Engineering Biostatistics

Preface v 1 Introduction 1 Chapter References 7 2 The Sample and Its Properties 9 2.1 Introduction 9 2.2 A MATLAB Session on Univariate Descriptive Statistics 10 2.3 Location Measures 12 2.4 Variability Measures 15 2.4.1 Ranks 24 2.5 Displaying Data 25 2.6 Multidimensional Samples: Fisher's Iris Data and Body Fat Data 29 2.7 Multivariate Samples and Their Summaries 35 2.8 Principal Components of Data 40 2.9 Visualizing Multivariate Data 45 2.10 Observations as Time Series 49 2.11 About Data Types 52 2.12 Big Data Paradigm 53 2.13 Exercises 55 Chapter References 70 3 Probability, Conditional Probability, and Bayes' Rule 73 3.1 Introduction 73 3.2 Events and Probability 74 3.3 Odds 85 3.4 Venn Diagrams 86 3.5 Counting Principles 88 3.6 Conditional Probability and Independence 92 3.6.1 Pairwise and Global Independence 97 3.7 Total Probability 97 3.8 Reassesing Probabilities: Bayes' Rule 100 3.9 Bayesian Networks 105 3.10 Exercises 111 Chapter References 130 4 Sensitivity, Specificity, and Relatives 133 4.1 Introduction 133 4.2 Notation 134 4.2.1 Conditional Probability Notation 138 4.3 Combining Two or More Tests 141 4.4 ROC Curves 144 4.5 Exercises 149 Chapter References 157 5 Random Variables 159 5.1 Introduction 159 5.2 Discrete Random Variables 161 5.2.1 Jointly Distributed Discrete Random Variables 166 5.3 Some Standard Discrete Distributions 169 5.3.1 Discrete Uniform Distribution 169 5.3.2 Bernoulli and Binomial Distributions 170 5.3.3 Hypergeometric Distribution 174 5.3.4 Poisson Distribution 177 5.3.5 Geometric Distribution 180 5.3.6 Negative Binomial Distribution 183 5.3.7 Multinomial Distribution 184 5.3.8 Quantiles 186 5.4 Continuous Random Variables 187 5.4.1 Joint Distribution of Two Continuous Random Variables 192 5.4.2 Conditional Expectation 193 5.5 Some Standard Continuous Distributions 195 5.5.1 Uniform Distribution 196 5.5.2 Exponential Distribution 198 5.5.3 Normal Distribution 200 5.5.4 Gamma Distribution 201 5.5.5 Inverse Gamma Distribution 203 5.5.6 Beta Distribution 203 5.5.7 Double Exponential Distribution 205 5.5.8 Logistic Distribution 206 5.5.9 Weibull Distribution 207 5.5.10 Pareto Distribution 208 5.5.11 Dirichlet Distribution 209 5.6 Random Numbers and Probability Tables 210 5.7 Transformations of Random Variables 211 5.8 Mixtures 214 5.9 Markov Chains 215 5.10 Exercises 219 Chapter References 232 6 Normal Distribution 235 6.1 Introduction 235 6.2 Normal Distribution 236 6.2.1 Sigma Rules 240 6.2.2 Bivariate Normal Distribution 241 6.3 Examples with a Normal Distribution 243 6.4 Combining Normal Random Variables 246 6.5 Central Limit Theorem 249 6.6 Distributions Related to Normal 253 6.6.1 Chi-square Distribution 254 6.6.2 t-Distribution 258 6.6.3 Cauchy Distribution 259 6.6.4 F-Distribution 260 6.6.5 Noncentral chi2, t, and F Distributions 262 6.6.6 Lognormal Distribution 263 6.7 Delta Method and Variance Stabilizing Transformations 265 6.8 Exercises 268 Chapter References 274 7 Point and Interval Estimators 277 7.1 Introduction 277 7.2 Moment Matching and Maximum Likelihood Estimators 278 7.2.1 Unbiasedness and Consistency of Estimators 285 7.3 Estimation of a Mean, Variance, and Proportion 288 7.3.1 Point Estimation of Mean 288 7.3.2 Point Estimation of Variance 290 7.3.3 Point Estimation of Population Proportion 294 7.4 Confidence Intervals 295 7.4.1 Confidence Intervals for the Normal Mean 296 7.4.2 Confidence Interval for the Normal Variance 299 7.4.3 Confidence Intervals for the Population Proportion . . . 302 7.4.4 Confidence Intervals for Proportions When X = 0 306 7.4.5 Designing the Sample Size with Confidence Intervals 307 7.5 Prediction and Tolerance Intervals 309 7.6 Confidence Intervals for Quantiles 311 7.7 Confidence Intervals for the Poisson Rate 312 7.8 Exercises 315 Chapter References 328 8 Bayesian Approach to Inference 331 8.1 Introduction 331 8.2 Ingredients for Bayesian Inference 334 8.3 Conjugate Priors 338 8.4 Point Estimation 340 8.4.1 Normal-Inverse Gamma Conjugate Analysis 343 8.5 Prior Elicitation 345 8.6 Bayesian Computation and Use of WinBUGS 348 8.6.1 Zero Tricks in WinBUGS 351 8.7 Bayesian Interval Estimation: Credible Sets 353 8.8 Learning by Bayes' Theorem 357 8.9 Bayesian Prediction 358 8.10 Consensus Means 362 8.11 Exercises 365 Chapter References 372 9 Testing Statistical Hypotheses 375 9.1 Introduction 375 9.2 Classical Testing Problem 377 9.2.1 Choice of Null Hypothesis 377 9.2.2 Test Statistic, Rejection Regions, Decisions, and Errors in Testing 379 9.2.3 Power of the Test 380 9.2.4 Fisherian Approach: p-Values 381 9.3 Bayesian Approach to Testing 382 9.3.1 Criticism and Calibration of p-Values 386 9.4 Testing the Normal Mean 388 9.4.1 z-Test 389 9.4.2 Power Analysis of a z-Test 389 9.4.3 Testing a Normal Mean When the Variance Is Not Known: t-Test 391 9.4.4 Power Analysis of t-Test 394 9.5 Testing Multivariate Mean: T-Square Test* 397 9.5.1 T-Square Test 397 9.5.2 Test for Symmetry 401 9.6 Testing the Normal Variances 402 9.7 Testing the Proportion 404 9.7.1 Exact Test for Population Proportions 406 9.7.2 Bayesian Test for Population Proportions 409 9.8 Multiplicity in Testing, Bonferroni Correction, and False Discovery Rate 412 9.9 Exercises 415 Chapter References 425 10 Two Samples 427 10.1 Introduction 427 10.2 Means and Variances in Two Independent Normal Populations 428 10.2.1 Confidence Interval for the Difference of Means 433 10.2.2 Power Analysis for Testing Two Means 434 10.2.3 More Complex Two-Sample Designs 438 10.2.4 A Bayesian Test for Two Normal Means 439 10.3 Testing the Equality of Normal Means When Samples Are Paired 443 10.3.1 Sample Size in Paired t-Test 448 10.3.2 Difference-in-Differences (DiD) Tests 449 10.4 Two Multivariate Normal Means 451 10.4.1 Confidence Intervals for Arbitrary Linear Combinations of Mean Differences 453 10.4.2 Profile Analysis With Two Independent Groups 454 10.4.3 Paired Multivariate Samples 456 10.5 Two Normal Variances 459 10.6 Comparing Two Proportions 463 10.6.1 The Sample Size 465 10.7 Risk Differences, Risk Ratios, and Odds Ratios 466 10.7.1 Risk Differences 466 10.7.2 Risk Ratio 467 10.7.3 Odds Ratios 469 10.7.4 Two Proportions from a Single Sample 473 10.8 Two Poisson Rates 476 10.9 Equivalence Tests 479 10.10 Exercises 483 Chapter References 500 11 ANOVA and Elements of Experimental Design 503 11.1 Introduction 503 11.2 One-Way ANOVA 504 11.2.1 ANOVA Table and Rationale for F-Test 506 11.2.2 Testing Assumption of Equal Population Variances . . . 509 11.2.3 The Null Hypothesis Is Rejected. What Next? 511 11.2.4 Bayesian Solution 516 11.2.5 Fixed- and Random-Effect ANOVA 518 11.3 Welch's ANOVA 518 11.4 Two-Way ANOVA and Factorial Designs 521 11.4.1 Two-way ANOVA: One Observation Per Cell 527 11.5 Blocking 529 11.6 Repeated Measures Design 531 11.6.1 Sphericity Tests 534 11.7 Nested Designs 535 11.8 Power Analysis in ANOVA 539 11.9 Functional ANOVA 545 11.10 Analysis of Means (ANOM) 548 11.11 Gauge R&R ANOVA 550 11.12 Testing Equality of Several Proportions 556 11.13 Testing the Equality of Several Poisson Means 557 11.14 Exercises 559 Chapter References 582 12 Models for Tables 585 12.1 Introduction 586 12.2 Contingency Tables: Testing for Independence 586 12.2.1 Measuring Association in Contingency Tables 591 12.2.2 Power Analysis for Contingency Tables 593 12.2.3 Cohen's Kappa 594 12.3 Three-Way Tables 596 12.4 Fisher's Exact Test 600 12.5 Stratified Tables: Mantel-Haenszel Test 603 12.5.1 Testing Conditional Independence or Homogeneity . . . 604 12.5.2 Odds Ratio from Stratified Tables 607 12.6 Paired Tables: McNemar's Test 608 12.7 Risk Differences, Risk Ratios, and Odds Ratios for Paired Tables 610 12.7.1 Risk Differences 610 12.7.2 Risk Ratios 611 12.7.3 Odds Ratios 612 12.7.4 Liddell's Procedure 617 12.7.5 Garth Test 619 12.7.6 Stuart-Maxwell Test 620 12.7.7 Cochran's Q Test* 626 12.8 Exercises 628 Chapter References 643 13 Correlation 647 13.1 Introduction 647 13.2 The Pearson Coefficient of Correlation 648 13.2.1 Inference About rho 650 13.2.2 Bayesian Inference for Correlation Coefficients 663 13.3 Spearman's Coefficient of Correlation 665 13.4 Kendall's Tau 667 13.5 Cum hoc ergo propter hoc 670 13.6 Exercises 671 Chapter References 677 14 Regression 679 14.1 Introduction 679 14.2 Simple Linear Regression 680 14.2.1 Inference in Simple Linear Regression 688 14.3 Calibration 697 14.4 Testing the Equality of Two Slopes 699 14.5 Multiple Regression 702 14.5.1 Matrix Notation 703 14.5.2 Residual Analysis, Influential Observations, Multicollinearity, and Variable Selection 709 14.6 Sample Size in Regression 720 14.7 Linear Regression That Is Nonlinear in Predictors 720 14.8 Errors-In-Variables Linear Regression 723 14.9 Analysis of Covariance 724 14.9.1 Sample Size in ANCOVA 728 14.9.2 Bayesian Approach to ANCOVA 729 14.10 Exercises 731 Chapter References 748 15 Regression for Binary and Count Data 751 15.1 Introduction 751 15.2 Logistic Regression 752 15.2.1 Fitting Logistic Regression 753 15.2.2 Assessing the Logistic Regression Fit 758 15.2.3 Probit and Complementary Log-Log Links 769 15.3 Poisson Regression 773 15.4 Log-linear Models 779 15.5 Exercises 783 Chapter References 798 16 Inference for Censored Data and Survival Analysis 801 16.1 Introduction 801 16.2 Definitions 802 16.3 Inference with Censored Observations 807 16.3.1 Parametric Approach 807 16.3.2 Nonparametric Approach: Kaplan-Meier or Product-Limit Estimator 809 16.3.3 Comparing Survival Curves 815 16.4 The Cox Proportional Hazards Model 818 16.5 Bayesian Approach 822 16.5.1 Survival Analysis in WinBUGS 823 16.6 Exercises 829 Chapter References 835 17 Goodness of Fit Tests 837 17.1 Introduction 837 17.2 Probability Plots 838 17.2.1 Q-Q Plots 838 17.2.2 P-P Plots 841 17.2.3 Poissonness Plots 842 17.3 Pearson's Chi-Square Test 843 17.4 Kolmogorov-Smirnov Tests 852 17.4.1 Kolmogorov's Test 852 17.4.2 Smirnov's Test to Compare Two Distributions 854 17.5 Cramér-von Mises and Watson's Tests 858 17.5.1 Rosenblatt's Test 860 17.6 Moran's Test 862 17.7 Departures from Normality 863 17.7.1 Ellimination of Unknown Parameters by Transformations 866 17.8 Exercises 867 Chapter References 876 18 Distribution-Free Methods 879 18.1 Introduction 879 18.2 Sign Test 880 18.3 Wilcoxon Signed-Rank Test 884 18.4 Wilcoxon Sum Rank Test and Mann-Whitney Test 887 18.5 Kruskal-Wallis Test 890 18.6 Friedman's Test 894 18.7 Resampling Methods 898 18.7.1 The Jackknife 898 18.7.2 Bootstrap 901 18.7.3 Bootstrap Versions of Some Popular Tests 908 18.7.4 Randomization and Permutation Tests 916 18.7.5 Discussion 919 18.8 Exercises 919 Chapter References 929 19 Bayesian Inference Using Gibbs Sampling - BUGS Project 931 19.1 Introduction 931 19.2 Step-by-Step Session 932 19.3 Built-in Functions and Common Distributions in WinBUGS 937 19.4 MATBUGS: A MATLAB Interface to WinBUGS 938 19.5 Exercises 942 Chapter References 943 Index 945