Engineering Biostatistics

BRANI VIDAKOVIC, PhD, is a Professor in the School of Industrial and Systems Engineering (ISyE) at Georgia Institute of Technology and Department of Biomedical Engineering at Georgia Institute of Technology/Emory University. Dr. Vidakovic is a Fellow of the American Statistical Association, Elected Member of the International Statistical Institute, an Editor-in-Chief of Encyclopedia of Statistical Sciences, Second Edition, and former and current Associate Editor of several leading journals in the field of statistics.

• Preface v1 Introduction 1Chapter References 72 The Sample and Its Properties 92.1 Introduction 92.2 A MATLAB Session on Univariate Descriptive Statistics 102.3 Location Measures 122.4 Variability Measures 152.4.1 Ranks 242.5 Displaying Data 252.6 Multidimensional Samples: Fisher’s Iris Data and Body Fat Data 292.7 Multivariate Samples and Their Summaries 352.8 Principal Components of Data 402.9 Visualizing Multivariate Data 452.10 Observations as Time Series 492.11 About Data Types 522.12 Big Data Paradigm 532.13 Exercises 55Chapter References 703 Probability, Conditional Probability, and Bayes’ Rule 733.1 Introduction 733.2 Events and Probability 743.3 Odds 853.4 Venn Diagrams 863.5 Counting Principles 883.6 Conditional Probability and Independence 923.6.1 Pairwise and Global Independence 973.7 Total Probability 973.8 Reassesing Probabilities: Bayes’ Rule 1003.9 Bayesian Networks 1053.10 Exercises 111Chapter References 1304 Sensitivity, Specificity, and Relatives 1334.1 Introduction 1334.2 Notation 1344.2.1 Conditional Probability Notation 1384.3 Combining Two or More Tests 1414.4 ROC Curves 1444.5 Exercises 149Chapter References 1575 Random Variables 1595.1 Introduction 1595.2 Discrete Random Variables 1615.2.1 Jointly Distributed Discrete Random Variables 1665.3 Some Standard Discrete Distributions 1695.3.1 Discrete Uniform Distribution 1695.3.2 Bernoulli and Binomial Distributions 1705.3.3 Hypergeometric Distribution 1745.3.4 Poisson Distribution 1775.3.5 Geometric Distribution 1805.3.6 Negative Binomial Distribution 1835.3.7 Multinomial Distribution 1845.3.8 Quantiles 1865.4 Continuous Random Variables 1875.4.1 Joint Distribution of Two Continuous Random Variables 1925.4.2 Conditional Expectation 1935.5 Some Standard Continuous Distributions 1955.5.1 Uniform Distribution 1965.5.2 Exponential Distribution 1985.5.3 Normal Distribution 2005.5.4 Gamma Distribution 2015.5.5 Inverse Gamma Distribution 2035.5.6 Beta Distribution 2035.5.7 Double Exponential Distribution 2055.5.8 Logistic Distribution 2065.5.9 Weibull Distribution 2075.5.10 Pareto Distribution 2085.5.11 Dirichlet Distribution 2095.6 Random Numbers and Probability Tables 2105.7 Transformations of Random Variables 2115.8 Mixtures 2145.9 Markov Chains 2155.10 Exercises 219Chapter References 2326 Normal Distribution 2356.1 Introduction 2356.2 Normal Distribution 2366.2.1 Sigma Rules 2406.2.2 Bivariate Normal Distribution 2416.3 Examples with a Normal Distribution 2436.4 Combining Normal Random Variables 2466.5 Central Limit Theorem 2496.6 Distributions Related to Normal 2536.6.1 Chi-square Distribution 2546.6.2 t-Distribution 2586.6.3 Cauchy Distribution 2596.6.4 F-Distribution 2606.6.5 Noncentral χ2, t, and F Distributions 2626.6.6 Lognormal Distribution 2636.7 Delta Method and Variance Stabilizing Transformations 2656.8 Exercises 268Chapter References 2747 Point and Interval Estimators 2777.1 Introduction 2777.2 Moment Matching and Maximum Likelihood Estimators 2787.2.1 Unbiasedness and Consistency of Estimators 2857.3 Estimation of a Mean, Variance, and Proportion 2887.3.1 Point Estimation of Mean 2887.3.2 Point Estimation of Variance 2907.3.3 Point Estimation of Population Proportion 2947.4 Confidence Intervals 2957.4.1 Confidence Intervals for the Normal Mean 2967.4.2 Confidence Interval for the Normal Variance 2997.4.3 Confidence Intervals for the Population Proportion . . . 3027.4.4 Confidence Intervals for Proportions When X = 0 3067.4.5 Designing the Sample Size with Confidence Intervals 3077.5 Prediction and Tolerance Intervals 3097.6 Confidence Intervals for Quantiles 3117.7 Confidence Intervals for the Poisson Rate 3127.8 Exercises 315Chapter References 3288 Bayesian Approach to Inference 3318.1 Introduction 3318.2 Ingredients for Bayesian Inference 3348.3 Conjugate Priors 3388.4 Point Estimation 3408.4.1 Normal-Inverse Gamma Conjugate Analysis 3438.5 Prior Elicitation 3458.6 Bayesian Computation and Use of WinBUGS 3488.6.1 Zero Tricks in WinBUGS 3518.7 Bayesian Interval Estimation: Credible Sets 3538.8 Learning by Bayes’ Theorem 3578.9 Bayesian Prediction 3588.10 Consensus Means 3628.11 Exercises 365Chapter References 3729 Testing Statistical Hypotheses 3759.1 Introduction 3759.2 Classical Testing Problem 3779.2.1 Choice of Null Hypothesis 3779.2.2 Test Statistic, Rejection Regions, Decisions, and Errors in Testing 3799.2.3 Power of the Test 3809.2.4 Fisherian Approach: p-Values 3819.3 Bayesian Approach to Testing 3829.3.1 Criticism and Calibration of p-Values 3869.4 Testing the Normal Mean 3889.4.1 z-Test 3899.4.2 Power Analysis of a z-Test 3899.4.3 Testing a Normal Mean When the Variance Is Not Known: t-Test 3919.4.4 Power Analysis of t-Test 3949.5 Testing Multivariate Mean: T-Square Test∗ 3979.5.1 T-Square Test 3979.5.2 Test for Symmetry 4019.6 Testing the Normal Variances 4029.7 Testing the Proportion 4049.7.1 Exact Test for Population Proportions 4069.7.2 Bayesian Test for Population Proportions 4099.8 Multiplicity in Testing, Bonferroni Correction, and False Discovery Rate 4129.9 Exercises 415Chapter References 42510 Two Samples 42710.1 Introduction 42710.2 Means and Variances in Two Independent Normal Populations 42810.2.1 Confidence Interval for the Difference of Means 43310.2.2 Power Analysis for Testing Two Means 43410.2.3 More Complex Two-Sample Designs 43810.2.4 A Bayesian Test for Two Normal Means 43910.3 Testing the Equality of Normal Means When Samples Are Paired 44310.3.1 Sample Size in Paired t-Test 44810.3.2 Difference-in-Differences (DiD) Tests 44910.4 Two Multivariate Normal Means 45110.4.1 Confidence Intervals for Arbitrary Linear Combinations of Mean Differences 45310.4.2 Profile Analysis With Two Independent Groups 45410.4.3 Paired Multivariate Samples 45610.5 Two Normal Variances 45910.6 Comparing Two Proportions 46310.6.1 The Sample Size 46510.7 Risk Differences, Risk Ratios, and Odds Ratios 46610.7.1 Risk Differences 46610.7.2 Risk Ratio 46710.7.3 Odds Ratios 46910.7.4 Two Proportions from a Single Sample 47310.8 Two Poisson Rates 47610.9 Equivalence Tests 47910.10 Exercises 483Chapter References 50011 ANOVA and Elements of Experimental Design 50311.1 Introduction 50311.2 One-Way ANOVA 50411.2.1 ANOVA Table and Rationale for F-Test 50611.2.2 Testing Assumption of Equal Population Variances . . . 50911.2.3 The Null Hypothesis Is Rejected. What Next? 51111.2.4 Bayesian Solution 51611.2.5 Fixed- and Random-Effect ANOVA 51811.3 Welch’s ANOVA 51811.4 Two-Way ANOVA and Factorial Designs 52111.4.1 Two-way ANOVA: One Observation Per Cell 52711.5 Blocking 52911.6 Repeated Measures Design 53111.6.1 Sphericity Tests 53411.7 Nested Designs 53511.8 Power Analysis in ANOVA 53911.9 Functional ANOVA 54511.10 Analysis of Means (ANOM) 54811.11 Gauge R&R ANOVA 55011.12 Testing Equality of Several Proportions 55611.13 Testing the Equality of Several Poisson Means 55711.14 Exercises 559Chapter References 58212 Models for Tables 58512.1 Introduction 58612.2 Contingency Tables: Testing for Independence 58612.2.1 Measuring Association in Contingency Tables 59112.2.2 Power Analysis for Contingency Tables 59312.2.3 Cohen’s Kappa 59412.3 Three-Way Tables 59612.4 Fisher’s Exact Test 60012.5 Stratified Tables: Mantel–Haenszel Test 60312.5.1 Testing Conditional Independence or Homogeneity . . . 60412.5.2 Odds Ratio from Stratified Tables 60712.6 Paired Tables: McNemar’s Test 60812.7 Risk Differences, Risk Ratios, and Odds Ratios for Paired Tables 61012.7.1 Risk Differences 61012.7.2 Risk Ratios 61112.7.3 Odds Ratios 61212.7.4 Liddell’s Procedure 61712.7.5 Garth Test 61912.7.6 Stuart–Maxwell Test 62012.7.7 Cochran’s Q Test∗ 62612.8 Exercises 628Chapter References 64313 Correlation 64713.1 Introduction 64713.2 The Pearson Coefficient of Correlation 64813.2.1 Inference About ρ 65013.2.2 Bayesian Inference for Correlation Coefficients 66313.3 Spearman’s Coefficient of Correlation 66513.4 Kendall’s Tau 66713.5 Cum hoc ergo propter hoc 67013.6 Exercises 671Chapter References 67714 Regression 67914.1 Introduction 67914.2 Simple Linear Regression 68014.2.1 Inference in Simple Linear Regression 68814.3 Calibration 69714.4 Testing the Equality of Two Slopes 69914.5 Multiple Regression 70214.5.1 Matrix Notation 70314.5.2 Residual Analysis, Influential Observations, Multicollinearity, and Variable Selection 70914.6 Sample Size in Regression 72014.7 Linear Regression That Is Nonlinear in Predictors 72014.8 Errors-In-Variables Linear Regression 72314.9 Analysis of Covariance 72414.9.1 Sample Size in ANCOVA 72814.9.2 Bayesian Approach to ANCOVA 72914.10 Exercises 731Chapter References 74815 Regression for Binary and Count Data 75115.1 Introduction 75115.2 Logistic Regression 75215.2.1 Fitting Logistic Regression 75315.2.2 Assessing the Logistic Regression Fit 75815.2.3 Probit and Complementary Log-Log Links 76915.3 Poisson Regression 77315.4 Log-linear Models 77915.5 Exercises 783Chapter References 79816 Inference for Censored Data and Survival Analysis 80116.1 Introduction 80116.2 Definitions 80216.3 Inference with Censored Observations 80716.3.1 Parametric Approach 80716.3.2 Nonparametric Approach: Kaplan–Meier or Product–Limit Estimator 80916.3.3 Comparing Survival Curves 81516.4 The Cox Proportional Hazards Model 81816.5 Bayesian Approach 82216.5.1 Survival Analysis in WinBUGS 82316.6 Exercises 829Chapter References 83517 Goodness of Fit Tests 83717.1 Introduction 83717.2 Probability Plots 83817.2.1 Q–Q Plots 83817.2.2 P–P Plots 84117.2.3 Poissonness Plots 84217.3 Pearson’s Chi-Square Test 84317.4 Kolmogorov–Smirnov Tests 85217.4.1 Kolmogorov’s Test 85217.4.2 Smirnov’s Test to Compare Two Distributions 85417.5 Cramér-von Mises and Watson’s Tests 85817.5.1 Rosenblatt’s Test 86017.6 Moran’s Test 86217.7 Departures from Normality 86317.7.1 Ellimination of Unknown Parameters by Transformations 86617.8 Exercises 867Chapter References 87618 Distribution-Free Methods 87918.1 Introduction 87918.2 Sign Test 88018.3 Wilcoxon Signed-Rank Test 88418.4 Wilcoxon Sum Rank Test and Mann–Whitney Test 88718.5 Kruskal–Wallis Test 89018.6 Friedman’s Test 89418.7 Resampling Methods 89818.7.1 The Jackknife 89818.7.2 Bootstrap 90118.7.3 Bootstrap Versions of Some Popular Tests 90818.7.4 Randomization and Permutation Tests 91618.7.5 Discussion 91918.8 Exercises 919Chapter References 92919 Bayesian Inference Using Gibbs Sampling – BUGS Project 93119.1 Introduction 93119.2 Step-by-Step Session 93219.3 Built-in Functions and Common Distributions in WinBUGS 93719.4 MATBUGS: A MATLAB Interface to WinBUGS 93819.5 Exercises 942Chapter References 943Index 945