Bayesian Biostatistics

Preface xiii Notation, terminology and some guidance for reading the book xvii Part I BASIC CONCEPTS IN BAYESIAN METHODS 1 Modes of statistical inference 3 1.1 The frequentist approach: A critical reflection 4 1.2 Statistical inference based on the likelihood function 10 1.3 The Bayesian approach: Some basic ideas 14 1.4 Outlook 18 2 Bayes theorem: Computing the posterior distribution 20 2.1 Introduction 20 2.2 Bayes theorem - the binary version 20 2.3 Probability in a Bayesian context 21 2.4 Bayes theorem - the categorical version 22 2.5 Bayes theorem - the continuous version 23 2.6 The binomial case 24 2.7 The Gaussian case 30 2.8 The Poisson case 36 2.9 The prior and posterior distribution of h(theta) 40 2.10 Bayesian versus likelihood approach 40 2.11 Bayesian versus frequentist approach 41 2.12 The different modes of the Bayesian approach 41 2.13 An historical note on the Bayesian approach 42 2.14 Closing remarks 44 3 Introduction to Bayesian inference 46 3.1 Introduction 46 3.2 Summarizing the posterior by probabilities 46 3.3 Posterior summary measures 47 3.4 Predictive distributions 51 3.5 Exchangeability 58 3.6 A normal approximation to the posterior 60 3.7 Numerical techniques to determine the posterior 63 3.8 Bayesian hypothesis testing 72 3.9 Closing remarks 78 4 More than one parameter 82 4.1 Introduction 82 4.2 Joint versus marginal posterior inference 83 4.3 The normal distribution with mu and sigma2 unknown 83 4.4 Multivariate distributions 89 4.5 Frequentist properties of Bayesian inference 92 4.6 Sampling from the posterior distribution: The Method of Composition 93 4.7 Bayesian linear regression models 96 4.8 Bayesian generalized linear models 101 4.9 More complex regression models 102 4.10 Closing remarks 102 5 Choosing the prior distribution 104 5.1 Introduction 104 5.2 The sequential use of Bayes theorem 104 5.3 Conjugate prior distributions 106 5.4 Noninformative prior distributions 113 5.5 Informative prior distributions 121 5.6 Prior distributions for regression models 129 5.7 Modeling priors 134 5.8 Other regression models 136 5.9 Closing remarks 136 6 Markov chain Monte Carlo sampling 139 6.1 Introduction 139 6.2 The Gibbs sampler 140 6.3 The Metropolis(-Hastings) algorithm 154 6.4 Justification of the MCMC approaches* 162 6.5 Choice of the sampler 165 6.6 The Reversible Jump MCMC algorithm* 168 6.7 Closing remarks 172 7 Assessing and improving convergence of the Markov chain 175 7.1 Introduction 175 7.2 Assessing convergence of a Markov chain 176 7.3 Accelerating convergence 189 7.4 Practical guidelines for assessing and accelerating convergence 194 7.5 Data augmentation 195 7.6 Closing remarks 200 8 Software 202 8.1 WinBUGS and related software 202 8.2 Bayesian analysis using SAS 215 8.3 Additional Bayesian software and comparisons 221 8.4 Closing remarks 222 Part II BAYESIAN TOOLS FOR STATISTICAL MODELING 9 Hierarchical models 227 9.1 Introduction 227 9.2 The Poisson-gamma hierarchical model 228 9.3 Full versus empirical Bayesian approach 238 9.4 Gaussian hierarchical models 240 9.5 Mixed models 244 9.6 Propriety of the posterior 260 9.7 Assessing and accelerating convergence 261 9.8 Comparison of Bayesian and frequentist hierarchical models 263 9.9 Closing remarks 265 10 Model building and assessment 267 10.1 Introduction 267 10.2 Measures for model selection 268 10.3 Model checking 288 10.4 Closing remarks 316 11 Variable selection 319 11.1 Introduction 319 11.2 Classical variable selection 320 11.3 Bayesian variable selection: Concepts and questions 325 11.4 Introduction to Bayesian variable selection 326 11.5 Variable selection based on Zellner's g-prior 333 11.6 Variable selection based on Reversible Jump Markov chain Monte Carlo 336 11.7 Spike and slab priors 339 11.8 Bayesian regularization 345 11.9 The many regressors case 351 11.10 Bayesian model selection 355 11.11 Bayesian model averaging 357 11.12 Closing remarks 359 Part III BAYESIAN METHODS IN PRACTICAL APPLICATIONS 12 Bioassay 365 12.1 Bioassay essentials 365 12.2 A generic in vitro example 369 12.3 Ames/Salmonella mutagenic assay 371 12.4 Mouse lymphoma assay (L5178Y TK+/.) 373 12.5 Closing remarks 374 13 Measurement error 375 13.1 Continuous measurement error 375 13.2 Discrete measurement error 382 13.3 Closing remarks 389 14 Survival analysis 390 14.1 Basic terminology 390 14.2 The Bayesian model formulation 394 14.3 Examples 397 14.4 Closing remarks 406 15 Longitudinal analysis 407 15.1 Fixed time periods 407 15.2 Random event times 417 15.3 Dealing with missing data 420 15.4 Joint modeling of longitudinal and survival responses 424 15.5 Closing remarks 429 16 Spatial applications: Disease mapping and image analysis 430 16.1 Introduction 430 16.2 Disease mapping 430 16.3 Image analysis 444 17 Final chapter 456 17.1 What this book covered 456 17.2 Additional Bayesian developments 456 17.3 Alternative reading 459 Appendix: Distributions 460 A.1 Introduction 460 A.2 Continuous univariate distributions 461 A.3 Discrete univariate distributions 477 A.4 Multivariate distributions 481 References 484 Index 509