited to model positively skewed environmental data sets. When a moderately skewed to highly skewed data set (uncensored data set or detected values in a left-censored data set) follows a gamma, as well as, a lognormal distribution, the use of a gamma distribution tends to result in more stable and realistic estimates of EPCs and BTVs (Examples 2-2 and 3-2, Chapters 2 and 3). Furthermore, when using a gamma distribution to compute decision statistics such as a UCL of the mean, one does not have to transform the data and back-transform the resulting UCL into the original scale. Let X(k+i) ------- In the above equation, %^ represents a chi-square random variable with 2k degrees of freedom (df), and Pi are the plotting positions (percentiles) obtained using the process described above. The process of computing plotting positions, pt, i:=l,2,...,n, for left-censored data sets with multiple DLs has been incorporated in ProUCL. The inverse chi-square algorithm function (AS91) from Best and Roberts (1975) has been used to compute the inverse chi-square percentage points, zoi, as given by the above equations. Using the OLS line (4-4) fitted to the (n - k) detected pairs, one can impute the k NDs resulting in a full data set of size n = k + (n - k). Notes about GROS for smaller values of k (e.g.. <): In the ProUCL 5.0 Technical Guide (and its earlier versions) and ProUCL software, a suggestion was made that GROS may not be used when the shape parameter, k is less than 0.1 or less than 0.5. However, during late 2014, some users pointed out that k should be higher. Therefore, the latest version of ProUCL 5.1 now suggests that GROS may not be used for values of k < 1.0. It should be pointed out that the GROS algorithm incorporated in ProUCL works well for values of k > 2. The GROS method incorporated in ProUCL does not appear to work well for smaller values of k or its MLE estimate, k (e.g., <\). The algorithm used to compute gamma quantiles is not efficient enough and does not perform well for smaller values of k. The developers thus far have not found time to look into this issue. In January 2015, the developers of ProUCL requested the statistical community (via the American Statistical Association's section on environmental statistics and/or personal communication) to provide code/algorithms which may be used to improve the computation of gamma quantiles for smaller values of k. For now, GROS may not be used when the data set with detected observations (used to compute OLS regression line) consists of outliers and/or is highly skewed (e.g., estimated values of k are small such as <=1.0). When the estimated value (MLE) of the shape parameter, k, based upon detected data is small (<= 1.0), or when the data set consists of many tied NDs at multiple DLs with a high percentage of NDs (>50%), the GROS tends to not perform well and often yields negative imputed NDs, due to outliers distorting the OLS regression. Since environmental concentration data are non-negative, one needs to replace the imputed negative values by a small positive value such as 0.1, 0.001. In ProUCL, negative imputed values are replaced by 0.01. The use of such imputed values tends to yield inflated values of sd, UCLs, andBTVestimates (e.g., UPLs, UTLs). Preferred Method: Alternatively, when detected data follow a gamma distribution, one can use KM estimates (described above) in gamma distribution based equations to compute UCLs (and other limits) which account for data skewness, unlike KM estimates when used in normal UCL equations. This hybrid gamma-KM method for computing upper limits is available in ProUCL 5.0/ProUCL 5.1. The details are provided in Section 4.6. The hybrid KM-gamma method yields reasonable UCLs and accounts for NDs as well as data skewness as demonstrated in Example 4-4. /v* Note: It is noted that when k >1, UCLs based upon the GROS method and gamma UCLs computed using KM estimates tend to yield comparable UCLs from practical a point of view. This can also be seen in Example 4-4 below. 139 ------- Example 4-4 (Oahu Data Set Continued): The detected data set of size 11 follows a gamma distribution as shown in Figure 4-4. The GROS data consisting of 11 detects and 13 imputed NDs also follows a gamma distribution as shown in Figure 4-5. Summary statistics and GROS UCLs are summarized in Table 4-3 following Figure 4-5. Since the data set is only mildly skewed all methods (GROS and Hybrid KM-Gamma) yield comparable results. Gamma Q-Q Plot (Statistics using Detected Data) for Arsenic Theoretical Quantiles of Gamma Distribution ToialNumberofData-24 N =11 Percent NDs = 54% Mean = 1.2X4 k star -1.701 8 theta star = 0.7255 Slope = 1.0345 Intscepl = -0 01 77 Correlation, R = Q9G38 Test Statistic = 0.254 Critical ValueED. 05] = 0.258 Data appear Gamma Distributed Distribution Test Suspect • Best Fit Line Figure 4-4. Gamma GOF Test on Detected Concentrations of the Oahu Data Set Gamma Q-Q Plot for Arsenic Statistics using ROS Gamma Imputed Estimates Theoretical Quantiles of Gamma Distribution Imputed NDs Displayed in smaller font Number o! NDs = 13 k star-1.8399 theta star = 0.5193 Slope = 1.0794 Correlation, R = 0.9752 Anderson-Darling Test Tsst Statistic = 0.480 Critical Value(O.G5] = 0.755 Data appear Gamma Distributed • Best Fit Line Figure 4-5. Gamma GOF Test on GROS Data Obtained Using the Oahu Data Set 140 ------- Table 4-3. Summary Statistics and UCL95 Based upon Gamma ROS data Minimum 0.119 Maximum 3.2 SD 0.758 k hat (MLE) 2.071 Theta hat (MLE) 0.461 nu hat {MLE) 99.41 MLE Mean (bias corrected) 0.956 MLE Sd (bias corrected) 3.704 .Approximate Chi Square Value (83.32, a) 67.S5 95% Gamma Approximate UCL (use when n>=50) 1.247 Mean 0.956 Median 0.7 CV 0.793 fc star {bias corrected M LE) 1.84 Theta star {bias corrected MLE) 0.519 nu star $>ias corrected) 88.32 .Adjusted Level of Significance$) 0.0392 Adjusted Chi Square Value (88.32, PS 66.38 35% Gamma Adjusted UCL Xise when n=:5C'; 1.271 Kaplan Meier (KM) Statistics Using Nomral Critical Values Mean 0.949 Standard Error of Mean 0.165 0.713 95% KM (BCA) UCL 1.192 1.231 95% KM (Percentile Bootstrap) UCL 1.219 1.22 95% KM Bootstrap! UCL 1.374 1.443 95% KM Chebyshev UCL 1.6S7 1.977 99% KM Chebyshev UCL 2.588 SD 95% KM K UCL 95% KM (z) UCL 90% KM Chebyshev UCL 97.5% KM Chebyshev UCL Gamma Kaplan-Meier (KM) Statistics k hat (KM) 1.771 nuhat(KM) 85.02 .Approximate Chi Square Value (85.02, a) 64.77 Adjusted Chi Square Value (85.02, (5) S3.53 95% Gamma Approximate KM-UCL (use when ns=50) 1.246 35% Gamma Adjusted KM-UCL Xise when n=:5C) 1.27 Suggested UCL to Use 35% KM J; UCL 1.231 ; Adjusted Gamma KM-UCL 1.27 95% GROS Adjusted Gamma UCL 1.271 ProUCL suggests using GROS UCL of 1.27. 4.6 A Hybrid KM Estimates and Distribution of Detected Observations Based Approach to Compute Upper Limits for Skewed Data Sets - New in ProUCL 5.0/ ProUCL 5.1 The KM method yields good estimates of the population mean and sd. Since it is hard to verify and justify the distribution of an entire left-censored data set consisting of detects and NDs with multiple DLs, it is suggested that the KM method be used to compute estimates of the mean, sd, and standard error of the mean. Depending upon the distribution and skewness of detected observations, one can use KM estimates in parametric upper limit computation formulae to compute upper limits including UCLs, UPLs, UTLs, and USLs. The use of this hybrid approach will yield more appropriate skewness adjusted upper limits than those obtained using KM estimates in normal distribution based UCL and UTL equations. Depending upon the distribution of detected data, ProUCLS.l (and its earlier version ProUCL 5.0) computes upper limits using KM estimates in parametric (normal, lognormal, and gamma) equations to compute the various upper limits. The use of this hybrid approach has also been suggested in Chapter 15 of EPA (2009) to compute upper limits using KM estimates in the lognormal distribution based equations to compute the various upper limits. ProUCL 5.1 and its earlier versions compute a 95% UCL of the mean based upon the KM method using: 1) the standard normal critical value, za and Student's t-critical value, ta,(n-i)', 2) bootstrap methods including the percentile bootstrap method, the bias-corrected accelerated (BCA) bootstrap method, and 141 ------- bootstrap-t method, and 3) the Chebyshev inequality. Additionally, when detected observations of a left- censored data set follow a gamma or a lognormal distribution, ProUCL 5.1 also computes KM UCLs and other upper limits using a lognormal or a gamma distribution. The use of these methods yields skewness adjusted upper limits. For a gamma distributed detected data, UCLs based upon the GROS and gamma distribution on KM estimates are generally in good agreement unless the data set is highly skewed (with estimated values of shape parameter, k50%) with NDs tied at multiple DLs. The various UCL computation formulae based upon KM estimates and incorporated in ProUCL 5.0/ProUCL 5.1 are described as follows. 4. 6. 1 Detected Data Set Follows a Normal Distribution Based upon Student's t-statistic, a 95% UCL of the mean based upon the KM estimates is as follows: KM UCL95 (t)=fi + t^^ ^ (4-8) The above KM UCL (t) represents a good estimate of the EPC when detected data are normally distributed or mildly skewed. However, KM UCLs, computed using a normal or t-critical value, do not account for data skewness. The various bootstrap methods for left-censored data described in Section 4.7 can also be used on KM estimates to compute UCLs of the mean. 4. 6. 2 Detected Data Set Follows a Gamma Distribution For highly skewed gamma distributed left-censored data with a large percentage of NDs and several NDs tied at multiple RLs, the GROS method tends to yield impractical, negative imputed values for NDs. It is also well known that the OLS estimates get distorted by outliers, therefore, GROS estimates and upper limits also get distorted when outliers are present in a data set. In order to avoid these situations, one can use the gamma distribution on KM estimates to compute the various upper limits provided the detected data follow a gamma distribution. Using the properties of the gamma distribution, an estimate of the shape parameter, k, is computed based upon a KM mean and a KM variance. The mean and variance of a gamma distribution are given as follows: Mean=£*0, and Variance = k*(f Substituting a KM mean, fiKM , and a KM variance, cr^ , in the above equations, an estimate of the shape parameter, k, is computed by using the following equation: Using /^ , cr^- , n, and kin equations (2-34) and (2-35), gamma distribution based approximate and adjusted UCLs of the mean can be computed. Similarly, for gamma distributed left-censored data sets with detected observations following a gamma distribution, KM mean and KM variance estimates can be used to compute gamma distribution based upper limits described in Chapter 3. ProUCL 5.0/ProUCL 5.1 computes gamma distribution and KM estimates based UCLs and upper limits to estimate BTVs when detected data follow a gamma distribution. 142 ------- Notes: It should be noted that the KM method does not require concentration data to be positive. In radio chemistry, the DLs (or minimum detectable concentration [MDC]) for the various radionuclides are often reported as negative values. Statistical models such as a gamma distribution cannot be used on data sets consisting of negative values. However, the hybrid gamma-KM method described above can be used on radionuclides data provided detected activities are all positive and follow a gamma distribution. One can compute KM estimates using the entire data sets consisting of negative NDs and detected positive values. Those KM estimates can be used to compute gamma UCLs described above provided fiKM >0. 4.6.3 Detected Data Set Follows a Lognormal Distribution The EPA RCRA (2009) guidance document suggests computing KM estimates on logged data and computing a lognormal H-UCL based upon the H-statistic. ProUCL computes lognormal and KM estimates based UCLs and upper limits to estimate BTVs when detected data follow a lognormal distribution. Like uncensored lognormally distributed data sets, for moderately skewed to highly skewed left-censored data sets, the use of a lognormal distribution on KM estimates tends to yield unrealistically high values of the various decision statistics; especially when the data sets are of sizes less than 30 to 50. Example 4-5 (Oahu Data Set Continued): It was noted earlier that the detected Oahu data set follows a gamma as well as a lognormal distribution. The hybrid normal, lognormal and gamma UCLs obtained using the KM estimates are summarized in Table 4-4 as follows. The hybrid Gamma UCL is 1.27, close to the UCL obtained using the GROS method of 1.271 (Example 4-4). The H-UCL as suggested in EPA (2009) is 1.155 which appears to be a little lower than the other LROS BCA bootstrap UCL of 1.308 (Table 4-2). 143 ------- Table 4-4. UCL95 Based on Hybrid KM Method and Normal, Lognormal and Gamma Distribution Kaplan-Meier (KM) Statistics using Normal Critical Values and either Nonparametric UCLs Mean 0.949 Standard Error of Mean 0.165 §713 95% KM (BCA) UCL 1.228 1 .231 95% KM (Percentile Bootstrap) UCL 1 .21 1 .22 95% KM Bootstrap t UCL 1 .363 1 .443 95% KM Chebyshev UCL 1 .667 1 .977 99% KM Chebyshev UCL 2.5&B SD 35% KM j; UCL 95% KM (z) UCL 30% KM Chebyshev UCL 97.5% KM Chebyshev UCL Gamma GOF Tests on Detected Observations Only A-D Test Statistic 0.787 Anderson-Darling GOF Test 5% A-D Critical Value 0.733 Detected Data Not Gamma Distributed at 5% Significance Level K-S Test Statistic 0.254 KDlmogrov-Smirnoff GOF 5% K-S Critical Value 0.258 Detected data appear Gamma Distributed at 5% Significance Level Detected data follow Appr Gamma Distribution at 5% Significance Level Gamma Statistics on Detected Data Only khat(MLE) 2.257 Theta hat (MLE) Q.548 nu hat (MLE) 49.65 MLE Mean (bias corrected) 1.236 k star (bias corrected MLE} 1.702 Theta star (bias corrected MLE} 0.727 nu star (bias corrected) 37.44 MLE Sd (bias corrected) 0.948 Gamma Kaplan-Meier (KM) Statistics k hat {KM} 1.771 nu hat (KM) 85.02 Approximate Chi Square Value (85.02. a) 54.77 Adjusted Chi Square Value (85.02. P) 63.53 95% Gamma Approximate KM-UCL (use when n>=50) 1.24S 35% Gamma .Adjusted KM-UCL (use when n«5CJ 1.27 UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognonnally Distributed KM Mean fogged) -0.236 35% H-UCL (KM -Log) 1 . 1 55 KM SD (logged) 0.547 95% Critical H Value (KM-Log) 2.023 KM Standard Error of Mean fogged) 0.137 Example 4-6. A real data set of size 55 with 18.8% NDs is considered next. The data set can be downloaded from the ProUCL website. The minimum detected value is 5.2 and the largest detected value is 79000, sd of detected logged data is 2.79 suggesting that the data set is highly skewed. The detected data follow a gamma as well as a lognormal distribution as shown in Figures 4-6 and 4-7. It is noted that GROS data set with imputed values follows a gamma distribution and LROS data set with imputed values follows a lognormal distribution (results not included). 144 ------- Lognormal Q-Q Plot (Statistics using Detected Data) for A-DL A-DL Toial Number of Dat. Number of NDs = 10 Max DL-124 18X Percent ND s Mean-7.031 Sd = 2.738 Slope = 2.789 Intercept-7 031 Correlation, R = 0.931 LJIiefors Test Test Statistic = 0.104 CriticalVa!ue(& 051-0.132 Data Appear Lognormal Theoretical Quantiles (Standard Normal) Figure 4-6. Lognormal GOF Test on Detected TRS Data Set Gamma Q-Q Plot (Statistics using Detected Data) for A-DL Theoretical Quantiles of Gamma Distribution AOL Total Number of Data = 55 Number otNDs = 1Q N = 45 Percent NDs = m Mean-10556.1867 k star = 0 301S theta star-34979.7238 Slope = 1 0745 Intercept--535.7060 Data appear Gam • Best Fit Line Figure 4-7. Gamma GOF Test on Detected TRS Data Set 145 ------- Table 4-5. Statistics and UCL95s Obtained Using Gamma and Lognormal Distributions ArDL Total Number of Observations Number of Detects Number of Distinct Detects Minimum Detect Maximum Detect 79000 Variance Detects 3.954E+S Mean Detects 10556 Median Detects 1940 Skewness Detects 2.632 Mean of Logged Detects 7.831 General Satisfies 55 45 45 5.2 Number of Distinct Observations Number of Non-Detects Number of Distinct Non-Detects Minimum Non-Detect Maximum Non-Detect Percent Non-Detects SD Detects CV Detects KLirtosis Detects SD of Logged Detects 53 10 8 3.8 124 18.18^; 13886 1.884 6.4% 2.7SS Kaplan Meier {KM) Statistics using Normal Critical Values and other Nonparametnc UCLs Mean 8638 Standard Error of Mean 2488 SD 18246 95% KM (BCA) UCL 13396 95% KM « UCL 12802 95% KM (Percentile Bootstrap} UCL 12792 35% KM (z) UCL 12731 35% KM Bootstrap t UCL 14509 30% KM Chebyshev UCL 16102 95% KM Chebyshev UCL 1S4S3 37.5% KM Chebyshev UCL 24176 93% KM Chebyshev UCL 33334 Gamma GOF Tests on Detected Observations Only A-D Test Statistic 0.591 Anderson-Darling GOF Test 5% A-D Critical Value 0.86 Detected data appear Gamma Distributed at 5% Significance Level K-S Test Statistic 0.115 Kolmogrov Smirnoff GOF 5% K-S Critical Value 0.143 Detected data appear Gamma Distributed at 5% Significance Level Detected data appear Gamma Distributed at 5% Significance Level Gamma Statistics on Detected Data Only k hat (NILE) 0.307 Theta hat (MLE) 34333 nu hat (MLE) 27.67 MLE Mean tbias corrected) 10556 k star (bias corrected MLE} 0.302 Theta star (bias corrected MLE) 34980 nu star (bias corrected} 27.16 MLESd{bias corrected) 19216 Gamma Kaplan-Meier (KM) Statistics k hat (KM) 0.224 nu hat (KM) 24.66 .Approximate Chi Square Value (24.66, a) 14.35 Adjusted Chi Square Value (24.66, P) 14.14 95% Gamma Approximate KM-UCL (use when ni=5D} 14844 95% Gamma Adjusted KM-UCL (use when n<50) 15066 146 ------- Table 4-5 (continued). Statistics and UCL95s Obtained Using Gamma and Lognormal Distributions Gamma ROS Statistics using Imputed Non-Deteets Minimum 0.1 Maximum 79000 SD 18415 khat(MLE) 0.138 Theta hat (MLE) 43697 nuhat(MLE) 21.74 MLE Mean (bias corrected) 8637 Approximate Chi Square Value (21 .S3. a] 12.26 35% Gamma .Approximate UCL fuse when n s=5C} 15426 Mean 8637 Median 588 CV 2.132 k star (bias corrected MLE} 0.139 Theta star (bias corrected MLE} 43402 nu star (bias corrected) 21 .S3 M LE Sd (bias corrected} 19361 Adjusted Level of Significance §} 0.0456 Adjusted Chi Square Value (21.83. p} 12.06 35% Gamma Adjusted UCL (use when n <50} 15675 lognoimal GOF Test on Detected Observations Only Shapiro Wilk Test Statistic 0.333 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value 0.345 Detected Data Not Lognormal at 5% Significance Level Ulliefors Test Statistic 0.104 Lilliefors GOF Test 5% Lilliefors Critical Value 0.132 Detected Data appear Lognormal at 5% Significance Level Detected Data appear Approximate Lognormal at 5% Significance Level Lognormal ROS Statistics Using Imputed Non-Detects Mean in Original Scale 8638 Mean in Log Scale 5.383 SD in Original Scale 18414 SD in Log Scale 3.331 35% Percentile Bootstrap UCL 12853 95% Bootstrap t UCL 15032 35"% t UCL {assumes normality of RO S data) 12733 95% BCA Bootstrap UCL 13904 35% H-UCL (Log ROS) 1855231 UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognormally Distributed KM Mean fogged) 6.03 95% H-UCL (KM -Log) 1173988 KM SD (logged) 3.286 95% Critical H Value (KM-Log) 5.7 KM Standard Error of Mean (logged) 0.449 Nonparametric Distribution Free UCL Statistics Detected Data appear Gamma Distributed at 5% Significance Level Suggested UCL to Use 35% KM (Chebyshev; UCL 19483 55% .Approximate Gamma KM-UCL 148^ 95 vi G RO S Approximate Gamma UC L 15^26 From the above table, it is noted that the percentile bootstrap method on LROS method as described in Helsel (2012) yields a lower value of the UCL95 = 12797, which is comparable to a KM (t)-UCL =12802. The student's t statistic based upper limits (e.g., KM (t)-UCL) do not adjust for data skewness; the two UCLs, bootstrap LROS UCL and KM(t)-UCL, appear to represent underestimates of the population mean. As expected, H-UCL on the other hand, resulted in impractically large UCL values (using both the LROS and KM methods). Based upon the data skewness, ProUCL suggested three UCLs (e.g., Gamma UCL = 15426) out of several UCL methods available in the literature and incorporated in ProUCL software. 147 ------- 4.6.3.1 Issues Associated with the Use ofLognormal distribution to Compute a UCL of Mean for Data Sets with Nondetects Some drawbacks associated with the use of the lognormal distribution based UCLs on data sets with NDs are discussed next. Example 4-7. Consider the benzene data set (Benzene-H-UCL-RCRA.xls) of size 8 used in Chapter 21 of the RCRA Unified Guidance document (EPA 2009). The data set consists of one ND value with DL of 0.5 ppb. In the RCRA guidance, the ND value was replaced by 0.5/2=0.25 to compute a lognormal H- UCL. In this example, lognormal 95% UCLs (H-UCLs) are computed replacing the ND by the DL (0.5) and also replacing the ND by DL/2=0.25. Normal and lognormal GOF tests using DL/2 for the ND value are shown in Figures 4-8 and 4-9 as follows. Benzene-ppb Normal Q-Q Plot for Benzene-ppb Bnnan»-ppb Statistics using DL/2 Substitution N^lacfNOs 1 1.1 0.5 05 0.25 • • M«.zai Sd-5.353 Slope * 4.032 Intercept -2. 931 : Correlation. R - 0 701 Shapro-Wilk Test EHactTeslVakie = 0.521 Critical Val(0.05] = 081 8 Data Not Normal Approx.Te$tVarue = 0.495 p-Value-1.8931E-5 NOTE: Proxy Methods are not recommended D Best Fit Line V .5 -1.0 -0.5 00 0.5 1.0 1.5 Theoretical Quantiles (Standard Normal) DL/2 Substituted NDs Displayed in smaller font Figure 4-8. Normal Q-Q Plot on Benzene Data with ND Replaced by DL/2 From the above Q-Q plot, it is easy to see that observation 16.1 ppb represents an outlier. The Dixon test on logged data suggests that 2.779 (=ln(16.1)) is an outlier and observation 16.1 is an outlier in the original scale. The outlier, 2.779 was accommodated by the lognormal distribution resulting in the conclusion that the data set follows a lognormal distribution (Figure 4-9). 148 ------- Lognormal Q-Q Plot for Benzene-ppb Statistics using DL/2 Substitution Theoretical Quantiles (Standard Normal) DL/2 Substituted NDs displayed in smaller font Number ol NDs - 1 Mean - 0.204 Sd = 1.257 Sbpe-1.266 Intercept - 0.204 Correlate. R = 0.937 Shapiro-WtikTest E^tTeslValue = 0.396 CrticalVal(0 051 = 0.818 Data Appear Lognormal Appro*. TeslValue = 0.88 pValue = 0.192 NOTE: Proxy Methods • Best Fit Line Figure 4-9. Lognormal Q-Q Plot on Benzene Data with ND Replaced by DL/2 4.6.3.1.1 Impact of Using DL and DL/2 for Nondetects on UCL95 Computations Lognormal distribution based H-UCLs computed by replacing ND by DL and by DL/2 are respectively given in Tables 4-6 and 4-7 below. Table 4-6. Lognormal 95% UCL (H-UCL) - Replacing ND by DL (=0.5) Lognormal GOF Test Shapiro Wilk Test Statistic 0.803 Shapiro Wilk Lognormal GOF Test 5% Shapiro Wilk Critical Value D.S18 Data Not Lognormal at 5% Significance Level Ulliefors Test Statistic 0.273 5% Lilliefors Critical Value 0.313 Lilliefors Lognormal GOF Test Data appear Lognormal at 5% Significance Level Data appear Approximate Lognormal at 5% Significance Level Lognormal Statistics Minimum of Logged Data -0.693 Maximum of Logged Data 2.779 Assuming Lognormal Distribution 95*1 H-UCL 13.S2 95% Chebyshev (MVU E) UC L 6.496 99% Chebyshev {MVUE} UCL 11.86 Mean of logged Data 0.29 SD of logged Data 1.152 90% Chebyshev (MVU E) UCL 5.191 97.5% Chebyshev (MVU E) UCL 8.306 149 ------- Table 4-7. Lognormal 95% UCL (H-UCL) - Replacing ND by DL/2 (=0.25) Lognoimal GOF Test Shapiro Wilk Test Statistic 0.896 Shapiro Wilk Lognormal GOF Test 5% Shapiro Wilk Critical Value 0.818 Data appear Lognormal at 5% Significance Level Ulliefors Test Statistic 0.255 Ljlliefors Lognormal GOF Test 5% Ljlliefors Critica! Value 0.313 Data appear Lognormal at 5% Significance Level Data appear Lognormal at 5% Significance Level Lognormal Statistics Minimum of Logged Data -1.386 Mean of logged Data 0.204 Maximum of Logged Data 2.779 SDof logged Data 1.257 Assuming Lognomial Distribution 95% H-UCL 18.86 90% Chebyshev {MVUE} UCL 5.514 95% Chebyshev (MVUE) UCL 6.952 37.5% Chebyshev {MVUE} UCL 8.348 99% Chebyshev {MVUE} UCL 12.87 Note: 95% H-UCL (with ND replaced by DL/2) computed by ProUCL is in agreement with results summarized in Chapter 21 of the RCRA Guidance (EPA 2009). However, it should be noted that the UCL computed using the DL for ND is 13.62, and the UCL computed using DL/2 for ND is 18.86. Substitution by DL/2 resulted in a data set with higher variability and a UCL higher than the one obtained using the DL method. These two UCLs differ considerably confirming that the use of substitution methods should be avoided. From results summarized above, it is noted that replacing NDs reported as ------- Table 4-9. Normal 95% UCL Computed by Replacing ND by DL = 0.5 Normal GOF Test Shapiro Wilk Test Statistic 0.814 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value 0.803 Data appear Normal at 5% Significance Level Lilliefors Test Statistic 0.269 Lilliefors GOF Test 5% Ulliefors Critical Value 0.335 Data appear Normal at 5% Significance Level Data appear Normal at 5% Significance Level Assuming Normal Distribution 95X Normal UCL 95X UCLs (Adjusted for Skewness) 3 E % Student s4 UC L 1.517 35% .Adjusted^ LT DC L (Chen-1995) 1.454 35% Modified^ UCL (Johnson-1978) 1.518 Table 4-10. Normal 95% UCL Computed by Replacing ND by DL/2 = 0.25 Normal GOF Test Shapiro Wilk Test Statistic 0.875 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value 0.803 Data appear Normal at 5% Significance Level Ulliefors Test Statistic 0.23S Ulliefors GOF Test 5% Ulliefors Critical Value 0.335 Data appear Normal at 5% Significance Level Data appear Normal at 5% Significance Level Assuming Normal Distribution 95% Normal UCL 95% UCLs (Adjusted for Skewness) 3E% Student s4 UCL 1.E1S 95% AdjustedCLT UCL (Chen-1335} 1.436 95% Modified^ UCL {Johnson-1978} 1.515 Note: The recommended UCL is the KM UCL= 1.523. It is noted that normal UCLs are not influenced by changing a single ND from 0.5 (UCL95=1.517) to 0.25 (UCL95=1.516). Normal UCL95s without the outlier appear to represent more realistic estimates of the EPC (population mean). The Lognormal UCL based upon the data set with the outlier represents the outlying value(s) rather than representing the population mean. 4.7 Bootstrap UCL Computation Methods for Left-Censored Data Sets The use of bootstrap methods has become popular with the easy access to fast personal computers. As described in Chapter 2, for full-uncensored data sets, repeated samples of size n are drawn with replacement (that is each xl has the same probability = lln of being selected in each of the N bootstrap replications) from the given data set of n observations. The process is repeated a large number of times, N (e.g., 1000-2000), and each time an estimate, 6 of 9 (e.g., mean) is computed. These estimates are used to compute an estimate of the SE of the estimate, 9 . Just as for the full uncensored data sets without any NDs, for left-censored data sets, the bootstrap resamples are obtained with replacement. An indicator variable, / (1 = detected value, and 0 = nondetected value), is tagged to each observation in a bootstrap sample (Efron 1981). Singh, Maichle, and Lee (EPA 2006) studied the performances, in terms of coverage probabilities, of four bootstrap methods for computing UCL95s for data sets with ND observations. The four bootstrap 151 ------- methods included the standard bootstrap method, the bootstrap-t method, the percentile bootstrap method, and the bias-corrected accelerated (BCA) bootstrap method (Efron and Tibshirani 1993; Manly 1997). Some bootstrap methods, as incorporated in ProUCL, for computing upper limits on left-censored data sets are briefly discussed in this section. 4.7.1 Bootstrapping Data Sets with Nondetect Observations As before, let xndi, xnd2, ..., xndk, Xk+i, Xk+2, ..., x» be a random sample of size n from a population (e.g., AOC, or background area) with an unknown parameter 9 such as the mean, //, or the pth upper percentile (used to compute bootstrap UTLs), xp, that needs to be estimated from the sampled data set with ND observations. Let 6 be an estimate of 0, which is a function of k ND and (n - k) detected observations. For example, the parameter, # , could be the population mean, //, and a reasonable choice for the estimate, 9, might be the robust ROS, gamma ROS, or KM estimate of the population mean. If the parameter, 0, represents the pth upper percentile, then the estimate, 9, may represent the pth sample percentile, x , based upon a full data set obtained using one of the ROS methods described above. The bootstrap method can then be used to compute a UCL of the percentile, also known as upper tolerance limit. The computations of upper tolerance limits are discussed in Chapter 5. An indicator variable, / (taking only two values: 1 and 0), is assigned to each observation (detected or nondetected) when dealing with left-censored data sets (Efron 1981; Barber and Jennison 1999). The indicator variables, 7,• :j:=l,2,...,n, represent the detection status of the sampled observations, Xj ;j: = 1, 2,..., n. A large number, N (1000, 2000) of two-dimensional bootstrap resamples, (xa, In ),j:=j: = 1, 2,..., N, and /': = 1, 2,..., n, of size n are drawn with replacement. The indicator variable, /, takes on a value = 1 when a detected value is selected and / = 0 if a nondetected value is selected. The two-dimensional bootstrap process keeps track of the detection status of each observation in a bootstrap re-sample. In this setting, the DLs are fixed as entered in the data set, and the number of NDs vary from bootstrap sample to bootstrap sample. There may be k\ NDs in the first bootstrap sample, fc NDs in the second sample, ..., and kx NDs in the TV"1 bootstrap sample. Since the sampling is conducted with replacement, the number of NDs, fa, i: = 1, 2, ..., N, in a bootstrap re-sample can take any value from 0 to n inclusive. This is typical of a Type I left-censoring bootstrap process. On each of the N bootstrap resample, one can use any of the ND estimation methods (e.g., KM, ROS) to compute the statistics of interest (e.g., mean, sd, upper limits). It is possible that all (or most) observations in a bootstrap re-sample are the same. This is specifically true, when one is dealing with small data sets. To avoid such situations (with all equal values) it is suggested that there be at least 15 to 20 (preferably more) observations in the data set. As noted in Chapter 2, it is not advisable to compute statistics based upon a bootstrap resample consisting of only a few detected values such as < 4-5. Let 9 be an estimate of 0 based upon the original left-censored data set of size n; if the parameter, 0 represents the population mean, then a reasonable choice for the estimate, 9, can be the sample ROS mean, or sample KM mean. Similarly, calculate the sd using one of these methods for left-censored data sets. The following two steps are common to all bootstrap methods incorporated in the ProUCL software. Step 1. Let (jCji, xi2, ... , xin) represent the /'th bootstrap resample of size n with replacement from the original left-censored data set (x\, x2, ..., x»). Note that an indicator variable (as mentioned above) is tagged along with each data value, taking values 1 (if a detected value is chosen) and 0 (if a ND is chosen in the resample). Compute an estimate of the mean (e.g., KM, and ROS) using the /'th bootstrap resample, /: = 1,2, ...,7V. 152 ------- Step 2. Repeat Step 1 independently TV times (e.g., N = 2000), each time calculating new estimates (e.g., KM estimates) of the population mean. Denote these estimates (e.g., KM means, and ROS means) by xl, X2, ..., XN . The bootstrap estimate of the population mean is given by the arithmetic mean, XB, of the TVestimates xt C/VROS means or 7VKM means). The bootstrap estimate of the standard error is given by: In general, a bootstrap estimate of 6 may be denoted by 9B (instead of XB). The estimate, 6B is the arithmetic mean of the TV bootstrap estimates (e.g., KM mean, or ROS mean) given by 6?., i:=l,2,...N. If the estimate, 9 , represents the KM estimate of, 6, then 6i (denoted by J. in the above paragraph) also represents the KM mean based upon the ith bootstrap resample. The difference, 0B -9 , provides an estimate of the bias of the estimate, 9 . After these two steps, a bootstrap procedure (percentile, BCA, or bootstrap-t) is used similarly to the conventional bootstrap procedure on a full uncensored data set as described in Chapter 2. Notes: Just like for small uncensored data sets, for small left-censored data sets (<8-10) with only a few distinct values (2 or 3), it is not advisable to use bootstrap methods. In these scenarios, ProUCL does not compute bootstrap limits. However, due to the complexity of decision tables and lack of enough funding, there could be some rare cases where ProUCL may recommend a bootstrap method based UCL which is not computed by ProUCL (due to lack of enough data). 4.7.1.1 UCL of Mean Based upon Standard Bootstrap Method Once the desired number of bootstrap samples and estimates has been obtained following the two steps described above, a UCL of the mean based upon the standard bootstrap method can be computed as follows. The standard bootstrap confidence interval is derived from the following pivotal quantity, t: (4-10) A (1 - a)*100% standard bootstrap UCL for #is given as follows: aaB (4-11) Here za is the upper a* critical value (quantile) of the standard normal distribution (SND). It is observed that the standard bootstrap method does not adequately adjust for skewness, and the UCL given by the above equation fails to provide the specified (1 - a)*100% coverage of the mean of skewed (e.g., lognormal and gamma) data distributions (populations). 153 ------- 4.7.1.2 UCL of Mean Based upon Bootstrap-t Method A (1 - a)* 100% UCL of the mean based upon the bootstrap-t method is given as follows. (4-12) It should be noted that the mean and sd used in equation (4-12) represent estimates (e.g., KM estimates, ROS estimates) obtained using original left-censored data set. Similarly, the ^-cutoff value used in equation (4-12) is computed using the pivotal ^-values based upon KM estimates or some other estimates obtained using bootstrap re-samples. Typically, for skewed data sets (e.g., gamma, lognormal), the 95% UCL based upon the bootstrap-t method performs better than the 95% UCLs based upon the simple percentile and the BCA percentile methods. However, the bootstrap-t method sometimes results in unstable and erratic UCL values, especially in the presence of outliers (Efron and Tibshirani 1993). Therefore, the bootstrap-t method should be used with caution. In case this method results in erratic unstable UCL values. The use of an appropriate Chebyshev inequality-based UCL is recommended. Additional suggestions on this topic are offered in Chapter 2. 4. 7. 1. 3 Percentile Bootstrap Method A detailed description of the percentile bootstrap method is given in Chapter 2. For left-censored data sets, sample means are computed for each bootstrap sample using a selected method (e.g., KM, ROS), which are arranged in ascending order. The 95% UCL of the mean is the 95th percentile and is given by: 95% Percentile - UCL = 95th%3c! ; /: = 1, 2, ..., N (4-13) For example, when N = 1000, a simple 95% percentile-UCL is given by the 950th ordered mean value given by ^(950) . It is observed that for skewed (lognormal and gamma) data sets, the BCA bootstrap method performs (described below) slightly better (in terms of coverage probability) than the simple percentile method. 4.7.1.4 Bias-Corrected Accelerated (BCA) Percentile Bootstrap Procedure Singh, Maichle and Lee (2006) noted that for skewed data sets, the BCA method does represent a slight improvement, in terms of coverage probability, over the simple percentile method. However, for moderately skewed to highly skewed data sets with the sd of log -transformed data >1, this improvement is not adequate and yields UCLs with a coverage probability lower than the specified coverage of 0.95. The BCA UCL for a selected estimation method (e.g., KM, ROS) is given by the following equation: (1- a) *100% UCLpRoc = BCA - UCL= x^ROC (4-14) Here XpROC is the 012 100th percentile of the distribution of statistics given byxPROC; i: = 1, 2, ..., N, and PROC is one of the many (e.g., KM, DL/2, ROS) mean estimation methods. Here 012 is given by the following probability statement: 154 ------- a2 = O z0 + ^ ^- (4-15) <1>(Z) is the standard normal cumulative distribution function and z(1 ~a) is the 100*(1 - a)th percentile of a standard normal distribution. Also, z0 (bias correction) and a (acceleration factor) are given as follows: z0=0 & \XPROC ,i < XPROC . , , , T ,AIS\ ,/: =1, 2, ..., N (4-16) \ > ------- gamma or a lognormal distribution however, the detected data set without the outlier follows a lognormal distribution. Table 4-lla. Statistics Computed Using Outlier=2982 Pyrene Total Number of Observations Number of Detects Number of Distinct Detects Minimum Detect Maximum Detect Variance Detects 189219 Mean Detects 190.1 Median Detects 103 Skewness Detects 6.282 Mean of Logged Detects 4.711 General 9atistics 56 45 39 31 2332 Number of Distinct Observations Number of Non-Defects Number of Distinct Non-Detects Minimum Non-Detect Maximum Non-Detect Percent Non-Detects SD Detects CV Detects Kurtosis Detects SD of Logged Detects Kaplan Meier (KM) Satisfies using Normal Critical Values and other Nonparametric UCLs Mean 164.1 Standard Error of Mean SD 3S3.4 35% KM (BCA) UCL 95% KM |) UCL 252.2 35% KM (Percentile Bootstrap) UCL 250.7 35% KM Bootstrap t UCL 322 35% KM Chebyshev UCL 432.9 33% KM Chebyshev UCL 35% KM (z) UCL 90% KM Chebyshev UCL 37.5% KM Chebyshev UCL Lognormal ROS Statistics Using Imputed Non-Detects Mean in Original Scale 163.2 SD in Original Scale 333.1 95% t UC L (assumes normality of ROS data) 251.1 95% BCA Bootstrap UCL 322.1 95%H-UCL{LogROS) 170.4 Mean in Log Scale SD in Log Scale 95% Percentile Bootstrap UCL 35% Bootstrap t UCL 44 11 8 28 174 13.64% 435 2.288 41 Q.8Q5 52.65 271.8 261 507,5 333.6 S87.3 4.537 S.843 262.S 5Q7.8 UCLs computed using the KM method and percentile bootstrap and t-statistic are 261 and 252.2. The corresponding UCLs obtained using the LROS method are 262.6 and 251.2, which appear to underestimate the population mean. The H-UCL based upon the LROS method is unrealistically lower (170.4) than the other UCLs. Depending upon the data skewness (sd of detected logged data =0.81), one can use the Chebyshev UCL95 (or Chebyshev UCL90) to estimate the EPC. Note that as expected, the presence of one outlier resulted in a bootstrap-t UCL95 significantly higher than the various other UCLs. Table 4-1 Ib has UCLs computed without the outlier. Exclusion of the outlier resulted in all comparable UCL values. Any of those UCLs can be used to estimate the EPC. 156 ------- Table 4-1 Ib. Statistics Computed without Outlier=2982 Pyrene General Statistics Total Number of Observations 55 Number of Distinct Observations 43 Number of Detects 44 Number of Non-Detects 11 Number of Distinct Detects 38 Number of Distinct Non-Detects 8 Minimum Detect 31 Minimum Non-Detect 28 Maximum Detect 453 Maximum Non-Detect 174 Variance Detects 8226 Percent Non-Detects 20% Mean Detects 126.6 SD Detects 90,7 Median Detects 103 CV Detects 0.716 Skewness Detects 1.735 Kurtosis Detects 3.483 Mean of Logged Detects 4.636 SD of Logged Detects 0.637 Kaplan-Meier (KM) Statistics using Normal Critical Values and other Nonparametric UCLs Mean 112.3 Standard Error of Mean 11.84 SD 86.03 S5°= KM iBCA) UCL 134 35% KM ft) UCL 132.7 35% KM {Percentile Bootstrap) UCL 132.4 95% KM (z) UCL 132,3 95% KM Bootstrap t UCL 135.3 3fl% KM Chebyshev UCL 148.4 35% KM Chebyshev UCL 164,5 37.5% KM Chebyshev UCL 186.8 33% KM Chebyshev UCL 230.7 Lognormal GOF Test on Detected Observations Only Shapiro Wilk Test Statistic Q.373 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value 0.344 Detected Data appear Lognormal at 5% Significance Level Ulliefors Test Statistic O.Q9€5 Lilliefors GOF Test 5% Lilliefors Critical Value 0.134 Detected Data appear Lognormal at 5% Significance Level Detected Data appear Lognormal at 5% Significance Level Lognormal ROS Statistics Using Imputed Non Detects Mean in Original Scale 112.4 Mean in Log Scale 4.43 SD in Original Scale 86.61 SD in Log Scale 0.677 95% t UCL {assumes normality of ROS data) 132 95% Percentile Bootstrap UCL 133 35% BCA Bootstrap UCL 135.7 35% Bootstrap t UCL 137 35%H-UCL{LogROS) 134.3 UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognormally Distributed KM Mean flogged) 4.431 95% H-UCL (KM-Log) 135 KM SD (logged) 0.676 95% Critical H Value (KM-Log) 2.013 KM Standard Error of Mean flogged) 0.0356 The data set is not highly skewed with sd = 0.64 of logged detected data. Most methods (including H- UCL) yield comparable results. Based upon data skewness, ProUCL recommends the use of a UCL95 based upon the KM BCA method (highlighted in blue in Table 4-1 Ib). 157 ------- 4.9 Saving Imputed NDs Using Stats/Sample Sizes Module of ProUCL Using this option, NDs are imputed based upon the selected distribution (normal, lognormal, or gamma) of the detected observations. Using the menu option, "Imputed NDs using ROS Methods" ProUCL 5.1 can be used to impute and save imputed NDs along with the original data in additional columns automatically generated by ProUCL. ProUCL assigns self-explanatory titles for those generated columns. This option is available in ProUCL for researchers and advanced users who want to experiment with the full data sets consisting of detected and imputed ND observations for other applications (e.g., ANOVA, PCA). 4.10 Parametric Methods to Compute UCLs Based upon Left-Censored Data Sets Some researchers have suggested that parametric methods such as the expectation maximization (EM) method and maximum likelihood method (MLE) cited earlier in this chapter would perform better than the GROS method for data sets with NDs. As reported in ProUCL guidance and on ProUCL generated output sheets, the developers do realize that the GROS method does not perform well when the shape parameter, k, or its MLE estimate is small (<1). The GROS method appears to work fine when k is large (> 2). However, for data sets with NDs and with many DLs, the developers are not sure if parametric methods such as the MLE method and the EM method perform better than the GROS method and other methods available in ProUCL. More research needs to be conducted to verify these statements. As noted earlier, it is not easy (perhaps not possible in most cases) to correctly assess the distribution of a data set containing NDs with multiple censoring points, a common occurrence in environmental data sets. If distributional assumptions are incorrect, the decision statistics computed using this incorrect distribution may also be incorrect. To the best of our knowledge, the EM method can be used on data sets with a single DL. Earlier versions of ProUCL (e.g., ProUCL 4.0, 2007) had some parametric methods including the MLE and RMLE methods; those methods were excluded from later versions of ProUCL due to their poor performances. The research in this area is limited; to the best of our knowledge, parametric methods (MLE and EM) for data sets with multiple censoring points are not well-researched. The enhancement of these parametric methods to accommodate left-censored data sets with multiple DLs will be a big achievement in environmental statistical literature. The developers will be happy to include contributed better performing methods in ProUCL. 4.11 Summary and Suggestions Most of the parametric methods including the MLE, the RMLE, and the EM method assume that there is only one DL. Like parametric estimates computed using uncensored data sets, MLE and EM estimates obtained using a left-censored data set are influenced by outliers, especially when a lognormal model is used. These issues are illustrated by an example as follows. Example 4-9: Consider a left-censored data set of size 25 with multiple censoring points: <0.24, <0.24, <1, <0.24, <15, <10, <0.24, <22, <0 .24, < 5.56, <6.61, 1.33, 168.6, 0.28, 0.47, 18.4, 0.48, 0.26, 3.29, 2.35, 2.46, 1.1, 51.97, 3.06, and 200.54. The data set appears to have 2 extreme outliers and 1 intermediate outlier as can be seen from Figure 4-10. From Figure 4-10 and the results of the Rosner outlier test performed on the data set, it can be concluded that the 3 high detected values represent 158 ------- outliers. The Shapiro-Wilk test results performed on detected data shown in Figure 4-11 (censored probability plot) suggest that the detected data set (with outliers) follows a lognormal distribution accommodating the outliers. Q-Q Plot for X * Reported values used for nondetects TowNumbwofData-a Number of N on-D elects = 1 1 180 150 120 X 30 60 30 0 - Number cf Detects -14 DetecledMean-32.4? Delected Sd = 66. 19 Slope (displayed data] - 34.52 ! E8 6 Intercept (displayed data)" 20.64 Correlation. R = 0.653 Q Best Fit Line 51 97 15 1«4 -1$ -1.2 -06 00 0.6 12 18 Theoretical Quantiles (Standard Normal) NDs Displayed in smaller font Figure 4-10. Exploratory Q-Q Plot to Identify Outliers Showing All Detects and Nondetects Lognormal Q-Q Plot for X Statistics Displayed using Censored Quantiles and Detected Data Tola! Number otDat=r-25 Number d Detects = 14 Minimum Censored Quamlile = 1.964 Percent NDs = 44% ManDL = 22 MearaofDeleclE = 1.259 SldvKol Detect! = 2.238 Slope = 2.64 Intercept = 0.57S Correlation, R = 0.994 Shapiro-Wilk Test ExactTest Sialistic = 0.333 Critical Vakje(0.05] = 0874 Dala Appear Lognwmal Appro*. Test Value = 0.901 p-Vakw-0.116 • Best Fit Line Theoretical Quantiles (Standard Normal) Only Quantiles for Detects Displayed Figure 4-11. Censored Q-Q Plot Showing GOF Test Results on Detected Log-transformed Data 159 ------- Statistics Computed with Outliers From File: N D -D ala-f or M LE 1 xfc General Statistics for Uncensaed Dataset Variable NumObs tt Missing Minimum Maximum Mean SD SEM MAD/Q.675 Skewness Kurtosis CV X 25 0 0.24 200,5 20.64 50.81 10.16 3.128 3.085 8.876 2.462 Percentiles for Uncensored Dataset Variable NumObs ft Missing lOXile 2UZi\e 252ile(Q1]50£ile(Q2]75Xile(Q3] W%\\e 90%ile 95£ile 99%ile X 25 0 0.24 0.256 0.28 2.35 10 15.68 39.88 145.3 182.9 Nonparametric estimates of the mean and sd using the KM method are summarized as follows. From File: ND Data for MLE-1 .xb General Statistics tor Censored Datasets (with NDs) using Kaplan Meet Method Variable NumObs ft Missing Num Ds NumNDs % NDs Min ND Max ND KM Mean KM Var KM SD KM CV X 25 0 14 11 44.00% 0.24 22 18.48 2528 50.28 2.72 MLE estimates of the mean and sd obtained using Minitab 16, UCL95, and a 95%-95% upper tolerance limit based upon a lognormal distribution are summarized as follows. ML estimates in log scale are: Parameter Location Scale Estimate -0.247900 2.71896 Standard Error 0.641686 0.530176 Upper Bound 0.807580 3.74710 Log Likelihood = -58. 151; MLE estimates in original raw scale are (back transformation): Mean = 31. 45, SE of mean = 43.1279, and UCL95 = 300.041 The inclusion of outliers has resulted in inflated estimates, mean = 31.45, UCL95 = 300.41, and a UTL95-95 = 346.54. The estimate of the mean based upon a data set with NDs should be smaller (e.g., KM mean = 18.48) than the mean estimate obtained using all NDs at their reported DLs, 20.64. For this left-censored data set, the MLE of the mean based upon a lognormal distribution is 31.45 which appears to be incorrect. 160 ------- Statistics Computed without Outliers Detected data without the 2 extreme outliers also follow a lognormal distribution. MLE estimates, UCL95, UTL95-95 computed without the outliers and lognormal distribution (using Minitab) are: Estimates in log scale are provided as follows: Standard 95% Upper Parameter Estimate Error Bound Location -0.561639 -0.561639 0.28616 Scale 2.02381 0.421546 2.85079 Log Likelihood = -38.56; MLE estimates in original raw scale are: Mean = 4.42, SE of mean = 3.688, and UCL95 = 17.433, and UTL95-95 = 63.42 • Substantial differences are noted in the UCL95s ranging from 300.04 to 17.43, and in the UTL95- 95s ranging from 346.54 to 63.42. It is not easy to verify the data distribution of a left-censored data set consisting of detects and NDs with multiple DLs, therefore some poor performing estimation methods including the parametric MLE methods and the Winsorization method are not retained in ProUCL 4.1 and higher versions. In ProUCL 5.1, emphasis is given on the use of nonparametric UCL computation methods and hybrid parametric methods based upon KM estimates which account for data skewness in the computation of UCL95s. It is recommended that one avoid the use of transformations to achieve symmetry while computing the upper limits based upon left-censored data sets. It is not easy to correctly interpret statistics computed in the transformed scale. Moreover, the results and statistics computed in the original scale do not suffer from transformation bias. When the sd of the log -transformed data, o, becomes >1.0, avoid the use of a lognormal model even when the data appear to be lognormally distributed. Its use often results in unrealistic statistics of no practical merit (Singh, Singh, and Engelhard 1997; Singh, Singh, and laci 2002). It is also recommended the user identifies potential outliers representing observations coming from population(s) different from the main dominant population and investigate them separately. Decisions about the disposition of outliers should be made by all interested members of the project team. It is recommended that the use of the DL/2 (t) UCL method be avoided, as the DL/2 UCL does not provide the desired coverage (for any distribution and sample size) for the population mean, even for censoring levels as low as 10% and 15%. This is contrary to the conjecture and assertion (EPA 2006a) made that the DL/2 method can be used for lower (< 20%) censoring levels. The coverage provided by the DL/2 (t) method deteriorates fast as the censoring intensity increases. The DL/2 (t) method is not recommended by the authors or developers of this document and ProUCL software. The use of the KM estimation method is a preferred method as it can handle multiple DLs. Therefore, the use of KM estimates is suggested for computing decision statistics based upon methods which adjust for data skewness. Depending upon the data set size, distribution of the detected data, and data skewness, the various nonparametric and hybrid KM UCL95 methods including KM (BCA), bootstrap-t KM UCL, 161 ------- Chebyshev KM UCL, and Gamma-KM UCL based upon the KM estimates provide good coverages for the population mean. Suggestions regarding the selection of a 95% UCL of the mean are provided to help the user select the most appropriate 95% UCL. These suggestions are based upon the results of the simulation studies summarized in Singh, Singh, and laci (2002) and Singh, Maichle, and Lee (2006). It is advised that the project team collectively determine which UCL will be most appropriate for their site project. For additional insight, the user may want to consult a statistician. 162 ------- CHAPTER 5 Computing Upper Limits to Estimate Background Threshold Values Based upon Data Sets Consisting of Nondetect (ND) Observations 5.1 Introduction As described in Chapter 3, a BTV considered in this chapter represents an upper threshold parameter (e.g., 95th) of the background population; which is used to perform point-by-point comparisons of onsite observations. Estimation of BTVs and comparison studies require the computation of UPLs and UTLs based upon left-censored data sets containing ND observations. Not much guidance is available in the statistical literature on how to compute UPLs and UTLs based upon left-censored data sets of varying sizes and skewness levels. Like UCLs, the use of Student's t-statistic and percentile bootstrap methods based UPLs and UTLs are difficult to defend for moderately skewed to highly skewed data sets with standard deviation (sd) of the log-transformed data exceeding 0.75-1.0. Since it is not easy to reliably perform GOF tests on left-censored data sets; emphasis is given on the use of distribution-free nonparametric methods including the KM, Chebyshev inequality, and other computer intensive bootstrap methods to compute upper limits needed to estimate BTVs. All BTV estimation methods for full uncensored data sets as described in Chapter 3 can be used on data sets consisting of detects and imputed NDs obtained using ROS methods (e.g., GROS and LROS). Moreover, all other comments about the use of substitution methods, disposition of outliers, and minimum sample size requirements as described in Chapter 4 also apply to BTV estimation methods for data sets with ND observations. 5.2 Treatment of Outliers in Background Data Sets with NDs Just like full uncensored data sets, a few outlying observations present in a left-censored data set tend to yield distorted estimates of the population parameters (means, upper percentiles, OLS estimates) of interest. OLS regression estimates (slope and intercept) become distorted (Rousseeuw and Leroy 1987; Singh and Nocerino 1995) by the presence of outliers. Specifically, in the presence of outliers, the ROS method performed on raw data (e.g., GROS) tends to yield unfeasible imputed negative values for ND observations. Singh and Nocerino (2002) suggested the use of robust regression methods to compute regression estimates needed to impute NDs based upon ROS methods. Robust regression methods are beyond the scope of ProUCL. It is therefore suggested that potential outliers be manually identified where they may be present in a data set before proceeding with the computation of the various BTV estimates as described in this chapter. As mentioned in earlier chapters, upper limits computed by including a few low probability high outliers tend to represent locations with those elevated concentrations rather than representing the main dominant population. It is suggested that relevant statistics be computed using data sets with outliers and without outliers for comparison. This extra step helps the project team to see the potential influence of outlier(s) on the various decision making statistics (e.g., UCLs, UPLs, UTLs); and helps the project team in making informative decisions about the disposition of outliers. That is, the project team and experts familiar with the site should decide which of the computed statistics (with outliers or without outliers) represent more accurate estimate(s) of the population parameters (e.g., mean, EPC, BTV) under consideration. 163 ------- A couple of classical outlier tests (Dixon and Rosner tests) are available in the ProUCL software. These tests can be used on data sets with or without ND observations. Additionally, one can use graphical displays such as Q-Q plots and box plots to visually identify high outliers in a left-censored data set. It should be pointed out, that for environmental applications, it is the identification of high outliers (perhaps representing contaminated locations and hot spots) that is important. The occurrence of ND (less than values) observations and other low values is quite common in environmental data sets, especially when the data are collected from a background or a reference area. For the purpose of the identification of high outliers, one may replace ND values by their respective DLs or half of the DLs or may just ignore them (especially when high reporting limits are associated with NDs) from the outlier tests. A similar approach can be used to generate graphical displays, Q-Q plots and histograms. Except for the identification of high outlying observations, the outlier test statistics, computed with NDs or without NDs, are not used in any of the estimation and decision making processes. Therefore, for the purpose of testing for high outliers, it does not matter how the ND observations are treated. 5.3 Estimating BTVs Based upon Left-Censored Data Sets This section describes methods for computing upper limits (UPLs, UTLs, USLs, upper percentiles) that may be used to estimate BTVs and other not-to-exceed levels from data sets with ND observations. Several Student's t-type statistic and normal z-scores based methods have been described in the literature (Helsel 2005; Millard and Neerchal 2002; USEPA 2007, 2010d, 2011) to compute UPLs and UTLs based upon statistics (e.g., mean, sd) obtained using MLE, KM, or ROS methods. The methods used to compute upper limits (e.g., UPL, UTL, and percentiles) based upon a Student's t-type statistic are also described in this chapter; however, the use of such methods is not recommended for moderately skewed to highly skewed data sets. These methods may yield reasonable upper limits (e.g., with proper coverage) for normally distributed and mildly skewed to moderately skewed data sets with the sd of the detected log-transformed data less than 1.0. Singh, Maichle, and Lee (EPA 2006) demonstrated that the use of the t-statistic and the percentile bootstrap method on moderately to highly skewed left-censored data sets yields UCL95s with coverage probabilities much lower than the specified CC, 0.95. A similar pattern is expected in the behavior and properties of the various other upper limits (e.g., UTLs, UPLs) used in the decision making processes of the USEPA. It is anticipated that the performance (in terms of coverages) of the percentile bootstrap and Student's t-type upper limits (e.g., UPLs, UTLs) computed using the KM and ROS estimates for moderately skewed to highly skewed left-censored data sets (sd of detected logged data >1) would also be less than acceptable. For skewed data sets, the use of the gamma distribution on KM estimates (when applicable) or nonparametric methods, which account for data skewness, is suggested for computing BTV estimates. A brief description of those methods is provided in the following sections. 5.3.1 Computing Upper Prediction Limits (UPLs) for Left-Censored Data Sets This section describes some parametric and nonparametric methods for computing UPLs for left-censored data sets. 5.3.1.1 UPLs Based upon Normal Distribution of Detected Observations and KM Estimates When detected observations in a data set containing NDs follow a normal distribution (which can be verified by using the GOF module of ProUCL), one may use the normal distribution on KM estimates to compute the various upper limits needed to estimate BTVs (also available in ProUCL 4.1). A (1 - a)* 100 164 ------- UPL for a future (or next) observation (observation not belonging to the current data set) can be computed using the following KM estimates based equation: +t ^2KM (l + l/n) (5-1) ((1-a ),(«-!)) Here t^^^^ is the critical value of the Student's t-distribution with (n-1) degrees of freedom (df). If the distributions of the site data and the background data are comparable, then a new (next) observation coming from the site population (e.g., site) should lie at or below the UPLi95 with probability 0.95. A similar equation can be developed for upper prediction limits for future k observations (described in Chapter 3) and the mean of k future observations (incorporated in ProUCL 5.0/ProUCL 5.1). 5.3.1.2 UPL Based upon the Chebyshev Inequality The Chebyshev inequality can be used to compute a reasonably conservative but stable UPL and is given as follows: UPL = Jc+[^((1/00-1)*(1+ 1/71)]^ (5-2) The mean, x , and sd, sX: used in the above equation are computed using one of the estimation methods (e.g., KM) for left-censored data sets. Just like the Chebyshev UCL, a UPL based upon the Chebyshev inequality tends to yield higher estimate of BTVs than the other methods. This is especially true when skewness is moderately mild (sd of log-transformed data is low < 0.75), and the sample size is large n > 30). It is advised to apply professional/expert judgment before using this method to compute a UPL. Specifically, for larger skewed data sets, instead of using a 95% UPL based upon the Chebyshev inequality, the user may want to compute a Chebyshev UPL with a lower CC (e.g., 85%, 90%) to estimate a BTV. ProUCL can compute a Chebyshev UPL (and all other UPLs) for any user specified CC in the interval [0.5, 1]. 5.3.1.3 UPLs Based upon ROS Methods As described earlier, ROS methods first impute k ND values using an OLS linear regression model (Chapter 4). This results in a full data set of size n. For ROS methods (gamma, lognormal), ProUCL generates additional columns consisting of (n - k) detected values and k imputed values of the k ND observations present in a data set. Once, the ND observations have been imputed, the user may use any of the available parametric and nonparametric BTVs estimation methods for full data sets (without NDs), as described in Chapter 3. Those BTV estimation methods are not repeated here. The users may want to review the behavior of the various ROS methods as described in Chapter 4. 5.3.1.4 UPLs when Detected Data are Gamma Distributed When detected data follow a gamma distribution, methods described in Chapter 3 can be used on KM estimates to compute gamma distribution based upper prediction limits for future k>l observations. These limits are described below when k=l. Wilson-Hilferty (WH) UPL = max OAy 165 ------- UPL = +1^^ * SyKM * Here t^^^^ is a critical value from the Student's t-distribution with («-l) degrees of freedom (df), and KM estimates are computed based upon the transformed y data as described in Chapter 3. All detects and NDs are transformed to y-space to compute the KM estimates. One of the advantages of using this method is that one does not have to impute NDs based upon the data distribution using LROS or GROS method. 5.3.1.5 UPLs when Detected Data are Lognormally Distributed When detected data follow a lognormal distribution, methods described in Chapter 3 can be used on KM estimates to compute lognormal distribution based upper prediction limits for future k>l observations. These limits are described below when k=l. An upper (1 - a)*100% lognormal UPL is given by the following equation: UPL = exp(y + f((1_«M»-D) *sy * Here t^^^^ is a critical value from Student's t-distribution with (n-\) df, and y and sy represent the KM mean and sd based upon the log-transformed data (detects and NDs), y. All detects and NDs are transformed to y-space to compute the KM estimates. 5.3.2 Computing Upper p*100% Percentiles for Left-Censored Data Sets This section briefly describes some parametric and nonparametric methods to compute upper percentiles based upon left-censored data sets. 5.3.2.1 Upper Percentiles Based upon Standard Normal Z-Scores In a left-censored data set, when detected data are normally distributed, one can use normal percentiles and KM estimates (or some other estimates such as ROS estimates) of the mean and sdto compute the/?* percentile given as given as follows: XP=£KM+ZP^I£L (5-3) Here zp is the p* 100th percentile of a standard normal, N (0, 1), distribution which means that the area (under the standard normal curve) to the left of zp is p. If the distributions of the site data and the background data are comparable, then an observation coming from a population (e.g., site) similar (comparable) to that of the background population should lie at or below the />*100% percentile, with probability/?. 166 ------- 5.3.2.2 Upper Percentiles when Detected Data are Lognormally Distributed When detected data follow a lognormal distribution, methods described in Chapter 3 can be used on the KM estimates to compute lognormal distribution based upper percentiles. The lognormal distribution based/?* percentile based upon KM estimates is given as follows: In the above equation, y and sy represent the KM mean and sd based upon the log-transformed data (detects and NDs), y. All detects and NDs are transformed to y-space to compute the KM estimates. 5.3.2.3 Upper Percentiles when Detected Data are Gamma Distributed When detected data are gamma distributed, gamma percentiles can be computed similarly using the HW and WH approximations to compute KM estimates. According to the WH approximation, the transformed detected data Y = X1/3 follow an approximate normal distribution; and according to the HW approximation, the transformed detected data Y = X1/4 follow an approximate normal distribution. Let y and sy represent the KM mean and sd of the transformed data (detects and NDs), y. The percentiles based upon the WH and HW transformations respectively are given as follows: xp = max I ( Alternatively, following the process described in Section 4.6.2, one can use KM estimates to compute KM estimates, k and 9 of the shape, k and scale, 6 parameters of the gamma distribution, and use the chi-square distribution to compute gamma percentiles using the equation: X = Y * 9 12, where Y follows a chi-square distribution with 2k degrees of freedom (df). This method does not require HW or WH approximations to compute gamma percentiles. Once an a* 100% percentile, ya = y(a) 2k, of a chi-square distribution with 2k dfis obtained, the a*100% percentile for a gamma distribution is computed using the equation: xa = ya * 912. ProUCL 5.1 computes gamma percentiles using this equation based upon KM estimates. 5.3.2.4 Upper Percentiles Based upon ROS Methods As noted in Chapter 4, all ROS methods first impute k ND values using an OLS linear regression (Chapter 4) assuming a specified distribution of detected observations. This process results in a full data set of size n consisting of k imputed NDs and (n-k) detected original values. For ROS methods (normal, gamma, lognormal), ProUCL generates additional columns consisting of the (n-k) detected values, and k imputed ND values. Once, the ND observations have been imputed, an experienced user may use any of the parametric or nonparametric percentile computation methods for full uncensored data sets as described in Chapter 3. Those methods are not repeated in this chapter. 167 ------- 5.3.3 Computing Upper Tolerance Limits (UTLs) for Left-Censored Data Sets UTL computation methods for data sets consisting of NDs are described in this section. 5.3.3.1 UTLs Based on KM Estimates when Detected Data are Normally Distributed Normal distribution based UTLs computed using KM estimates may be used when the detected data are normally distributed (can be verified using GOF module of ProUCL) or moderately to mildly skewed, with the sd of log-transformed detected data, a, less than 0.5-0.75. An upper (1 - a)* 100% tolerance limit with tolerance or coverage coefficient, p, is given by the following statement: (5-4) Here K = K (n, a, p) is the tolerance factor used to compute upper tolerance limits and depends upon the sample size, n, CC = (1- a), and the coverage proportion = p. The K critical values are based upon the non-central t-distribution, and have been tabulated extensively in the statistical literature (Hahn and Meeker 1991). For samples of sizes larger than 30, one can use Natrella's approximation (1963) to compute the tolerance factor, K = K (n, a, p). 5.3.3.2 UTLs Based on KM Estimates when Detected Data are Lognormally Distributed When detected data follow a lognormal distribution, methods described in Chapter 3 can be used on KM estimates to compute lognormal distribution based upper tolerance limits. An upper (1 - a,)* 100% tolerance limit with tolerance or coverage coefficient, p, is given by the following statement: UTL = The K factor in the above equation is the same as the one used to compute the normal UTL; y and sy represent the KM mean and sd based upon the log-transformed data. All detects and NDs are transformed to y-space to compute KM estimates. 5.3.3.3 UTLs Based on KM Estimates when Detected Data are Gamma Distributed According to the WH approximation, the transformed detected data Y = X1/3 follow an approximate normal distribution; and according to the HW approximation, the transformed detected data Y = X1/4 follow an approximate normal distribution when detected X data are gamma distributed. Let y and sy represent the KM mean and sd based upon transformed data (detects and NDs), Y. Using the WH approximation, the gamma UTL (in original scale, X), is given by: UTL = max f 0, (j7 + K * sy )3 Similarly, using the HW approximation, the gamma UTL in original scale is given by: UTL=(y+K*sy}* 168 ------- 5.3.3.4 UTLs Based upon ROS Methods As noted in Chapter 4, all ROS methods first impute k ND values using an OLS linear regression line assuming a specified distribution of detected and nondetected observations. This process results in a full data set of size n consisting of k imputed NDs and (n-k) detected original values. For ROS methods (normal, gamma, lognormal), ProUCL generates additional columns consisting of the (n-k) detected values, and k imputed ND values. Once, the ND observations have been imputed, an experienced user may use any of the parametric or nonparametric UTL computation methods for full data sets as described in Chapter 3 . Those methods are not repeated in this chapter. Note: In the Stats/Sample Sizes module, using the General Statistics option for data sets with NDs, for information and summary purposes, percentiles are computed using detects and nondetects, where reported DLs are used for NDs. Those percentiles do not account for NDs. However, KM method based upper limits such as the UTL95-95 account for NDs; therefore, sometimes, a UTL95-95 computed based upon a ND method (e.g., KM method) may be lower than the 95% percentile computed using the General Statistics option of Stats/Sample Sizes module. 5. 3. 4 Computing Upper Simultaneous Limits (USLs) for Left-Censored Data Sets Parametric and nonparametric USL computation methods for are described as follows. 5.3.4.1 USLs Based upon Normal Distribution of Detected Observations and KM Estimates When detected observations follow a normal distribution (can be verified by using the GOF module of ProUCL), one can use the normal distribution on KM estimates to compute a USL95 . A one-sided (1 - a) 100% USL providing (1 - a) 100% coverage for all sample observations is given by: USL= Here (db2a)2 is the critical value of Max (Mahalanobis Distances) for 2*a level of significance. 5.3.4.2 USLs Based upon Lognormal Distribution of Detected Observations and KM Estimates When detected data follow a lognormal distribution, methods described in Chapter 3 can be used on the KM estimates to compute lognormal distribution based USLs. Let y and sy represent the KM mean and sd of the log-transformed data (detects and NDs), y; a (1 - a) 100% USL is given by as follows: 5. 3. 4. 3 USLs Based upon Gamma Distribution of Detected Observations and KM Estimates According to the WH approximation, the transformed detected data Y = X1/3 follow an approximate normal distribution; and according to the HW approximation, the transformed detected data Y = X1/4 169 ------- follow an approximate normal distribution. Let y and sy represent the KM mean and sd of the transformed data (detects and NDs), y. A gamma distribution based (using WH approximation), one- sided (1 - a) 100% USL is given by: A gamma distribution based (HW approximation) one-sided (1 - a) 100% USL is given as follows: 5.3.4.4 USLs Based upon ROS Methods Once, the ND observations have been imputed, one can use parametric or nonparametric USL computation methods for full data sets as described in Chapter 3. Example 5-1 (Oahu Data Set). The detected data are only moderately skewed (sd of logged detects = 0.694) and follow a lognormal as well as a gamma distribution. The various upper limits computed using ProUCL 5.1 are listed in Tables 5-1 through 5-3 as follows. Table 5-1. Nonparametric and Normal Upper Limits Using KM Estimates Arsenic Total Number of Observations Number of Distinct Observations Number of Detects Number of Distinct Detects Minimum Detect Maximum Detect Variance Detected Mean Detected General Statistics 24 10 11 a 0.5 3.2 0.331 1.236 Mean of Detected Logged Data -0.0255 Number of Missing Observations 0 Number of Non-Detects 13 Number of Distinct Non-Detects 3 Minimum Non-Detect 0.9 Maximum Non-Detect 2 Percent Non-Detects 54.17% SD Detected Q.3S5 SD of Detected Logged Data 0.694 Critical Values for Background Threshold Values (BTVs) Tolerance Factor K (For UTL) 2.309 d2max for USL) 2.644 170 ------- Table 5-1 (continued). Nonparametric and Normal Upper Limits Using KM Estimates Normal GOF Test on Detects Only Shapiro Wilk Test Statistic §.777 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value Q.85 Data Not Normal at 5% Significance Level IJIIiefors Test Statistic 0.273 Lilliefors GOF Test 5% Lilliefors Critical Value 0.2S7 Data Not Normal at 5*1 Significance Level Data Not Normal at 5% Significance Level Kaplan Meier (KM) Background Statistics Assuring Normal Distribution Mean 8.949 SD 0.713 35% UTL95% Coverage 2.595 95%KMIJPL|) 2.196 95%KMChebyShevUPL 4.121 30% KM Percentile ------- Table 5-2 (continued). Upper Limits Using GROS, KM Estimates and Gamma Distribution of Detected Data Minimum 0.119 Maximum 3.2 SD 0.758 k hat (HLE) 2.071 Theta hat {MLE} 0.461 nu hat {MLE} 33.41 MLE Mean |bias corrected) 0.956 35% Percentile of Chisquare (2k) 8.364 95% Percentile 2.32S Mean Median CV k star (bias corrected MLE} Theta star (bias corrected MLE} nu star {bias corrected) MLE Sd {bias corrected) 30% Percentile 93% Percentile The following statistics are computed using Gamma ROS Statistics on Imputed Data Upper Limits using Wilson Hirferty (WH) and Hawkins Wbdey (HW) Methods WH HW WH 95% Approx. Gamma UTL with 35% Coverage 3.149 3.239 35% Approx. Gamma UPL 2,384 95% Gamma USL 3.676 3.915 The following statistics are computed using gamma distribution and KM estimates Upper Limits using Wilson Hilferty {WH) and Hawkins Wbdey (HW) Methods k hat (KM) 1.771 WH HW 35% Approx. Gamma UTL with 95% Coverage 2.661 2.685 95% Gamma LISL 3.051 3.107 0.956 0.7 0.793 1.84 0,519 88.32 0.704 1.835 3,291 HW 2.436 nu hat {KM} 85.02 WH HW 35% Approx. Gamma UPL 2.087 2.077 Table 5-3. Upper Limits Using LROS method and KM Estimates and Lognormal Distribution of Detected Data Lognormal GOF Test on Detected Observations Only Shapiro Wilk Test Statistic 0.86 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value 0.85 Detected Data appear Lognormal at 5% Significance Level Lilliefons Test Statistic 0.223 Lilliefore GOF Test 5% Lilliefors Critical Value 0.267 Detected Data appear Lognormal at 5% Significance Level Detected Data appear Lognormal at 5% Significance Level Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Detects Mean in Original Scale 0.972 SD in Original Scale 0.718 35% UTL35% Coverage 3.032 95% Bootstrap {%} UTL95% Coverage 3.2 9-0% Percentile (z) 1.686 99% Percentile {z} 3.0€2 Mean in Log Scale -0.209 SD in Log Scale 0.571 95% BCA UTL35% Coverage 3.2 35% UPL ft) 2.202 35% Percentile {z) 2.075 95% USL 3.671 Statistics using KM estimates on Logged Data and Assuming Lognormal Distribution KM Mean of Logged Data -0.236 95% KM UTL fLognormal)35% Coverage 2.792 KM SD of Logged Data 0.547 95% KM UPL (Lognormal} 2.056 35% KM Percentile Lognormal {z} 1.942 95% KM USL {Lognormal} 3.354 172 ------- Example 5-2 A real data set of size 55 with 18.8% NDs is considered next. This data was used in Chapter 4 to illustrate the differences in UCLs computed using a lognormal and a gamma distribution. This data set is considered here to illustrate the merits of the gamma distribution based upper limits. It can be seen that the detected data follow a gamma as well as a lognormal distribution. The minimum detected value is 5.2 and the largest detected value is 79000. The sd of the detected logged data is 2.79 suggesting that the detected data set is highly skewed. Relevant statistics and upper limits including a UPL95, UTL95- 95, and UCL95 have been computed using both the gamma and lognormal distributions. The gamma GOF Q-Q plot is shown as follows. Gamma Q-Q Plot (Statistics using Detected Data) for A-DL A-DL Total Number oIOata = ffi Number o!NDs = 10 MaxDL=124 No.45 Percent NDs = 18* Mean- IfJbSG !«;.' k star = 0.3018 Iheta stai - 34373.7238 Slope=1.0745 Intercept = -5357060 Correlalion,R = 0.9e44 Andeison-DailirgTest Tesl Statistic = 0.591 CriticalValue(0 051 = 0.860 Data appeal Gamma DiitiibU • Beit F* Line Theoretical Quantiles of Gamma Distribution Summary Statistics for Data Set of Example 5-2 ArDL Total Number of Observations Number of Distinct Observations Number of Detects Number of Distinct Detects Minimum Detect Maximum Detect Variance Detected Mean Detected Mean of Detected Logged Data General Statistics 55 53 45 45 5.2 7SCCC 3.954E+S 10556 7.031 Critical Values for Background Threshold Values Tolerance Factor K (For UTL) 2.036 Number of Missing Observations 0 Number of Non-Detects 1 0 Number of Distinct Non-Detects S Minimum Non-Detect 3.S Maximum Non-Detect 124 Percent Non-Detects 18.1 8*4 SD Detected 19SS6 SD of Detected Logged Data 2.738 (BTVs) d2maxfforUSL) 2.994 Mean of detects (=10556) reported above ignores all 18.18% NDs. 173 ------- KM Method Based Estimates of the Mean, SE of the Mean, and sd Mean 8638 SD 18246 Standard Error of Mean 2488 KM mean (= 8638) reported above accounts for 18.18% NDs reported in the data set. Notes: Direct estimate of KM sd = 18246 Indirect Estimate of KM sd (Helsel 2012) = 18451.5 The gamma GOF test results on detected data and various upper limits including UCLs obtained using the GROS method and gamma distribution on KM estimates are provided in Table 5-4; and the lognormal GOF test results on detected data and the various upper limits obtained using the LROS method and lognormal distribution on KM estimates are provided in Table 5-5. Table 5-6 is a summary of the main upper limits computed using the lognormal and gamma distribution of the detected data. Table 5-4. Upper Limits Using GROS, KM Estimates and Gamma Distribution of Detected Data Gamma GOF Tests on Detected Observations Only A-D Test Statistic 0.531 Anderson-Darling GOF Test 5% A-D Critical Value 0.86 Detected data appear Gamma Distributed at 5% Significance Level K-S Test Statistic 0115 Kblmogruv-Smimoff GOF 5% K-S Critical Value 0.143 Detected data appear Gamma Distributed at 5% Significance Level Detected data appear Gamma Distributed at 5X Significance Level Gamma Statistics on Detected Data Only k hat (MLE) 0.307 Theta hat (MLE) 34333 nu hat (MLE) 27.67 MLE Mean {bias corrected) 18556 MLE Sd {bias corrected) 13216 k star jbias corrected MLE) 0.302 Theta star {bias corrected MLE) 34980 nu star (bias corrected) 27.16 35% Percentile of Chisquare (2k) 2.756 Upper Limits Computed Using Gamma ROS Method Gamma ROS Statistics using Imputed Non-Detects 1.121 Mean 8642 Median 588 CV 2.13 k star {bias corrected MLE) 0.246 Theta star (bias corrected MLE) 35133 nu star teas corrected) 27.01 M LE Sd {bias corrected) 17440 90% Percentile 25972 35% Percentile 42055 33% Percentile 84376 The following statistics are computed using Gamma ROS Statistics on Imputed Data Upper Limits using Wilson Hitferty (WH) and Hawkins Wijdey {HW) Methods WH HW WH HW 95% Apprax. Garnrna UTL with 95% Coverage 47429 54346 95% Apprax. Gamma UPL 33332 35476 Minimum Maximum 7500-0 SD 18412 k hat (MLE) Theta hat (MLE) 35001 nu hat (MLE) 27.16 MLE Mean {bias corrected) 5""; Percentile of Chisquare (2k) 0.247 8642 2.33 174 ------- Upper Limits Computed Using Gamma Distribution and KM Estimates The following statistics are computed using gamma distribution aid KM estimates Upper Limits using Wilson HiHerty (WH) and Hawkins Wudey (HW) Methods k hat (KM) 0.224 nu hat (KM) 24.66 WH HW WH HW 95'% Apprax. Gannma UTL with 95% Coverage 46378 54120 95'% Approx. Gamma UPL 32961 35195 95% UCL of the Mean Based upon GROS Method Approximate Chi Square Value (27.01 .a) 1S. 16 95% Gamma Approximate UCL [use when ri>=5D) 14M5 .Adjusted Level of Significance f} 0.345S Adjusted Chi Square Value (27.01, P) 15.93 95% Gamma Adjusted UCL (use when n<5<0) 14651 95% UCL of the Mean Using Gamma Distribution on KM Estimates Gamma Kaplan-Meier (KM) Satisfies k hat (KM) 0.224 nu hat (KM) 24.66 Approximate Chi Square Value (24.66, a] 14.35 Adjusted Chi Square Value (24.66. (3) 14.14 95% Gamma Approximate KM-UCL (use when n;=5G} 14844 95%Gamma Adjusted KM-UCL (use when n<50) 15066 Table 5-5. Upper Limits Using LROS and KM Estimates and Lognormal Distribution of Detected Data Lognormal GOF Test on Detected Observations Only Shapiro Wilk Test Statistic 0.939 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value 0.945 Data Not Lognormal at 5% Significance Level Lilliefors Test Statistic 0.104 Lilliefore GOF Test 5% Ulliefors Critical Value 0.132 Detected Data appear Lognormal at 5% Significance Level Detected Data appear Approximate Lognormal at 5% Significance Level Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Detects Mean in Original Scale 863-8 Mean in Log Scale 5.983 SD in Original Scale 18414 SD in Log Scale 3.391 95% UTL95% Coverage 394791 95% BCA UTL95% Coverage 77530 95% Bootstrap (%} UTL95% Coverage 77530 95% UPL ft} 121584 90% Percentile (z) .3-0572 95% Percentile (z) 104784 99% Percentile (z) 1056400 95% USL 10156719 Statistics using KM estimates on Logged Data and Assuming Lognormal Distribution KM Mean of Logged Data 6.03 95% KM UTL (Lognormal)95% Coverage 334181 KM SD of Logged Data 3.286 95% KM UPL (Lognormal) 106741 95% KM Percentile Lognomial (z} 92417 175 ------- 95% UCL of the mean Using LROS and Lognormal Distribution on KM Estimates Methods Lognormal ROS Statistics Using Imputed Non-Detects Mean in Original Scale B63S Mean in Log Scale 5.SB3 SD in Original Scale 18414 SD in Log Scale 3.391 95% t UC L (assumes normality of RO S data) 12793 95% BCA Bootstrap UC L 13762 955; H-UCL [Log ROS) 1B55231 955; Percentile Bootstrap UCL 12676 955; Bootstrap! UCL 14659 UCLs using Lognormal Distribution and KM Estimates when Detected data are Lognormally Distributed KM Mean togged) 6.03 955; H-UCL (KM-Log) 11735BS KM SD togged) 3.286 95%Critical H Value (KM-Log) 5.7 KM Standard Error of Mean Dogged) 0.449 Nonparametric upper percentiles are: 9340 (80%), 25320 (90%), 46040 (95%), and 77866 (99%). Other upper limits, based upon the gamma and lognormal distribution, are described in Table 5-6. All computations have been performed using the ProUCL software. In the following Table 5-6, method proposed/described in the literature have been cited in the Reference Method of Calculation column. Table 5-6. Summary of Upper Limits Computed using Gamma and Lognormal Distribution of Detected Data: Sample Size = 55, No. of NDs=10, % NDs = 18.18% Upper Limits Min (detects) Max (detects) Mean (KM) Mean (ROS) 95%Percentile(ROS) UPL95 (ROS) UTL95-95 (ROS) UPL95 (KM) UTL95-95 (KM) UCL95 (ROS) UCL (KM) Gamma Distribution Result 5.2 79,000 8,638 8,642 42,055 33,332 47,429 32,961 46,978 14,445 14,844 Reference/ Method of Calculation ~ ~ ~ ~ ~ WH- ProUCL WH- ProUCL WH-ProUCL WH-ProUCL ProUCL ProUCL Lognormal Distribution Result 1.65 11.277 6.3 8,638 104,784 121,584 394,791 106,741 334,181 14,659 12,676 1,173,988 Reference/ Method of Calculation Logged Logged Logged ~ ~ Helsel(2012),EPA2009 Helsel(2012),EPA2009 EPA (2009) EPA(2009) bootstrap-t, ProUCL 5.0 percentile bootstrap, Helsel (2012) H-UCL, KM mean and sd on logged data - EPA (2009) The statistics listed in Tables 5-4 and 5-5, and summarized in Table 5-6 demonstrate the need and merits of using the gamma distribution for computing practical and meaningful estimates (upper limits) of the decision parameters (e.g., mean, upper percentile) of interest. Example 5.3. The benzene data set (Benzene-H-UCL-RCRA.xls) of size 8 used in Chapter 21 of the RCRA Unified Guidance document (EPA 2009) was used in Section 4.6.3.1 to address some issues associated with the use of lognormal distribution to compute a UCL of mean for data sets with 176 ------- nondetects. The benzene data set is used in this example to illustrate similar issues associated with the computation of UTLs and UPLs based upon lognormal distribution using substitution methods. Lognormal distribution based upper limits using ROS and KM methods are summarized in Table 5-7. Table 5-7. Lognormal 95%-95% Upper Limits based upon LROS and KM Estimates Lognormal GOF Test on Detected Observations Only Shapiro Wilk Test Statistic 0.829 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value 0.803 Detected Data appear Lognormal at 5% Significance Level Lilliefors Test Statistic 0.304 Ljlliefors GOF Test 5% Ulliefors Critical Value 0.335 Detected Data appear Lognormal at 5% Significance Level Detected Data appear Lognonnal at 5% Sgnificanee Level Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Detects Mean in Original Scale 2.913 Mean in Log Scale 0.0326 SD in Original Scale 5.364 SD in Log Scale 1.443 95% UTL95% Coverage 109.2 95% BCA UTL95% Coverage 1S.1 95% Bootstrap {%} UTL95% Coverage 1S.1 95% UPL ------- Table 5-9. Lognormal Distribution Based Upper Limits using DL/2 (=0.25) for ND Mean of logged Data 0.204 SD of logged Data 1.257 Lognormal GOF Test Shapiro Wilk Test Statistic 0.836 Shapiro Wilk Lognormal GOF Test 5% Shapiro Wilk Critical Value 0.818 Data appear Lognormal at 5% Significance Level Ulliefors Test Statistic 0.255 Ulliefors Lognormal GOF Test 5% Ulliefors Critical Value 0.313 Data appear Lognormal at 5% Significance Level Data appear Lognormal at 5% Significance Level Background Statistics assuming Lognormal Distribution 95%UTLwith 35% Coverage 67.44 90% Pereentile (z) S.142 95%UPLft) 15,34 35% Pereentile {2} 9.S93 miJSL 15.73 99% Pereentile (z) 22.85 Note: Even though UPLs and UTLs computed using the lognormal distribution do not suffer from transformation bias, a minor increase in the sd of logged data (from 1.152 to 1.257 above) causes a significant increase in upper limits, especially in UTLs (from 52.5 to 67.44) computed using a small data set (<15-20). This is particularly true when the data set consists of outliers. Impact of Outlier, 16.1 ppb on the Computations of Upper Limits Benzene data set without the outlier, 16.1 ppb, follows a normal distribution, and normal distribution based upper limits without the outlier 16.1 are summarized as follows in Tables 5-10 (KM estimates), 5- 11 (ND by DL), and 5-12 (ND by DL/2). Table 5-10 Normal Distribution Based Upper Limits Computed Using KM estimates Normal GOF Test on Detects Only Shapiro Wilk Test Statistic 0.847 Shapiro Wilk GOF Test 5*4 Shapiro Wilk Critical Value 0.788 Detected Data appear Normal at 5% Significance Level Ulliefors Test Statistic Q.255 Ulliefors GOF Test 5°i Ulliefors Critical Value Q.362 Detected Data appear Normal at 5% Significance Level Detected Data appear Normal at 5% Significance Level Kaplan Meier (KM) Background Statistics Assuming Normal Distribution Mean 1.086 SD 0.544 95% UTL95% Coverage 2.933 95%KMUPLft) 2.215 95% KM Chebyshev LI P L 3.519 30% KM Pereentile bj 1.732 95% KM Pereentile (z) 1.98 99% KM Pereentile (z) 2,35 95% KM USL 2.139 178 ------- Table 5-11 Normal Distribution Based Upper Limits Computed using DL for ND Normal GOF Test Shapiro Wilk Test Statistic 0.814 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value O.S03 Data appear Normal at 5% Significance Level Ulliefors Test Statistic 0.263 Lilliefors GOF Test 5% Ulliefors Critical Value 0.335 Data appear Normal at 5% Significance Level Data appear Normal at 5% Significance Level Background Statistics Assuming Normal Distribution 95%UTLwith 95% Coverage 3.081 90% Percentile (z) 1.838 35%UPL|) 2.305 95% Percentile (z) 2.052 95%USL 2.224 99% Percentile (z) 2.452 Note: DL (=0.5) has been used for the ND value (does not accurately account for its ND status). Therefore, upper limits are slightly higher than those computed using KM estimates. Table 5-12 Normal Distribution Based Upper Limits Computed using DL/2 for ND Normal GOF Test Shapiro Wilk Test Statistic Q.875 Shapiro Wilk GOF Test 5% Shapiro Wilk Critical Value Q.SQ3 Data appear Normal at 5% Significance Level Lilliefors Test Statistic Q.236 Lilliefors GOF Test 5% Ulliefors Critical Value 0.335 Data appear Normal at 5% Significance Level Data appear Normal at 5X Significance Level Background Statistics Assuming Normal Distribution 95%UTLwith 95% Coverage 3.2Q6 30% Percentile (z) 1.863 35%UPL{t) 2.36S 35% Percentile (z) 2.094 35% USL 2.2S 93% Percentile (z) 2.526 Note: DL/2 (=0.25) has been used for the ND value (does not accurately account for its ND status). The use of DL/2 has increased the variance slightly which causes a slight increase in the various upper limits. Therefore, upper limits are slightly higher than those computed using KM estimates and using DL for the ND value. Based upon the benzene data set, normal UTL95-95 (= 2.93) computed using KM estimates appears to represent a more realistic estimate of background threshold value. 179 ------- Example 5-4: The manganese (Mn) data set used in Chapter 15 of the Unified RCRA Guidance (2009) has been used here to demonstrate how LROS method generates elevated BTVs. Summary statistics are summarized as follows. General Statistics for Censored Datasets (with NDs) using Kaplan Meief Method Variable NumObs tt Missing Num Ds NumNDs % NDs Min ND Max ND KM Mean KM Vai KM SD KM CV Mn 25 0 19 6 24.00% 2 5 General Statistics for Raw Dataset using Detected Data Onry 19.87 641 25.32 1274 Variable NumObs tt Missing Minimum Maximum Mean Median Mn 19 0 3.3 106.3 25.46 12.6 Var 752.7 SD 27.44 MADALETSSkewness CV 9.34 1.942 1.079 Percentiles using all D elects (Ds) and N on-Detects (NDs) Variable NumObs tt Missing IQXile 20%ile 25Xile(Q1] 50Xile(Q2) 75Xile(Q3) SOXrle 90Xile 95%ile 99Xile Mn 25 0 2.52 5 5 10 21.6 25.06 50.52 72.48 99.32 The detected data follow a lognormal distribution, the maximum value in the data set is 106, and using the LROS method (robust ROS method), one gets a 99% percentile = 183.4, and a UTL of 175. These statistics are summarized in Table 5-13. The detected data also follows a gamma distribution. Gamma-KM method based upper limits are summarized as follows. The Gamma UTL95-95s (KM) are 92.5 (WH) and 99.32 (HW) and the 99% percentiles are: 94.42 (WH) and 101.8 (HW). The Gamma UTL (KM) appears to represent a reasonable estimate of BTV. These BTV estimates are summarized in Table 5-14. Table 5-13 LROS and Lognormal KM Method Based Upper Limits Background Lognormal ROS Statistics Assuming Lognormal Distribution Using Imputed Non-Detects Mean in Original Scale 19.83 Mean in Log Scale 2.277 SDinOriginalS cale 25.87 S D in Log S cale 1.261 175.6 95% BCA UTL95% Coverage 106.3 106.3 95%UPL(t) 88.06 43.1 85% Percentile (z) 77.64 183.4 95%USL 280.4 95% UTL95% Coverage 95% Bootstrap (%) UTL95% Coverage 90% Percentile (z) 99% Percentile (z) Statistics using KM estimates on Logged Data and Assuming Lognormal Drsfcrburjon KM Mean of Logged Data 2.309 95% KM UTL (Lognormal)95% Coverage 151 KM S D of Logged D ata 1.182 95% KM U PL (Lognormal) 79.12 95% KM Percentile Lognormal (z) 70.31 95% KM USL (Lognormal) 234.1 180 ------- Table 5-14 Gamma KM Method Based Upper Limits The following statistics are computed using gamma distribution and KM estuiates U pper Limits using Wilson H ilferty (WH) and H awkins Wixley (HWJ Methods k hat (KM) 0.616 mj hat (KM) 30.79 WH HW WH HW 95?; Approx. Gamma LITL with 95-t Coverage 92.4 99.32 95?; Approx. Gamma UPL 63.96 65.76 35% KM Gamma Percentile 59.5 60.7 95%Gamma(JSL 115.8 128.4 Notes: Even though one can argue that there is no transformation bias when computing lognormal distribution based UTLs and UPLs, the use of a lognormal distribution on data with or without NDs often yields inflated values which are not supported by the data set used to compute them. Therefore, its use including LROS method should be avoided. Before using a nonparametric BTV estimate, one should make sure that the detected data do not follow a known distribution. When dealing with a data set with NDs, it is suggested to account for NDs and determine the distribution of detected values instead of using a nonparametric UTL as used in Example 17-4 on page 17-21 of Chapter 17 of the EPA Unified Guidance, 2009. If detected data follow a parametric distribution, one may want to compute a UTL using that distribution and KM estimates; this approach will account for data variability instead defaulting to higher order statistics. Summary and Recommendation • It is recommended that occasional low probability outliers not be used in the computation of decision making statistics. The decision making statistics (e.g., UCLs, UTLs, UPLs) should be computed using observations representing the main dominant population. The use of a lognormal distribution should be avoided in computing upper limits (UCLs, UTLs, UPLs) based upon data sets with sd of detected logged data for moderately skewed to highly skewed data sets of sizes smaller than 20-30. It is reasonable to state that, like uncensored data sets without NDs, the minimum sample size requirement increases as the skewness increases. • The project team should collectively make a decision about the disposition of outliers. It is often helpful to compute decision statistics (upper limits) and hypothesis test statistics twice: once including outliers, and once without outliers. By comparing the upper limits computed with and without outliers, the project team can determine which limits are more representative of the site conditions under investigation. 5.4 Computing Nonparametric Upper Limits Based upon Higher Order Statistics For full data sets without any discernible distribution, nonparametric UTLs and UPLs are computed using higher order statistics. Therefore, when the data set consists of enough detected observations, and if some of those detected data are larger than all of the NDs and the DLs, ProUCL computes USLs, UTLs, UPLs, and upper percentiles by using nonparametric methods as described in Chapter 3. Since, nonparametric UTLs, UPLs, USLs, and upper percentiles are represented by higher order statistics (or by some value in between higher order statistic obtained using linear interpolation) every effort should be made to make sure that those higher order statistics do not represent observations coming from population(s) other than the main dominant (e.g., background) population under study. 181 ------- CHAPTER 6 Single and Two-sample Hypotheses Testing Approaches Both single-sample and two-sample hypotheses testing approaches are used to make cleanup decisions at polluted sites, and compare constituent concentrations of two (e.g., site versus background) or more (GW in MWs) populations. This chapter provides guidance on when to use single-sample hypothesis test and when to use two-sample hypotheses approaches. These issues were also discussed in Chapter 1 of this Technical Guide. For interested users, this chapter presents a brief description of the mathematical formulations of the various parametric and nonparametric hypotheses testing approaches as incorporated in ProUCL. ProUCL software provides hypotheses testing approaches for data sets with and without ND observations. For data sets containing multiple nondetects, a new two-sample hypothesis test, the Tarone- Ware (T-W; 1978) test has been incorporated in the current ProUCL, versions 5.0 and 5.1. The developers of ProUCL recommend supplementing statistical test results with graphical displays. It is assumed that the users have collected an appropriate amount of good quality (representative) data, perhaps based upon data quality objectives (DQOs). The Stats/Sample Sizes module can be used to compute DQOs based sample sizes needed to perform the hypothesis tests described in this chapter. 6.1 When to Use Single Sample Hypotheses Approaches When pre-established background threshold values and not-to-exceed values (e.g., USGS background values, Shacklette and Boerngen 1984) exist, there is no need to extract, establish, or collect a background or reference data set. Specifically, when not-to-exceed action levels or average cleanup standards are known, one-sample hypotheses tests can be used to compare onsite data with known and pre-established threshold values, provided enough onsite data needed to perform the hypothesis tests are available. When the number of available site observations is less than 4-6, one might perform point-by-point site observation comparisons with a BTV; and when enough onsite observations (> 8 to 10, more are preferable) are available, it is suggested to use single-sample hypothesis testing approaches. Some recent EPA guidance documents (EPA 2009) also recommend the availability of at least 8-10 observations to perform statistical inference. Some minimum sample size requirements related to hypothesis tests are also discussed in Chapter 1 of this Technical Guide. Depending upon the parameter (e.g., the average value, juo, or a not-to-exceed action level, Ao), representing a known threshold value, one can use single-sample hypothesis tests for the population mean (t-test, sign test) or single-sample tests for proportions and percentiles. Several single-sample tests listed below are available in ProUCL 5.1 and its earlier versions. One-Sample t-Test: This test is used to compare the site mean,/^, with some specified cleanup standard, Cs (jUo), where Cs represents a specified value of the true population mean, /u. The Student's t- test or UCL of the mean is used (assuming normality of site data, or when the sample size is larger than 30, 50, or 100) to verify the attainment of cleanup levels at a polluted sites (EPA 1989a, 1994). Note that the large sample size requirement («= 30, 50, or 100) depends upon the data skewness. Specifically, as skewness increases measured in terms of the sd, a, of the log-transformed data, the large sample size requirement also increases to be able to apply the normal distribution and Student's t-statistic, due to the central limit theorem (CLT). One-Sample Sign Test or Wilcoxon Signed Rank (WSR) Test: These tests are nonparametric tests which can also handle ND observations, provided all NDs and therefore their associated DLs are less than the 182 ------- specified threshold value, Cs. These tests are used to compare the site location (e.g., median, mean) with some specified cleanup standard, Cs, representing the similar location measure. One-Sample Proportion Test or Percentile Test: When a specified cleanup standard, Ao, such as a preliminary remediation goal (PRG), or a compliance limit (CL) represents an upper threshold value of a constituent concentration distribution rather than the mean threshold value, /n, a test for proportion or a test for percentile (e.g., UTL95-95, UTL95-90) can be used to compare exceedances to the actionable level. The proportion, p, of exceedances of A0 by site observations are compared to some pre-specified allowable proportion, Po, of exceedances. One scenario where this test may be applied is following remediation activities at an AOC. The proportion test can also handle NDs provided all NDs are below the action level, A0. It is beneficial to use DQO-based sampling plans to collect an appropriate amount of data. In any case, in order to obtain reasonably reliable estimates and compute reliable test statistics, an adequate amount of representative site data (at least 8 to 10 observations) should be made available to perform the single- sample hypotheses tests listed above. As mentioned before, if only a small number of site observations are available, instead of using hypotheses testing approaches, point-by-point site concentrations may be compared with the specified action level, Ao. Individual point-by-point observations are not to be compared with the average cleanup or threshold level, Cs. The estimated sample mean, such as a UCL95, is compared with a threshold representing an average cleanup standard. 6.2 When to Use Two-Sample Hypotheses Testing Approaches When BTVs, not-to-exceed values, and other cleanup standards are not available, then site data are compared directly with the background data. In such cases, a two-sample hypothesis testing approach is used to perform site versus background comparisons provided enough data are available from each of the two populations. Note that this approach can be used to compare concentrations of any two populations including two different site areas or two different MWs. The Stats/Sample Sizes module of ProUCL can be used to compute DQO-based sample sizes for two-sample parametric and nonparametric hypothesis testing approaches. While collecting site and background data, for better representation of populations under investigation, one may also want to account for the size of the background area (and site area for site samples) in sample size determinations. That is, a larger number (>10 to 15) of representative background (or site) samples may need to be collected from larger background (or site) areas to capture the greater inherent heterogeneity/variability typically present in larger areas. The two-sample hypotheses approaches are used when the site parameters (e.g., mean, shape, distribution) are compared with the background parameters (e.g., mean, shape, distribution). Specifically, two-sample hypotheses testing approaches can be used to compare the average (also medians or upper tails) constituent concentrations of two or more populations such as the background population and the potentially contaminated site areas. Several parametric and nonparametric two-sample hypotheses testing approaches, including Student's t-test, the Wilcoxon-Mann-Whitney (WMW) test, Gehan's test, and the T-W test are included in ProUCL 5.1. Some details of those methods are described in this chapter for interested users. It is recommended that statistical results and test statistics be supplemented with graphical displays, such as the multiple Q-Q plots and side-by-side box plots as graphical displays do not require any distributional assumptions and are not influenced by outlying observations and NDs. Data Types: Analytical data sets collected from the two (or more) populations should be of the same type obtained using similar analytical methods and sampling equipment. Additionally, site and background data should be all discrete or all composite (obtained using the same design, pattern, and number of 183 ------- increments), and should be collected from the same medium (soil) at comparable depth levels (e.g., all surface samples or all subsurface samples) and time (e.g., during the same quarter in groundwater applications). Good sample collection methods and sampling strategies are described in Gerlach, R. W., and J. M. Nocerino (2003) and the ITRC Technical Regulatory guidance document (2012). 6.3 Statistical Terminology Used in Hypotheses Testing Approaches The first step in developing a hypothesis test is to state the problem in statistical terminology by developing a null hypothesis, Ho, and an alternative hypothesis, HA. These hypotheses tests result in two alternative decisions: acceptance of the null hypothesis or the rejection of the null hypothesis based on the computed hypothesis test statistic (e.g., t-statistic, WMW test statistic). The statistical terminologies including error rates, hypotheses statements, Form 1, Form 2, and two-sided tests, are explained in terms of two-sample hypotheses testing approaches. Similar terms apply to all parametric and nonparametric single-sample and two-sample hypotheses testing approaches. Additional details may be found in EPA guidance documents (2002b, 2006b), and MARSSIM (2000) or in statistical text books including Bain and Engelhardt (1992), Hollander and Wolfe (1999), and Hogg and Craig (1995). Two forms, Form 1 and Form 2, of the statistical hypothesis test are useful for environmental applications. The null hypothesis in the first form (Form 1) states that the mean/median concentration of the potentially impacted site area does not exceed the mean/median of the background concentration. The null hypothesis in the second form (Form 2) of the test is that the concentrations of the impacted site area exceed the background concentrations by a substantial difference, S, with S>0. Formally, let Xi, X2, ..., Xn represent a random sample of size n collected from Population 1 (e.g., downgradient MWs or a site AOC) with mean (or median) fix, and Yi, Y2, ..., Ym represent a random sample of size m from Population 2 (upgradient MWs or a background area) with mean (or median) /JY. Let A = /Jx- JUY represent the difference between the two means (or medians). 6.3.1 Test Form 1 The null hypothesis (Ho): The mean/median of Population 1 (constituent concentration in samples collected from potentially impacted areas (or monitoring wells)) is less than or equal to the mean/median of Population 2 (concentration in samples collected from background (or upgradient wells) areas) with Ho: A < 0. The alternative hypothesis (HA). The mean/median of Population 1 (constituent concentration in samples collected from potentially impacted areas) is greater than the mean of Population 2(background areas) with HA: A > 0. When performing this form of hypothesis test, the collected data should provide statistically significant evidence that the null hypothesis is false leading to the conclusion that the site mean/median does exceed background mean/median concentration. Otherwise, the null hypothesis cannot be rejected based on the available data, and the mean/median concentration found in the potentially impacted site areas is considered equivalent and comparable to that of the background areas. 184 ------- 6.3.2 Test Form 2 The null hypothesis (Ho): The mean/median of Population 1 (constituent concentration in potentially impacted areas) exceeds the mean/median or Population 2 (background concentrations) by more than S units. Symbolically, the null hypothesis is written as Ho: A > S, where S>0. The alternative hypothesis (HA): The mean/median of Population 1 (constituent concentration in potentially impacted areas) does not exceed the mean/median of Population 2 (background constituent concentration) by more than S (HA: A < S). Here, S is the background investigation level. When S>0, Test Form 2 is called Test Form 2 with substantial difference, S. Some details about this hypothesis form can be found in the background guidance document for CERCLA sites (EPA 2002b). 6.3.3 Selecting a Test Form The test forms described above are commonly used in background versus site comparison evaluations. Therefore, these test forms are also known as Background Test Form 1 and Background Test Form 2 (EPA, 2002b). Background Test Form 1 uses a conservative investigation level of A = 0, but relaxes the burden of proof by selecting the null hypothesis that the constituent concentrations in potentially impacted areas are not statistically greater than the background concentrations. Background Test Form 2 requires a stricter burden of proof, but relaxes the investigation level from 0 to S. 6.3.4 Errors Rates and Confidence Levels Due to the uncertainties that result from sampling variation, decisions made using hypotheses tests will be subject to errors, also known as decision errors. Decisions should be made about the width of the gray region, A, and the degree of decision errors that is acceptable. There are two ways to err when analyzing sampled data (Table 6-1) to derive conclusions about population parameters. Type I Error: Based on the observed collected data, the test may reject the null hypothesis when in fact the null hypothesis is true (a false positive or equivalently a false rejection). This is a Type I error. The probability of making a Type I error is often denoted by a (alpha); and Type II Error: On the other hand, based upon the collected data, the test may fail to reject the null hypothesis when the null hypothesis is in fact false (a false negative or equivalently a false acceptance). This is called Type II error. The probability of making a Type II error is denoted by /? (beta). Table 6-1. Hypothesis Testing: Type I and Type II Errors Decision Based on Sample Data Ho is not rejected Ho is rejected Actual Site Condition Ho is True Correct Decision: (1 - a) Type I Error: False Positive (a) Ho is not true Type II Error: False Negative ((3) Correct Decision: (1 - (3) 185 ------- The acceptable level of decision error associated with hypothesis testing is defined by two key parameters: confidence level and power. These parameters are related to two error probabilities, a and/?. Confidence level 100(1- oc)%: As the confidence level is lowered (or alternatively, as a is increased), the likelihood of committing a Type I error increases. Power 100(1 - fi)%: As the power is lowered (or alternatively, as /? is increased), the likelihood of committing a Type II error increases. Although a range of values in the interval (0, 1) can be selected for these two parameters, as the demand for precision increases, the number of samples and the associated cost (sampling and analytical cost) will generally also increase. The cost of sampling is often an important determining factor in selecting the acceptable level of decision errors. However, unwarranted cost reduction at the sampling stage may incur greater costs later in terms of increased threats to human health and the environment, or unnecessary cleanup at a site area under investigation. The number of samples, and hence the cost of sampling, can be reduced but at the expense of a higher possibility of making decision errors that may result in the need for additional sampling, or increased risk to human health and the environment. There is an inherent tradeoff between the probabilities of committing a Type I or a Type II error, a simultaneous reduction in both types of errors can only occur by increasing the number of samples. If the probability of committing a false positive error is reduced by increasing the level of confidence associated with the test (in other words, by decreasing a), the probability of committing a false negative is increased because the power of the test is reduced (increasing /?). The choice of a determines the probability of the Type I error. The smaller the a-value, the less likely to incorrectly reject the null hypothesis (H0). However, a smaller value for a also means lower power with decreased probability of detecting a difference when one exists. The most commonly used a value is 0.05. With a = 0.05, the chance of finding a significance difference that does not really exist is only 5%. In most situations, this probability of error is considered acceptable. Suggested values for the Two Types of Error Rates: Typically, the following values for error probabilities are selected as the minimum recommended performance measures: • For the Background Test Form 1, the confidence level should be at least 80% (a = 0.20) and the power should be at least 90% (J3 = 0.10). • For the Background Test Form 2, the confidence level should be at least 90% (a = 0.10) and the power should be at least 80% (ft = 0.20). Seriousness of the Two Types of Error Rates: • When using the Background Test Form 1, a Type I error (false positive) is less serious than a Type II error (false negative). This approach favors the protection of human health and the environment. To ensure that there is a low probability of committing a Type II error, a Test Form 1 statistical test should have adequate power at the right edge of the gray region. • When the Background Test Form 2 is used, a Type II error is preferable to committing a Type I error. This approach favors the protection of human health and the environment. The choice of the hypotheses used in the Background Test Form 2 is designed to be protective of human health and the environment by requiring that the data contain evidence of no substantial contamination. 186 ------- 6.4 Parametric Hypotheses Tests Parametric statistical tests assume that the data sets follow a known statistical distribution (mostly normal); and that the data sets are statistically independent with no expected spatial and temporal trends in the data sets. Many statistical tests (e.g., two-sample t-test) and models are only appropriate for data that follow a particular distribution. Statistical tests that rely on knowledge of the form of the population distribution of data are known as parametric tests. The most commonly used distribution for tests involving environmental data is the normal distribution. It is noted that GOF tests which are used to determine data set's distribution (e.g., S-W test for normality) often fail if there are not enough observations, if the data contain multiple populations, or if there is a high proportion of NDs in the collected data set. Tests for normality lack statistical power for small sample sizes. In this context, a sample consisting of less than 20 observations may be considered a small sample. However, in practice, many times it may not be possible, due to resource constraints, to collect data sets of sizes greater than 10. This is especially true for background data sets, as the decision makers often do not want to collect many background samples. Sometimes they want to make cleanup decisions based upon data sets of sizes even smaller than 10. Statistics computed based upon small data sets of sizes < 5 cannot be considered reliable to derive important decisions affecting human health and the environment. 6.5 Nonparametric Hypotheses Tests Statistical tests that do not assume a specific statistical form for the data distribution(s) are called distribution-free or nonparametric statistical tests. Nonparametric tests have good test performance for a wide variety of distributions, and their performances are not unduly affected by NDs and the outlying observations. In two-sample comparisons (e.g., t-test), if one or both of the data sets fail to meet the test for normality, or if the data sets appear to come from different distributions with different shapes and variability, then nonparametric tests may be used to perform site versus background comparisons. Typically, nonparametric tests and statistics require larger size data sets to derive correct conclusions. Several two-sample nonparametric hypotheses tests, the WMW test, Gehan test, and Tarone-Ware (T-W) test, are available in ProUCL. Like the Gehan test, the T-W test is used for data sets containing NDs with multiple RLs. The T-W test was new in ProUCL 5.0 and is also included in ProUCL 5.1. The relative performances of different testing procedures can be assessed by comparing, />-values associated with those tests. The p-value of a statistical test is defined as the smallest value of a (level of significance, Type I error) for which the null hypothesis would be rejected based upon the given data sets of sampled observations. The />-value of a test is sometimes called the critical level or the significance level of the test. Whenever possible, critical values and/"-values have been computed using the exact or approximate distribution of the test statistics (e.g., GOF tests, t-test, Sign test, WMW test, Gehan test, M- K trend test). Performance of statistical tests is also compared based on their robustness. Robustness means that the test has good performance for a wide variety of data distributions, and that its performance is not significantly affected by the occurrence of outliers. Not all nonparametric methods are robust and resistant to outliers. Specifically, nonparametric upper limits used to estimate BTVs can get affected and misrepresented by outliers. This issue has been discussed earlier in Chapter 3 of this Technical Guide. • If a parametric test for comparing means is applied to data from a non-normal population and the sample size is large, then a parametric test may work well, provided that the data sets are not heavily skewed. For heavily skewed data sets, the sample size requirement associated with the CLT can become quite large, sometimes larger than 100. A brief simulation study elaborating on the sample 187 ------- size requirements to apply the CLT on skewed data sets is given in Appendix B. For moderately skewed (Chapter 4) data sets, the CLT ensures that parametric tests for the mean will work because parametric tests for the mean are robust to deviations from normal distributions as long as the sample size is large. Unless the population distribution is highly skewed, one may choose a parametric test for comparing means when there are at least 25-30 data points in each group. • If a nonparametric test for comparing means is applied on a data set from a normal population and the sample size is large, then the nonparametric test will work well. In this case, the p-values tend to be a little too large, but the discrepancy is small. In other words, nonparametric tests for comparing means are only slightly less powerful than parametric tests with large samples. • If a parametric test is applied on a data set from a non-normal population and the sample size is small (< 20 data points), then the/"-value may be inaccurate because the CLT does not apply in this case. • If a nonparametric test is applied to a data set from a non-normal population and the sample size is small, then the /"-values tend to be too high. In other words, nonparametric tests may lack statistical power with small samples. Notes: It is suggested that the users supplement their test statistics and conclusions by using graphical displays for visual comparisons of two or more data sets. ProUCL software has side-by-side box plots and multiple Q-Q plots that can be used to graphically compare two or more data sets with and without ND observations. 6.6 Single Sample Hypotheses Testing Approaches This section describes the mathematical formulations of parametric and nonparametric single-sample hypotheses testing approaches incorporated in ProUCL software. For the sake of interested users, some directions to perform these hypotheses tests are described as follows. The directions are useful when the user wants to manually perform these tests. 6.6.1 The One-Sample t-Testfor Mean The one-sample t-test is a parametric test used for testing a difference between a population (site area, AOC) mean and a fixed pre-established mean level (cleanup standard representing a mean concentration level). The Stats/Sample Sizes module of ProUCL can be used to determine the minimum number of observations needed to achieve the desired DQOs. The collected sample should be a random sample representing the AOC under investigation. 6.6.1.1 Limitations and Robustness of One-Sample t-Test The one-sample t-test is not robust in the presence of outliers and may not yield reliable results in the presence of ND observations. Do not use this test when dealing with data sets containing NDs. Some nonparametric tests described below may be used in cases where NDs are present in a data set. This test may yield reliable results when performed on mildly or moderately skewed data sets. Note that levels of skewness are discussed in Chapters 3 and 4. The use of a t-test should be avoided when data are highly skewed (sd of log-transformed data exceeding 1, 1.5), even when the data set is of a large size such as «=100. 188 ------- 6.6.1.2 Directions for the One-Sample t-Test Let Xj, x2, . . . , xn represent a random sample (analytical results) of size, n, collected from a population (AOC). The use of the One-Sample t-Test requires that the data set follows a normal distribution; that is when using a typical software package (e.g., Minitab), the user needs to test for the normality of the data set. For the sake of users and to make sure that users do not skip this step, ProUCL verifies normality of the data set automatically. STEP 1: Specify an average cleanup goal or action level, fjto (G), and choose one of the following combination of null and alternative hypotheses: Form 1: Ho: site p <^ovs. HA: site ^ > ^o Form 2: Ho: site ^i>^o vs. HA: site p < jUo Two-Sided: Ho: site ju = jUo vs. HA: site ju ^ jUo. Form 2 with substantial difference, S: Ho: site ^>^o+Svs. HA: site p < ^o + S, here S> 0. STEP 2: Calculate the test statistic: '.= In the above equation, S is assumed to be equal to "0", except for Form 2 with substantial difference. STEP 3: Use Student's t-table (ProUCL computes them) to find the critical value tn-i, i-a Conclusion: Form 1: If to > tn-i,a, then reject the null hypothesis that the site population mean is less than the cleanup level, ^io Form 2: If to < -tn-i,a, then reject the null hypothesis that the site population mean exceeds the cleanup level, ^io Two-Sided: If \to > tn-i, 0/2, then reject the null hypothesis that the site population mean is same as the cleanup level, jUo Form 2 with substantial difference, S: Ifto< -tn-i, i-a, then reject the null hypothesis that the site population mean is more than the cleanup level, po + the substantial difference, S. Here, tn-i,a represents the critical value from t-distribution with (n-\) degrees of freedom (df) such that the area to the right of tn-i,a under the t-distribution probability density function is a. 6.6.1.3 P -values In addition to computing critical values (some users still like to use critical values for a specified a), ProUCL computes exact or approximate /^-values. A />-value is the smallest value for which the null hypothesis is rejected in favor of the alternative hypotheses. Thus, based upon the given data set, the null hypothesis is rejected for all values of a (the level of significance) greater than or equal to the />-value. 189 ------- The details of computing a p-value for a t-test can be found in any statistical text book such as Daniel (1995). ProUCL computes /^-values for t-tests associated with each form of the null hypothesis. Specifically, if the computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis based upon the collected data set. 6.6.1.4 Relation between One-Sample Tests and Confidence Limits of the Mean or Median There has been some confusion among the users whether to use a LCL or a UCL of the mean to determine if the remediated site areas have met the cleanup standards. There is a direct relation between one sample hypothesis tests and confidence limits of the mean or median. For example, depending upon the hypothesis test form, a t-test is related to the upper or lower confidence limit of the mean, and a Sign test is related to the confidence limits of the median. In confirmation sampling, either a one sample hypothesis test (e.g., t-test, WSR test) or a confidence interval of the mean (e.g., LCL, UCL) can be used. Both approaches result in the same conclusion. These relationships have been illustrated for the t-test and the LCLs and upper UCLs for normally distributed data sets. The use of a UCL95 to determine if a polluted site has attained the cleanup standard, HO, after remediation is very common. If a UCL95 < HO, then it is concluded that the site meets the standard. The conclusion based upon the UCL or LCL, or the interval (LCL, UCL) is derived from hypothesis test statistics. For an example, while using a 95% lower confidence limit (LCL95), one is testing hypothesis test Form 1, and when using UCL95, one is testing hypothesis Form 2. For a normally distributed data set: x,,x2, . . . , xn ( e.g., collected after excavation), the UCL95 and LCL95 are given as follows: UCL95 = x+tn_^05 Objective: Does the site average, H, meet the cleanup level, /^o? Form 1: Ho: site H HO Form 2: Ho: site H>HO vs. HA: site H < Ho Two-Sided: Ho: site H = HO vs. a HA: site H ^ Ho- Based upon the t-test, conclusions are: Form 1: If t> tn-i, o.os, then reject the null hypothesis in favor of the alternative hypothesis Form 2: If to < -tn-i, o.os, then reject the null hypothesis in favor of the alternative hypothesis Two-Sided: If \to > tn-i, 0.025, then reject the null hypothesis that the site population mean is same as the cleanup level Here tn-i, o.os represents a critical value from the right tail of the t-distribution with («-l) degrees of freedom such that area to right of tn-i, o.os is 0.05. 190 ------- For Form 1, we have: Reject Ho ift>tn.i,o.os , that is reject the null hypothesis when Equivalently reject the null hypothesis and conclude that site has not met the cleanup standard when x - tn_l 005*sd/ 4n > fJ0; or when LCL95>cleanup goal, po. The site is concluded dirty when LCL95> po. For Form 2, we have: Reject Ho if t< -tn-i,o.o5 , that is reject the null hypothesis when Equivalently reject the null hypothesis and conclude that site meets the cleanup standard when x + tn_1005 * sd 1 4n < ju0 ; or when UCL95 < ju0. The site is concluded clean when UCL95< jUo. 6.6.2 The One-Sample Test for Proportions The one-sample test for proportions represents a test for evaluating the difference between the population proportion, P, and a specified threshold proportion, P0. Based upon the sampled data set and sample proportion,/), of exceedances of a pre-specified action level, Ao, by the n sample observations (e.g., onsite observations); the objective is to determine if the population proportion (of exceedances of the threshold value, Ao) exceeds the pre-specified proportion level, Po. This proportion test is equivalent to a sign test (described next), when Po = 0.5. The Stats/Sample Sizes module of ProUCL can be used to determine the minimum sample size needed to achieve pre-specified DQOs. 6.6.2.1 Limitations and Robustness Normal approximation to the distribution of the test statistic is applicable when both (nPo) and n (1- Po) are at least 5. For smaller data sets, ProUCL uses the exact binomial distribution (e.g., Conover, 1999) to compute the critical values when the above statement is not true. The Proportion test may also be used on data sets with ND observations, provided all ND values (DLs, reporting limits) are smaller than the action level, Ao. 6.6.2.2 Directions for the One-Sample Test for Proportions Let xh x2, . . . , xn represent a random sample (data set) of size, n, from a population (e.g., the site (e.g., exposure area) under investigation. Let Ao represent a compliance limit or an action level to be met by site 191 ------- data. It is expected (e.g., after remediation) that the proportion of site observations exceeding the action level, Ao, is smaller than the specified proportion, P0. Let B = number of site values in the data set exceeding the action level, Ao. A typical observed sample value of B (based upon a data set) is denoted by b. It is noted that the random variable, B follows a binomial distribution (BD) ~ B(«, P) with n equal to the number of trials and P being the unknown population proportion (probability of success). Under the null hypothesis, the variable B follows a binomial distribution (BD) ~ B(«, Po). The sample proportion, p=b/n = (number of site values in the sample > Ao)/n STEP 1: Specify a proportion threshold value, Po, and state the following null hypotheses: Form 1: H0: P Po Form 2: H0: P>Po vs. HA: P < Po Two-Sided: H0: P = Po vs. HA: P+Po STEP 2: Calculate the test statistic: + C-P (6-2) Where c = Ip0(i-p0y V /n -0.5 n V'P > ° x(# of site values > A0) and p = — 0.5.n n Here c is the continuity correction factor for use of the normal approximation. Large Sample Normal Approximation STEP 3: Typically, one should use BD (as described above) to perform this test. However, when both (nPo) and n (1- Po) are at least 5, a normal (automatically computed by ProUCL) approximation may be used to compute the critical values (z-values) and/"-values. STEP 4: Conclusion described for the approximate test based upon the normal approximation: Form 1: If ZQ > za, then reject the null hypothesis that the population proportion, P, of exceedances of action level, Ao, is less than the specified proportion, Po. Form 2: If ZQ < -za, then reject the null hypothesis that the population proportion, P, is more than the specified proportion, Po. Two-Sided: If ZQ > z«/2, then reject the null hypothesis that the population proportion, P, is the same as the specified proportion, P0. Here, z« represents the critical value of a standard normal variable, Z, such that area to the right of z« under the standard normal curve is a. 192 ------- P-Values Based upon a Normal Approximation As mentioned before, a/"-value is the smallest value for which the null hypothesis is rejected in favor of the alternative hypotheses. Thus, based upon the given data, the null hypothesis is rejected for all values of a (the level of significance) greater than or equal to the /"-value. The details of computing a/"-value for the proportion test based upon large sample normal approximation can be found in any statistical text book such as Daniel (1995). ProUCL computes large sample /"-values for the proportion test associated with each form of null hypothesis. 6.6.2.3 Use of the Exact Binomial Distribution for Smaller Samples ProUCL 5.0 also performs the proportion test based upon the exact binomial distribution when the sample size is small and one may not be able to use the normal approximation as described above. ProUCL 5.0 checks for the availability of appropriate amount of data, and performs the tests using a normal approximation or the exact binomial distribution accordingly. STEP 1: When the sample size is small (e.g., < 30), and either (nPo), or n (1 - Po) is less than 5, one should use the exact BD to perform this test. ProUCL 5.0 performs this test based upon the BD, when the above conditions are not satisfied. In such cases, ProUCL 5.0 computes the critical values and/"-values based upon the BD and its cumulative distribution function (CDF). The probability statements concerning the computation of/"-values can be found in Conover (1999). STEP 2: Conclusion Based upon the Binomial Distribution Form 1: Large values of B cause the rejection of the null hypothesis. Therefore, reject the null hypothesis, when B > b. Here b is obtained using the binomial cumulative probabilities based upon a BD (n, Po). The critical value, b (associated with a) is given by the probability statement: P(B>b) = a, or equivalently, P(B < b) = (1 - a). Since B is a discrete binomial random variable, the level, a may not be exactly achieved by the critical value, b. Form 2: For this form, small values of B will cause the rejection of the null hypothesis. Therefore, reject the null hypothesis, when B < b. Here b is obtained using the binomial cumulative probabilities based upon a BD(«, Po). The critical value, b is given by the probability statement: P(B) = a. As mentioned before, since B is a discrete binomial random variable, the level, a may not be exactly achieved by the critical value, b. Two-Sided Alternative: The critical or the rejection region for the null hypothesis is made of two areas, one in the right tail (of area ~ a2) and the other in the left tail (with area ~ ai), so that the combined area of the two tails is approximately, a = a; + 0.2. That is for this hypothesis form, both small values and large values of B will cause the rejection of the null hypothesis. Therefore, reject the null hypothesis, when B < bi or B > \)2 Typically 0.1 and 0.2 are roughly equal, and in ProUCL, both are chosen to be equal to a /2; bi and b2 are given by the probability statements: P (B < bi) ~ a/2, and P(B > bj) ~ a/2. B being a discrete binomial random variable, the level, a may not be exactly achieved by the critical values, bi and b2. P-Values Based upon Binomial Distribution as Incorporated In ProUCL: The probability statements for computing a p-value for a proportion test based upon BD can be found in Conover (1999). Using the BD, ProUCL computes /"-values for the proportion test associated with each form of null hypothesis. If the computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis based upon the collected data set used in the computations. There are some variations in the literature 193 ------- regarding the computation of p-values for a proportion test based upon the exact BD. Therefore, the p- value computation procedure as incorporated in ProUCL 5.0 is described below. Let b be the calculated value of the binomial random variable, B under the null hypothesis. ProUCL 5.0 computes the/>-values using the following probability (Prob) statements: Form 1: p-value = Prob(B > b) Form 2: p-value = Prob(B < b) Two-sided Alternative: For b>(n- b): .P-value = 2* Prob(B < b) For b<(n- b): /'-value = 2*Prob(B > b) 6.6.3 The Sign Test The Sign test is used to detect a difference between the population median and a fixed cleanup goal, C (e.g., representing the desired median value). Like the WSRtest, the Sign test can also be used on paired data to compare the location parameters of two dependent populations. This test makes no distributional assumptions. The Sign test is used when the data are not symmetric and the sample size is small (EPA, 2006). The Stats/Sample Sizes module of ProUCL can be used to determine minimum number of observations needed to achieve pre-specified DQOs associated with the Sign test. 6.6.3.1 Limitations and Robustness Like the Proportion test, the Sign test can also be used on data sets with NDs, provided all values reported as NDs are smaller than the cleanup level/action level, C. For data sets with NDs, the process to perform a Sign test is the same as that for data sets without NDs, provided DLs associated with all NDs are less than the cleanup level. Per EPA guidance document (2006), all NDs exceeding the action level are discarded from the computation of Sign test statistic; also all observations, detects and NDs equal to the action level are discarded from the computation of the Sign test statistic. Discarding of observations (detects and NDs) will have an impact on the power of the test (reduced power). ProUCL has the Sign test for data sets with NDs as described in USEPA (2006). However, the performance of the Sign test on data sets with NDs requires some evaluation. 6.6.3.2 Sign Test in the Presence ofNondetects A principal requirement when applying the sign test is that the cleanup level, C, should be greater than the largest ND value; in addition all observations (detects and NDs) equal to the action level and all NDs greater than or equal to the action level are discarded from the computation of the Sign test statistic. 6.6.3.3 Directions for the Sign Test Let xb x2, . . . , xn represent a random sample of size n collected from a site area under investigation. As before, S > 0 represents the substantial difference used in Form 2 hypothesis tests. STEP 1: Letjux be the site population median. State the following null and the alternative hypotheses: 194 ------- Form 1: H0: JUX C Form 2: H0: JUX>C vs. HA: JUX < C Two-Sided: H0: jux = C vs. HA: JUX^C Form 2 with substantial difference, S: H0: jux >C+Svs. HA: jux < C + S STEP 2: Calculate the n differences, dt = x. -C . If some of the 4=0, then reduce the sample size until all the remaining \dt\>0. This means that all observations (detects and NDs) tied at C are ignored from the computation. Compute the binomial random variable, B representing the number oft/. > 0, /': = l,2,...,n. Note that under the null hypothesis, the binomial random variable, B follows a binomial distribution (BD) ~ BD (n, 5/2) where n represents the reduced sample size after discarding observations as described above. Thus, one can use the exact BD to compute the critical values and /"-values associated with this test. STEP 3: For n < 40, ProUCL computes the exact BD based test statistic, B; and For n > 40, one may use the approximate normal test statistic given by, B-n-S (6.3) The substantial difference, S =0, except for Form 2 hypotheses with substantial difference. STEP 4: For n < 40, one can use the BD table as given in EPA (2006). These critical values are automatically computed by ProUCL) to calculate the critical values. For n > 40, use the normal approximation and the associated normal z critical values. STEP 5: Conclusion when n < 40 (following EPA 2006): Form 1: If B > BUFFER (n, 2a), then reject the null hypothesis that the population median is less than the cleanup level, C. Form 2: If B < BUFFER (n, 2a), then reject the null hypothesis that the population median is more than the cleanup level. Two-Sided: If B > BUFFER (n, a) or B < BUFFER (n, a) - 1, then reject the null hypothesis that the population median is comparable to the cleanup level, C. Form 2 with substantial difference, S: If B < BUFFER (n, 2a), then reject the null hypothesis that the population median is more than the cleanup level, C + substantial difference, S. ProUCL calculates the critical values andp-values based upon the BD (n, ^2) for both small samples and large samples. Conclusion: Large Sample Approximation when n>40 195 ------- Form 1: If zo > za, then reject the null hypothesis that population median is less than the cleanup level, C. Form 2: If zo <- za, then reject the null hypothesis that the population median is greater than the cleanup level, C. Two-Sided: If zo > 7.0/2, then reject the null hypothesis that the population median is comparable to the cleanup level, C. Form 2 with substantial difference, S: If zo <- za, then reject the null hypothesis that the population median is more than the cleanup level, C + substantial difference, S. Here, za represents the critical value of a standard normal distribution (SND) such that area to the right of za under the standard normal curve is a. P-Values for One-Sample Sign Test ProUCL calculates the critical values and p-values based upon: the BD(«, Yi) for small data sets; and normal approximation for larger data sets as described above. 6.6.4 The Wilcoxon Signed Rank Test The Wilcoxon Signed Rank (WSR) test is used for evaluating the difference between the location parameter (mean or median) of a population and a fixed cleanup standard such as C, with Cs representing a location value. It can also be used to compare the medians of paired populations (e.g., placebo versus treatment). Hypotheses about parameters of paired populations require that data sets of equal sizes are collected from the two populations. 6.6.4.1 Limitations and Robustness For symmetric distributions, the WSR test appears to be more powerful than the Sign test. However, WSR test tends to yield incorrect results in the presence of many tied values. On data sets with NDs, the process to perform a WSR test is the same as that for data sets without NDs once all NDs are assigned some surrogate value. However, like the Sign test, not much guidance is available in the literature for performing WSR test on data sets consisting of ND observations. The WSR test for data sets with NDs as described in USEPA (2006) and incorporated in ProUCL requires further investigation especially when multiple DLs with NDs exceeding the detects are present in the data set. For data sets with NDs with a single DL, DL, a surrogate value of DL/2 is used for all ND values (EPA, 2006). The presence of multiple DLs makes this test less powerful. It is suggested not to use this test when multiple DLs are present with NDs exceeding the detected values. Per EPA (2006) guidance, when multiple DLs are present, then all detects and NDs less than the largest DL may be censored which tends to reduce the power of the test. In ProUCL 5.0, all NDs including the largest ND value are replaced by half of their respective reporting limit values. All detected values are used as reported. 196 ------- 6.6.4.2 Wilcoxon Signed Rank (WSR) Test in the Presence ofNondetects Following the suggestions made in the EPA guidance document (2006), ProUCL uses the following process to perform WSR test in the presence of NDs. • For left-censored data sets with a single DL (it is preferred to have all detects greater than the NDs), it is suggested (EPA, 2006) to replace all NDs by DL/2. This suggestion (EPA, 2006) has been used in the WSR test as incorporated in ProUCL software. Specifically, if there are k ND values with the same DL, then they are considered as "ties" and are assigned the average rank for this group. • The presence of multiple DLs makes this test less powerful. When multiple DLs are present, then all NDs are replaced by half of their respective DLs. All detects are used as reported. 6.6.4.3 Directions for the Wilcoxon Signed Rank Test Let xh x2, . . . , xn represent a random sample of size, n collected from a site area under investigation, and C represent the cleanup level. STEP 1: State/select one of the following null hypotheses: Form 1: Ho: Site location < C vs. HA: Site location > C Form 2: Ho: Site location > C vs. HA: Site location < C Two-Sided: Ho: Site location = C vs. HA: Site location ^ C Form 2 with substantial difference, S: Ho: Site location >C+Svs. Ha: Site location < C + S, here S > 0. STEP 2: Calculate the deviations, dt = xt—C. If some dt =0, then reduce the sample size until all \dt\ > 0. That is, ignore all observations withe/. =0. STEP 3: Rank the absolute deviations, \dt\, from smallest to the largest. Assign an average rank to the tied observations. STEP 4: Let R be the signed rank of \dt\, where the sign of Ri is determined by the sign ofdt. STEP 5: Test statistic calculations: For n < 20, compute T+ = \, Rt , where T+ is the sum of the positive signed ranks. {iJJjX)} For n > 20, use a normal approximation and compute the test statistic given by z0 = , \ [ (6-4) Vv Here var \T+ j is the variance of T+ and is given by 197 ------- var (T+ ) = y^ t,(tt2 -1); g = number of tied groups. V / 24 48^;V; >* B P STEP 6: Conclusion when n < 20: Form 1: Larger values of the test statistic, T+, will cause the rejection of the Form 1 null hypothesis. That is if T+ > — - wa = W(i-a), then reject the null hypothesis that the location parameter is less than the cleanup level, C. Form 2: Smaller values of the test statistic will cause the rejection of the Form 2 null hypothesis. If r+ < wa, then reject the null hypothesis that the location parameter is greater than the cleanup level, C. 77(77 + 1) Two-Sided Alternative: If T+ > — — waj2 or T+ < wa!2 , then reject the null hypothesis that the location parameter is comparable to the action level, C. Form 2 with substantial difference, S: lfT+ 20: Form 1: Ifzo> za, then reject the null hypothesis that location parameter is less than the cleanup level, C. Form 2: If zo < - za, then reject the null hypothesis that the location parameter is greater than the cleanup level, C. Two-Sided: If zo > Za/2, then reject the null hypothesis that the location parameter is comparable to the cleanup level, C. Form 2 with substantial difference, S: If zo <- za, then reject the null hypothesis that the location parameter is more than the cleanup level, C + the substantial difference, S. It should be noted that WSR can be used to compare medians (means when data are symmetric) of two correlated (paired) data sets. 198 ------- Notes: The critical values, wa as tabulated in EPA (2006b) have been programmed in ProUCL. For smaller data sets with n < 20 the/?-values are computed using the BD; and for larger data sets with n > 20 the normal approximation is used to compute the critical values and/"-values. Example 6-1: Consider the aluminum and thallium concentrations of the real data set used in Example 2- 4 of Chapter 2. Please note that the aluminum data set follows a normal distribution and the thallium data set does not follow a discernible distribution. One-sample t-test (Form 2), Proportion test (2-sided) and WRS test (Form 1) results are shown below. Single-sample t-Test, Ho.- Aluminum Mean Concentration > 10000 Date/Time of Computation From File Full Precision Confidence Coefficient Substantial Difference .Action Level 3/9/2-013 8:46:40 AM SuperFund^ds OFF 95% Q.OQQ 10QQQ.QOQ Selected Null Hypothesis Mean ?= Action Level (Form 2) Aftemative Hypothesis Mean < the Action Level Aluminum One Sample t-Test HO: Sample Mean >= 10000 (Form 2) Raw Statistics Number of Valid Observations 24 Number of Distinct Observations 24 Minimum 1710 Maximum 16200 Mean 7789 Median 7012 SD 4264 SE of Mean 870.4 Test Value -2.54 Degrees of Freedom 23 Critical Value (0.05) -1.714 P-Value 0.00915 Conclusion with Alpha = 0.05 Reject HO. Conclude Mean < 10000 P-Value < Alpha (0.05) Conclusion: Reject the null hypothesis and conclude that mean aluminum concentration <10000. 199 ------- Single-sample Proportion Test Ho'. Proportion, P, of exceedances by thallium values exceeding the action level of 0.2 is equal to 0.1, vs. HA'. Proportion of exceedances is not equal to 0.1. Confidence Coefficient 95% User Specified Proportion 0.100 (PQ of Exceedances of Action Level) Action/compliance Limit 0.200 Select Null Hypothesis Sample Proportion, P of Exceedances of Action Level = User Specified Proportion (2 Sided Alternative) Alternative Hypothesis Sample Proportion, P of Exceedances of Action Level o User Specified Proportion Thallium One Sample Proportion Test Raw Statistics Number of Valid Observations 24 Number of Distinct Observations 18 Minimum 0.0€6 Maximum 0.456 Mean 0.147 Median 0.07 SD 0.133 SE of Mean 0.0271 Number of Exceedances 6 Sample Proportion of Exceedances 0.25 HO: Sample Proportion = 0.1 .Approximate P-Value 0.0349 Conclusion with Alpha = 0.05 Reject HO. Conclude Sample Proportion <> 0.1 Conclusion: Proportion of thallium concentrations exceeding 0.2 is not equal to 0.1. 200 ------- Single-sample WRS Test Ho'. Median of thallium concentrations <0.2 Confidence Coefficient Substantial Difference Action Level 95% 0.006 0.200 Selected Null Hypothesis Mean/Median <= Action Level (Form 1} .Alternative Hypothesis Mean/Median > the Action Level Thallium One Sample Wilcoxon Signed Rank Test Raw Statistics Number of Valid Observations 24 Number of Distinct Observations 18 Minimum 0.066 Maximum 0.456 Mean 0.147 Median 0.07 SD 0.133 SE of Mean 0.0271 Number Above Action Level 6 Number Equal Action Level 0 Number Below Action Level 18 T-plus 93 T-minus 207 HO: Sample Mean/Median <= 0.2 (Form 1) Large Sample z-Test Statistic -1.644 Critical Value (0.05) 1.645 P-Value 0.95 Conclusion with Alpha = 0.05 Do Not Reject HO. Conclude Mean/Median <= 0.2 P-Value > Alpha (0.05) Conclusion: Do not reject the null hypothesis and conclude that median of thallium concentrations < 0.2. Example 6-2: Consider the blood lead-levels data set discussed in the environmental literature (Helsel, 2013). The data set consists of several NDs. The box plot shown in Figure 6-1 suggests that median of lead concentrations is less than the action level. The WSR tests the null hypothesis: Median lead concentrations in blood > action level of 0.1 Box PloS for Blood Pb Figure 6-1. Box Plot of Lead in Blood Data Comparing Pb Concentrations with the Action Level of 0.1 201 ------- Hood_Pb One Sample Wilcoxon Signed Rank Test Raw Statistics Number of Valid Data 27 Number of Distinct Data 13 Number of Non-Detects 19 Number of Detects 8 Percent Non-Detects 70.37% Minimum Non-detect 0.0137 Maximum Non-detect O.Q2 Minimum Detect 0.0235 Maximum Detect 0.2S3 Mean of Detects 0.107 HO: Sample Median >= 0.1 (Form 2) Median of Detects O.Q776 SD of Detects 0.0911 , - , T . -,.... ,-~ Large Sample z-Test Statistic -3.66/ Median of Processed Data used in WSR 0.01 Critical Value {0.05} -1.645 Number Above .Action Level 4 P-Value 12291E-4 Number Equal .Action Level 0 Number Below .Action Level 23 Conclusion with Alpha = 0.05 T-plus 39 Reject HO. Conclude Mean/Median < 0.1 T-minus 339 P-Value < Alpha (0.05) Conclusion: Both the graphical display and the WSR test suggest that median of lead concentrations in blood is less than 0.1. 6.7 Two-sample Hypotheses Testing Approaches The use of parametric and nonparametric two-sample hypotheses testing approaches is quite common in environmental applications including site versus background comparison studies. Several of those approaches for data sets with and without ND observations have been incorporated in the ProUCL software. Additionally some graphical methods (box plots and Q-Q plots) for data sets with and without NDs are also available in ProUCL to visually compare two or more populations. Student's two-sample t-test is used to compare the means of the two independently distributed normal populations such as the potentially impacted site area and a background reference area. Two cases arise: 1) the variances (dispersion) of the two populations are comparable, and 2) the variances of the two populations are not comparable. Generally, a t-test is robust and not sensitive to minor deviations from the assumptions of normality. 6.7.1 Student's Two-sample t-Test (Equal Variances) 6.7.1.1 Assumptions and their Verification Xi, X2, ..., Xn represent site samples and Yi, ¥2, ... , Ym represent background samples that are collected at random from the two independent populations. The validity of random samples and independence assumptions may be confirmed by reviewing the procedures described in EPA (2006b). Let X and Y represent the sample means of the two data sets. Using the GOF tests (available in ProUCL 5.0 under 202 ------- Statistical Tests Module), one needs to verify that the two data sets are normally distributed. If both m and n are large (and the data are mildly to moderately skewed), one may make this assumption without further verification (due to the CLT). If the data sets are highly skewed (skewness discussed in Chapters 3 and 4), the use of nonparametric tests such as the WMW test supplemented with graphical displays is preferable. 6.7.1.2 Limitations and Robustness The two-sample t-test with equal variances is fairly robust to violations of the assumption of normality. However, if the investigator has tested and rejected normality or equality of variances, then nonparametric procedures such as the WMW may be applied. This test is not robust to outliers because sample means and standard deviations are sensitive to outliers. It is suggested that a t-test not be used on log- transformed data sets as a t-test on log -transformed data tests the equality of medians and not the equality of means. For skewed distributions there are significant differences between mean and median. The Student's t- test assumes the equality of variances of the two populations under comparison; if the two variances are not equal and the normality assumption of the means is valid, then the Satterthwaite's t-test (described below) can be used. In the presence of NDs, it is suggested to use a Gehan test or T-W (new in ProUCL 5.0) test. Sometimes, users tend to use a t-test on data sets obtained by replacing all NDs by surrogate values, such as respective DL/2 values, or DL values. The use of such methods can yield incorrect results and conclusions. The use of substitution methods (e.g., DL/2) should be avoided. 6. 7. 1. 3 Guidance on Implementing the Student 's Two-sample t-Test The number of site (Population 1), n and background (Population 2), m measurements required to conduct the two-sample t-test should be calculated based upon appropriate DQO procedures (EPA [2006a, 2006b]). In case, it is not possible to use DQOs, or to collect as many samples as determined using DQOs, one may want to follow the minimum sample size requirements as described in Chapter 1. The Stats/Sample Sizes module of ProUCL can be used to determine DQOs based sample sizes. ProUCL also has an F-test to verify the equality of two variances. ProUCL automatically performs this test to verify the equality of two dispersions. The user should review the output for the equality of variances test conclusions before using one of the two tests: Student's t-test or Satterthwaite's t-test. If some measurements appear to be unusually large compared to the majority of the measurements in the data set, then a test for outliers (Chapter 7) should be conducted. Once any identified outliers have been investigated to determine if they are mistakes or errors and, if necessary, discarded, the site and background data sets should be re-tested for normality using formal GOF tests and normal Q-Q plots. The project team should decide the proper disposition of outliers. In practice, it is advantageous to carry out the tests on data sets with and without the outliers. This extra step helps the users to assess and determine the influence of outliers on the various test statistics and the resulting conclusions. This process also helps the users in making appropriate decisions about the proper disposition (include or exclude from the data analyses) of outliers. 6. 7. 1. 4 Directions for the Student 's Two-sample t-Test X2, . . . , Xn represent a random sample collected from a site area (Population 1) and Yi, Y2, . . . , Ym represent a random data set collected from another independent population such as a background population. The two data sets are assumed to be normally distributed or mildly skewed. 203 ------- STEP 1: State the following null and the alternative hypotheses: Form 1: H0: /Jx -/JY < 0 vs. HA: /Jx -/JY > 0 Form 2: H0: fJx —fJY > 0 vs. HA: fJx — fJY < 0 Two-Sided: H0: fJx — fJY = 0 vs. HA: fJx —fJY ^ 0 Form 2 with substantial difference, S: H0: /Jx -/^ > S vs. HA: jUx -fa < S STEP 2: Calculate the sample mean X and the sample variance Sx for the site (e.g., Population 1, Sample 1) data and compute the sample mean Y and the sample variance Sy for the background data (e.g., Population 2, Sample 2). STEP 3: Determine if the variances of the two populations are equal. If the variances of the two populations are not equal, use the Satterthwaite's test. Calculate the pooled sd, Sp and the t-test statistic, to. (6-5) (x-T)-s to = - - , ' (6-6) Here S = 0, except when used in Form 2 hypothesis with substantial difference, S > 0. STEP 4: Compute the critical value tm+n-2,i-a such that 100(1 - a) % of the t-distribution with (m + n - 2) dfis below tm+n.2,i-a. STEP 5: Conclusion: Form 1: Ifto> tm+n-2, i-a, then reject the null hypothesis that the site population mean is less than or equal (comparable) to the background population mean. Form 2: If to < -tm+n-2, i-a, then reject the null hypothesis that the site population mean is greater than or equal to the background population mean. Two-Sided: If to > tm+n-2,1-0/2, then reject the null hypothesis that the site population mean comparable to the background population mean. Form 2 with substantial difference, S: If to <- tm+n-2, i- a, then reject the null hypothesis that the site mean is greater than or equal to the background population mean + the substantial difference, S. 6.7.2 The Satterthwaite Two-sample t-Test (Unequal Variances) Satterthwaite's t-test is used to compare two population means when the variances of the two populations are not equal. It requires the same assumptions as the two-sample t-test (described above) except for the assumption of equal variances. 204 ------- 6.7.2.1 Limitations and Robustness In the presence of NDs, replacement by a surrogate value such as the DL or DL/2gives biased results. As mentioned above, the use of these substitution methods should be avoided. Instead the use of nonparametric tests such as the Gehan test or Tarone-Ware test is suggested when the data sets consist of NDs. In cases where the assumptions of normality of means are violated, the use of nonparametric tests such as the WMW test is preferred. 6.7.2.2 Directions for the Satterthwaite Two-sample t-Test LetXi, X2, . . . , Xn represent random site (Population 1) samples and Yi, Y2, . . . , Ym represent random background (Population 2) samples collected from two independent populations. STEP 1: State the following null and the alternative hypotheses: Form 1: H0: jUx - ]UY < 0 vs. HA: jUx - ]UY > 0 Form 2: H0: jux - JUY > 0 vs. HA: jUx - jUY < 0 Two-Sided: Ho: fj.x — fJ.Y = 0 vs. HA: £ix —//7 ^ 0 Form 2 with substantial difference, S: Ho: jUx — jL^^Svs. HA: jUx — Jur 0. STEP 4: Use a t-table (ProUCL computes them) to find the critical value ti-a such that 100(1 - a)% of the t-distribution with df degrees of freedom is below ti-a, where the Satterthwaite's Approximation for dfis given by: df = SX | SY n m (6-8) n (n-V) m (m-V) STEP 5: Conclusion: Form 1: If to > % i-a, then reject the null hypothesis that the site (Population 1) mean is less than or equal (comparable) to the background (Population 2) mean. 205 ------- Form 2: \£to< -% i-a, then reject the null hypothesis that the site (Population 1) mean is greater than or equal to the background (Population 2) mean. Two-Sided: If to > % 1-0/2, then reject the null hypothesis that the site (Population 1) mean is comparable to the background (Population 2) mean. Form 2 with substantial difference, S: If to < -% ;- a, then reject the null hypothesis that the site (Population 1) mean is greater than or equal to the background (Population 2) mean + the substantial difference, S. P-Values for Two-sample t-Test A /rvalue is the smallest value for which the null hypothesis is rejected in favor of the alternative hypotheses. Thus, based upon the given data, the null hypothesis is rejected for all values of a (the level of significance) greater than or equal to the /"-value. ProUCL computes (based upon an appropriate t- distribution) /^-values for two-sample t-tests associated with each form of the null hypothesis. If the computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis based upon the collected data set used in the various computations. 6.8 Tests for Equality of Dispersions This section describes a test that verifies the assumption of the equality of two variances. This assumption is needed to perform a simple two-sample Student's t-test described above. 6.8.1 The F- Test for the Equality of Tw o- Variances An F-test is used to verify whether the variances of two populations are equal. Usually the F-test is employed as a preliminary test, before conducting the two-sample t-test for the equality of two means. The assumptions underlying the F-test are that the two-samples represent independent random samples from two normal populations. The F-test for equality of variances is sensitive to departures from normality. There are other statistical tests such as the Levene's test (1960) which also tests the equality of the variances of two normally distributed populations. However, the inclusion of the Levene test will not add any new capability to the software. Therefore, taking the budget constraints into consideration, the Levene's test has not been incorporated in the ProUCL software. Moreover, it should be noted that, although it makes sense to first determine if the two variances are equal or unequal, this is not a requirement to perform a t-test. The t-distribution based confidence interval or test for Hi - fi2 based on the pooled sample variance does not perform better than the approximate confidence intervals based upon Satterthwaite's test. Hence testing for the equality of variances is not required to perform a two-sample t-test. The use of Welch-Satterthwaite's or Cochran's method is recommended in all situations (see, for example, F. Hayes [2005]). 6.8.1.1 Directions for the F-Test Let Xl5 X2, . . . , Xn represent the n data points from site (Population 1) and Y1; Y2, . . . , Ym represent the m data points from background (Population 2). To manually perform an F-test, one can proceed as follows: STEP 1: Calculate the sample variances S% (for the X's) and Sy (for the Y's) 206 ------- 22 22 STEP 2: Calculate the variance ratios Fx= SX/SY and FY= SY fsx • Let F equal the larger of these two values. If F = Fx, then let k = n - 1 and q = m - 1. If F = Fy, then let k = m - 1 and q = n- 1. STEP 3: Using a table of the F- distribution (ProUCL 5.0 computes them), find a cutoff, U =f1_a/2(k, q) associated with the F distribution with k and q degrees of freedom for some significance level, a. If the calculated F value > U, conclude that the variances of the two populations are not equal. P-Values for Two-sample Dispersion Test for Equality of Variances ProUCL computes /"-values for the two-sample F-test based upon an appropriate F-distribution. If the computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis based upon the collected data sets. Example 6-3: Consider a real manganese data set collected from an upgradient well (Well 1) and two downgradient MWs (Wells 2 and 3). The side-by-side box plots comparing concentrations of the three wells are shown in Figure 6-2. The two-sample t-test comparing the manganese concentrations of the two downgradient MWs are summarized in Table 6-2. Box Plot for Mn Figure 6-2. Box Plots Comparing Concentrations of Three Wells: One Upgradient and Two Downgradient 207 ------- Table 6-2. T-Test Comparing Mn in MW8 vs. MW9 HQ: Mean Mn concentrations of MW 8 and MW9 are comparable Selected Null Hypothesis Sample 1 Mean = Sample 2 Mean (Two Sided Aftemative) Alternative Hypothesis Sample 1 Mean <> Sample 2 Mean Sample 1 Data: Mn-89{8) Sample 2 Data: Mn-89(9) Raw Statistics Number of Valid Observations Number of Distinct Observations Minimum Maximum Mean Median SD SE of Mean Sample 1 vs Sample 2 Two-Sample t-Test Sample 1 16 16 1270 4600 1998 1750 B3S.S 209.7 Sample 2 16 15 1050 3080 1968 2055 500.2 125 HO: Mean of Sample 1 = Mean of Sample 2 t-Test Value 0.123 0.123 Method DF Pooled (Equal Variance) 30 Welch-Satterthwaite (Unequal Variam 24.5 Pooled SD: 690.548 Conclusion with Alpha = D.G5C Student t [Pooled): Do Not Reject HO. Conclude Sample 1 = Sample 2 Welch-Satterthwalte: Do Not Reject H2. Conclude Sample 1 = Sample 2 Lower C.Val Upper C.V'al t (0.025) t (0.975} P-Value -2.042 2.042 0.903 -2.0€4 2.064 0.903 Test of Equality of Variances Variance of Sample 1 703523 Variance of Sample 2 250190 Numerator DF Denominator DF 15 15 Conclusion with .Alpha = 0.05 Two variances appear to be equal F-Test Value 2.812 P-Value 0,054 Conclusion: The variances of the two populations are comparable, both the t-test and Satterthwaite test lead to the conclusion that there are no significant differences in the mean manganese concentrations of the two downgradient monitoring wells. 208 ------- 6.9 Nonparametric Tests When the data do not follow a discernible distribution, the use of parametric statistical tests may lead to inaccurate conclusions. Additionally, if the data sets contain outliers or ND values, an additional level of uncertainty is faced when conducting parametric tests. Since most environmental data sets tend to consist of observations from two or more populations including some outliers and ND values, it is unlikely that the current wide-spread use of parametric tests is justified, given that these tests may be adversely affected by outliers and by the assumptions made for handling ND values. Several nonparametric tests have been incorporated in ProUCL that can be used on data sets consisting of ND observations with single and multiple DLs. 6.9.1 The Wilcoxon-Mann-Whitney (WMW) Test The Mann-Whitney (M-W) (or WMW) test (Bain and Engelhardt, 1992) is a nonparametric test used for determining whether a difference exists between the site and the background population distributions. This test is also known as the WRS test. The WMW test statistic tests whether or not measurements (location, central) from one population consistently tend to be larger (or smaller) than those from the other population based upon the assumption that the dispersion/shapes of the two distributions are roughly the same (comparable). 6.9.1.1 Advantages and Disadvantages The main advantage of the WMW test is that the two data sets are not required to be from a known type of distribution. The WMW test does not assume that the data are normally distributed, although a normal distribution approximation is used to determine the critical value of the WMW test statistic for large sample sizes. The WMW test may be used on data sets with NDs provided the DL or the reporting limit (RL) is the same for all NDs. If NDs with multiple DLs are present, then the largest DL is used for all ND observations. Specifically, the WMW test handles ND values by treating them as ties. Due to these constraints, other tests such as the Gehan test and theTarone-Ware test are better suited to perform two- sample tests on data sets consisting of NDs. The WMW test is more resistant to outliers than two-sample t-tests discussed earlier. It should be noted that the WMW test does not place enough weight on the larger site and background measurements. This means, a WMW may lead to the conclusion that two populations are comparable even when the observations in the right tail of one distribution (e.g., site) are significantly larger than the right tail observations of the other population (e.g., background). Like all other tests, it is suggested that the WMW test results be supplemented with graphical displays. 6.9.1.2 WMW Test in the Presence ofNondetects If there are t ND values with a single DL, then they are considered as "ties" and are assigned the average rank for this group. If more than one DL is present in the data set, then WMW test censors all of the observations below the largest DL, and are treated as NDs at the largest DL. This of course results in loss of power associated with WMW test. 209 ------- 6. 9. 1. 3 WMW Test Assumptions and Their Verification The underlying assumptions of the WMW test are: • The soil sample measurements obtained from the site and background areas are statistically and spatially independent (not correlated). This assumption requires: 1) that an appropriate probability- based sampling design strategy be used to determine (identify) the sampling locations of the soil samples for collection, and 2) those soil sampling locations are spaced far enough apart that a spatial correlation among concentrations at different locations is not likely to be present. The probability distribution of the measurements from a site area (Population 1) is similar to (e.g., including variability, shape) the probability distribution of measurements collected from a background or reference area (Population 2). The assumption of equal variances of the two regions: site vs. background should also be evaluated using descriptive statistics and graphical displays such as side-by-side box plots. The WMW test may result in an incorrect conclusion if the assumption of equality of variability is not met. 6.9.1.4 Directions for the WMW Test when the Number of Site and Background Measurements is small (n < 20 or m <20) i, X-2, . . . ,Xn represent systematic and random site samples (Group 1, Sample 1) and Yi, ¥2, . . . , Ym represent systematic and random background samples (Group 2, Sample 2) collected from two independent populations. It should be noted that instead of 20, some texts suggest to use 10 as a small sample size for the two populations. STEP 1: Let fj.x represent site (Population 1) median and JUY represent the background (Population 2) median. State the following null and the alternative hypotheses: Form I: Ho. jux- JUY <0 vs.HA: JUX-JUY >0 Form 2: H0: JUX-JUY > 0 vs. HA: JUX-JUY < 0 Two-Sided: H0: jux- JUY = 0 vs. HA: jux- JUY ^0 Form 2 with substantial difference, S: H0: JUX-JUY >S vs. HA: JUX-JUY ------- • If a few less-than values (NDs) occur (say, < 10%), and if all such values are less than the smallest detected measurement in the pooled data set, then treat all NDs as tied values at the reported DL or at an arbitrary (when no DL is reported) value less than the smallest detected measurement. Assign the average of the ranks that would otherwise be assigned to these tied less- than values (the same procedure as for tied detected measurements). Today with the availability of advanced technologies and instruments, instead of reporting NDs as less-than values, NDs are typically reported at DL levels below which the instrument cannot accurately measure the concentrations present in a sample. The use of DLs is particularly helpful when NDs are reported with multiple DLs (RLs). • If between 10% and 40% of the pooled data set are reported as NDs, and all are less than the smallest detected measurement, then one may use the approximate WMW test procedure described below provided enough (e.g., n > 10 and m > 10) data are available. However, the use of the WMW test is not recommended in the presence of multiple DLs or RLs with NDs larger than the detected values. STEP 3: Calculate the sum of the ranks of the n site measurements. Denote this sum by Ws and then calculate the Mann-Whitney (M-W), ^/-statistic as follows: U = Ws-n(n + l)/2 (6-9) The test proposed by Wilcoxon based upon the rank sum, Ws is called the WRS test. The test based upon the f/-statistic given by (6-9) was proposed by Mann and Whitney and is called the WMW test. These two tests are equivalent tests and yield the same results and conclusions. ProUCL outputs both statistics; however the conclusions are derived based upon the U-statistic and its critical and p-values. Mean and variance of the U-statistic are given as follows: Notes: Note the difference between the definitions of U and Ws. Obviously the critical values for Ws and U are different. However, critical values for one test can be derived from the critical values of the other test by using the relationship given by the above equation (6-9). These two tests (WRS test and WMW test) are equivalent tests, and the conclusions derived by using these test statistics are equivalent. For data sets of small sizes (with m or n <20), ProUCL computes exact as well as normal distribution based approximate critical values. For large samples with n and m both greater than 20, ProUCL computes normal distribution based approximate critical values and/>-values. STEP 4: For specific values of n, m, and a, find an appropriate WMW critical value, wa, from the table as given in EPA (2006) and also in Daniel (1995). These critical values have been incorporated in the ProUCL software. STEP 5: Conclusion: Form 1 : lfU>nm-wa, then reject the null hypothesis that the site population median is less than or equal to the background population median. 211 ------- Form 2: If U< wa, then reject the null hypothesis that the site population median is greater than or equal to the background population median. Two-Sided: If U > nm - Wa/2 or U< Wa/2, then reject the null hypothesis that the site population median (location) is comparable to that of the background population median (location). Form 2 with substantial difference, S: If U< wa, then reject the null hypothesis that the site population median is greater than or equal to the background population median + the substantial difference, S. S takes a positive value only for this form of the hypothesis with substantial difference, in all other forms of the null hypothesis, S = 0. P-Values for Two-sample WMW Test for Small Samples For small samples, ProUCL computes only approximate (as computed for large samples) />-values for the WMW test. Details of computing approximate /"-values are given in the next section for larger data sets. If the computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis based upon the collected data set. 6.9.1.5 Directions for the WMW Test when the Number of Site and Background Measurements is Large (n > 20 andm > 20) It should be noted that some texts suggest that both n and m needs to be >10 to be able to use the large sample approximation. ProUCL uses large sample approximations when «>20 and m>20. STEP 1: As before, let fj.x represent the site and JUY represent the background population medians (means). State the following null and the alternative hypotheses: Form 1: H0: JUX-JUY < 0 vs. HI: jux-juY > 0 Form 2: H0: jux- JUY > 0 vs. HI: jux- JUY < 0 Two-Sided: H0: JUX-JUY = 0 vs. Hi: jux-juY ^0 Form 2 with substantial difference, S: Ho: jux — jUY > S vs. jux — JUY < S Note that when the Form 2 hypothesis is used with substantial difference, S, the value S is added to all observations in the background data set before ranking the combined data set of size (n+rri). For data sets with NDs, the Form 2 hypothesis test with substantial difference, S is not incorporated in ProUCL 5.0. STEP 2: List and rank the pooled set of n + m site and background measurements from smallest to largest, keeping track of which measurements came from the site and which came from the background area. Assign the rank of 1 to the smallest value among the pooled data, the rank of 2 to the second smallest value among the pooled data, and so forth. All observations tied at a give value, x0, are assigned the average rank of the observations tied at x0. The same process is used for all tied values. • The WMW test is not recommended when many NDs observations with multiple DLs and /or NDs exceeding the detected values are present in the data sets. Other tests such as the T-W and Gehan tests also available in ProUCL 5.0 are better suited for data sets consisting of many NDs with multiple DLs and/or NDs exceeding detected values. 212 ------- • It should however be noted these nonparametric tests (WMW test, Gehan test, and T-W test) assume that the shape (variability) of the two data distributions (e.g., background and site) are comparable. If this assumption is not met, these tests may lead to incorrect test statistics and conclusions. STEP 3: Calculate the sum of the ranks of the site (Population 1) measurements. Denote this sum by Ws. ProUCL 5.1 computes the WMW test statistics by adjusting for tied observations using equation (6-11); that is the large sample variance of the WMW test statistic is computed using equation (6-11) which adjusts forties. STEP 4: When no ties are present, calculate the approximate WMW test statistic, Zo as follows: ( nm(n + 12 The above test statistic, Z0 is equivalent to the following approximate Z0 statistic based upon the Mann- Whitney f/-statistic: U-nm/2 nm (n + m + i) .12 When ties are present in the combined data set of size (n+rri), the adjusted large sample approximate test value, Z0 is computed by using the following equation: = (nm 10 andn > 10 Let Xi, X2, . . . , Xn represent data points from the site population and Yi, ¥2, . . . , Ym represent background data from the background population. Like the WMW test, this test also assumes that the variabilities of the two distributions (e.g., background vs. Site, MW1 vs. MW2) are comparable. Since we are dealing with data sets consisting of many NDs, the use of graphical methods such as the side-by-side box plots and multiple Q-Q plots is also desirable to compare the spread/variability of the two data distributions. For data sets of sizes larger than 10 (recommended), a test based upon normal approximations is described in the following. STEP 1: Let jux represent the site and JUY represent the background population medians. State the following null and the alternative hypotheses: Form 1: H0: jux-juY < 0 vs. HA: JUX-JUY > 0 216 ------- Form 2: H0: jux- JUT > 0 vs. HA: JUX-JUT < 0 Two-Sided: H0: jux- JUY = 0 vs. HA: jux- JUY ^0 For data sets with NDs, the Form 2 hypothesis test with substantial difference, S is not incorporated in ProUCL 5.0/5.1. The user may want to adjust their background data sets accordingly to perform this hypothesis test form. STEP 2: List the combined m background and n site measurements, including the ND values, from smallest to largest, where the total number of combined samples is N = m + n. The DLs associated with the ND (or less-than values) observations are used when listing the TV data values from smallest to largest. STEP 3: Determine the N ranks, Ri, R2, ... , Rn, for the N ordered data values using the method described in the example given below. STEP 4: Compute the N scores, a(Ri), a(R2), ... , a(Rn), using the formula a(Ri) = 2R, - N - 7, where / is successively set equal to 7, 2, ... , N. STEP 5: Compute the Gehan statistic, G, as follows: G = (6-12) mn Where s ' or u=° /z; = 7 if the ith datum is from the site population hi = 0 if the ith datum is from the background population N = n + m a(Rt) = 2 R,•- N-l, as indicated above. STEP 6: Use the normal z-table to get the critical values. STEP 7: Conclusion based upon the approximate normal distribution of the G-statistic: Form 1: If G > z;_a, then reject the null hypothesis that the site population median is less than or equal to the background population median. Form 2: If G <- z;_«, then reject the null hypothesis that the site population median is greater than or equal to the background population median. Two-Sided: If |G| >z;_o/2, then reject the null hypothesis that the site population median is same as the background population median. 217 ------- P-Values for Two-sample Gehan Test For the Gehan's test, p-values are computed using a normal approximation for the Gehan's G-statistic. The /"-values can be computed using the simple procedure as used for computing large sample /"-values for the two-sample nonparametric WMW test. ProUCL computes p-values for the Gehan test for each form of the null hypothesis. If the computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis based upon the collected data set used in the various computations. 6.9.3 Tarone-Ware (T-W) Test Like the Gehan test, the T-W test (1978) is a nonparametric test which can be used to test for the differences between the distributions of two populations (e.g., two sites, site versus background, two monitoring wells) when the data sets have multiple censoring points and DLs. The T-W test as described below has been incorporated in ProUCL 5.0 and 5.1. It is noted that the Gehan and T-W tests yield comparable test results. 6.9.3.1 Limitations and Robustness The T-W test can be used when the background and/or site data sets contain multiple NDs with different DLs and NDs exceeding detected values. • If the censoring mechanisms are different for the site and background data sets, then the test results may be an indication of this difference in censoring mechanisms (e.g., high DLs due to dilution effects) rather than an indication that the null hypothesis is rejected. • Like the Gehan test, the T-W test can be used when many ND observations or multiple DLs may be present in the two data sets; conclusions derived using this test may not be reliable when dealing with samples of small sizes (<10). Like the Gehan test, the T-W test described below is based upon the normal approximation of the T-W statistic and should be used when enough (e.g., m>10 and n>10) site and background (or monitoring well) data are available. 6.9.3.2 Directions for the Tarone-Ware Test when m > 10 andn > 10 Let Xi, X2, . . . , Xn represent n data points from the site population and Yi, Y2, . . . , Ym represent sample data from the background population. Like the Gehan test, this test also assumes that the variabilities of the two data distributions (e.g., background vs. site, monitoring wells) are comparable. One may use exploratory graphical methods to informally verify this assumption. Graphical displays are not affected by NDs and outlying observations. STEP 1: Let jux represent the site and //7 represent the background population medians. The following null and alternative hypotheses can be tested: Form 1: H0: jux- JUY < 0 vs. HA: jux- JUY > 0 Form 2: H0: jux-juY > 0 vs. HA: JUX-JUY < 0 Two-Sided: H0: jux- JUY = 0 vs. HA: jux- JUY ^0 218 ------- STEP 2: Let TV denote the number of distinct detected values in the combined background and site data set of size (n+rri) including the ND values. Arrange the N distinct detected measurements in the combined data set in ascending order from smallest to largest. Note that N will be less than n+m. Let Zj < z2 < z3 <...< ZN represent N distinct ordered detected values in the data set of size, (n+m). STEP 3: Determine the N ranks, Ri, R2, ..., RN, for the N ordered distinct detected data values: Zj < z2 < z3 <... < ZN in the combined data set of size (n+rri). STEP 4: Count the number, «,, 1=1,2, ..., TV of detects and NDs (reported as DLs or reporting limits) less than or equal to z; in the combined data set of size (n+rri). For each distinct detected value, z, compute ct = number of detects exactly equal to z*; 1=1,2,... .N STEP 5: Repeat Step 4 on the site data set. That is count the number, mt ,1=1,2,... .N of detects and NDs (reported as DLs or reporting limits) less than or equal to z* in site data set of size, («). Also, for each distinct detected value, z,, compute dt = number of detects in the site data set exactly equal to zt; i=l,2, ....N. Finally, compute, li,1=1,2, ....N, the number of detects and NDs (reported as DLs or reporting limits) less than or equal to z* in background data set of size (rri). STEP 6: Compute the expected value and variance of detected values in the site data set of size, n, using the following equations: ESite(Detectioii) = ci *mt /nt (6-13) VSlte (Detection) = c, * (n, - c, )m,l, /(n? (n, -1)) (6-14) STEP 7: Compute the normal approximation of the TW test statistic using the following equation: T-W= l=l , (6-15) STEP 8: Conclusion based upon the approximate normal distribution of the T-W statistic: Form 1 : If T-W> z;_«, then reject the null hypothesis that the site population median is less than or equal to the background population median. Form 2: If T-W<- zi.a, then reject the null hypothesis that the site population median is greater than or equal to the background population median. Two-Sided: If \T-W\ >z 1.0/2, then reject the null hypothesis that the site population median is same as the background population median. 219 ------- P-Values for Two-sample T-W Test Critical values and p-values for the T-W test are computed following the same procedure as used for the Gehan test. ProUCL computes normal distribution based approximate critical values and /"-values for the T-W test for each form of the null hypothesis. If the computed p-value is smaller than the specified value of, a, the conclusion is to reject the null hypothesis based upon the data set used in the computations. Example 6-5. The copper (Cu) and zinc (Zn) concentrations data with NDs (from Millard and Deverel 1988) collected from groundwater of the two zones, Alluvial Fan and Basin Trough, is used to perform the Gehan and T-W tests using ProUCL 5.0. Box plots comparing Cu in the two zones are shown in Figure 6-3 and box plots comparing Zn concentrations in the two zones are shown in Figure 6-4. Box Plot for Cu Figure 6-3. Box plots Comparing Cu in Two Zones: Alluvial Fan versus Basin Trough Box Plot for Zn Figure 6-4. Box Plots Comparing Zn in Two Zones: Alluvial Fan versus Basin Trough 220 ------- Table 6-4. Gehan Test Comparing the Location Parameters of Copper (Cu) in Two Zones H0: Cu concentrations in two zones. Alluvial Fan and Basin Trough, are comparable Selected Null Hypothesis Sample 1 Mean/Median = Sample 2 Mean/Median (Two Sided Alternative} Alternative Hypothesis Sample 1 Mean/Median <> Sample 2 Mean/Median Sample 1 Data: Cu (alluvial fan) Sample 2 Data: Cu (basin trough) Raw Statistics Number of Valid Data Number of Missing Observations Number of Non-Detects Number of Detect Data Minimum Non-Detect Maximum Non-Detect Percent Non-detects Minimum Detect Maximum Detect Mean of Detects Median of Detects SD of Detects Sample 1 65 3 17 48 1 20 26.15% 1 20 4.14S 2 4.005 Sample 2 43 1 14 35 1 15 28.57% 1 23 5.229 3 5.214 Sample I vs Sample 2 Gehan Test HO: Mean of Sample 1 = Mean erf background Gehan 2 Test Value -1.372 Lower Critical z {0.025} -1.96 Upper Critical z (0.975) 1.96 P-Value 0.17 Conclusion with jftJpha = 0.05 Do Not Reject HO, Cooduefe Sample 1 = Sample 2 P-Value >= alpha (0.05) Conclusion: Based upon the box plots shown in Figure 6-3 and the Gehan test summarized in Table 6-4, the null hypothesis is not rejected, and it is concluded that the mean/median Cu concentrations in groundwater from the two zones are comparable. 221 ------- Table 6-5. Tarone-Ware Comparing Location Parameters of Zinc Concentrations H0: Zn concentrations in groundwaters of Alluvial Fan = groundwaters of Basin Trough Selected Null Hypothesis Sample 1 Mean/Median = Sample 2 Mean/Median (Two Sided Alternative) Alternative Hypothesis Sample 1 Mean/Median <> Sample 2 Mean/Median Sample 1 Data: Zn{alluvial fan) Sample 2 Data: Zn (basin trough) Raw Statistics Sample 1 Sample 2 Number of Valid Data 67 50 Number of Missing Observations 1 0 Number of Non-Detects 16 4 Number of Detects 51 46 Minimum Non-Detect 3 3 Maximum Non-Detect 10' 10 Percent Non-detects 23.88% 8.00°; Minimum Detect 5 3 Maximum Detect 620 90 Mean of Detects 27.88 23.13 Median of Detects 11 20 SD of Detects 85.02 19.03 Sample 1 vs Sample 2 Tanme-Ware Ted: HO: Mean/Median of Sample 1 = Mean/Median of Sample 2 TW Statistic -2.113 Lower TW Critical Value{0.025) -1.96 Upper TW Critical Value {0.975} 1.9€ P-Value Conclusion with Alpha = 0,05 Reject HO, Conclude Sample 1 <> Sample 2 P-Value < alpha (0.05) Conclusion: Based upon the box plots shown in Figure 6-4 and the T-W test results summarized in Table 6-5, the null hypothesis is rejected, and it is concluded that the Zn concentrations in groundwaters of two zones are not comparable (p-value = 0.0346). 222 ------- CHAPTER 7 Outlier Tests for Data Sets with and without Nondetect Values Due to resource constraints, it is not possible (nor needed) to sample an entire population (e.g., reference area) of interest under investigation; only parts of the population are sampled to collect a random data set representing the population of interest. Statistical methods are then used on sampled data sets to draw conclusions about the populations under investigation. In practice, a sampled data set can consist of some wrong/incorrect values, which often result from transcription errors, data-coding errors, or instrument breakdown errors. Such wrong values could be outlying (well-separated, coming from 'low' probability far tails), with respect to the rest of the data set; these outliers need to be fixed and corrected (or removed) before performing a statistical method. However, a sampled data set can also consist of some correct measurements that are extremely large or small relative to the majority of the data, and therefore those low probability extreme values are suspected of misrepresenting the main dominant background population from which they were collected. Typically, correct extreme values represent observations coming from population(s) other than the main dominant population; and such observations are called outliers with respect to the main dominant population. In practice, the boundaries of an environmental population (background) of interest may not be well- defined and the selected population actually may consist of areas (concentrations) not belonging to the main dominant population of interest (e.g., reference area). Therefore, a sampled data set may consist of outlying observations coming from population(s) not belonging to the main dominant background population of interest. Statistical tests based on parametric methods generally are more sensitive to the existence of outliers than are those based on nonparametric distribution-free methods. It is well-known (e.g., Rousseeuw and Leroy 1987; Barnett and Lewis 1994; Singh and Nocerino 1995) that the presence of outliers in a data set distorts the computations of all classical statistics (e.g., sample mean, sd, upper limits, hypotheses test statistics, GOF statistics, OLS regression estimates, covariance matrices, and also outlier test statistics themselves) of interest. Outliers also lead to both Types I and Type II errors by distorting the test statistics used for hypotheses testing. Statistics computed using a data set with outliers lack statistical power to address the objective/issue of interest (e.g., use of a BTV to identify contaminated locations). The use of such distorted statistics (e.g., two-sample tests, UCL95, UTL95-95) may lead to incorrect cleanup decisions which may not be cost-effective or protective of human health and the environment. A distorted estimate (e.g., UCL95) computed by accommodating a few low probability outliers (coming from far tails) tends to represent the population area represented by those outliers and not the main dominant population of interest. It is also well-known that classical outlier tests such as the Rosner Test suffer from masking effects (Huber 1981; Rousseeuw and Leroy 1987; Barnett and Lewis 1994; Singh and Nocerino 1995, and Marona, Martin, and Yohai 2006); this is especially true when outliers are present in clusters of data points and /or the data set represents multiple populations. Masking means that the presence of some outliers hides the presence of other intermediate outliers. The use of robust and resistant outlier identification methods is recommended in the presence of multiple outliers. Several modern robust outlier identification methods exist in the statistical literature cited above. However, robust outlier identification procedures are beyond the scope of the ProUCL software and this technical guidance document. In order to compute robust and resistant estimates of the population parameters of interest (e.g., EPCs, BTVs), EPA NERL-Las Vegas, NV developed a multivariate statistical software package, Scout 2008, Version 223 ------- 1.0 (EPA 2009d) consisting of several univariate and multivariate robust outlier identification and estimation methods. Scout software can be downloaded from the following EPA website: http://archivc.cpa.gov/csd/archivc-scoutAvcb/html/ 7.1 Outliers in Environmental Data Sets In addition to representing contaminated locations, outliers in an environmental data set occur due to non- random, random and seasonal fluctuations in the environment. Outliers tests identify statistical outliers present in a data set. The variabilities of data sets originating from environmental applications are much higher than the variabilties of data sets collected from other applications such as the biological and manufacturing processes, therefore, in environmental applications, not all outliers identified by a statistcial test may represent real physical outliers. Typically, extreme statistical outliers in a data set represent non-random situations potentially representing impacted locations; extreme outliers should not be included in statistical evaluations. Mild and intermediate statistical outliers may be present due to random natural fluctuations and variability in the environment; those outlying observations may be retained in statistical evaluations such as estimating BTVs. Based upon site CSM and expert knowledge, the project team should make these determinations. The use of graphical displays is very helpful in distingushing between extreme statistical outliers (real physical outliers) and intermediate statistical outliers. It is suggested that outlier tests be supplemented with exploratory graphical displays such as Q-Q plots and box plots (Johnson and Wichern 2002; Hoaglin, Moseteller and Tukey 1983). ProUCL has several of these graphical methods which can be used to identify multiple outliers potentially present in a data set. Graphical displays provide additional insight into a data set that cannot be revealed by tests statistics (e.g., Rosner test, Dixon test, S-W test). Graphical displays help identify observations that are much larger or smaller than the bulk (majority) of the data. The statistical tests alone cannot determine whether a statistical outlier should be investigated further. Based upon historical and current site and regional information, graphical displays, and outlier test results, the project team and the decision makers should decide about the proper disposition of outliers to include or not to include them in the computation of the various decision making statistics such as UCL95 and UTL95-95. Performing statistical analyses twice on the same data set, once using the full data set with outliers and once using the data set without high/extreme outliers coming from the far tails, helps the project team in determining the proper disposition of those outliers. Several examples illustrating these issues have been discussed in this technical guidance document (e.g., Chapters 2 through 5). Some Notes Note 1: In practice, extreme outliers represent: 1) low probability observations possibly coming from the extreme far tails of the distribution of the main population under consideration, with low to negligible probability, or 2) observations coming from population(s) different from the main dominant population of interest. On a normal exploratory Q-Q plot, observations well-separated (sticking out, significantly higher than the majority of the data) from the majority of observations represent extreme physical outliers; and the presence of a few high outlying observations distorts the normality of a data set. That is, many data sets follow a normal distribution after the removal of identified outliers. Note 2 (about Normality): Rosner and Dixon outlier tests require normality of a data set without the suspected outliers. Literature about these outlier tests is somewhat confusing and users tend to believe that the original data (with outliers) should follow a normal distribution. A data set with outliers very seldom follow a normal distribution as the presence of outliers tends to destroy the normality of a data set. 224 ------- Note 3: Methods incorporated in ProUCL can be used on any data set with or without NDs, and with or without the outliers. In the past, some practitioners have mis-stated that ProUCL software is restricted and can be used only on data sets without outliers. Just like any other software, it is not a requirement to exclude outliers before using any of the statistical methods incorporated in ProUCL. However, it is the intent of the developers of the ProUCL software to inform the users on how the inclusion of a. few low probability outliers can yield distorted UCL95; UPLs, UTLs, as well as other statistics. The outlying observations should be investigated separately to determine the reasons for their occurrences (e.g., errors or contaminated locations). It is suggested that statistics are computed with and without the outliers followed by evaluation of the potential impact of outliers on the decision making processes. 7.2 Outliers and Normality The presence of outliers in a data set destroys the normality of the data set (Wilks 1963; Barnett and Lewis 1994; Singh and Nocerino 1995). It is highly likely that a data set which contains outliers will not follow a normal distribution unless the outliers are present in clusters. The classical outlier tests, Dixon and Rosner tests, assume that the data set without the suspected outliers follow a normal distribution; that is for both Rosner and Dixon tests, the data set representing the main body of the data obtained after removing the outliers, and not the original data set with outliers needs to follow a normal distribution. There appears to be some confusion among some practitioners (Helsel and Gilroy 2012) who mistakenly assume that one can perform Dixon and Rosner tests only when the data set, including outliers, follows a normal distribution, which is only rarely true. As noted earlier, a lognormal model tends to accommodate outliers (Singh, Singh, and Engelhardt 1997), and a data set with outliers can follow a lognormal distribution. This does not imply that the outlier potentially representing the impacted location does not exist! Those impacted locations may need further investigations. Outlier tests should be performed on raw data, as the cleanup decision needs to be made based upon concentration values in the raw scale and not in the log-scale or some other transformed scale (e.g., cube root). Outliers are not known in advance. ProUCL has normal Q-Q plots which can be used to get an idea about the number of outliers or mixture populations potentially present in a data set. This can help a user to determine the suspected number of outliers needed to perform the Rosner test. Since the Dixon and Rosner tests may not identify all potential outliers present in a data set, the data set obtained, even without the identified outliers, may not follow a normal distribution. Over the last 25 years, several modern iterative robust outlier identification methods have been developed (Rousseeuw and Leroy 1987; Singh and Nocerino 1995) which are beyond the scope of ProUCL. Some of those methods are available in the Scout 2008 version 1.0 software (EPA 2009d). 7.3 Outlier Tests for Data Sets without Nondetect Observations A couple of classical outlier tests discussed in the environmental literature (EPA 2006b, and Gilbert 1987) and included in ProUCL software are described as follows. It is noted that these classical tests suffer from masking effects and may fail to identify potential outliers present in a data set. This is especially true when multiple outliers or multiple populations (e.g., various AOCs of a site) may be present in a data set. Such scenarios can be revealed by using exploratory graphical displays including Q-Q and box plots. 7.3.1 Dixon's Test Dixon's Extreme Value test (1953) can be used to test for statistical outliers when the sample size is less than or equal to 25. Initially, this test was derived for manual computations. This test is described here for historical reasons. It is noted that Dixon's test considers both extreme values that are much smaller than 225 ------- the rest of the data (Case 1) and extreme values that are much larger than the rest of the data (Case 2). This test assumes that the data without the suspected outlier are normally distributed; therefore, one may want to perform a test for normality on the data without the suspected outlier. However, since the Dixon test may not identify all potential outliers present in a data set, the data set obtained after excluding the identified outliers may still not follow a normal distribution. This does not imply that the identified extreme value does not represent an outlier. 7. 3. 1. 1 Directions for the Dixon 's Test Steps described below are provided for interested users, as ProUCL performs all of the operations described as follows: STEP 1: Let X(i), X(2), . . . , X(n) represent the data ordered from smallest to largest. Check that the data without the suspect outlier are normally distributed. If normality fails, then apply a different outlier identification method such as a robust outlier identification procedure. Avoid the use of a data transformation, such as a log-transformation, to achieve normality so that the data meet the criteria to use the Dixon test. All cleanup and remediation decisions are made based upon the data set in raw scale. Therefore, outliers, perhaps representing isolated contaminated locations, should be identified in the original scale. As mentioned before, the use of a log -transformation tends to hide and accommodate outliers (instead of identifying them). STEP 2: X(i) is a potential outlier (Case 1): Compute the test statistic, C, where X(n) ~ for8V> forl4------- 7.3.2 Rosner's Test An outlier test developed by Rosner (1975, 1983) can be used to identify up to 10 outliers in data sets of sizes > 25. The details of the test can be found in Gilbert (1987). Like the Dixon test, the critical values associated with the Rosner test are computed using the normal distribution of the data set without the k (<10) suspected outliers. The assumption here is that the data set without the suspected outliers follows a normal distribution, as a data set with outliers tends not to follow a normal distribution. A graphical display, such as a Q-Q plot, can be used to identify suspected outliers needed to perform the Rosner test. Like the Dixon test, the Rosner test also suffers from masking. 7.3.2.1 Directions for the Rosner's Test To apply Rosner's test, first determine an upper limit, r0, on the number of outliers (r0 < 10), then order the r0 extreme values from most extreme to least extreme. Rosner's test statistic is computed using the sample mean and sample sd. STEP 1: Let Xi, X2, . . . , Xn represent the ordered data points. By inspection, identify the maximum number of possible outliers, ro. Check that the data are normally distributed (without outliers). A data set with outliers seldom passes the normality test. STEP 2: Compute the sample mean, r , and the sample sd, s, for all the data. Label these values J(0) and 5(0), respectively. Determine the value that is farthest from J(0) and label this observation j/0). Delete _y(0) from the data and compute the sample mean, labeled xm, and the sample sd, labeled sm. Then determine the observation farthest from x(~Y) and label this observationj/1-1. Delete _y(1) and compute x(2) and s(2). Continue this process until r0 extreme values have been eliminated. After carrying out the above process, we have: r*(0),s(0), v(0)l; \x(l},s(l}, v(1)l: ..., \x(r^l),s(r^l), v^^l where , -x(0)2 ,and /° is the farthest value x(f). n-i j=\ \n-y=i The above formulae for ;tw and s^ assume that the data have been re-numbered after each outlying observation is deleted. y(r-l) _ j(r-l) I STEP 3: To test if there are "r" outliers in the data, compute: Rr = — and compare Rr to i3 the critical value Ar in the tables from any statistical literature. If Rr> hr, conclude that there are r outliers. First, test if there are r0 outliers (compare Rr _\ to Ar_j). If not, then test if there are r0 - 1 outliers (compare Rr _2 to Xr _2). If not, then test if there are ro - 2 outliers, and continue, until either it is determined that there are a certain number of outliers or that there are no outliers. 227 ------- 7.4 Outlier Tests for Data Sets with Nondetect Observations In environmental studies, identification of detected high outliers, coming from the right tail of the data distribution and potentially representing impacted locations, is important as locations represented by those extreme high values may require further investigation. Therefore, for the purpose of the identification of high outliers, one may replace the NDs by their respective DLs, DL/2, or may just ignore them (especially when elevated DLs are associated with NDs and/or when the number of detected values is large) from any of the outlier test (e.g., Rosner test) computations, including the graphical displays such as Q-Q plots. Both of these procedures, ignoring NDs or replacing them by DL/2, for identification of outliers are available in ProUCL for data sets containing NDs. Like uncensored full data sets, outlier tests on data sets with NDs should be supplemented with graphical displays. ProUCL can be used to generate Q-Q plots and box plots for data sets with ND observations. Notes: Outlier identification procedures represent exploratory tools and are used for pre-processing of a data set to identify outliers or multiple populations that may be present in a data set. Except for the identification of high outlying observations, the outlier identification statistics, computed with NDs or without NDs, are not used in any of the estimation and decision making process. Therefore, for the purpose of the identification of high outliers, it should not matter how the ND observations are treated. To compute test statistics (e.g., Gehan test) and decision statistics (e.g., UCL95, UTL95-95), one should follow the procedures as described in Chapters 4 through 6. Example 7-1. Consider a lead data set of size 10 collected from a Superfund site. The site data set appears to have some outliers. Since the data set is of small size, only the Dixon test can be used to identify outliers. The normal Q-Q plot of the lead data is shown in Figure 7-1 below. Figure 7-1 immediately suggests that the data set has some outliers. The Dixon test cannot directly identify all outliers present in a data set, only robust methods can identify multiple outliers. Multiple outliers may be identified one at a time iteratively by using the Dixon test on data sets after removing outliers identified in previous iterations. However, due to masking, the iterative process based upon the Dixon test may or may not be able to identify multiple outliers. Q-Q Plot for OS_Lead Intercept = 268.2 Correlation, R = 0.67 Theoretical Quantiles (Standard Normal) Figure 7-1. Normal Q-Q Plot Identifying Outliers 228 ------- Table 7-1. Dixon Outlier Test Results for Site Lead Data Set Dixon's Outlier Test for OS Lead Number of Observations = 10 10% critical value: 0.409 5% critical value: 0.477 1% critical value: 0.597 t. Observation Value 1940 is a Potential Outlier (Upper Tail)? Test Statistic: 0.836 For 10% significance level. 194Q is an outlier. For 5% significance level, 194C is an outlier. For 1% significance level. 194(1 is an outlier. 2. Observation Value 19.7 is a Potential Outlier (Lower Tail)? Test Statistic: 0.013 For 10% significance level, 19.7 is not an outlier. For 5% significance level, 19.7 is not an outlier. For 1 % significance level, 19.7 is not an outlier. Example 7-2. Consider She's (1997) pyrene data set of size n=56 with 11 NDs. The Rosner test results on data without the 11 NDs are summarized in Table 7-2, and the normal Q-Q plot without NDs is shown in Figure 7-2 below. 3000 Q-Q Plot for Pyrene Nondetects not displayed 2500 2000 a c 1000 500 ' 0 » ....?.- ' ' ' • 459 3GB ^ Pyrene Total Number of Data-SE Number of Non-Deleete*11 Number of Detects -45 Detected Mean-190.1 Defected Sd = 435 Slope (displayed data] -224.4 Intercept (displayed data^ 130.1 Conelalion, R = 0 506 D Best Fit Line -1.6 -1.2 -0.6 0.0 0.6 1.2 1.8 Theoretical Quantiles (Standard Normal) NDs Displayed in smaller font Figure 7-2. Normal Q-Q Plot of Pyrene Data Set Excluding NDs 229 ------- Table 7-2. Rosner Test Results on Pyrene Data Set Excluding NDs Rosner's Outlier Test for 10 Outliers in Pyrene Total N 56 Number NDs 11 Number Detects 45 Mean of Detects 190.1 SD of Detects 435 Number of data 45 Number of suspected outliers 10 s not included in the following: tt 1 2 3 4 5 6 7 8 9 10 Mean 190.1 12S.6 118.9 111S 109.1 104.6 100.3 9668 93.3 9061 Potential sd outlier 430.1 2982 90.7 459 75.7 333 68.74 306 62.43 289 56.1 273 4965 238 4478 222 40.17 190 37.21 187 Obs. Number 45 44 43 42 41 40 39 3B 37 36 Test value 6.491 3.665 2.828 2.796 2.SS1 3.001 2.773 2.798 2.408 2.59 Critical Critical value (5%) value (1%) 3.09 3.44 3.08 343 3.07 3.41 306 3.05 3.038 3.026 3.014 3.002 2.99 3.4 3.39 3.378 3366 3354 3342 3.33 | For 5:i significance level, there are 2 Potential Outliers 2982. 459 For 1 *= Signrficance Level, there are 2 Potential Outliers 2982. 459 Example 7-3. Consider the aluminum data set of size 28 collected from a Superfund site. The normal Q- Q plot is shown in Figure 7-3 below. Figure 7-3 suggests that there are 4 outliers (at least the observation=30,000) present in the data set. The Rosner test results are shown in Table 7-3. Due to masking, the Rosner test could not even identify the outlying observation of 30,000. Q-Q Plot for Aluminum Slope-7193 Intercept-102E Corte^fon, R = Theoretical Quantiles (Standard Normal) Figure 7-3. Normal Q-Q Plot of Aluminum Concentrations 230 ------- Table 7-3. Rosner Test Results on Pyrene Data Set Excluding NDs Rosner's Outlier Test for Aluminum Mean 10284 Standard Deviation 7449 Number of data 28 Number of suspected outliers 10 8 1 2 3 4 5 6 7 8 9 10 Mean 10284 9553 S359 8358 7789 7423 70€1 5Sfi9 S377 S102 sd 7315 6490 5822 5050 42S4 3356 3637 3215 3000 2S12 Potential outlier 30000 25000 240(10 22000 162QO 15400 15300 12500 11600 10700 Obs. Number 26 25 27 28 10 13 6 14 21 12 Test Critical Critical value value (5%) value (1 %} 2.695 2,38 2.584 2.702 1.973 2.016 2.2S5 1.814 1.741 1.635 2.88 2.86 2.84 2.82 2.8 2.776 2.752 2.728 2.704 2.S8 3.2 3.18 3.16 3.14 3.11 3.082 3.054 3.026 2.998 2.97 For 5% Significance Level, there is no Potential Outlier For 1 ;= Significance Level, there is no Potential Outlier As mentioned earlier, there are robust outlier identification methods which can be used to identify multiple outliers/multiple populations present in a data set. Several of those methods are incorporated in Scout 2008 (EPA 2009d). A couple of formal (with test statistics) robust graphs based upon the PROP influence function and MCD method (Singh and Nocerino 1995) are shown in Figures 7-4 and 7-5. The details of these methods are beyond the scope of ProUCL. The two graphs suggest that there are several outliers present including the elevated value of 30,000. All observations exceeding the horizontal lines displayed at critical values of the Largest Mahalanobis Distance (MD) (Wilks 1963; Barnett and Lewis 1994) represent outliers. 231 ------- Index Plot of MDs using PROP Estimate Initial Eitlmatai: OKG Influence Aipns G UK MDDiitnbution: Bell Figure 7-4. Robust Index Plot of MDs Based Upon the PROP Influence Function Index Plot of MDs using MCD Estimate Initul S.teof V • r •h'v«lueof ||n*pt1)f2)-tE lmti*l Sublet* = 500 Bed Retained Subieti = 11 Index of Observations Figure 7-5. Robust Index Plot of MDs Based upon the MCD Method 232 ------- CHAPTER 8 Determining Minimum Sample Sizes for User Specified Decision Parameters and Power Assessment This chapter describes mathematical formulae used to determine data quality objectives (DQOs)-based minimum sample sizes required by estimation, and hypothesis testing approaches used to address statistical issues for environmental projects (EPA 2006a, 2006b). The sample size determination formulae for estimation of the unknown population parameters (e.g., mean, percentiles) depend upon the pre-specified values of the decision parameters: CC, (1-a), and the allowable error margin, A, between the estimate and the unknown true population parameter. For example, if the environmental problem requires the calculation of the minimum number of samples required to estimate the true unknown population mean, A would represent the maximum allowable difference between the estimate of the sample mean and the unknown population mean. Similarly, for hypotheses testing approaches, sample size determination formulae depend upon the pre-specified values of the decision parameters chosen while defining and describing the DQOs associated with an environmental project. The decision parameters associated with hypotheses testing approaches include the Type I false positive error rate, a; and the Type II false negative error rate, /?=! -power; and the allowable width, A, of the gray region. For values of the parameter of interest (e.g., mean, proportion) lying in the gray region, the consequences of committing the two types of errors described in Chapter 6 are not significant from both the human health and the cost effectiveness points of view. Even though the same symbol, A, has been used to denote the allowable error margin in an estimate (e.g.. of mean) and the width of the gray region associated with the various hypothesis testing approaches, there are differences in the meanings of the error margin and width of the gray region. A brief description of these terminology is provided in this chapter. The user is advised to consult the already existing EPA guidance documents (EPA 2006a, 2006b; MARSSIM 2000) for the detailed description of the terms with interpretation used in this chapter. Both parametric (assuming normality) and nonparametric (distribution free) DQOs-based sample size determination formulae as described in EPA guidance documents (MARSSIM 2000; EPA 2002c, 2006a, 2006b, and 2009) are available in the ProUCL software. These formulae yield minimum sample sizes needed to perform statistical methods meeting pre-specified DQOs. The Stats/ Sample Sizes module of ProUCL has the minimum sample size determination methods for most of the parametric and nonparametric one-sided and two-sided hypotheses testing approaches available in ProUCL. ProUCL includes the DQOs-based parametric minimum sample size formula to estimate the population mean, assuming that the sample mean follows a normal distribution or assuming that the criteria is met due to the CLT]. ProUCL outputs a non-negative integer as the minimum sample size. This minimum sample size is calculated by rounding the value, obtained by using a sample size formula, upward. For all sample size determination formulae incorporated in ProUCL, it is implicitly assumed that samples (e.g., soil, groundwater, sediment samples) are randomly collected from the same statistical population (e.g., AOC or MW), and therefore the sampled data (e.g., analytical results) represent independently and identically distributed (i.i.d) observations from a single statistical population. During the development of the Stats/Sample Sizes module of ProUCL, emphasis was given to assure that the module is user friendly with a straight forward unambiguous mechanism (e.g., graphics user interface [GUIs]) to input desired 233 ------- decision parameters (e.g., a, /? error rates, width, A of the gray region) needed to compute the minimum sample size for a selected statistical application. Most of the sample size formulae available in the literature and incorporated in ProUCL) require an estimate (e.g., preliminary from other sites and pilot studies or based upon actual collected data) of the population variability. In practice, the population variance, ------- size computed in retrospect, the user may want to collect additional samples to assure that the test achieves the desired power. • It should be pointed out that there could be differences in the sample sizes computed in the two different stages due to the differences in the values of the estimated variability. Specifically, the preliminary estimate of the variance computed using information from similar sites could be significantly different from the variance computed using the available data already collected from the study area under investigation which will yield different values of the sample size. Sample size determination methods in ProUCL can be used for both stages. The only difference will be in the input value of the standard deviation/variance. It is the users' responsibility to input a correct value for the standard deviation during the two stages. 8.1 Sample Size Determination to Estimate the Population Mean In exposure and risk assessment studies, a UCL95 of the population mean is used to estimate the EPC term. Listed below are several variations of methods available in the literature to compute the minimum sample size, n, needed to estimate the population mean with specified confidence coefficient (CC), (1 - a), and allowable/tolerable error margin (allowable absolute difference between the estimate and the parameter), A in an estimate of the mean. 8. 1. 1 Sample Size Formula to Estimate Mean without Considering Type II (ft) Error Rate The sample size can be computed using the following normal distribution based equation (when population variance is known), « = ------- normal distribution (which can be assumed due to the CLT). ProUCL does not compute minimum sample sizes required to estimate the population median. While estimating the mean, the symbol A represents the allowable error margin (+/-) in the mean estimate. For example for A = 10, the sample size is computed to assure that the error in the estimate will be within ±10 units of the true unknown population mean with specified CC of (1 -a). For estimation of the mean, the most commonly used formula to compute the sample size, n, is given by (8-2) above; however, under normal theory, the use of t-distribution based formula (8-3) is more appropriate to compute n. It is noted that the difference between the sample sizes obtained using (8-2) or (8-3) is not significant. They usually differ by only 2 to 3 samples (Blackwood 1991; Singh, Singh, and Engelhardt 1999). It is a common practice to address this difference by using the following adjusted formula (Kupper and Hafner 1989; Bain and Engelhardt 1991) to compute the minimum sample size needed to estimate the mean for specified CC, (1 - a), and margin of error, A. « = AWA2 + zi2W2 (8-4) To be able to use a normal (instead oft-critical value) distribution based critical value, as used in (8-4), a similar adjustment factor is used in other sample size formulae described in the following sections (e.g., two-sample t-test, WRS test). This adjustment is also used in various sample size formulae described in EPA guidance documents (MARSSIM 2000; EPA 2002c, 2006a, 2006b). ProUCL uses equation (8-4) to compute sample sizes needed to estimate the population mean for specified values of CC, (1- a), and error margin, A. An example illustrating the sample size determination to estimate the mean is given as follows. Example 8-1. Sample Size for estimation of the mean (CC = 0.95, s = 25, error margin, A = 10) I i Sample Size for Estimation of Mean Based on Specified Values of Decision Paiameters/DQQs (DataQuaRy Objectives] Date^Time of Computation 2/26^2010 12:12:37 PM User Selected Options Confidence Coefficient 95% Allowable Error Margin 10 Estimate of Standard Deviation 25 Approximate Minimum Sample Size 95% Confidence Coefficient: 26 8.1.2 Sample Size Formula to Estimate Mean with Consideration to Both Type I (a) and Type II (ft) Error Rates This scenario corresponds to the single-sample hypothesis testing approach. For specified decision error rates, a and ft, and width, A, of the gray region, ProUCL can be used to compute the minimum sample size based upon the assumption of normality. ProUCL also has nonparametric minimum sample size determination formulae to perform Sign and WSR tests. The nonparametric Sign test and WSR test are used to perform single sample hypothesis tests for the population location parameter (mean or median). A brief description of the standard terminology used in the sample size calculations associated with hypothesis testing approaches is described first as follows. 236 ------- a = False Rejection Rate (Type I Decision Error), i.e., the probability of rejecting the null hypothesis when in fact the null hypothesis is true /? = False Acceptance Rate (Type II Decision Error), i.e., the probability of not rejecting the null hypothesis when in fact the null hypothesis is false z;_« = a value from a standard normal distribution for which the proportion of the distribution to the left of this value is 1 - a zi-p = a value from a standard normal distribution for which the proportion of the distribution to the left of this value is 1 - /? A = width of the gray region (specified by the user); in a gray region, decisions are "too close to call", a gray region is that area where the consequences of making a decision error (Type I or Type II) are relatively minor. The user is advised to note the difference between the gray region (associated with hypothesis testing approaches) and error margin (associated with estimation approaches). Example illustrating the above terminology: Let the null and alternative hypotheses be: H0: p < Cs, and HA: /J > Cs. The width, A, of the gray region for this one sided alternative hypothesis is A = jUi - Cs, where Cs is the cleanup standard specified in the null hypothesis, and jUi (>G) represents an alternative value belonging to the parameter value set determined by the alternative hypothesis. Note that the gray region lies to the right (e.g., see Figure 8-1) of the cleanup standard, Cs, and for all values of ju in the interval, (Cs, jUi], with length of the interval = width of gray region= A = jUi - Cs. The consequences of making an incorrect decision (e.g., accepting the null hypothesis when in fact it is false) will be minor. 8.2 Sample Sizes for Single-Sample Tests 8.2.1 Sample Size for Single-Sample t-test (Assuming Normality) This section describes formulae to determine the minimum number of samples, n, needed to conduct a single-sample t-test, for 1-sided as well as two-sided alternatives, with pre-specified decision error rates and width of the gray region. This hypothesis test is used when the objective is to determine whether the mean concentration of an AOC exceeds an action level (AL); or to verify the attainment of a cleanup standard, Cs (EPA 1989a). In the following, s represents an estimate (e.g., an initial guess, historical estimate, or based upon expert knowledge) of the population sd, a. Three cases/forms of hypothesis testing as incorporated in ProUCL are described as follows: 237 ------- 8.2.1.1 Case I (Right-Sided Alternative Hypothesis, Form 1) Ho: site mean, n AL or a Cs Gray Region: Range of the mean concentrations where the consequences of deciding that the site mean is less than the AL when in fact it is greater (that is a dirty site is declared clean) are not significant. The upper bound of the gray region, A, is defined as the alternative mean concentration level, //; (> Cs), where the human health and environmental consequences of concluding that the site is clean (when in fact it is not clean) are relatively significant. The false acceptance error rate, /?, is associated with this upper bound (jj.i) of the gray region: A=,U;_ Cs. These are illustrated in Figure 8-1 below (EPA 2006a). A similar explanation of the gray region applies to other single-sample Form 1 right-sided alternative hypotheses tests (e.g., Sign test, WSRtest) considered later in this chapter. •3 — M *. .= 2 ^ = .i w 1 0.9 - 0.8 0.7 - 0.6 - 0.5- 0.4 0.3 - 0.1- 0.1 - I Baseline I i i 0 40 Alternative i I Tolerable False Rejection Decision EnqrBates Tolerable False Acceptance Decision Error Rates Gray Region Relatively Large Decision Error Rates are Considered Tolerable 140 160 180 200 Action Level True Value of the Parameter (Mean Concentration, ppm) Diagram Where the Alternative Condition Exceeds the Action Level Figure 8-1. Gray Region for Right-Sided (Form 1) Alternative Hypothesis Tests (EPA 2006a) 8.2.1.2 Case II (Left-Sided Alternative Hypothesis, Form 2) Ho: site mean, p >AL or Cs vs. HA: site mean, p ------- (when in fact it is not dirty) would be costly requiring unnecessary cleaning of a site. The false acceptance rate, ft, is associated with that lower bound (pii) of the gray region, A= Cs - //;. These are illustrated in Figure 8-2. A similar explanation of the gray region applies to other single-sample left-sided (left-tailed) alternative hypotheses tests including the Sign test and WSR test. l- 0.9- 0.8- 0.7- 0.6- 0.5- 0.4- 0.3- 0.2- 0.1- 0 • Alteniative ^_^: Baseline Gray Region Relatively Large Decision Error Rites are Considered Tolerable Tolerable False Rejection Decision Error Rates Tolerable False Acceptance Decision En or Rates 20 40 60 80 100 120 140 160 180 200 Action Level True Value of the Parameter (Menu Concentration, ppm) Diagram Where the Alternative Condition Falls Below the Action Level Figure 8-2. Gray Region for Left-Sided (Form 2) Alternative Hypothesis Tests (EPA 2006a) The minimum sample size, n, needed to perform the single-sample one-sided t-test (both Forms 1 and 2 described above) is given by (8-5) 8.2.1.3 Case III (Two-Sided Alternative Hypothesis) Ho: site mean, p = Cs; vs. HA: site mean, p ^ Cs The minimum sample size for specified performance (decision) parameters is given by: 239 ------- (8-6) _a_ A = width of the gray region, A= abs (Cs- jUi), abs represents the absolute value operation. In this case, the gray region represents a two-sided region symmetrically placed around the mean concentration level equal to Cs, or AL; consequences of committing the two types of errors in this gray region would be minor (not significant). A similar explanation of the gray region applies to other single- sample two-sided (two-tailed) alternative hypotheses tests such as the Sign test and WSRtest. In equations (8-5) and (8-6), the computation of the estimated variance, s2 depends upon the project stage. Specifically, s2 = a preliminary estimate of the population variance (e.g., estimated from similar sites, pilot studies, expert opinions) which is used during the planning stage; or s2 = actual sample variance of the collected data to be used when assessing the power of the test in retrospect based upon collected data. Note: ProUCL outputs the estimated variance based upon the collected data on single sample t-test output sheet; ProUCL 5.1 sample size GUI draws users' attention to input an appropriate estimate of variance, the user should input an appropriate value depending upon the project stage/data availability. The following example: "Sample Sizes for Single-sample t-Test" discussed in Guidance on Systematic Planning Using the Data Quality Objective Process (EPA 2006a, page 49) is used here to illustrate the sample size determination for a single-sample t-test. For specified values of the decision parameters, the minimum number of samples is given by n > 8.04. For a one-sided alternative hypothesis, ProUCL computes the minimum sample size to be 9 (rounding up), and a sample size of 1 1 is computed for a two- sided alternative hypothesis. Example 8-2. Sample Size for Single-sample t-Test Sample Sizes (a = 0.05, ft = 0.2, s = 10.41, A = 10) ! Sample Sizes for Single Sample t Test Based on Specified Values of Decision Paiameters/DQOs (Data Quafty Objectives) D ate/T ime of Computation 2/26/2010 12:41:58 PM User Selected Options False Rejection Rate [Alpha] 0.05 False Acceptance R ate [B eta] 0.2 Width of Grav Region [Delta] 10 E stimate of S tandard D eviation 10.41 Approximate Minimum Sample Size Single Sided Alternative Hypothesis: 9 Two Sided Alternative Hypothesis: 11 5.2.2 Single Sample Proportion Test This section describes formulae used to determine the minimum number of samples, n, needed to compare an upper percentile or proportion, P, with a specified proportion, P0 (e.g., proportion of 240 ------- exceedances, proportion of defective items/drums, proportion of observations above the specified AL), for user selected decision parameters. The details are given in EPA guidance document (2006a). Sample size formulae for three forms of the hypotheses testing approach are described as follows. 5.2.2.1 Case I (Right-Sided Alternative Hypothesis, Form 1) Ho: population proportion < specified value (Po) vs. HA: population proportion > specified value (Po) Gray Region: Range of true proportions where the consequences of deciding that the site proportion, P, is less than the specified proportion, P0, when in fact it is greater (that is a dirty site is declared clean) are not significant. The upper bound of the gray region, A, is defined as the alternative proportion, Pi (> Po), where the human health and environmental consequences of concluding that the site is clean (when in fact it is not clean) are relatively significant. The false acceptance error rate, /?, is associated with this upper bound (Pi) of the gray region (A=P;_/V- 8.2.2.2 Case II (Left-Sided Alternative Hypothesis, Form 2) Ho: population proportion > specified value (Po) vs. HA: population proportion < specified value (Po) Gray Region: Range of true proportions where the consequences of deciding that the site proportion, P, is greater than or equal to the specified proportion, Po, when in fact it is smaller (a clean site is declared dirty) are not considered significant. The lower bound of the gray region is defined as the alternative proportion, Pi (< P0), where the consequences of concluding that the site is dirty (when in fact it is not dirty) would be costly requiring unnecessary cleaning of a clean site. The false acceptance rate, (3, is associated with that lower bound (Pi) of the gray region (A= Po- Pi). The minimum sample size, n, for the single-sample proportion test (for both cases I and II) is given by n = P -P •M •'O (8-7) 8.2.2.3 Case III (Two-Sided Alternative Hypothesis) Ho: population proportion = specified value (Po)vs. HA: population proportion^ specified value (Po) The following procedure is used to determine the minimum sample size needed to conduct a two-sided proportion test. a = P ~P •M •'O when Pi = Po + A; and P -P •M •'O for right-sided alternative; for left-sided alternative; 241 ------- when Pi = Po - A Po = specified proportion Pi = outer bound of the gray region. A = width of the gray region = \Po - Pi =abs (Po - Pi) The sample size, n, for two-sided proportion test (Case III) is given by n = max(a,b) (8-8) An example illustrating the single-sample proportion test is considered next. This example: "Sample Sizes for Single-sample Proportion Test" is also discussed in EPA 2006a (page 59). For this example, for the specified decision parameters, the number of samples is given by n > 365. However, ProUCL computes the sample size to be 419 for the right-sided alternative hypothesis, 368 for the left-sided alternative hypothesis, and 528 for the two-sided alternative hypothesis. Example 8-3. Output for Single-Sample proportion test sample size (a = 0.05, /? = 0.2, Po = 0.2, A = 0.05) j I Sample Sizes for Single Sample Proportion Test Based on Specified Values of Decision Parameters/DQOs (Data QuaEty Objectives) D ate/T ime of Computation 2/26/2010 12:50:52 PM User Selected Options False Rejection Rate [Alpha] 0.05 False Acceptance R ate [B eta] 0.2 Width of G ray R egion [D elta] 0.05 Proportion/Action Level [PO] 0.2 Approximate Minimum Sample Size Right Sided Alternative Hypothesis: 419 Left Sided Alternative Hypothesis: 368 Two Sided Alternative Hypothesis: max(471,528) Notes: The correct use of the Sample Size module, to determine the minimum sample size needed to perform a proportion test, requires that the users have some familiarity with the single-sample hypothesis test for proportions. Specifically the user should input feasible values for the specified proportion, Po, and width, A, of the gray region. The following example shows the output screen when unfeasible values are selected for these parameters. Example 8-4. Output - Single-sample Proportion Test Sample Sizes (a = 0.05, ft = 0.2, P0 = 0.7, A = 0.8) i Sample Sizes for Single Sample Proportion Test Based on Specified Values of Decision Pararneters/DQOs (Data Quafty Objectives) Date/Time of Computation 2/26/2010 12:55:51 PM User Selected Options False Rejection Rate [Alpha] 0.05 False Acceptance Rate [Beta] 0.2 Width of Gray Region [Delta] 0.8 Proportion/Action Level [PO] 0.7 Approximate Minimum Sample Size Right Sided Alternative Hypothesis: Not Feasible - Please check your Decision Pararneters/DOOs Left Sided Alternative Hypothesis: Not Feasible - Please check your Decision Parameters/DQQs Two Sided Alternative Hypothesis: Not Feasible - Please check your Decision Pararneters/DQOs 242 ------- 8.2.3 Nonparametric Single-sample Sign Test (does not require normality) The purpose of the single-sample nonparametric Sign test is to test a hypothesis involving the true location parameter (mean or median) of a population against an AL or Cs without assuming normality of the underlying population. The details of sample size determinations for nonparametric tests can be found inConover(1999). 8. 2. 3. 1 Case I (Right-Sided Alternative Hypothesis) Ho: population location parameter < specified value, Cs vs. HA: population location parameter > specified value, Cs A description of the gray region associated with the right-sided Sign test is given in Section 8.2.1.1. 8. 2. 3. 2 Case II (Left-Sided Alternative Hypothesis) Ho: population location parameter > specified value, Cs vs. HA: population location parameter < specified value, Cs A description of the gray region associated with this left-sided Sign test is given in Section 8.2. 1 .2. The minimum sample size, n, for the single-sample one-sided (both left-sided and right-sided) Sign test is given by the following equation: - - i /o r,\ n = — - - — — - , where (8-9) 4(SignP-0.5) SignP = <&\ — (8-10) {sdj A = width of the gray region sd = an estimate of the population (e.g., reference area, AOC, survey unit) standard deviation Some guidance on the selection of an estimate of the population sd, a, is given in Section 8.1.1 above. ------- n = ( 4(SignP-0.5) In the following example, ProUCL computes the sample size to be 35 for a single-sided alternative hypothesis and 43 for a two-sided alternative hypothesis for default values of the decision parameters. Note: Like the parametric t-test, the computation of the standard deviation (sd) depends upon the project stage. Specifically, sd2 (used to compute P in equation (8-10)) = a preliminary estimate of the population variance (e.g., estimated from similar sites, pilot studies, expert opinion) which is used during the planning stage; and sd2 (used to compute P) = sample variance computed using the actual collected data to be used when assessing the power of the test in retrospect based upon the collected data. ProUCL outputs the sample variance based upon the collected data on the Sign test output sheet; and ProUCL 5.1 sample size GUI draws user's attention to input an appropriate estimate, sd2, the user should input an appropriate value depending upon the project stage/data availability. Example 8-5. Output for Single-sample Sign Test Sample Sizes (a = 0.05, ft = 0.1, sd = 3, A = 2) I Sample Sizes for Single Sample Sign Test Based on SpecifiedValues of Decision Parameters/DQOs [Data Quaiy Objectives] Date/Time of Computation 2/26/201012:15:27 PM User Selected Options False Rejection Rate [Alpha] 0.05 False Acceptance Rate [Beta] 0.1 Width of G ray R egion [D elta] 2 Estimate of Standard Deviation 3 Approximate Minimum Sample Size Single Sided Alternative Hypothesis: 35 Two Sided Alternative Hypothesis: 43 8.2.4 Nonparametric Single Sample Wilcoxon Sign Rank (WSR) Test The purpose of the single WSR test is similar to that of the Sign test described above. This test is used to compare the true location parameter (mean or median) of a population against an AL or Cs without assuming normality of the underlying population. The details of this test can be found in Conover (1999) andEPA(2006a). 8.2.4.1 Case I (Right-Sided Alternative Hypothesis) Ho: population location parameter < specified value, Cs vs. HA: population location parameter > specified value, Cs A description of the gray region associated with this right-sided test is given in Section 8.1.2.1. 244 ------- 8.2.4.2 Case II (Left-Sided Alternative Hypothesis) Ho: population location parameter > specified value, Cs vs. HA: population location parameter < specified value, Cs A description of the gray region associated with this left-sided (left-tailed) test is given in Section 8.1.2.2. The minimum sample size, n, needed to perform the single-sample one-sided (both left-sided and right- sided) WSRtest is given as follows. n = \.\6 £* 2 J (8-11) Where: sd2 = a preliminary estimate of the population variance which is used during the planning stage; and sd2 = actual sample variance computed using the collected data to be used when assessing the power of the test in retrospect based upon collected data Note: ProUCL 5.0 sample size GUI draws user's attention to input an appropriate estimate, sd2; the user should input an appropriate value depending upon the project stage/data availability. 8.2.4.3 Case III (Two-Sided Alternative Hypothesis) Ho: population location parameter = specified value, Cs;vs. HA: population location parameter =£ specified value, Cs A description of the gray region associated with the two-sided WSRtest is given in Section 8.1.2.3. The sample size, n, needed to perform the single-sample two-sided WSRtest is given by: Zl ~"+Zl " 7' " (8-12) A" 2 V J Where: sd2 = a preliminary estimate of the population variance (e.g., estimated from similar sites) which is used during the planning stage; and sd2 = sample variance computed using actual collected data to be used to assess the power of the test in retrospect. Note: ProUCL 5.0 sample size GUI draws user's attention to input an appropriate estimate, sd2, the user should input an appropriate value depending upon the project stage/data availability. 245 ------- The following example: "Sample Sizes for Single-sample Wilcoxon Signed Rank Test" is discussed in the EPA 2006a (page 65). ProUCL computes the sample size to be 10 for a one-sided alternative hypothesis, and 14 for a two-sided alternative hypothesis. Example 8-6. Output for Single-sample WSR Test Sample Sizes (a = 0.1, ft = 0.2, sd = 130, A = 100) i j Sample Sizes (or Single Sample WilcoHon Signed Rank Ted B ased on S pecified Values of D ecision Paiameters/D Q Q s (D ata Q uaAy Objectives) D ate/T ime of Computation 2/26/2010 1:13:58 PM User Selected Options False Rejection Rate [Alpha] 0.1 False Acceptance Rate [Beta] 0.2 Width of Q ray R egion [D elta] 100 Estimate of Standard Deviation 130 Approximate Minimum Sample Size Single Sided Alternative Hypothesis: 10 Two Sided Alternative Hypothesis: 14 8.3 Sample Sizes for Two-Sample Tests for Independent Sample This section describes minimum sample size determination formulae needed to compute sample sizes (same number of samples (n=m) from two populations) to compare the location parameters of two populations (e.g., reference area vs. survey unit, two AOC, two MW) for specified values of the decision parameters. ProUCL computes sample sizes for one-sided as well as two-sided alternative hypotheses. The sample size formulae described in this section assume that samples are collected following the simple random or systematic random sampling (e.g., EPA 2006a) approaches. It is also assumed that samples are collected randomly from two independently distributed populations (e.g., two different uncorrelated AOCs); and samples (analytical results) collected from each of population represent independently and identically distributed observations from their respective populations. 8.3.1 Parametric Two-sample t-test (Assuming Normality) The details of the two-sample t-test can be found in Chapter 6 of this ProUCL Technical Guide. 8.3.1.1 Case I (Right-Sided Alternative Hypothesis) Ho: site mean, jUi < background mean, jU2VS. HA: site mean, jUi > background mean, jU2 Gray Region: Range of true concentrations where the consequences of deciding the site mean is less than or equal to the background mean (when in fact it is greater), that is, a dirty site is declared clean, are relatively minor. The upper bound of the gray region is defined as the alternative site mean concentration level, jUi (> jU2), where the human health, and environmental consequences of concluding that the site is clean (or comparable to background) are relatively significant. The false acceptance rate, /?, is associated with the upper bound of the gray region, A. 246 ------- 8. 3. 1 2 Case // (Left-Sided Alternative Hypothesis) Ho: site mean, jUi > background mean, jU2VS. HA: site mean, jUi < background mean, jU2 Gray Region: Range of true mean values where consequences of deciding the site mean is greater than or equal to the background mean (when in fact it is smaller); that is, a clean site is declared a dirty site, are considered relatively minor. The lower bound, //; (< ^2) of the gray region, is defined as the concentration where consequences of concluding that the site is dirty would be too costly, potentially requiring unnecessary cleanup. The false acceptance rate is associated with the lower bound of the gray region. The minimum sample sizes (equal sample sizes for both populations) for the two-sample one-sided t-test (both cases I and II described above) are given by: m = n = 2(zl_a+zl_p)2 ^ +^ (8-13) The decision parameters used in equations (8-13) and (8-14) have been defined earlier in Section 8.1.1.2. A = width (e.g., difference between two means) of the gray region Sp = a preliminary estimate of the common population standard deviation, a, of the two populations (discussed in Chapter 6). Some guidance on the selection of an estimate of the population sd, a, is given above in Section 8.1.2. Sp = pooled standard deviation computed using the actual collected data to be used when assessing the power of the test in retrospect. 8.3.1.3 Case III (Two-Sided Alternative Hypothesis) Ho: site mean, pi = background mean, /Li2vs. HA: site mean, //; ^ background mean, /j.2 The minimum sample sizes for specified decision parameters are given by: (8-14) The following example: "Sample Sizes for Two-sample t Test" is discussed in the EPA 2006a guidance document (page 68). According to this example, for the specified decision parameters, the minimum number of samples from each population comes out to be m = n > 4.94. ProUCL computes minimum sample sizes for the two populations to be 5 (rounding up) for the single sided alternative hypotheses and 7 for the two-sided alternative hypothesis. Note: Sp represents the pooled estimate of the populations under comparison. During the planning stage, the user inputs a preliminary estimate of variance while computing the minimum sample sizes; and while assessing the power associated with the t-test, the user inputs the pooled standard deviation, Sp, computed using the actual collected data. 247 ------- Sp = a preliminary estimate of the common population standard deviation (e.g., estimated from similar sites, pilot studies, expert opinion) which is used during the planning stage; and Sp = pooled standard deviation computed using the collected data to be used when assessing the power of the test in retrospect. ProUCL outputs the pooled standard deviation, Sp, based upon the collected data on the two sample t-test output sheet; ProUCL 5.1 sample size GUI draws user's attention to input an appropriate estimate of the standard deviation, the user should input an appropriate value depending upon the project stage/data availability. Example 8-7. Output for two-sample t-test sample sizes (a = 0.05, ft = 0.2, sp = 1.467, A = 2.5) j Sample Sizes for Two Sample (Test Based on Specified Values of Decision Parameters/DQOs(Data QuaHy Objectives) D ate/T ime of Computation 2/26/2010 1:17:57 PM User Selected Options False R ejection R ate [Alpha] 0.05 False Acceptance R ate [B eta] 0.2 Width of G ray R egion [D elta] 2.5 Estimate of Pooled SD 1.467 Approximate Minimum Sample Size Single Sided Alternative Hypothesis: 5 Two Sided Alternative Hypothesis: 7 8.3.2 Wilcoxon-Mann-Whitney (WMW) Test (Nonparametric Test) The details of the two-sample nonparametric WMW can be found in Chapter 6; this test is also known as the two-sample WRS test. 8.3.2.1 Case I (Right-Sided Alternative Hypothesis) Ho: site median < background median vs. HA: site median > background median The gray region for the WMW Right-Sided alternative hypothesis is similar to that of the two-sample li- test described in Section 8.1.3.1. 8.3.2.2 Case II (Left-Sided Alternative Hypothesis) Ho: site median > background median vs. HA: site median < background median The gray region for the WMW left-sided alternative hypothesis is similar to that of two-sample t-test described in Section 8.1.3.2. The sample sizes n and m, for one-sided two-sample WMW tests are given by 248 ------- (8-15) Here: sd2 =a preliminary estimate of the common variance, ------- Example 8-8. Output for Two-sample WMW Test Sample Sizes (a = 0.05, ft = 0.1, s = 3, A = 2) ! Sample Sizes for Two SampleWilcoxon Mann Whitney Test Based on Specified Values of Decision Paramelers/DQQs (Data Quafcy Objectives) Date/Time of Computation 2/26/201012:18:47 PM User Selected Options False R ejection R ate [Alpha] 0.05 False Acceptance R ate [B eta] 0.1 Width of G ray R egion [D elta] 2 Estimate of Standard Deviation 3 Approximate Minimum Sample Size Single Sided Alternative Hypothesis: 46 Two Sided Alternative Hypothesis: 56 8. 3. 3 Sample Size for WMW Test Suggested by Noether(198 7) For the two-sample WRS test (WMW test), the MARSSIM guidance document (EPA 2000) uses the following combined sample size formula suggested by Noether (1987). The combined sample size, N=(m+n) equation for the one-sided alternative hypothesis defined in Case I (Section 8.3.2.1) and Case II (Section 83.2.2) above is given as follows: -- N = m + n = - - — , where 3(P-0.5) A = Width of the gray region sd = an estimate of the common standard deviation of the two populations. P = ------- Example: An example illustrating these sample size calculations is discussed as follows. In the following example, ProUCL computes the sample size to be 46 for the single sided alternative hypothesis and 56 for the two sided alternative hypothesis when the user selects the default values of the decision parameters. Using Noether's formula (as used in MARSSIM document), the combined sample size, N= m + n (assuming m = n) is 87 for the single sided alternative hypothesis, and 107 for the two sided alternative hypothesis. Output for two sample WMW Test sample sizes (a = 0.05, /? = 0.1, s = 3, A = 2) Sample Sizes for Two S ample WilcoKon-M ann-WHney Test B ased on S pecified Values of D ecision Parameters/D Q 0 s (D ata Q uafty Objectives) D ate/T ime of Computation 7/23/2010 11:58:40 AM User Selected Options False R ejection R ate [Alpha] 0.05 False Acceptance R ate [B eta] 0.1 Width of Gray Region [Delta] 2 Estimate of Standard Deviation 3 Approximate Minimum Sample Size Single Sided Alternative Hypothesis: 46 Two Sided Alternative Hypothesis: 56 MARSSIM WRS Test (Noether, 1987) Approximate Minimum Sample Size Single Sided Alternative Hypothesis: 87 Two Sided Alternative Hypothesis: 107 8.4 Acceptance Sampling for Discrete Objects ProUCL can be used to determine the minimum number of discrete items that should be sampled, from a lot consisting of n discrete items, to accept or reject the lot (drums containing hazardous waste) based upon the number of defective items (e.g., mean contamination above an action level, not satisfying a characteristic of interest) found in the sampled items. This acceptance sampling approach is specifically useful when the sampling is destructive, that is an item needs to be destroyed (e.g., drums need to be sectioned) to determine if the item is defective or not. The number of items that need to be sampled is determined for the allowable number of defective items, d= 0, 1, 2, ...,«. The sample size determination is not straight forward as it involves the use of the beta and hypergeometric distributions. Several researchers (Scheffe and Tukey 1944; Laga and Likes 1975; Hahn and Meeker 1991) have developed statistical methods and algorithms to compute the minimum number of discrete objects that should be sampled to meet specified (desirable) decision parameters. These methods are based upon nonparametric tolerance limits. That is, computing a sample size so that the associated UTL will not exceed the acceptance threshold of the characteristic of interest. The details of the terminology and algorithms used for acceptance sampling of lots (e.g., a batch of drums containing hazardous waste) can be found in the RCRA guidance document (EPA 2002c). In acceptance sampling, sample sizes based upon the specified values of decision parameters can be computed using the exact beta distribution (Laga and Likes 1975) or the approximate chi-square distribution (Scheffe and Tukey 1944). Exact as well as approximate algorithms have been incorporated 251 ------- in ProUCL 4.1 and higher versions of ProUCL. It is noted that the approximate and exact results are often in complete agreement for most values of the decision parameters. A brief description now follows. 8.4.1 Acceptance Sampling Based upon Chi-square Distribution The sample size, n, for acceptance sampling using the approximate chi-square distribution is given by: m-\ ( (l + p) ] 2f , n = ~j~ + \ 4/1 \ *«(2/M) (8'17) v V / J Where: m = number of non-conforming defective items (always >\,m = \ implies '0' exceedance rule) p=\- proportion proportion = pre-specified proportion of non-conforming items a = 1 - confidence coefficient, and Xa 2m = th£ cumulative percentage point of a chi-square distribution with 2m df, the area to the left of xl,im is «. 8.4.2 Acceptance Sampling Based upon Binomial/Beta Distribution Let x be a random variable with arbitrary continuous probability density function f(x). Let xi p\ = \-a (8-18) The statement given by (8-18) implies that the interval (xr, xn+l_s) contains at least a proportion, p, of the distribution with the probability, (1 - a). The interval, (xr,xn+l_s), whose endpoints are the rth smallest and sth largest observations in a sample size of n, is a nonparametric 100p% tolerance interval with a confidence coefficient of (1 - a), and xr and xn+i-s are the lower and upper tolerance limits respectively. xn+l-s The variable z = f(x)dx has the following beta probability density function: x, g(z) = — -z™(\-z)m-\ 0------- The probability P (z >p) can be expressed in terms of binomial distribution as follows: n}ptl-p"-t (8-20) For given values of m, p and a, the minimum sample size, n, for acceptance sampling is obtained by solving the inequality: P(z>p)>l-a (8-21) Where: #2 = number of non-conforming items (always greater than 1) p = 1 -proportion proportion = pre-specified proportion of non-conforming items; and a = 1 - confidence coefficient. An example output generated by ProUCL is given as follows. Example 8-9. Output Screen for Sample Sizes for Acceptance Sampling (default options) ! Acceptance Sampling for Pre-specified Proportion of Non-confomwig Items Based on Specified Values of Decision Paiameters/DQOs D ate/T ime of Computation 2/26/2010 12:20:34 PM User Selected Options Confidence Coefficient 0.95 Pre-specified proportion of non-conforming items in the lot 0.05 Number of allowable non-conforming items in the lot 0 Approximate Minimum Sample Size Exact Binomial/Beta Distribution 59 Approximate Chisquare Distribution (Tukey-Scheffe) 59 253 ------- CHAPTER 9 Oneway Analysis of Variance Module Both parametric and nonparametric Oneway Analysis of Variance (ANOVA) methods are available in ProUCL 5.0 under the Statistical Tests module. A brief description of Oneway ANOVA is described in this chapter. 9.1 Oneway Analysis of Variance (ANOVA) In addition to the two-sample hypothesis tests, ProUCL software has Oneway ANOVA to compare the location (mean, median) parameters of more than two populations (groups, treatments, monitoring wells). Both classical and nonparametric ANOVA are available in ProUCL. Classical Oneway ANOVA assumes the normality of all data sets collected from the various populations under comparison; classical ANOVA also assumes the homoscedasticity of the populations that are being compared. Homoscedasticity means that the variances (spread) of the populations under comparisons are comparable. Classical Oneway ANOVA represents a generalization of the two-sample t-test (Chapter 6). ProUCL has GOF tests to evaluate the normality of the data sets but a formal F-test to compare the variances of more than two populations has not been incorporated in ProUCL. The users may want to use graphical displays such as side-by-side box plots to compare the spreads present in data sets collected from the populations that are being compared. A nonparametric Oneway ANOVA test: Kruskal-Wallis (K-W) test is also available in ProUCL. The K-W test represents a generalization of the two-sample WMW test described in Chapter 6. The K-W test does not require the normality of the data sets collected from the various populations/groups. However, for each group, the distribution of the characteristic of interest should be continuous and those distributions should have comparable shapes and variabilities. 9.1.1 General Oneway ANOVA Terminology Statistical terminology used in Oneway ANOVA is described as follows: g: number of groups, populations, treatments under comparison /': an index used for the ith group, / = 1, 2, ... , g nf. number of observations in the ith group j: an index used for they* observation in a group; for the ith,j = 1,2, ...,«, xtf. the j* observation of the response variable in the /'* group n: total number of observations= nl+n2+... x. . = sum of all observations in the ith group j=i J. = mean of the observations collected from the ith group x = mean of all, nt (the observations) jut = true (unknown) mean of the /'th group 254 ------- In Oneway ANOVA, the null hypothesis, Ho, is stated as: the g groups under comparison have equal means (medians) and that any differences in the sample means/medians are due to chance. The alternative hypothesis, HA is stated as: the means/medians of the g groups are not equal. The decision to reject or accept the null hypothesis is based upon a test statistic computed using the available data collected from the g groups. 9.2 Classical Oneway ANOVA Model The ANOVA model is represented by a regression model in which the predictor variables are the treatment or group variables. The Oneway ANOVA model is given as follows: xtj=»t+etj (9-1) Where ^ is the population mean (or median) of the ith group, and errors, e^, are assumed to be independently and normally distributed with mean = 0 and with a constant variance, a2. All observations in a given group have the same expectation (mean) and all observations have the same variance regardless of the group. The details of Oneway ANOVA can be found in most statistical books including the text by Kunteretal. (2004). The null and the alternative hypotheses for Oneway ANOVA are given as follows: HA : At least one of the means (or medians) is not equal to others Based upon the available data collected from the g groups, the following statistics are computed. ProUCL summarizes these results in an ANOVA Table. • Sum of Squares Between Groups is given by: g , 99 —'Vwfv—vl (Q 7^ 00Between Groups £^ ' \ ' I \y~^> i=l • Sum of Squares Within Groups is given by: 99 =VV(r -rV CQ T> ^Within Groups Zj Zj V W ') \ ' • Total Sum of Squares is given by: (9-4) Between Groups Degrees of Freedom (df): g-1 Within Groups df. n-g 255 ------- etween Groups • Total df. n-1 • Mean Squares Between Groups is given by: ss, yl fC^ L ^ Between Groups ~ ^ \^~^7 • Mean Squares Within Groups: TI v-r> Within Groups ,„ ,, Within Groups \ J n-g • Scale estimate is given by: S = ^MSmthm Groups (9-7) • R2 is given by: ee 7?2 = 1 mthin Groups ,„ „, OO ^ ' • Decision statistic, F, is given by: T^ t^, <• <• Between Groups /r. r.\ F Statistic = — (9-9) Within Groups Under the null hypothesis, the F-statistic given in equation (9-9) follows the F(g.i), (n-g) distribution with (g-1) and (n-g) degrees of freedom, provided the data sets collected from the g groups follow normal distributions. ProUCL software computes/>-values using the F distribution, F(g.i), (n-g). Conclusion: The null hypothesis is rejected for all levels of significance, a >/rvalue. 9.3 Nonparametric Oneway ANOVA (Kruskal-Wallis Test) Nonparametric Oneway ANOVA or the K-W test (Kruskal and Wallis 1952, Hollander and Wolfe 1999) represents a generalization of the two-sample WMW, test which is used to compare the equality of medians of two groups. Like the WMW test, analysis for the K-W test is also conducted on ranked data, therefore, the distributions of the g groups under comparisons do not have to follow a known statistical distribution (e.g., normal). However, distributions of the g groups should be continuous with comparable shapes and variabilities. Also the g groups should represent independently distributed populations. 256 ------- The null and alternative hypotheses are defined in terms of medians, /w* of the g groups: (9-10) Hn :m, = 01 HA : At least one of the g medians is not equal to others While performing the K-W test, all n observations in the g groups are arranged in ascending order with the smallest observation receiving the smallest rank and the largest observation getting the highest rank. All tied observations receive the average rank of those tied observations. K-W Test on Data Sets with NDs: It should be noted that the K-W test may be used on data sets with NDs provided all NDs are below the largest detected value. All NDs are considered as tied observations irrespective of reporting limits (RLs) and receive the same rank. However, the performance of the K-W test on data sets with NDs is not well studied; therefore, it is suggested that the conclusion derived using the K-W test statistics be supplemented with graphical displays such as side-by-side box plots. Side-by- side box plots can also be used as an exploratory tool to compare the variabilities of the g populations based upon the g data sets collected from those populations. The K-W ANOVA table displays the following information and statistics: • Mean Rank of the ith Group, R^ : Average of the ranks (in the combined data set of size, «) of the Hi observations in the ith group. • Overall Mean Rank, R : Average of the ranks of all n observations. • Z-value of each group are computed using the following equation (Standardized Z): Z,= ., *- ., (9-H) 12 n = total number of observations = nl+n2+ ... + n Hi = observation in the ith group g = number of groups Zj given by (9-11) represents standardized normal deviates. The Z, can be used to determine the significance of the difference between the average rank of the ith group and the overall average rank, R, of the combined data set of sized n. • Kruskal-Wallis H-Statistic (without ties) is given by: - (9-12) n(n + l) 257 ------- K-W H-Statistic adjusted for ties is given by: H H adj-ties i & S',3-', n -n (9-13) Where tt = number of tied values in ith group For large values of n, the H-statistic given above follows an approximate chi-square distribution with (g- 1) degrees of freedom, /'-values associated with the H-statistic given by (9-12) and (9-13) are computed by using a chi-square distribution with (g-1) degrees of freedom. The /"-values based upon a chi-square approximation test are fairly accurate when the number of observations, n, is large such as > 30. Conclusion: The null hypothesis is rejected in favor of the alternative hypothesis for all levels of significance, a >/>-value. Example 9-1. Consider Fisher's famous Iris data set (Fisher 1936) with 3 iris species. The classical Oneway ANOVA results comparing petal widths of 3 iris species are summarized as follows. Classical Oneway ANOVA Date/Time erf Computation 3/2/2013 1:25:29 PM From File FULLIRISjds Full Precision OFF pt-width Group Obs 1 50 2 50 3 50 Grand Statistics (All data) 153 Mean SD Variance 0.246 0.105 0.0111 1.326 0.198 0.0391 2.026 0.275 0.07.54 1.133 0.762 0.581 Classical One-Way Analysis of Variance Table Source SS DOF Between Groups 80.41 2 Within Groups 6.157 147 Total 36.57 149 MS V.R.(FStat) P-Value 40.21 360 0 0.0419 Pooled Standard Deviation 0.205 R-Sq 0.929 Note: A p-value <= 0.05 (or some other selected level) suggests that there are significant differences in mean/median characteristics of the various groups at 0.05 or other selected level of significance A p-value > 0.05 (or other selected level) suggests that mean/median characteristics of the various groups are comparable. 258 ------- Example 9-2 (Iris Data). The K-W Oneway ANOVA results comparing petal widths of 3 iris species are summarized as follows. Nonparametric Oneway ANGVATKnlskai-Wallis Test) Date/Time of Computation 3/2/2013 123:12 P M From File FLJLLIRISjds Full Precision OFF pt-width Group 1 2 3 Overall Obs 50 50 50 150 Median 0.2 1.3 2 1,3 Ave Rank 25,5 76.43 124.5 75,5 Z -3.9€7 0.195 9.771 K-W(H-Stat) DOF P-Value {Apprax. Chisquare) 129.9 2 0 131.2 2 0 {Adjusted for Ties} Note: A p value <= 0.05 (or some other selected level) suggests that there are significant differences in mean/median characteristics of the various groups at 0.05 or other selected level of significance A p value > 0.05 (or other selected level) suggests that mean/median characteristics of the various groups are comparable 259 ------- CHAPTER 10 Ordinary Least Squares Regression and Trend Analysis Trend tests and ordinary least squares (OLS) regression methods are used to determine trends (e.g., decreasing, increasing) in time series data sets. Typically, OLS regression is used to determine linear relationships between a dependent response variable and one or more predictor (independent) variables (Draper and Smith 1998); however statistical inference on the slope of the OLS line can also be used to determine trends in the time series data used to estimate an OLS line. A couple of nonparametric statistical tests, the Mann-Kendall (M-K) test and the Theil-Sen (T-S) test to perform trend analysis have also been incorporated in ProUCL 5.0/ProUCL 5.1. Methods to perform trend analysis and OLS Regression with graphical displays are available under the Statistical Tests module of ProUCL 5.1. In environmental monitoring studies, OLS regression and trend tests can be used on time series data sets to determine potential trends in constituents' concentrations over a defined period of time. Specifically, the OLS regression with time or a simple index variable as the predictor variable can be used to determine a potential increasing or decreasing trend in mean concentrations of an analyte over a period of time. A significant positive (negative) slope of the regression line obtained using the time series data set with predictor variable as a time variable suggests an upward (downward) trend. A brief description of the classical OLS regression as function of the time variable, T(t), is described as follows. It should however be noted that the OLS regression and associated graphical displays can be used to determine a linear relation for any pair of dependent variable, Y, and independent variable, X. The independent variable does not have to be a time variable. 10.1 Ordinary Least Squares Regression The linear regression model for a response variable, Y and a predictor (independent) variable, t is given as follows: Y = bn + b,t + e: (10-1) E[Y] = b0+ bj = mean response at t In (10-1), variable e is a random variable representing random measurement error in the response variable, Y (concentrations). The error variable, e, is assumed to follow a normal distribution, N (0, o2), with mean 0 and unknown variance, a2. Let (tit yj; i: =1, 2,....n represent the paired data set of size n, where yt is the measured response when the predictor variable, t =t\. It is noted that multiple observations may be collected at one or more values of the prediction variable, t. Using the regression model (10-1) on this data set, we have: l ! (10-2) E[yt ] = b0+ blti = mean response when t = tt 260 ------- For each fixed value, tt of the predictor variable, t, the random error,et is normally distributed with NfO,^2). Random errors, et, are independently distributed. Without the random error, e, all points will lie exactly on the population regression line estimated by the OLS line. The OLS estimates of the intercept, b0 and slope, bi are obtained by minimizing the residual sum of squares. The details of deriving the OLS estimates, b0 and \ of the intercept and slope can be found in Draper and Smith (1998). The OLS regression method can be used to determine increasing or decreasing trends in the response variable Y (e.g., constituent concentrations in a MW) over a time period (e.g., quarters during a 5 year time period). A positive statistically significant slope estimate suggests an upward trend and a statistically significant negative slope estimate suggests a downward or decreasing trend in the mean constituent concentrations. The significance of the slope estimate is determined based upon the normal assumption of the distribution of error terms, ef, and therefore, of responses, yit i:=l,2,...,n ProUCL computes OLS estimates of parameters bo and bf, performs inference about the slope and intercept estimates, and outputs the regression ANOVA table including the coefficient of determination, R2, and estimate of the error variance, a2. Note that R2 represents the square of the Pearson correlation coefficient between the dependent response variable, y, and the independent predictor variable, t. ProUCL also computes confidence intervals and prediction intervals around the OLS regression line; and can be used to generate scatter plots of n pairs, (t, y), displaying the OLS regression line, confidence interval for mean responses, and prediction interval band for individual observations (e.g., future observations). General OLS terminology and sum of squares computed using the collected data are described as follows: (10-3) The OLS estimates of slope and intercept are given as follows: =Sty/Stt; and (10-4) The estimated OLS regression line is given by: y = b0 + bj and error estimates also called residuals are given by ef =yi—yi', i = 1,2,...., n . It should be noted that for each /', j>. represents the mean response at value, tt of the predictor variable, t, for i:=l,2,... ,n. The residual sum of squares is given by: SSE = £(yt-yt? (10-5) z=l 261 ------- Estimate of the error variance, a2, and variances of the OLS estimates, b0 and 1\ are given as follows: 0; and 3) or Ho: bv =0 vs. the alternative, Hi: bv < 0 . Under the null hypothesis, the test statistic is obtained by dividing the regression estimate by its SE: t = bJSE(b^) (10-7) Under normality of the responses, yt (and the random errors, e,), the test statistic given in (10-7) follows a Student's t-distribution with (n-2) degrees of freedom (df). A similar process is used to perform inference about the intercept, b0 of the regression line. The test statistic associated with the OLS estimate of the intercept, b0 also follows a Student's t-distribution with (n-2) degrees of freedom. P -values: ProUCL computes and outputs t-distribution based />-values associated with the two-sided alternative hypothesis, Hi: 4^ ^0. The /"-values are displayed on the output sheet as well as on the regression graph generated by ProUCL. Note: ProUCL displays residuals including standardized residuals on the OLS output sheet. Those residuals can be imported (copying and pasting) in an excel data file to assess the normality of those OLS residuals. The parametric trend evaluations based upon the OLS slope (significance, confidence interval) are valid provided the OLS residuals are normally distributed. Therefore, it is suggested that the user assesses the normality of OLS residuals before drawing trend conclusions using a parametric test based upon the OLS slope estimate. When the assumptions are not met, one can use graphical displays and nonparametric trend tests, M-K and T-S tests, to determine potential trends in time series data set. 262 ------- 10.1.1 Regression ANOVA Table The following statistics are displayed on the regression ANOVA table. Sum of Squares Regression (SSR): SSR represents that part of the variation in the response variable, Y, which is explained by the regression model, and is given by: Sum of Squares Error (SSE): SSE represents that part of the variation in the response variable, Y, which is attributed to random measurement errors, and is given by: Sum of Squares Total (SST): SST is the total variation present in the response variable, Y and is equal to the sum of SSR and SSE. SST = yj-y2=SSR + SSE (10-9) z=l Regression Degrees of Freedom (df): 1 (1 predictor variable) Error df. n-2; and Total df. n-1 Mean Sum of Squares (MS) Regression (MSR): is given by SSR divided by the regression df which is equal to 1 in the present scenario with only one predictor variable. MSR = SSR Mean Sum of Squares Error (MSB): is given by SSE divided by the error degrees of freedom MSE=SSE_ n-2 MSE represents an unbiased estimate of the error variance, ------- MSE (HMO, P -value: The overall p-value associated with the regression model is computed using the Fi,(n-2) distribution of the test- statistic given by equation (10-10). R2: represents the variation explained in the response variable, Y, by the regression model, and is given b: (10-11) SST Adjusted R square (Adjusted R2): The adjusted R2 is considered a better measure of the variation explained in the response variable, Y, and is given by: R2 =1 (n~l}SSE ad]usted (n-2JSST 10.1.2 Confidence Interval and Prediction Interval around the Regression Line ProUCL also computes confidence and prediction intervals around the regression line and displays these intervals along with the regression line on the scatter plot of the paired data used in the OLS regression. ProUCL generates, when selected, a summary table displaying these intervals and residuals. Confidence Interval (LCL. UCL): represents a band within which the estimated mean responses, j>. , are expected to fall with specified confidence coefficient, (1-a). Upper and lower confidence limits (LCL and UCL) are computed for each mean response estimate, j>. , observed at value, tt, of the predictor variable, t. These confidence limits are given by: Where the estimated standard deviation, sd(yf) , of the mean response, j). , is given by: Sd[y,]= MSE(- + ^-^-)-i = l,2,...,n \ n sa A confidence band can be generated by computing the confidence limits given by (10-12) for each value, tt of the predictor variable, t; i: =1,2, ...n. Prediction Limits (LPL. UPL): represents a band within which a predicted response (and not the mean response), j>0, for a specified new value, to ,of the predictor variable, t, is expected to fall. Since the variances of the individual predicted responses are higher than the variances of the mean responses, a prediction band around the OLS line is wider than the confidence band. The LPL and UPL comprising the prediction band are given by: 264 ------- ± t , with = (10-13) Where the estimated standard deviation, sd(y0~), of a new response, y0 ,(or the individual response for existing observations) is given by: n S Like the confidence band, a prediction band around the OLS line can be generated by computing the prediction limits given by (10-13) for each value, t\, of the predictor variable, t, and also other values oft (within the experiment range) for which the response, y, was not observed. Notes: Unlike M-K and T-S trend tests, multiple observations may be collected at one or more values of the predictor variable. Specifically, OLS can be performed on data sets with multiple measurements collected at one or more values of the predictor variable (e.g., sampling time variable, t). Example 10-1. Consider the time series data set for sulfate as described in RCRA Guidance (EPA 2009). The OLS graph with relevant test statistics is shown in Figure 10-1 below. The positive slope estimate, 33.12, is significant with ap-value of 0 suggesting that there is an upward trend in sulfate concentrations. Classical Regression Slope Intercept Scale Estimate P-value (Reg) P-vatue (Slope) 22 33.1230 -2,503 27S8 07372 0.8586 47.4355 0.0000 SDofS Staroiardized S Approxi mate p-val ue Confidence Coefficient Red = Prediction Interval 137.0000 35.3977 5.2546 Figure 10-1. OLS Regression of Sulfate as a Function of Time 265 ------- Number Reported ^-values) 22 Dependendant Variable Sulfate Independent Variable Date Regression Estimates and Inference Table Parameter Estimates Std. Error T-values p-values intercept -2503 410.7 -€.095 5.8853E-6 Date 33.12 4.422 7.49 3.1763E-7 OLS ANOVA TaWe Source of Variation SS DOF MS F-Value P-Value Regression 12S230 1 126230 56.1 Bror 45003 20 2250 Total 171233 21 R Square 0.737 Adjusted R Square 0.724 Sqrt(MSE) = Scale 47.44 10.2 Trend Analysis Time Series Data Set: When the predictor variable, t, represents a time variable (or an index variable), the data set ftt, yj; i:=l,2,....n is called a time series data set, provided values of the variable, t, satisfy: ti------- Handling Nondetects: The trend module in ProUCL 5.1 does not recognize a nondetect column consisting of zeros and ones. For data sets consisting of nondetects with varying DLs, one can replace all NDs with half of the lowest DL (DL/2) or by replacing all NDs by a single value lower than the lowest DL. When multiple DLs are present in a data set, the use of substitution methods should be avoided. Replacing NDs by their respective DLs or by their DL/2 values is like performing trend test on DLs or on DL/2s, especially when the percentage of NDs present in the data set is high. 10.2.1 Mann-Kendall Test The M-K trend test is a nonparametric test which is used on a time series data set, (tit yt); i:=l,2,... .n as described earlier. As a nonparametric procedure, the M-K test does not require the underlying data to follow a specific distribution. The M-K test can be used to determine increasing or decreasing trends in measurement values of the response variable, y, observed during a certain time period. If an increasing trend in measurements exists, then the measurement taken first from any randomly selected pair of measurements should, on average, have a lower response (concentration) than the measurement collected at a later point. The M-K statistic, S, is computed by examining all possible distinct pairs of measurements in the time series data set and scoring each pair as follows. It should be noted that for a measurement data set of size, n, there are n(n-l)/2 distinct pairs, (yj,yt) withj>/, which are being compared. • If an earlier measurement, yt, is less in magnitude than a later measurement, j;, then that pair is assigned a score of 1; • If an earlier measurement value is greater in magnitude than a later value, the pair is assigned a score of-1; and • Pairs with identical (y* = yj) measurements values are assigned a score of 0. The M-K test statistic, S, equals the sum of scores assigned to all pairs. The following conclusions are derived based upon the values of the M-K statistic, S. • A positive value of S implies that a majority of the differences between earlier and later measurements are positive suggesting the presence of a potential upward and increasing trend overtime. • A negative value for S implies that a majority of the differences between earlier and later measurements are negative suggesting the presence of a potential downward/decreasing trend. • A value of S close to zero indicates a roughly equal number of positive and negative scores assigned to all possible distinct pairs, (yj,yt) withy>/, suggesting that the data do not exhibit any evidence of an increasing or decreasing trend. When no trend is present in time series measurements, positive differences in randomly selected pairs of measurements should balance negative differences. In other words, the expected value of the test statistic S, E[S], should be close to '0' when the measurement data set does not exhibit any evidence of a trend. To account for randomness and inherent variability in measurements, the statistical significance of the M- K test statistic is determined. The larger the absolute value of S, the stronger the evidence for a real increasing or decreasing trend. The M-K test in ProUCL can be used to test the following hypotheses: 267 ------- Null Hypothesis, Ho: Data set does not exhibit sufficient evidence of any trends (stationary measurements) vs. • HA: Data set exhibits an upward trend (not necessarily linear); or • HA: Data set exhibits a downward trend(not necessarily linear); or • HA: Data set exhibits a trend (two-sided alternative - (not necessarily linear)). Under the null hypothesis of no trend, it is expected that the mean value of S =0; that is E[S] =0. Notes: The M-K test in ProUCL can be used for testing a two-sided alternative, HA, stated above. For a two-sided alternative hypothesis, the />-values (exact as well as approximate) reported by ProUCL need to be doubled. 10.2.1.1 Large Sample Approximation for M-K Test When the sample size n is large, the exact critical values for the statistic S are not readily available. However, as a sum of identically-distributed random quantities, the distribution of S tends to approximately follow a normal distribution by the CLT. The exact /"-values for the M-K test are available for sample sizes up to 22 and have been incorporated in ProUCL. For samples of sizes larger than 22, a normal approximation to S is used. In this case, a standardized ^-statistic, denoted by Z is computed by using the expected mean value and sd of the test statistic, S. The sd ofS, sd(S) is computed using the following equation: (10-14) Where n is the sample size, g represents the number of groups of ties (if any) in the data set, and /}• is the number of tied observations in the f1 group of ties. If no ties or NDs are present, the equation reduces to the simpler form: (10-15) The standardized S statistic denoted by Z for an increasing (or decreasing) trend is given as follows: Z = ^^- ifS>0; sd(S) Z = 0 if S = 0;and (10-16) ifs<0 sd(S) Like the S statistic, the sign of Z determines the direction of a potential trend in the data set. A positive value of Z suggests an upward (increasing) trend and a negative value of Z suggests a downward or decreasing trend. The statistical significance of a trend is determined by comparing Z with the critical 268 ------- value, z«, of the standard normal distribution; where za represents that value such that the area to the right of za under the standard normal curve is a. 10.2.1.2 Step-by-Step Procedure to perform the Mann-Kendall Test The M-K test does not require the availability of an event or a time variable. However, if graphical trend displays (e.g., T-S line) are desired, the user should provide the values for a time variable. When a time or an event variable is not provided, ProUCL generates an index variable and displays the time-series graph using the index variable. Step 1. Order the measurement data: yi, y?, ...., yn by sampling event or time of collection. If the numerical values of data collection times (event variable) are not known, the user should enter data values according to the order they were collected. Next, compute all possible differences between pairs of measurements, (yj-yi) for/ > /'. For each pair, compute the sign of the difference, defined by: sgr 0 Step 2. Compute the M-K test statistic, S, given by the following equation: (10-18) In the above equation the summation starts with a comparison of the very first sampling event against each of the subsequent measurements. Then the second event is compared with each of the samples taken after it (i.e., the third, fourth, and so on). Following this pattern is probably the most convenient way to ensure that all distinct pairs have been considered in computing S. For a sample of size n, there will be «(«-l)/2 distinct pairs, (i,j) withy>/. Step 3. For n<23, the tabulated critical levels, acp (tabulated />-values) given in Hollander and Wolfe (1999), have been incorporated in ProUCL. If S > 0 and a > acp, conclude there is statistically significant evidence of an increasing trend at the a significance level. If S < 0 and a> acp, conclude there is statistically significant evidence of a decreasing trend. If a < acp, conclude that data do not exhibit sufficient evidence of any significant trend at the a level of significance . Specifically, the M-K test in ProUCL tests for one-sided alternative hypothesis as follows: Ho: no trend vs. HA: upward trend or Ho: no trend vs. HA: downward trend ProUCL computes tabulated /"-values (for sample sizes <23) based upon the sign of the M-K statistic, S, as follows: 269 ------- IfS>0, the tabulated p-value (acp) is computed for Ho: no trend, vs. HA: upward trend IfS<0, the tabulated p-value (acp) is computed for Ho: no trend vs. HA. downward trend If the p-value is larger than the specified a (e.g., 0.05), the null hypothesis of no trend is not rejected. Step 4. For n > 22, large sample normal approximation is used for S, and a standardized S is computed. Under the null hypothesis of no trend, E(S) =0, and the sd is computed using equations (10-14) or (10-15). When ties are present, sd(S) is computed by adjusting for ties as given in (10-14). Standardized S, denoted by Z is computed using equation (10-16). Step 5. For a given significance level (a), the critical value za is determined from the standard normal distribution. If Z >0, a critical value and p-value are computed for Ho: no trend, vs. HA: upward trend. If Z<0, a critical value and p-value are computed for Ho: no trend vs. HA: downward trend If the p-value is larger than the specified a (e.g., 0.05), the null hypothesis of no trend is not rejected. Specifically, compare Z against this critical value, za. If Z>0 and Z > za, conclude there is a statistically significant evidence of an increasing trend at an a-level of significance. If Z<0 and Z < -za, conclude there is statistically significant evidence of a decreasing trend. If neither exists, conclude that the data do not exhibit sufficient evidence of any significant trend. For large samples, ProUCL computes the p-value associated with Z. Notes: As mentioned, the M-K test in ProUCL can be used for testing a two-sided alternative, HA stated above. For a two-sided alternative hypothesis, /"-values (both exact and approximate) reported by ProUCL need to be doubled. Example 10-2. Consider a nitrate concentration data set collected over a period of time. The objective is to determine if there is a downward trend in nitrate concentrations. No sampling time event values were provided. The M-K test has been used to establish a potential trend in nitrate concentrations. However, if the user also wants to see a trend graph, ProUCL generates an index variable and displays the trend graph along with OLS line and the T-S nonparametric line (based upon the index variable) as shown in Figure 10-2 below. Figure 10-2 displays all the statistics of interest. The M-K trend statistics are summarized as follows. 270 ------- Mann-Kendall Trend Test Analysis User Selected Options Date/Time of Computation From Rle Full Precision Confidence Coefficient Level of Significance 3/2/2013 3:45:38 PM Trend-data forNitrate_a jds OFF 0.95 0.05 Nitrate General Satisfies Number Values Number Values Missing Number Values Reported (n) Minimum Maximum Mean Geometric Mean Median Standard Deviation 2M 2 202 9.312 19.96 14.29 14.2 13.96 1.688 Mann-Kendall Test Test Value (S) -4684 Critical Value (0.05) -1.645 Standard Deviation of S 960.5 Standardized Value of S -4.876 Approximate p-value 5.4240E-7 Statistically significant evidence of a decreasing trend at the specified level of significance. Mann-Kendall Trend Test Mann-Kendall Trend Analysis Confidence Coefficient Level of Significance StaraJard Deviation of S Standardized Value of S Test Value (SI Appx. Critical Value (0.05) 202.0000 03500 0.0500 -1.6449 OOCOO OLS Regres sion Ljne (Blue) OLS Regression Slope -0.0102 OLS Regression. Intercept 15.3308 Theil-Sen Trend line (I Thai-Sen Slope Theil-Sen Intercept Statistically significant evidence -0.0093 14.9034 Generated Index Figure 10-2. Trend Graph with M-K Test Results and OLS Line and Nonparametric Theil-Sen Line 271 ------- 10.2.2 Theil - Sen Line Test The details of T-S test can be found in Hollander and Wolfe (1999). The T-S test represents a nonparametric version of the parametric OLS regression analysis and requires the values of the time variable at which the response measurements were collected. The T-S procedure does not require normally distributed trend residuals and responses as required by the OLS regression procedure. It is also not critical that the residuals be homoscedastic (having equal variance over time). For large samples, even a relatively mild to modest slope of the T-S trend line can be statistically significantly different from zero. It is best to first identify whether or not a significant trend (slope) exists, and then determine how steeply the concentration levels are increasing (or decreasing) over time for a significant trend. New in ProUCL 5.1: This latest ProUCL 5.1 version computes yhat values and residuals based upon the Theil-Sen nonparametric regression line. ProUCL outputs the slope and intercept of the T-S trend line, which can be used to compute residuals associated with the T-S regression line. Unlike the M-K test, actual concentration values are used in the computation of the slope estimate associated with the T-S trend test. The test is based upon the idea that if a simple slope estimate is computed for every pair (n(n-l)/2 pairs in all) of distinct measurements in the sample (known as the set of pairwise slopes), the average of this set of n(n-l)/2 slopes would approximate the true unknown slope. Since the T-S test is a nonparametric test, instead of taking an arithmetic average of the pairwise slopes, the median slope value is used as an estimate of the unknown population slope. By taking the median pairwise slope instead of the mean, extreme pairwise slopes - perhaps due to one or more outliers or other errors - are ignored and have little or negligible impact on the final slope estimator. The T-S trend line is also nonparametric because the median pairwise slope is combined with the median concentration value and the median of the time values to construct the final trend line. Therefore, the T-S line estimates the change in median concentration over time and not the mean as in linear OLS regression; the parametric OLS regression line described in Section 10.1 estimates the change in the mean concentration overtime (when the dependent variable represents the time variable). Averaging of Multiple Measurements at Sampling Events: In practice, when multiple observations are collected/reported at one or more sampling events (times), one or more pairwise slopes may become infinite, resulting in a failure to compute the T-S test statistic. In such cases, the user may want to pre- process the data before using the T-S test. Specifically, to assure that only one measurement is available at each sampling event, the user pre-processes the time series data by computing average, median, mode, minimum, or maximum of the multiple observations collected at those sampling events. The T-S test in ProUCL 5.1 provides the option of averaging multiple measurements collected at the various sampling events. This option also computes M-K test and OLS regression statistics using the averages of multiple measurements collected at the various sampling event. Note: The OLS regression and M-K test can be performed on data sets with multiple measurements taken at the various sampling time events. However, often it is desirable to use the averages (or median) of measurements taken at the various sampling events to determine potential trends present in a time-series data set. 272 ------- 10.2.2. 1 Step-by-Step Procedure to Compute Theil-Sen Slope Step 1. Order the data set by sampling event or time of collection of those measurements. Let yi, j2, ...,yn represent ordered measurement values. Consider all possible distinct pairs of measurements, (yit y}) for/ > /'. For each pair, compute the simple pairwise slope estimate given by: j , . m = — — for j For a time-series data set of size n, there are N=n(n-\)/2 such pairwise slope estimates, /%. If a given observation is a ND, one may use half of the DL or the RL as its estimated concentration. Alternatively, depending upon the distribution of detected values (also called the censored data set), the users may want to use imputed estimates of ND values obtained using the GROS or LROS method. Step 2. Order the N pairwise slope estimates, /% from the smallest to the largest and re-label them as m(\), m(2),..., m(N). Determine the T-S estimate of slope, Q, as the median value of this set of TV ordered slopes. Computation of the median slope depends on whether N is even or odd. The median slope is computed using the following algorithm: Q = (10-19) .. ,r if N = even Step 3. Arrange the n measurements in ascending order from smallest to the largest value: y(l), y(2), ... , y(n). Determine the median measurement using the following algorithm: y= ( \ / (10-20) K(n/2) +^((n+2)/2) )/ ._f \ { > ^ >V if n = even Similarly, compute the median time, t of the n ordered sampling times: t\, h, to tn by using the same median computation algorithm as used in (10-19) and (10-20). Step 4. Compute the T-S trend line using the following equation: 10.2.2.2 Large Sample Inference for Theil - Sen Test Based upon Normal Approximation As described in Step 2 above, order the N pairwise slope estimates, /% in ascending order from smallest to the largest: #?(!), m(2),..., m(N). Compute S given in (10-18) and its sd given below: 273 ------- (10-21) ProUCL can be used to test the following hypotheses: Ho: Data set does not exhibit sufficient evidence of any trends (stationary measurements) vs. I. HA: Data set exhibits a trend (two-sided alternative) II. HA: Data set exhibits an upward trend; or III. HA: Data set exhibits a downward trend. Case I. Testing for the null hypothesis, Ho.- Time series data set does not exhibit any trend, vs. the two- sided alternative hypothesis, HA: Data Set exhibits a trend. • Compute the critical value, Ca using the following equation: Ca=Za/Sd(S) Compute Mi andA/2 as: and M, = Obtain the Mf largest and Mf largest slopes, (^(MI) ) and yn(M^ j , from the set consisting of all n(n-l)/2 slopes. Then the probability of the T-S slope, Q, lying between these two slopes is given by the statement: On ProUCL output, (m(M^ j is labeled as LCL and (m(M^J is labeled as UCL. • Conclusion: If 0 belongs to the interval, (m,M),m,M)), conclude that T-S test slope is insignificant; that is, conclude that there is no significant trend present in the time series data set. Cases II and III: Test for an upward (downward) trend with Null hypothesis, Ho.- Time series data set does not exhibit any trend, vs. the alternative hypothesis, HA: data set exhibits an upward (downward) trend. • For specified level of significance, a, compute the following: Ca=Za*sd(S) 'N-Cn and M9 = 274 ------- • Obtain the Mf largest and Mf largest slopes, (jn(M^\ and (m(M^\from the set consisting of all n(n-l)/2 slopes. • Conclusion: If WMf) ) > 0 , then the data set exhibits a significant upward trend. If y^(M2) } < 0 5 then the data set exhibits a significant downward trend. Example 10-3. Time series data (time event, concentration) were collected from several groundwater MWs on a Superfund site. The objective is to determine potential trends present in concentration data collected quarterly from those wells over a period of time. Some missing sampling events (quarters) are also present. ProUCL handles the missing values, computes trend test statistics and generates a time series graph along with the OLS and T-S lines. Theil-Sen Trend Line and OLS Regression Line 0 LS Regression Line (Hue) OLS Regression Slope 34.0343 OLS Regression Intercept -2.585 1296 TheM-Sen Trend line (Red) Theil-S£fl Slope 31.4286 Thai-Sen Intercept -2.349.1429 M2 276260 633740 20.0000 LCL of Slope UCLd Slope 4215S7 Figure 10-3. Time Series Plot and OLS and Theil-Sen Results with Missing Values The Excel output sheet, generated by ProUCL and showing all relevant results, is shown as follows: Approximate inference for Theil-Sen Trend Test Mann-Kendall Statistic IS) 72 Standard Deviation of S 1 S.24 Standardized Value at S Vd -Amiss General Statistics Number of Events Number Values Observations Number Values Missing Number Values Reported in) Minimum Maximum Mean Geometric Mean Median Standard Deviation 14 16 2 14 450 700 536. S 533.6 525 62.99 3.SS3 Approximate p^alue 4.9562E-5 Number of Slopes Theil-Sen Slope 51 31.43 Theil-Sen Intercept -2349 MT 3D.5 One-sided 95% lower limit of Slope 21.36 95% LCL of Slope (0.025) 20 35% UCL of Slope (0.975) 42.16 Statistically significant evidence of an increasing trend at the specified level of significance. 275 ------- Notes: As with other statistical tests (e.g., Shapiro-Wilk and Lilliefors GOF tests for normality), it is very likely, that based upon a given data set, the three trend tests described here will lead to different trend conclusions. It is important that the user verifies the underlying assumptions required by these tests (e.g., normality of OLS residuals). A parametric OLS slope test is preferred when the underlying assumptions are met. Conclusions derived using nonparametric tests supplemented with graphical displays are preferred when OLS residuals are not normally distributed. These tests can also yield different results when the data set consists of missing values and/or there are gaps in the time series data set. It should be pointed out that an OLS line (therefore slope) can become significant even by the inclusion of an extreme value (e.g., collected after skipping of several intermediate sampling events) extending the domain of the sampling events time interval. For example, a perfect OLS line can be generated using two points at two extreme ends resulting in a significant slope; whereas nonparametric trend tests are not as influenced by such irregularities in the data collection and sampling events. In such circumstances, the user should draw a conclusion based upon the site CSM, expert and historical site knowledge and expert opinions. 10.3 Multiple Time Series Plots The Time Series Plot option of the Trend Analysis module can generate time series plots for multiple groups/wells comparing concentration levels of those groups over a period of time. Time series plots are also useful for comparing concentrations of a MW during multiple periods (every 2 years, 5 years, ...) collected quarterly, semi-annually. This option can also handle missing sampling events. However, the number of observations in each group should be the same, sharing the same time event variable (if provided). An example time series plot comparing concentrations of three MWs during the same period of time is shown as follows. Time-Series Trend Analysis 6 iverits.'Time Periods Missing Values OL Regression Line (Blue) h l-Sen Trend LJne (Red) Time-Series Trend Analysis Th l LS Regress!, LS Regress LS Regression Slope 5.533.1131 LS Regression Intercept 23,392.5536 heil-Sen Slope 3,623.7500 heil-Sen Intercept 25,178,1250 LS Regression Slope .S Regression Intercept heil-Sen Slope heil-Ser Intercept -2.3839 548716 -6.1500 61.9000 Figure 10-4. Time Series Plot Comparing Concentrations of Multiple Wells over a Period of Time This option is specifically useful when the user wants to compare the concentrations of multiple groups (wells) and the exact sampling event dates are not available (data only option). The user may just want to graphically compare the time-series data collected from multiple groups/wells during several quarters (every year, every 5 years, ...). Each group (e.g., well) defined by a group variable must have the same number of observations and should share the same sampling event values (when available). That is the 276 ------- number of sampling events and values (e.g., quarter ID, year ID, etc.) for each group (well) must be the same for this option to work. However, the exact sampling dates (not needed to use this option) in the various quarters (years) do not have to be the same as long as the values of the sampling quarters (1,3,5,6,7,9, etc.) used in generating the time-series plots for the various groups (wells) match. Using the geological and hydrological information, this kind of comparison may help the project team in identifying non-compliance wells (e.g., with upward trends in constituent concentrations) and associated reasons. 277 ------- CHAPTER 11 Background Incremental Sample Simulator (BISS) Simulating BISS Data from a Large Discrete Background Data The Background Incremental Sample Simulator (BISS) module was incorporated in ProUCLS.O at the request of the Office of Superfund Remediation and Technology Innovation (OSRTI). However, this module is currently under further investigation and research, and therefore it is not available for general public use. This module may be released in a future version of the ProUCL software, along with strict conditions and guidance for how it is applied. The main text for this chapter is not included in this document for the release to general public. Only a brief placeholder write-up is provided here. It is assumed that the user is familiar with the incremental sampling methodology (ISM) ITRC (2012) document and terminologies associated with the ISM approach. Those terminologies (e.g., sample support, decision unit [DU], replicated etc.) are not described in this chapters. The following scenario describes the site or project conditions under which the BISS module could be useful: Suppose there is a long history of soil sample collection at a site. In addition to having a large amount of site data, a robust background data set (at least 30 samples from verified background locations), has also been collected. Comparison of background data to on-site data has been, and will continue to be, an important part of this project's decision-making strategy. All historical data is from discrete samples, including the background data. There is now a desire to switch to incremental sampling for the site. However, guidance for incremental sampling makes it clear that it is inappropriate to compare discrete sample results to incremental sample results. That includes comparing a site's incremental results directly to discrete background results. One option is to recollect all background data in the form of incremental samples from background decision units (DUs) that are designed to match site DUs in geology, area, depth, target soil particle size, number of increments, increment sample support. If project decision-making uses a BTV strategy to compare site DU results one at a time against background, then an appropriate number (the default is no less than 10) of background DU incremental samples would need to be collected to determine the BTV for the population of background DUs. However, if the existing discrete background data show background concentrations to be low (in comparison to site concentrations) and fairly consistent relative standard deviation, RSD <1, there is a second option described as follows. When a robust discrete background data set that meets the above conditions already exists, the following is an alternative to automatically recollecting ALL background data as incremental samples. Step 1. Identify 3 background DUs and collect at least 1 incremental sample from each for a minimum of 3 background incremental samples. Step 2. Enter the discrete background data set (n > 30) and the >3 background incremental samples into the BISS module (the BISS module will not run unless both data sets are entered). • The BISS module will generate a specified (default is 7) simulated incremental samples from the discrete data set. 278 ------- • The module will then run a t-test to compare the simulated background incremental data set (e.g., with n = 7) to the actual background incremental data set (n > 3). o If the t-test finds no difference between the 2 data sets, the BISS module will combine the 2 data sets and determine the statistical distribution, mean, standard deviation, potential UCLs and potential BTVs for the combined data set. Only this information will be supplied to the general user. The individual values of the simulated incremental samples will not be provided. o If the t-test finds a difference between the actual and simulated data sets, the BISS module will not combine the data sets nor provide a BTV. o In both cases, the BISS module will report summary statistics for the actual and simulated data sets. Step 3. If the BISS module reported out statistical analyses from the combined data set, select the BTV to use with site DU incremental sample results. Document the procedure used to generate the BTV in project reports. If the BISS module reported that the simulated and actual data sets were different, the historical discrete data set cannot be used to simulate incremental results. Additional background DU incremental samples will need to be collected to obtain a background DU incremental data set with the number of results appropriate for the intended use of the background data set. The objective of the BISS module is to take advantage of the information provided by the existing background discrete samples. The availability of a large discrete data set collected from the background areas with geological formations and conditions comparable to the site DU(s) of interest is a requirement for successful application of this module. There are fundamental differences between incremental and discrete samples. For example, the sample support (defined in ITRC [2012]) of discrete and incremental samples are very different. Sample support has a profound effect on sample results so samples with different sample supports should not be compared directly, or compared with great caution. Since incremental sampling is a relatively new approach, the performance of the BISS module requires further investigation. If you would like to try this strategy for your project, or if you have questions, contact Deana Crumbling, crumbling.deana@epa.gov. 279 ------- 280 ------- APPENDIX A Simulated Critical Values for Gamma GOF Tests, the Anderson- Darling Test and the Kolmogorov-Smirnov Test & Summary Tables of Suggestions and Recommendations for UCL95S Updated Critical Values of Gamma GOF Test Statistics (New in ProUCL 5.0) For values of the gamma distribution shape parameter, k < 0.2, critical values of the two gamma empirical distribution tests (EDF) GOF tests: Anderson-Darling (A-D) and Kolmogorov Smirnov (K-S) tests incorporated in ProUCL 4.1 and earlier versions have been updated in ProUCL 5.0. Critical values incorporated in earlier versions of ProUCL were simulated using the gamma deviate generation algorithm (Whittaker 1974) available at the time and with the source code provided in the book Numerical Recipes in C, the Art of Scientific Computing (Press et al. 1990). It is noted that the gamma deviate generation algorithm available at the time was not very efficient, especially for smaller values of the shape parameter, k < 0.1. For small values of the shape parameter, k, significant discrepancies were found in the critical values of the two gamma GOF test statistics obtained using the two gamma deviate generation algorithms: Whitaker (1974) and Marsaglia and Tsang (2000). Even though, discrepancies were identified in critical values of the two GOF tests for value of k < 0.1, for comparison purposes, critical values of the two tests have also been re-generated for k=0.2. For values of k < 0.2, critical values for the two gammas EDF GOF tests have been re-generated and tables of critical values of the two gamma GOF tests have been updated in this Appendix A. Specifically, for values of the shape parameter, k (e.g., k < 0.2), critical values of the two gamma GOF tests have been generated using the more efficient gamma deviate generation algorithm as described in Marsaglia and Tsang (2000) and Best (1983). Detailed description about the implementation of Marsaglia and Tsang's algorithm to generate gamma deviates can be found in Kroese, Taimre, and Botev (2011). It is noted that for values of k > 0.1, the simulated critical values obtained using Marsaglia and Tsang's algorithm (2000) are in general agreement with the critical values of the two GOF test statistics incorporated in ProUCL 4.1 for the various values of the sample size considered. Therefore, those critical values for values of k > 0.2 have not been updated in tables as summarized in this Appendix A. The developers double checked the critical values of the two GOF tests by using MatLab to generate gamma deviates. Critical values obtained using MatLab code are in general agreement with the newly simulated critical values incorporated in critical value tables summarized in this appendix. Simulation Experiments The simulation experiments performed are briefly described here. The experiments were carried out for various values of the sample size, n = 5(25)1, 30(50)5, 60(100)10, 200(500)100, and 1000. Here the notation n=5(25)l means that n takes values starting at 5 all the way up to 25 at increments of 1 each; n=30(50)5 means that n takes values starting at 30 all the way up to 50 at increments of 5 each, and so on. Random deviates of sample size n were generated from a gamma, (k, 0), population. The considered values of the shape parameter, k, are: 0.025, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, and 50.0. These values of k cover a wide range of values of skewness, 2/Vk. The distributions of the Kolmogorov-Smirnov (K-S) test statistic, D, and the Anderson-Darling (A-D) test statistic, A2, do not depend upon the scale 281 ------- parameter, 6, therefore, the scale parameter, 6, has been set equal to 1 in all of the simulation experiments. A typical simulation experiment can be described in the following four steps. Step 1. Generate a random sample of the specified size, n, from a gamma, G (k, 1), distribution. For values of k>0.2, the algorithm as outlined in Whittaker (1974) was used to generate the gamma deviates; and for values of k < 0.2, Marsaglia and Tsang's algorithm (2000) has been used to generate gamma deviates. Step 2. For each generated sample, compute the MLEs of k and 6 (Choi and Wette 1969), and the K-S and the A-D test statistics (Anderson and Darling, 1954; D'Agostino and Stephens 1986; Schneider and Clickner 1976) using the incomplete gamma function (details can be found in Chapter 2 of this document). Step 3. Repeat Steps 1 and 2, a large number (iterations) of times. For values ofk>0.2, 20,000 iterations were used to compute critical values. However, since generation of gamma deviates are quite unstable for smaller values of k (<0.1), 500,000 iterations have been used to obtain the newly generated critical values of the two test statistics based upon Marsaglia and Tsang's algorithm. Step 4. Arrange the resulting test statistics in ascending order. Compute the 90%, 95%, and 99% percentiles of the K-S test statistic and the A-D test statistic. The resulting raw 10%, 5%, and 1% critical values for the two tests are summarized in Tables 1 through 6 as follows. The critical values as summarized in Tables 1-6 are in agreement (up to 3 significant digits) with all available exact or asymptotic critical values (note that critical values of the two GOF tests are not available for values of k<\). It is also noted that the critical values for the K-S test statistic are more stable than those for the A-D test statistic. It is hoped that the availability of the critical values for the GOF tests for the gamma distribution will result in the frequent use of more practical and appropriate gamma distributions in environmental and other applications. Note on computation of the gamma distribution based decision statistics and critical values: While computing the various decision statistics (e.g., UCL and BTVs), ProUCL uses biased corrected estimates, kstar, K , and theta star, 0* (described in Section 2.3.3) of the shape, k, and scale, 0, parameters of the gamma distribution. It is noted that the critical values for the two gamma GOF tests reported in the literature (D'Agostino and Stephens 1986; Schneider and Clickner 1976; Schneider 1978) were computed using the MLE estimates, k and 0, of the two gamma parameters, k and$. Therefore, the critical values of A-D and K-S tests incorporated in ProUCL have also been computed using the MLE estimates: khat, k, and theta hat, 0, of the two gamma parameters, k and 0. 282 ------- Table A-l. Critical Values for A-D Test Statistic for Significance Level = 0.10 n\k 0.025 0.05 0.1 0.2 0.5 1 2 5 10 20 50 5 6 7 8 9 10 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 60 70 80 90 100 200 300 400 500 1000 0.919726 0.923855 0.924777 0.928382 0.928959 0.930055 0.934218 0.934888 0.935586 0.936246 0.937456 0.937518 0.937751 0.938503 0.938587 0.939277 0.940150 0.941743 0.943737 0.945107 0.947909 0.947922 0.948128 0.948223 0.949613 0.951013 0.951781 0.952429 0.953464 0.955133 0.956040 0.957279 0.802558 0.819622 0.829767 0.834365 0.840361 0.847992 0.864609 0.866151 0.866978 0.869658 0.870368 0.871858 0.874119 0.874483 0.875008 0.875990 0.876204 0.882689 0.885557 0.885878 0.887142 0.887286 0.890153 0.891061 0.891764 0.892197 0.892833 0.893123 0.893406 0.898383 0.898554 0.898937 0.715363 0.735533 0.746369 0.758146 0.765446 0.771909 0.792009 0.795984 0.796929 0.799900 0.800417 0.801716 0.803861 0.804803 0.805412 0.806629 0.807918 0.811964 0.814862 0.817072 0.817778 0.818568 0.820774 0.822280 0.823067 0.823429 0.824216 0.826133 0.826715 0.827845 0.827995 0.828584 0.655580 0.670716 0.684718 0.694671 0.701756 0.707396 0.727067 0.727392 0.729339 0.731904 0.732093 0.733548 0.735995 0.736736 0.737239 0.738236 0.738591 0.741572 0.743736 0.747438 0.748890 0.749399 0.749930 0.750605 0.751452 0.752461 0.752765 0.753696 0.754433 0.755130 0.755946 0.757750 0.612 0.625 0.635 0.641 0.648 0.652 0.663 0.665 0.666 0.668 0.67 0.669 0.671 0.67 0.671 0.672 0.673 0.674 0.676 0.677 0.677 0.677 0.679 0.679 0.68 0.68 0.681 0.682 0.682 0.683 0.683 0.684 0.599 0.61 0.618 0.624 0.629 0.632 0.642 0.642 0.644 0.643 0.645 0.645 0.646 0.646 0.645 0.647 0.648 0.65 0.65 0.651 0.651 0.652 0.652 0.653 0.654 0.654 0.654 0.654 0.655 0.655 0.655 0.655 0.594 0.603 0.609 0.616 0.62 0.623 0.63 0.632 0.632 0.634 0.633 0.633 0.634 0.636 0.635 0.635 0.636 0.637 0.638 0.637 0.639 0.64 0.64 0.641 0.641 0.642 0.642 0.642 0.641 0.641 0.643 0.643 0.591 0.599 0.607 0.612 0.614 0.618 0.624 0.626 0.626 0.626 0.626 0.627 0.628 0.628 0.629 0.628 0.629 0.629 0.631 0.631 0.632 0.632 0.632 0.633 0.633 0.634 0.633 0.634 0.634 0.635 0.635 0.635 0.589 0.599 0.606 0.61 0.613 0.616 0.622 0.624 0.623 0.623 0.625 0.626 0.626 0.627 0.627 0.627 0.627 0.628 0.629 0.629 0.63 0.63 0.631 0.63 0.631 0.631 0.631 0.631 0.633 0.633 0.632 0.632 0.589 0.598 0.604 0.609 0.613 0.615 0.621 0.622 0.623 0.624 0.624 0.624 0.626 0.625 0.625 0.626 0.626 0.627 0.628 0.628 0.628 0.629 0.629 0.63 0.63 0.629 0.63 0.631 0.631 0.631 0.631 0.631 0.588 0.598 0.605 0.608 0.612 0.614 0.621 0.621 0.622 0.623 0.624 0.624 0.624 0.625 0.625 0.625 0.625 0.626 0.627 0.628 0.629 0.629 0.629 0.63 0.629 0.63 0.63 0.63 0.63 0.631 0.631 0.63 283 ------- Table A-2. Critical Values for K-S Test Statistic for Significance Level = 0.10 n\k 0.025 0.050 0.10 0.2 0.50 1.0 2.0 5.0 10.0 20.0 50.0 5 6 7 8 9 10 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 60 70 80 90 100 200 300 400 500 1000 0.382954 0.359913 0.336053 0.315927 0.300867 0.286755 0.238755 0.232063 0.225072 0.218863 0.213757 0.209044 0.204615 0.199688 0.195776 0.192131 0.188048 0.172990 0.160170 0.150448 0.142187 0.135132 0.123535 0.114659 0.107576 0.101373 0.096533 0.068958 0.056122 0.048635 0.043530 0.030869 0.377607 0.352996 0.329477 0.312018 0.296565 0.283476 0.237248 0.228963 0.222829 0.216723 0.211493 0.205869 0.201904 0.197629 0.193173 0.189663 0.185450 0.169910 0.158322 0.148475 0.140171 0.133619 0.122107 0.113414 0.106191 0.100267 0.095061 0.067898 0.055572 0.048048 0.042949 0.030621 0.370075 0.343783 0.321855 0.305500 0.290030 0.276246 0.231259 0.224049 0.218089 0.212018 0.206688 0.202242 0.197476 0.193503 0.188985 0.185566 0.181905 0.166986 0.155010 0.145216 0.137475 0.130496 0.119488 0.110949 0.104090 0.097963 0.093359 0.066258 0.054295 0.047103 0.042053 0.029802 0.358618 0.332729 0.312905 0.295750 0.280550 0.268807 0.223045 0.216626 0.211438 0.205572 0.201002 0.196004 0.191444 0.187686 0.182952 0.179881 0.176186 0.161481 0.150173 0.140819 0.133398 0.126836 0.116212 0.107529 0.100923 0.095191 0.090566 0.064542 0.052716 0.045745 0.040913 0.028999 0.346 0.319 0.301 0.284 0.27 0.257 0.214 0.208 0.202 0.197 0.192 0.187 0.183 0.179 0.175 0.172 0.169 0.155 0.144 0.135 0.127 0.121 0.111 0.103 0.096 0.091 0.086 0.062 0.05 0.044 0.039 0.028 0.339 0.313 0.294 0.278 0.264 0.251 0.209 0.203 0.197 0.192 0.187 0.183 0.179 0.175 0.171 0.168 0.165 0.151 0.14 0.132 0.124 0.118 0.108 0.1 0.094 0.089 0.084 0.06 0.049 0.043 0.038 0.027 0.336 0.31 0.29 0.274 0.26 0.248 0.206 0.2 0.194 0.189 0.184 0.18 0.176 0.172 0.169 0.165 0.162 0.149 0.138 0.13 0.122 0.116 0.107 0.099 0.093 0.088 0.083 0.059 0.048 0.042 0.038 0.027 0.334 0.307 0.288 0.272 0.258 0.246 0.204 0.198 0.193 0.188 0.183 0.179 0.175 0.171 0.167 0.164 0.161 0.147 0.137 0.128 0.121 0.115 0.106 0.098 0.092 0.087 0.082 0.059 0.048 0.042 0.037 0.026 0.333 0.307 0.288 0.271 0.257 0.245 0.204 0.198 0.192 0.187 0.182 0.178 0.174 0.17 0.167 0.163 0.16 0.147 0.136 0.128 0.121 0.115 0.105 0.098 0.092 0.086 0.082 0.058 0.048 0.042 0.037 0.026 0.333 0.307 0.287 0.271 0.257 0.245 0.203 0.197 0.192 0.187 0.182 0.178 0.174 0.17 0.166 0.163 0.16 0.147 0.136 0.128 0.121 0.115 0.105 0.097 0.091 0.086 0.082 0.058 0.048 0.041 0.037 0.026 0.333 0.307 0.287 0.271 0.257 0.245 0.203 0.197 0.192 0.187 0.182 0.178 0.174 0.17 0.166 0.163 0.16 0.147 0.136 0.128 0.121 0.115 0.105 0.097 0.091 0.086 0.082 0.058 0.048 0.041 0.037 0.026 284 ------- Table A-3. Critical Values for A-D Test Statistic for Significance Level = 0.05 n\k 0.025 0.05 0.1 0.2 0.5 1 2 5 10 20 50 5 6 7 8 9 10 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 60 70 80 90 100 200 300 400 500 1000 1.151052 1.163733 1.164504 1.164753 1.165715 1.165767 1.166499 1.166685 1.168544 1.168987 1.169801 1.169916 1.170231 1.170651 1.170815 1.171897 1.173062 1.174361 1.174900 1.177053 1.178564 1.178640 1.179045 1.179960 1.180934 1.183445 1.183507 1.184370 1.186474 1.186711 1.186903 1.188089 0.993916 1.015175 1.027713 1.033965 1.039023 1.051305 1.072701 1.072764 1.074729 1.076805 1.078026 1.080724 1.082101 1.083139 1.084161 1.085896 1.086184 1.095072 1.095964 1.097870 1.099630 1.100960 1.103255 1.105666 1.106509 1.106661 1.107269 1.108491 1.112771 1.113282 1.114064 1.114697 0.867326 0.892648 0.910212 0.926242 0.936047 0.945231 0.971851 0.976822 0.979261 0.982322 0.983408 0.985352 0.988749 0.989794 0.990147 0.991640 0.991848 1.000576 1.000838 1.004925 1.006416 1.007896 1.009514 1.013808 1.014011 1.015090 1.015433 1.018998 1.019934 1.020022 1.020267 1.020335 0.775584 0.801734 0.822761 0.835780 0.847305 0.855135 0.883252 0.883572 0.885946 0.889231 0.891016 0.892498 0.895978 0.896739 0.897642 0.898680 0.899874 0.903940 0.907253 0.909633 0.911353 0.912084 0.914286 0.914724 0.914808 0.915898 0.917512 0.920264 0.920502 0.920551 0.921806 0.923848 0.711 0.736 0.752 0.762 0.771 0.777 0.793 0.796 0.798 0.8 0.803 0.803 0.805 0.804 0.805 0.806 0.807 0.809 0.812 0.813 0.813 0.814 0.816 0.817 0.819 0.818 0.818 0.821 0.822 0.823 0.822 0.824 0.691 0.715 0.728 0.736 0.743 0.748 0.763 0.763 0.766 0.767 0.769 0.768 0.77 0.771 0.769 0.772 0.773 0.775 0.776 0.779 0.777 0.78 0.779 0.78 0.782 0.783 0.783 0.784 0.784 0.785 0.785 0.785 0.684 0.704 0.715 0.724 0.73 0.736 0.747 0.75 0.749 0.753 0.752 0.752 0.754 0.756 0.755 0.755 0.756 0.758 0.76 0.759 0.761 0.763 0.763 0.763 0.763 0.765 0.765 0.766 0.766 0.766 0.767 0.768 0.681 0.698 0.71 0.719 0.723 0.729 0.739 0.741 0.742 0.743 0.742 0.745 0.745 0.746 0.747 0.746 0.747 0.746 0.75 0.751 0.753 0.754 0.753 0.754 0.754 0.755 0.754 0.756 0.757 0.757 0.756 0.757 0.679 0.698 0.708 0.715 0.722 0.725 0.737 0.739 0.739 0.739 0.741 0.742 0.743 0.744 0.744 0.744 0.745 0.745 0.748 0.748 0.748 0.75 0.751 0.751 0.75 0.752 0.752 0.751 0.755 0.754 0.753 0.753 0.679 0.697 0.707 0.716 0.721 0.725 0.735 0.737 0.738 0.739 0.74 0.741 0.743 0.74 0.742 0.742 0.743 0.744 0.747 0.747 0.748 0.748 0.749 0.749 0.751 0.75 0.75 0.751 0.751 0.751 0.752 0.752 0.678 0.697 0.708 0.715 0.721 0.724 0.734 0.735 0.737 0.738 0.74 0.739 0.741 0.743 0.741 0.742 0.742 0.744 0.745 0.746 0.747 0.748 0.748 0.749 0.748 0.751 0.75 0.75 0.752 0.752 0.752 0.75 285 ------- Table A-4. Critical Values for K-S Test Statistic for Significance Level = 0.05 n\k 0.025 0.05 0.1 0.2 0.5 1 2 5 10 20 50 5 6 7 8 9 10 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 60 70 80 90 100 200 300 400 500 1000 0.425015 0.393430 0.367179 0.348874 0.331231 0.315236 0.262979 0.255659 0.247795 0.240719 0.235887 0.229517 0.224925 0.219973 0.215140 0.211022 0.207233 0.187026 0.176132 0.165449 0.156286 0.148646 0.135915 0.126014 0.118350 0.111619 0.106157 0.070489 0.061746 0.053335 0.047696 0.034028 0.416319 0.384459 0.361553 0.342809 0.325179 0.311210 0.260524 0.251621 0.244721 0.237832 0.232558 0.227125 0.221654 0.217725 0.212869 0.208355 0.204154 0.187026 0.174396 0.163501 0.154614 0.146991 0.134711 0.124810 0.116873 0.110232 0.104696 0.074659 0.061067 0.052747 0.047419 0.033719 0.405292 0.374897 0.353471 0.335397 0.317725 0.303682 0.253994 0.246493 0.240192 0.233566 0.227223 0.222103 0.217434 0.212415 0.207622 0.203870 0.200009 0.183312 0.170208 0.159727 0.151477 0.143731 0.131391 0.122186 0.114417 0.107708 0.102748 0.072990 0.059533 0.051917 0.046238 0.032830 0.388127 0.364208 0.342709 0.323081 0.308264 0.294373 0.245069 0.238415 0.231881 0.226194 0.220341 0.214992 0.209979 0.205945 0.201004 0.197443 0.193701 0.177521 0.165130 0.154749 0.146553 0.139040 0.127762 0.118044 0.111066 0.104276 0.099320 0.070805 0.057851 0.050257 0.044893 0.031659 0.372 0.349 0.327 0.309 0.294 0.281 0.234 0.227 0.221 0.215 0.21 0.205 0.2 0.196 0.192 0.188 0.184 0.169 0.157 0.148 0.139 0.132 0.121 0.113 0.105 0.1 0.095 0.067 0.055 0.048 0.043 0.03 0.364 0.341 0.32 0.301 0.287 0.274 0.228 0.221 0.215 0.209 0.204 0.199 0.195 0.191 0.187 0.183 0.18 0.165 0.153 0.144 0.136 0.129 0.118 0.11 0.103 0.097 0.092 0.065 0.054 0.047 0.042 0.03 0.36 0.336 0.315 0.297 0.282 0.27 0.224 0.218 0.212 0.206 0.201 0.196 0.192 0.188 0.184 0.18 0.177 0.162 0.151 0.141 0.133 0.127 0.116 0.108 0.101 0.095 0.091 0.064 0.053 0.046 0.041 0.029 0.358 0.333 0.313 0.295 0.28 0.267 0.222 0.216 0.21 0.204 0.199 0.194 0.19 0.186 0.182 0.178 0.175 0.16 0.149 0.14 0.132 0.126 0.115 0.107 0.1 0.094 0.09 0.064 0.052 0.045 0.041 0.029 0.358 0.332 0.312 0.294 0.279 0.267 0.222 0.215 0.209 0.203 0.199 0.194 0.189 0.185 0.182 0.178 0.175 0.16 0.149 0.139 0.132 0.125 0.115 0.106 0.1 0.094 0.089 0.064 0.052 0.045 0.04 0.029 0.357 0.332 0.311 0.294 0.279 0.266 0.221 0.215 0.209 0.203 0.198 0.193 0.189 0.185 0.181 0.178 0.174 0.16 0.148 0.139 0.132 0.125 0.114 0.106 0.099 0.094 0.089 0.064 0.052 0.045 0.04 0.029 0.357 0.332 0.311 0.293 0.279 0.266 0.221 0.214 0.208 0.203 0.198 0.193 0.189 0.185 0.181 0.177 0.174 0.16 0.148 0.139 0.131 0.125 0.114 0.106 0.099 0.094 0.089 0.063 0.052 0.045 0.04 0.029 286 ------- Table A-5. Critical Values for A-D Test Statistic for Significance Level = 0.01 n\k 0.025 0.05 0.1 0.2 0.5 1 2 5 10 20 50 5 6 7 8 9 10 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 60 70 80 90 100 200 300 400 500 1000 1.749166 1.751877 1.752404 1.752700 1.758051 1.759366 1.762174 1.763292 1.763403 1.763822 1.764890 1.765012 1.765021 1.765611 1.765703 1.766530 1.766655 1.771265 1.772614 1.772920 1.774318 1.775401 1.777021 1.780583 1.782174 1.786462 1.788600 1.789565 1.791785 1.796178 1.799037 1.810595 1.518258 1.543508 1.556906 1.561426 1.567347 1.575002 1.593432 1.596448 1.599618 1.599735 1.603396 1.604198 1.604737 1.605233 1.609641 1.609644 1.609908 1.617605 1.620179 1.622877 1.624156 1.630356 1.630972 1.634413 1.636678 1.637946 1.639307 1.640278 1.640656 1.641470 1.642244 1.642639 1.258545 1.305996 1.332339 1.358108 1.372050 1.384541 1.418705 1.422813 1.425118 1.435826 1.441772 1.443435 1.446116 1.448791 1.449964 1.451442 1.451659 1.462230 1.465890 1.468763 1.469148 1.471192 1.474981 1.477148 1.481082 1.483922 1.484231 1.486139 1.489654 1.491079 1.491158 1.492652 1.068746 1.123216 1.162744 1.187751 1.210845 1.218849 1.263841 1.273189 1.273734 1.274053 1.278280 1.279990 1.281092 1.284002 1.288792 1.289696 1.290311 1.295794 1.296988 1.304213 1.308833 1.311004 1.312242 1.313856 1.315184 1.316508 1.318003 1.318714 1.322935 1.323876 1.328415 1.328852 0.945 0.99 1.019 1.044 1.058 1.071 1.1 1.112 1.11 1.116 1.115 1.118 1.126 1.119 1.125 1.126 1.127 1.133 1.136 1.138 1.141 1.142 1.144 1.145 1.15 1.149 1.149 1.156 1.154 1.158 1.155 1.157 0.905 0.946 0.979 0.99 1.007 1.018 1.048 1.047 1.053 1.054 1.059 1.056 1.057 1.062 1.059 1.065 1.064 1.072 1.072 1.076 1.074 1.079 1.079 1.079 1.085 1.086 1.085 1.089 1.09 1.093 1.089 1.092 0.89 0.928 0.951 0.97 0.984 0.994 1.018 1.019 1.023 1.027 1.026 1.031 1.031 1.036 1.034 1.035 1.038 1.044 1.045 1.046 1.048 1.053 1.054 1.055 1.055 1.056 1.054 1.059 1.058 1.057 1.057 1.06 0.883 0.918 0.944 0.961 0.967 0.981 1.002 1.007 1.008 1.015 1.013 1.016 1.017 1.023 1.017 1.02 1.021 1.023 1.027 1.03 1.036 1.034 1.032 1.038 1.036 1.038 1.042 1.041 1.043 1.043 1.047 1.043 0.882 0.916 0.938 0.955 0.968 0.977 0.999 1.004 1.004 1.006 1.01 1.012 1.013 1.014 1.02 1.015 1.017 1.023 1.025 1.027 1.03 1.029 1.032 1.031 1.033 1.034 1.035 1.031 1.038 1.039 1.04 1.035 0.879 0.911 0.935 0.956 0.969 0.975 0.997 1 1.003 1.005 1.006 1.005 1.013 1.011 1.012 1.012 1.014 1.019 1.021 1.023 1.026 1.028 1.029 1.031 1.032 1.031 1.033 1.032 1.033 1.035 1.034 1.036 0.879 0.912 0.938 0.953 0.967 0.973 0.999 0.999 1 1.003 1.008 1.009 1.008 1.013 1.013 1.013 1.013 1.018 1.018 1.022 1.024 1.025 1.03 1.028 1.029 1.033 1.032 1.033 1.031 1.034 1.034 1.031 287 ------- Table A-6. Critical Values for K-S Test Statistic for Significance Level = 0.01 n\k 0.025 0.050 0.10 0.2 0.50 1.0 2.0 5.0 10.0 20.0 50.0 5 6 7 8 9 10 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 60 70 80 90 100 200 300 400 500 1000 0.495311 0.464286 0.437809 0.412467 0.390183 0.373002 0.310445 0.302682 0.294519 0.285220 0.277810 0.271994 0.266096 0.260430 0.254210 0.249574 0.246298 0.220685 0.208407 0.196230 0.185995 0.176191 0.161519 0.149283 0.139831 0.132254 0.126224 0.085150 0.073232 0.063283 0.056181 0.040020 0.482274 0.454103 0.426463 0.404538 0.383671 0.368362 0.307559 0.298348 0.289320 0.280990 0.275460 0.268927 0.262728 0.256537 0.252405 0.246722 0.242298 0.222267 0.206958 0.193613 0.183011 0.173662 0.158802 0.148241 0.138103 0.130746 0.123308 0.088338 0.072401 0.062708 0.056147 0.039807 0.467859 0.441814 0.411589 0.392838 0.375103 0.358647 0.300791 0.290148 0.283394 0.276126 0.269173 0.261936 0.256686 0.251727 0.245607 0.240947 0.236164 0.217254 0.202296 0.188617 0.179728 0.170513 0.155658 0.144542 0.135441 0.127231 0.121414 0.086339 0.071096 0.061239 0.054822 0.038938 0.449435 0.423777 0.398890 0.379962 0.361937 0.348328 0.289751 0.280643 0.274722 0.265561 0.260992 0.253878 0.247915 0.242711 0.236271 0.233143 0.228867 0.209442 0.194716 0.182935 0.173141 0.163792 0.150458 0.139590 0.131479 0.123253 0.117441 0.083391 0.068521 0.059235 0.053042 0.036987 0.431 0.402 0.38 0.36 0.343 0.328 0.274 0.266 0.259 0.252 0.246 0.24 0.235 0.23 0.225 0.221 0.216 0.199 0.185 0.173 0.164 0.156 0.143 0.132 0.124 0.117 0.111 0.079 0.065 0.056 0.05 0.036 0.421 0.391 0.369 0.349 0.333 0.318 0.266 0.258 0.251 0.245 0.238 0.233 0.228 0.223 0.218 0.214 0.21 0.193 0.179 0.168 0.158 0.151 0.138 0.128 0.12 0.114 0.108 0.077 0.063 0.054 0.049 0.035 0.414 0.385 0.362 0.344 0.327 0.312 0.261 0.253 0.246 0.24 0.234 0.228 0.223 0.219 0.215 0.21 0.206 0.189 0.176 0.165 0.156 0.148 0.136 0.126 0.118 0.111 0.106 0.075 0.062 0.053 0.048 0.034 0.41 0.382 0.36 0.34 0.323 0.309 0.258 0.251 0.244 0.237 0.232 0.226 0.221 0.216 0.212 0.208 0.204 0.187 0.174 0.163 0.154 0.146 0.134 0.124 0.117 0.11 0.105 0.074 0.061 0.053 0.047 0.034 0.41 0.381 0.358 0.339 0.323 0.308 0.257 0.25 0.243 0.236 0.231 0.225 0.22 0.216 0.211 0.207 0.203 0.186 0.173 0.162 0.154 0.146 0.134 0.124 0.116 0.11 0.104 0.074 0.061 0.053 0.047 0.033 0.408 0.38 0.357 0.339 0.322 0.308 0.257 0.249 0.242 0.236 0.23 0.225 0.22 0.215 0.211 0.207 0.203 0.186 0.173 0.162 0.153 0.146 0.133 0.124 0.116 0.109 0.104 0.074 0.061 0.053 0.047 0.033 0.408 0.38 0.357 0.338 0.322 0.307 0.256 0.249 0.242 0.236 0.23 0.225 0.219 0.215 0.21 0.206 0.203 0.185 0.172 0.162 0.153 0.145 0.133 0.124 0.116 0.11 0.104 0.074 0.06 0.053 0.047 0.033 288 ------- DECISION SUMMARY TABLES Table A-7. Skewness as a Function of a (or its MLE, sy = 6), sd of \og(X) Standard Deviation of Skewness Logged Data a < 0.5 Symmetric to mild skewness 0.5 < a < 1.0 Mild skewness to moderate skewness 1.0 < a < 1.5 Moderate skewness to high skewness 1.5 < a < 2.0 High skewness 9 0 < < ^ 0 Very high skewness (moderate probability of ~ ' outliers and/or multiple populations) > - „ Extremely high skewness (high probability of ~ ' outliers and/or multiple populations) Table A-8. Summary Table for the Computation of a 95% UCL of the Unknown Mean, fii, of a Gamma Distribution k (Skewness 0 , „. 0 Sample Size, n Suggestion Bias Adjusted) Approximate gamma 95% UCL (Gamma KM or k* > 1.0 n>=50 GROS) ,-* ,n Adjusted gamma 95% UCL (Gamma KM or GROS) K ^1.0 H ~~J(J £* Jt. 95% UCL based upon bootstrap-t ~ ' or Hall's bootstrap method* Adjusted gamma 95% UCL (Gamma KM) if k* <1.0 n>\5,n<50 available, otherwise use approximate gamma 95% UCL(Gamma KM) k* <1.0 n > 50 Approximate gamma 95% UCL (Gamma KM) *In case the bootstrap-t or Hall's bootstrap methods yield erratic, inflated, and unstable UCL values, the UCL of the mean should be computed using an adjusted gamma UCL. 289 ------- Table A-9. Summary Table for the Computation of a 95% UCL of the Unknown Mean, of a Lognormal Population /v a a <0.5 0.5 < a <1.0 1.0 < a <1.5 1.5 < a <2.0 2.0 < £ <2.5 2.5 < a <3.0 3.0< <7<3.5** A . f\ — ** a >3.5 Sample Size, n For all n For all n n<25 n>25 n<20 20<«<50 «>50 «<20 20<«<50 50<«<70 «>70 «<30 30<«<70 70<«< 100 n> 100 «<15 15<«<50 50<«< 100 100<«<150 n> 150 For all n Suggestions Student's t, modified-t, or H-UCL H-UCL 95% Chebyshev (Mean, Sd) UCL H-UCL 97.5% or 99% Chebyshev (Mean, Sd) UCL 95% Chebyshev (Mean, Sd) UCL H-UCL 99% Chebyshev (Mean, Sd) UCL 97.5% Chebyshev (Mean, Sd) UCL 95% Chebyshev (Mean, Sd) UCL H-UCL 99% Chebyshev (Mean, Sd) 97.5% Chebyshev (Mean, Sd) UCL 95% Chebyshev (Mean, Sd) UCL H-UCL Bootstrap-t or Hall's bootstrap method* 99% Chebyshev(Mea«, Sd) 97.5% Chebyshev (Mean, Sd) UCL 95% Chebyshev (Mean, Sd) UCL H-UCL Use nonparametric methods* *In the case that Hall's bootstrap or bootstrap-t methods yield an erratic unrealistically large UCL value, UCL of the mean may be computed based upon the Chebyshev inequality: Chebyshev (Mean, Sd) UCL ** For highly skewed data sets with a exceeding 3.0, 3.5, it is suggested that the user pre-process the data. It is very likely that the data include outliers and/or come from multiple populations. The population partitioning methods may be used to identify mixture populations present in the data set. 290 ------- Table A-10. Summary Table for the Computation of a 95% UCL of the Unknown Mean, fii, Based upon a Skewed Data Set (with all Positive Values) without a Discernible Distribution, Where a is the sd of Log-transformed Data /*, a a <0.5 0.5 < a <1.0 1.0 < a < 1.5 1.5 < a <2.0 2.0 < a <2.5 2.5 < a <3.0 3.0<£<3.5** 3.5« Sample Size, n For all n For all n For all n n<20 20 ------- 292 ------- APPENDIX B Large Sample Size Requirements to use the Central Limit Theorem on Skewed Data Sets to Compute an Upper Confidence Limit of the Population Mean As mentioned earlier, the main objective of the ProUCL software funded by the USEPA is to compute accurate and defensible decision statistics to help the decision makers in making reliable decisions which are cost-effective, and protective of human health and the environment. ProUCL software is based upon the philosophy that rigorous statistical methods can be used to compute the correct estimates of the population parameters (e.g., site mean, background percentiles) and decision making statistics including the upper confidence limit (UCL) of the population mean, the upper tolerance limit (UTL), and the upper prediction limit (UPL) to help decision makers and project teams in making decisions. The use and applicability of a statistical method (e.g., Student's t-UCL, CLT-UCL, adjusted gamma-UCL, Chebyshev UCL, bootstrap-t UCL) depend upon data size, data skewness, and data distribution. ProUCL computes decision statistics using several parametric and nonparametric methods covering a wide-range of data variability, skewness, and sample size. A couple of UCL computation methods described in the statistical text books (e.g., Hogg and Craig, 1995) based upon the Student's t-statistic and the Central Limit Theorem (CLT) alone cannot address all scenarios and situations commonly occurring in the various environmental studies. Moreover, the properties of the CLT and Student's t-statistic are unknown when NDs with varying DLs are present in a data set - a common occurrence in data sets originating from environmental applications. The use of a parametric lognormal distribution on a lognormally distributed data set tends to yield unstable impractically large UCLs values, especially when the standard deviation (sd) of the log- transformed data is greater than 1.0 and the data set is of small size such as less than 30-50 (Hardin and Gilbert 1993; Singh, Singh, and Engelhardt, 1997). Many environmental data sets can be modeled by a gamma as well as a lognormal distribution. Generally, the use of a gamma distribution on gamma distributed data sets yields UCL values of practical merit (Singh, Singh, and laci 2002). Therefore, the use of gamma distribution-based decision statistics such as UCLs, upper prediction limits (UPLs), and UTLs should not be dismissed just because it is easier to use a lognormal model. The advantages of computing the gamma distribution-based decision statistics have been discussed in Chapters 2 through 5 of this technical guidance document. Since many environmental decisions are made based upon a 95% UCL (UCL95) of the population mean, it is important to compute UCLs and other decision making statistics of practical merit. In an effort to compute correct and appropriate UCLs of the population mean and other decision making statistics, in addition to computing the Student's t statistic and the CLT based decision statistics (e.g., UCLs, UPLs), significant effort has been made to incorporate rigorous statistical methods based UCLs in ProUCL software covering a wide-range of data skewness and sample sizes (Singh, Singh, and Engelhardt 1997; Singh, Singh, and laci 2002). It is anticipated that the availability of the statistical limits in the ProUCL covering a wide range of environmental data sets will help decision makers in making more informative and defensible decisions at Superfund and RCRA sites. It is noted that even for skewed data sets, practitioners tend to use the CLT or Student's t-statistic based UCLs of the mean for samples of sizes 25-30 (large sample rule-of-thumb to use CLT). However, this rule-of-thumb does not apply to moderately skewed to highly skewed data sets, specifically when a (sd of 293 ------- the log-transformed data) starts exceeding 1. It should be noted that the large sample requirement depends upon the skewness of the data distribution under consideration. The large sample requirement for the sample mean to follow an approximate normal distribution increases with skewness. It is noted that for skewed data sets, even samples of size greater 100 may not be large enough for the sample mean to follow an approximate normal distribution (Figures B-l through B-7 below) and the UCLs based upon the CLT and Student's t statistics fail to provide the desired 95% coverage of the population mean for samples of sizes as large as 100 as can be seen in Figures B-l through B-7. Noting that the Student's t-UCL and the CLT-UCL fail to provide the specified coverage of the population mean of skewed distributions, several researchers, including Chen (1995), Johnson (1978), Kleijnen, Kloppenburg, and Meeuwsen (1986), and Sutton (1993), proposed adjustments for data skewness in the Student's t statistic and the CLT. They suggested the use of a modified-t-statistic and skewness adjusted CLT for positively skewed distributions (for details see Chapter 2 of this Technical Guide). From statistical theory, the CLT yields UCL results slightly smaller than the Student's t-UCL and the adjusted CLT, and the Student's t-statistic yield UCLs smaller than the modified t-UCLs (details in Chapter 2 of this document). Therefore, only the modified t-UCL has been incorporated in the simulation results described in the following. Specifically, if a UCL95 based upon the modified t-statistic fails to provide the specified coverage to the population mean, then the other three UCL methods, Student's t- UCL, CLT-UCL, and the adjusted CLT-UCL, will also fail to provide the specified coverage of the population mean. The simulation graphs summarized in this appendix suggest that the skewness adjusted UCLs such as the Johnson's modified-t UCL (and therefore Student's t-UCL and CLT-UCL) do not provide the specified coverage to the population mean even for mildly to moderately skewed (a in [0.5, 1.0]) data sets. The coverage of the population mean provided by these UCLs becomes worse (much smaller than the specified coverage) for highly skewed data sets. The graphical displays, shown in Figures B-l through B-7, cover mildly, moderately, and highly skewed data sets. Specifically, Figures B-l through B-7 compare the UCL95 of the mean based upon parametric and nonparametric bootstrap methods and also UCLs computed using the modified-t UCL for mildly skewed (G(5,50), LN(5,0.05)); moderately skewed (G(2,50), LN(5,1)); and highly skewed (G(0.5, 50), G(l,50), and LN(5,1.5)) data distributions. From the simulation results presented in Figures B-l through B-7, it is noted that for skewed distributions, as expected the UCLs based on the modified t-statistic (and therefore UCLs based upon the CLT and the Student's t-statistic) fail to provide the desired 95% coverage of the population mean of gamma distributions: G(0.5,50), G(l,50), G(2,50); and of lognormal distributions: LN(5,0.5), LN(5,1), LN(5,1.5) for samples of sizes as large as 100; and the large sample size requirement increases as the skewness increases. The use of the CLT-UCL and Student's t-UCL underestimate the population mean/ EPC for most skewed data sets. 294 ------- u D Q~- ID m >, .Q ® O) ns ,*«* BB - CO o i <0 no y db - l™ W a. •to ra B2 - m \... €> > 0 7S - 74 - [ x /V /' 1 — '$$— Max Test — * — Modified-t —A— 96% Chebysfiev — • — BoDtstrap-t — *— Bootstrap BCA — o — Approximate Gamma —B— Adjusted Gamma i i I 10 20 30 40 50 60 7D 00 90 100 Figure B-l. Graphs of Coverage Probabilities by 95% UCLs of the mean of G (£=0.50, 9=50) 295 ------- O D 10 en 4) CB £ 90 c Q. v m a o O 82 - 78 n 0 10 20 30 70 • Max Test • Modified-t • 95% Chebyshev • Bootstrap-! • Bootstrap BCA -Approximate Gamma -Adjusted Gamma 90 40 50 60 Figure B-2. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(£=1.00, 9=50) 100 Coverage Percentage by 95% UCLs D CD OD CQ CO J CO CU -fr QD I f>— /h^^ / -*-g H—-~— -____ n / ^s— __-»- —•— :zzzzrrrrrr^jrrz===^^ / __ »— ^_____ ^^^^_ j>_ ^ _______^--- — — "/ — '^— Max Test — *— Modified-t & 95% Chebyshev — •— Bootstrap-t — *r- Bootstrap BCA — e— Approximate Gamma — B— Adjusted Gamma ~+ 0 10 20 30 40 50 80 70 80 90 100 Sample Size Figure B-3. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(£=2.00, 9=50) 296 ------- 1DO - w _j O 05 rt QJ IS o o 92 - BB - Max Test Modified-t 95% Chebyshev Baotstrap-t Bootstrap BCA Approximate Garnma Adjusted Gamma 0 10 20 30 70 90 40 5Q BO Figure B4. Graphs of Coverage Probabilities by 95% UCLs of Mean of G(£=5.00, 9=50) 100 100 - o 3 Ol ft "c o 80 - CD 10 « 3 B5 • Max Test • Modified-! • 95% MVUE Chebahev • Hall's Bootstrap • Bootstrap BCA • H-Statistic UCL 0 10 30 7D eo ao 40 50 BO Figure B-5. Graphs of Coverage Probabilities by UCLs of Mean of LN(|i=5, o=0.5) 100 297 ------- 100 ^ o D ® O) **»* 0> u 1^ 'It Q. O) (0 O O 85 - GO - 75 - 70 • Max Test • Modified-t • 95% MVUE Chebshev • 97.5 MVUE Chebyshev • Hall's Bootstrap • Bootstrap BCA • H-Statistic UCL 1D 2D 30 70 80 BO 40 50 BO Sample ure B-6. Graphs of Coverage Probabilities by UCLs of Mean of LN(|i=5, o=1.0) 100 100 - o D 10 *-* 94 - 88 - €1 m a 5* O CJ 76 - 70- 64 - SB - Test - 95% MVUE Chebshev -97.5%MVUEChebyshe\ • 99% MVUE Chebyshev • Hall's Bootstrap • Bootstrap BCA • H-Statistlc UCL 0 10 20 30 40 50 60 7D 60 90 100 Figure B-7. Graphs of Coverage Probabilities by UCLs of Mean of LN(|i=5, o=1.5) 298 ------- 299 ------- REFERENCES Aitchison, J. and Brown, J.A.C. 1969. The Lognormal Distribution, Cambridge: Cambridge University Press. Anderson, T.W. and Darling, D. A. 1954. Test of goodness-of-fit. Journal of American Statistical Association, Vol. 49, 765-769. Bain, L.J., and Engelhardt, M. 1991. 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A Note on Inference for the Mean Parameter of the Gamma Distribution. Statistics Probability Letters, Vol. 17, 61-66. 110 ------- 311 ------- 112 ------- &EPA United States Environmental Protection Agency Office of Research and Development (8101R) Washington, DC 20460 Official Business Penalty for Private Use$300 EPA/600/R-07/041 May 2010 www.epa.gov Please make all necessary changes on the below label, detach or copy, and return to the address in the upper left-hand corner. If you do not wish to receive these reports CHECK HERE D ; detach, or copy this cover, and return to the address in the upper left-hand corner. PRESORTED STANDARD POSTAGE & FEES PAID EPA PERMIT No. G-35 V Recycled/Recyclable Printed with vegetable-based ink on paper that contains a minimum of 50% post-consumer fiber content processed chlorine free -------