Statistical Applications for Environmental Analysis and Risk Assessment

Joseph Ofungwu, PhD, is an environmental professional with over eighteen years of hands-on experience in environmental practice, including contaminant impact analysis, human health and ecological risk assessment, pollutant fate and transport modeling in ambient air, soil, ground and surface water. Dr. Ofungwu is also Visiting Assistant Professor with the Urban Environmental Systems Management Program at Pratt Institute and teaches statistics courses for professional engineer license maintenance requirements.

• Preface xviiAcknowledgments xix1 Introduction 11.1 Introduction and Overview 11.2 The Aim of the Book: Get Involved! 21.3 The Approach and Style: Clarity, Clarity, Clarity 3Part I Basic Statistical Measures and Concepts 52 Introduction to Software Packages Used in This Book 72.1 R 82.1.1 Helpful R Tips 92.1.2 Disadvantages of R 102.2 ProUCL 102.2.1 Helpful ProUCL Tips 112.2.2 Potential Deficiencies of ProUCL 122.3 Visual Sample Plan 122.4 DATAPLOT 132.4.1 Helpful Tips for Running DATAPLOT in Batch Mode 132.5 Kendall–Thiel Robust Line 142.6 Minitab® 142.7 Microsoft Excel 153 Laboratory Detection Limits, Nondetects, and Data Analysis 173.1 Introduction and Overview 173.2 Types of Laboratory Data Detection Limits 183.3 Problems with Nondetects in Statistical Data Samples 193.4 Options for Addressing Nondetects in Data Analysis 203.4.1 Kaplan–Meier Estimation 213.4.2 Robust Regression on Order Statistics 223.4.3 Maximum Likelihood Estimation 234 Data Sample, Data Population, and Data Distribution 254.1 Introduction and Overview 254.2 Data Sample Versus Data Population or Universe 264.3 The Concept of a Distribution 274.3.1 The Concept of a Probability Distribution Function 284.3.2 Cumulative Probability Distribution and Empirical Cumulative Distribution Functions 314.4 Types of Distributions 344.4.1 Normal Distribution 344.4.1.1 Goodness-of-Fit (GOF) Tests for the Normal Distribution 404.4.1.2 Central Limit Theorem 484.4.2 Lognormal, Gamma, and Other Continuous Distributions 494.4.2.1 Gamma Distribution 514.4.2.2 Logistic Distribution 514.4.2.3 Other Continuous Distributions 524.4.3 Distributions Used in Inferential Statistics (Student’s t, Chi-Square, F) 534.4.3.1 Student’s t Distribution 534.4.3.2 Chi-Square Distribution 554.4.3.3 F Distribution 574.4.4 Discrete Distributions 574.4.4.1 Binomial Distribution 574.4.4.2 Poisson Distribution 61Exercises 645 Graphics for Data Analysis and Presentation 675.1 Introduction and Overview 675.2 Graphics for Single Univariate Data Samples 685.2.1 Box and Whiskers Plot 685.2.2 Probability Plots (i.e., Quantile–Quantile Plots for Comparing a Data Sample to a Theoretical Distribution) 725.2.3 Quantile Plots 795.2.4 Histograms and Kernel Density Plots 825.3 Graphics for Two or More Univariate Data Samples 865.3.1 Quantile–Quantile Plots for Comparing Two Univariate Data Samples 865.3.2 Side-by-Side Box Plots 895.4 Graphics for Bivariate and Multivariate Data Samples 915.4.1 Graphical Data Analysis for Bivariate Data Samples 915.4.2 Graphical Data Analysis for Multivariate Data Samples 955.5 Graphics for Data Presentation 985.6 Data Smoothing 1055.6.1 Moving Average and Moving Median Smoothing 1055.6.2 Locally Weighted Scatterplot Smoothing (LOWESS or LOESS) 1085.6.2.1 Smoothness Factor and the Degree of the Local Regression 1095.6.2.2 Basic and Robust LOWESS Weighting Functions 1095.6.2.3 LOESS Scatterplot Smoothing for Data with Multiple Variables 112Exercises 1136 Basic Statistical Measures: Descriptive or Summary Statistics 1156.1 Introduction and Overview 1156.2 Arithmetic Mean and Weighted Mean 1166.3 Median and Other Robust Measures of Central Tendency 1176.4 Standard Deviation, Variance, and Other Measures of Dispersion or Spread 1196.4.1 Quantiles (Including Percentiles) 1216.4.2 Robust Measures of Spread: Interquartile Range and Median Absolute Deviation 1246.5 Skewness and Other Measures of Shape 1246.6 Outliers 1346.6.1 Tests for Outliers 1356.7 Data Transformations 139Exercises 141Part II Statistical Procedures for Mostly Univariate Data 1437 Statistical Intervals: Confidence, Tolerance, and Prediction Intervals 1457.1 Introduction and Overview 1457.2 Confidence Intervals 1467.2.1 Parametric Confidence Intervals 1517.2.1.1 Parametric Confidence Interval around the Arithmetic Mean or Median for Normally Distributed Data 1517.2.1.2 Lognormal and Other Parametric Confidence Intervals 1537.2.2 Nonparametric Confidence Intervals Around the Mean, Median, and Other Percentiles 1547.2.3 Parametric Confidence Band Around a Trend Line 1647.2.4 Nonparametric Confidence Band Around a Trend Line 1667.3 Tolerance Intervals 1687.3.1 Parametric Tolerance Intervals 1697.3.2 Nonparametric Tolerance Intervals 1707.4 Prediction Intervals 1737.4.1 Parametric Prediction Intervals for Future Individual Values and Future Means 1757.4.2 Nonparametric Prediction Intervals for Future Individual Values and Future Medians 1767.5 Control Charts 178Exercises 1788 Tests of Hypothesis and Decision Making 1818.1 Introduction and Overview 1818.2 Basic Terminology and Procedures for Tests of Hypothesis 1828.3 Type I and Type II Decision Errors, Statistical Power, and Interrelationships 1908.4 The Problem with Multiple Tests or Comparisons: Site-Wide False Positive Error Rates 1938.5 Tests for Equality of Variance 195Exercises 1999 Applications of Hypothesis Tests: Comparing Populations, Analysis of Variance 2019.1 Introduction and Overview 2019.2 Single Sample Tests 2029.2.1 Parametric Single-Sample Tests: One-Sample t-Test and One-Sample Proportion Test 2039.2.2 Nonparametric Single-Sample Tests: One-Sample Sign Test and One-Sample Wilcoxon Signed Rank Test 2059.2.2.1 Nonparametric One-Sample Sign Test 2069.2.2.2 Nonparametric One-Sample Wilcoxon Signed Rank Test 2089.3 Two-Sample Tests 2089.3.1 Parametric Two-Sample Tests 2109.3.1.1 Parametric Two-Sample t-Test for Independent Populations 2109.3.1.2 Parametric Two-Sample t-Test for Paired Populations 2149.3.2 Nonparametric Two-Sample Tests 2169.3.2.1 Nonparametric Wilcoxon Rank Sum Test for Two Independent Populations 2169.3.2.2 Nonparametric Gehan Test for Two Independent Populations 2209.3.2.3 Nonparametric Quantile Test for Two Independent Populations 2219.3.2.4 Nonparametric Two-Sample Paired Sign Test and Paired Wilcoxon Signed Rank Test 2229.4 Comparing Three or More Populations: Parametric ANOVA and Nonparametric Kruskal–Wallis Tests 2279.4.1 Parametric One-Way ANOVA 2289.4.1.1 Computation of Parametric One-Way ANOVA 2309.4.2 Nonparametric One-Way ANOVA (Kruskal–Wallis Test) 2359.4.3 Follow-Up or Post Hoc Comparisons After Parametric and Nonparametric One-Way ANOVA 2389.4.4 Parametric and Nonparametric Two-Way and Multifactor ANOVA 244Exercises 25510 Trends, Autocorrelation, and Temporal Dependence 25710.1 Introduction and Overview 25710.2 Tests for Autocorrelation and Temporal Effects 25810.2.1 Test for Autocorrelation Using the Sample Autocorrelation Function 25910.2.2 Test for Autocorrelation Using the Rank Von Neumann Ratio Method 26110.2.3 An Example on Site-Wide Temporal Effects 26410.3 Tests for Trend 26510.3.1 Parametric Test for Trends—Simple Linear Regression 26610.3.2 Nonparametric Test for Trends—Mann–Kendall Test and Seasonal Mann–Kendall Test 27110.3.3 Nonparametric Test for Trends—Theil–Sen Trend Test 27310.4 Correcting Seasonality and Temporal Effects in the Data 27910.4.1 Correcting Seasonality for a Single Data Series 28010.4.2 Simultaneously Correcting Temporal Dependence for Multiple Data Sets 28110.5 Effects of Exogenous Variables on Trend Tests 282Exercises 285Part III Statistical Procedures for Mostly Multivariate Data 28711 Correlation, Covariance, Geostatistics 28911.1 Introduction and Overview 28911.2 Correlation and Covariance 29011.2.1 Pearson’s Correlation Coefficient 29211.2.2 Spearman’s and Kendall’s Correlation Coefficients 29411.3 Introduction to Geostatistics 30011.3.1 The Variogram or Covariogram 30011.3.2 Kriging 30211.3.3 A Note on Data Sample Size and Lag Distance Requirements 311Exercises 31212 Simple Linear Regression 31512.1 Introduction and Overview 31512.2 The Simple Linear Regression Model 31612.2.1 The True or Population X–Y Relationship 31712.2.2 The Estimated X–Y Relationship Based on a Data Sample 32012.3 Basic Applications of Simple Linear Regression 32412.3.1 Description and Graphical Review of the Data Sample for Regression 32412.3.1.1 Computing the Regression 32512.3.1.2 Interpreting the Regression Results 32612.4 Verify Compliance with the Assumptions of Conventional Linear Regression 33212.4.1 Assumptions of Linearity and Homoscedasticity 33212.4.2 Assumption of Independence 33412.4.3 Exogeneity Assumption, Normality of the Y Errors, and Absence of Outliers 33712.5 Check the Regression Diagnostics for the Presence of Influential Data Points 33912.6 Confidence Intervals for the Predicted Y Values 34312.7 Regression for Left-Censored Data (Non-detects) 344Exercises 34913 Data Transformation Versus Generalized Linear Model 35113.1 Introduction and Overview 35113.2 Data Transformation 35213.2.1 General Approach for Data Transformations 35513.2.2 The Ladder of Powers 35713.2.3 The Bulging Rule and Data Transformations for Regression Analysis 35913.2.4 Facilitating Data Transformations Using Box–Cox Methods 36613.2.5 Back-Transformation Bias and Other Issues with Data Transformation 36713.2.5.1 Logarithmic Transformations 36913.2.5.2 Other Transformations 37013.2.6 Transformation Bias Correction 37113.3 The Generalized Linear Model (GLM) and Applications for Regression 37413.3.1 Components of the Generalized Linear Model and Inherent Limitations 37413.3.2 Estimation and Hypothesis Tests of Significance for GLM Parameters 37613.3.3 Deviance, Null Deviance, Residual Deviance, and Goodness of Fit 37713.3.4 Diagnostics for GLM 37913.3.5 Procedural Steps for Regression with GLM in R 38013.4 Extension of Data Transformation and Generalized Linear Model to Multiple Regression 38513.4.1 Data Transformation for Multiple Regression 38513.4.2 Generalized Linear Models for Multiple Regression 387Exercises 38714 Robust Regression 39114.1 Introduction and Overview 39114.2 Kendall–Theil Robust Line 39314.2.1 Computation of the Kendall–Theil Robust Line Regression 39314.2.2 Test of Significance for the Kendall–Theil Robust Line 39614.2.3 Bias Correction for Y Predictions by the Kendall–Theil Robust Line 39714.3 Weighted Least Squares Regression 39814.3.1 Procedure for Weighted Least Squares Regression for Known Variances of the Observations 39914.4 Iteratively Reweighted Least Squares Regression 40514.4.1 The Iteratively Reweighted Least Squares Procedure 40914.5 Other Robust Regression Alternatives: Bounded Influence Methods 41214.5.1 Least Absolute Deviation or Least Absolute Values 41214.5.2 Quantile Regression 41314.5.3 Least Median of Squares 41314.5.4 Least Trimmed Squares 41414.6 Robust Regression Methods for Multiple-Variable Data 416Exercises 41715 Multiple Linear Regression 41915.1 Introduction and Overview 41915.2 The Need for Multiple Regression 42015.3 The Multiple Linear Regression (MLR) Model 42115.4 The Estimated Multivariable X–Y Relationship Based on a Data Sample 42215.5 Assumptions of Multiple Linear Regression 43015.5.1 Linearity of the Relationship Between the Dependent and Explanatory Variables 43115.5.2 Absence of Multicollinearity Among the Explanatory Variables 43315.5.2.1 Potential Remedies for Multicollinearity 43615.5.3 Homoscedasticity or Constancy of Variance of the Y Population Errors 43915.5.4 Statistical Independence of the Y Population Errors 44115.5.5 Exogeneity Assumption, Normality of the Y Errors, and Absence of Outliers 44515.5.6 Absence of Variability or Errors in the Explanatory Variables 44615.6 Hypothesis Tests for Reliability of the MLR Model 44715.6.1 ANOVA F Test for Overall Significance of the Regression 44715.6.1.1 A Note on ANOVA Tables 44815.6.2 Partial t and Partial F Tests for Individual Regression Coefficients 45215.6.3 Complete and Reduced Models 45215.7 Confidence Intervals for the Regression Coefficients and Predicted Y Values 45715.8 Coefficient of Multiple Correlation (R), Multiple Determination (R2), Adjusted R2, and Partial Correlation Coefficients 45815.8.1 Coefficient of Multiple Correlation (R) 45815.8.2 Coefficient of Multiple Determination (R2) and Adjusted R2 45915.8.3 Partial Correlations and Squared Partial Correlations 46015.9 Regression Diagnostics 46215.10 Model Interactions and Multiplicative Effects 46715.10.1 The Multiple Linear Regression Interaction Model 46715.10.2 Hypothesis Tests of the Interaction Terms for Significance 468Exercises 47416 Categorical Data Analysis 47716.1 Introduction and Overview 47716.2 Types of Variables and Associated Data 47816.2.1 Quantitative Variables 47916.2.2 Qualitative Variables 47916.3 One-Way Analysis of Variance Regression Model 48016.3.1 Interpretation of the Regression Results and ANOVA F-Test for Overall Significance of the Regression Model 48516.4 Two-Way Analysis of Variance Regression Model with No Interactions 48616.5 Two-Way Analysis of Variance Regression Model with Interactions 49016.6 Analysis of Covariance Regression Model 491Exercises 49917 Model Building: Stepwise Regression and Best Subsets Regression 50117.1 Introduction and Overview 50117.2 Consequences of Inappropriate Variable Selection 50217.3 Stepwise Regression Procedures 50517.3.1 Advantages and Disadvantages of Stepwise Procedures 51217.4 Subsets Regression 513Exercises 52218 Nonlinear Regression 52518.1 Introduction and Overview 52518.2 The Nonlinear Regression Model 52618.3 Assumptions of Nonlinear Least Squares Regression 528Exercises 545Part IV Statistics in Environmental Sampling Design and Risk Assessment 54719 Data Quality Objectives and Environmental Sampling Design 54919.1 Introduction and Overview 54919.2 Sampling Design 55019.3 Sampling Plans 55019.3.1 Simple Random Sampling 55219.3.2 Systematic Sampling 55419.3.3 Other Sampling Designs 55619.4 Sample Size Determination 55719.4.1 Types I and II Decision Errors 55819.4.2 Variance and Gray Region 55919.4.3 Width of the Gray Region 56019.4.4 Computation of the Recommended Minimum Sample Size for Estimating the Population Mean or Median 56119.4.4.1 Minimum Sample Size for Computing UCL95 on the Mean for Normally Distributed Data 56219.4.4.2 Minimum Sample Size for Computing UCL95 on the Median for Nonnormally Distributed Data 56419.4.5 Computation of the Recommended Minimum Sample Size for Comparing a Population Mean or Median with a FixedThreshold Value 56519.4.6 Computation of the Recommended Minimum Sample Size for Comparing the Population Means or Medians for Two Populations 568Exercises 56920 Determination of Background and Applications in Risk Assessment 57120.1 Introduction and Overview 57120.2 When Background Sampling is Required and When it is not 57220.3 Background Sampling Plans 57220.4 Graphical and Quantitative Data Analysis for Site Versus Background Data Comparisons 57320.5 Determination of Exposure Point Concentration and Contaminants of Potential Concern 583Exercises 58521 Statistics in Conventional and Probabilistic Risk Assessment 58721.1 Introduction and Overview 58721.2 Conventional or Point Risk Estimation 58821.3 Probabilistic Risk Assessment Using Monte Carlo Simulation 594Exercises 598Appendix A: Software Scripts 599Appendix B: Datasets 603References 609Answers for Exercises 613Index 619