CiNii 図書 - Bayesian biostatistics

The growth of biostatistics has been phenomenal in recent years and has been marked by considerable technical innovation in both methodology and computational practicality. One area that has experienced significant growth is Bayesian methods. The growing use of Bayesian methodology has taken place partly due to an increasing number of practitioners valuing the Bayesian paradigm as matching that of scientific discovery. In addition, computational advances have allowed for more complex models to be fitted routinely to realistic data sets. Through examples, exercises and a combination of introductory and more advanced chapters, this book provides an invaluable understanding of the complex world of biomedical statistics illustrated via a diverse range of applications taken from epidemiology, exploratory clinical studies, health promotion studies, image analysis and clinical trials. Key Features: Provides an authoritative account of Bayesian methodology, from its most basic elements to its practical implementation, with an emphasis on healthcare techniques. Contains introductory explanations of Bayesian principles common to all areas of application. Presents clear and concise examples in biostatistics applications such as clinical trials, longitudinal studies, bioassay, survival, image analysis and bioinformatics. Illustrated throughout with examples using software including WinBUGS, OpenBUGS, SAS and various dedicated R programs. Highlights the differences between the Bayesian and classical approaches. Supported by an accompanying website hosting free software and case study guides. Bayesian Biostatistics introduces the reader smoothly into the Bayesian statistical methods with chapters that gradually increase in level of complexity. Master students in biostatistics, applied statisticians and all researchers with a good background in classical statistics who have interest in Bayesian methods will find this book useful.

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 1.1.1 The classical statistical approach 4 1.1.2 The P-value as a measure of evidence 5 1.1.3 The confidence interval as a measure of evidence 8 1.1.4 An historical note on the two frequentist paradigms 8 1.2 Statistical inference based on the likelihood function 10 1.2.1 The likelihood function 10 1.2.2 The likelihood principles 11 1.3 The Bayesian approach: Some basic ideas 14 1.3.1 Introduction 14 1.3.2 Bayes theorem - discrete version for simple events 15 1.4 Outlook 18 Exercises 19 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( ) 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 Exercises 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 3.3.1 Characterizing the location and variability of the posterior distribution 47 3.3.2 Posterior interval estimation 49 3.4 Predictive distributions 51 3.4.1 The frequentist approach to prediction 52 3.4.2 The Bayesian approach to prediction 53 3.4.3 Applications 54 3.5 Exchangeability 58 3.6 A normal approximation to the posterior 60 3.6.1 A Bayesian analysis based on a normal approximation to the likelihood 60 3.6.2 Asymptotic properties of the posterior distribution 62 3.7 Numerical techniques to determine the posterior 63 3.7.1 Numerical integration 63 3.7.2 Sampling from the posterior 65 3.7.3 Choice of posterior summary measures 72 3.8 Bayesian hypothesis testing 72 3.8.1 Inference based on credible intervals 72 3.8.2 The Bayes factor 74 3.8.3 Bayesian versus frequentist hypothesis testing 76 3.9 Closing remarks 78 Exercises 79 4 More than one parameter 82 4.1 Introduction 82 4.2 Joint versus marginal posterior inference 83 4.3 The normal distribution with and 2 unknown 83 4.3.1 No prior knowledge on and 2 is available 84 4.3.2 An historical study is available 86 4.3.3 Expert knowledge is available 88 4.4 Multivariate distributions 89 4.4.1 The multivariate normal and related distributions 89 4.4.2 The multinomial distribution 90 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 4.7.1 The frequentist approach to linear regression 96 4.7.2 A noninformative Bayesian linear regression model 97 4.7.3 Posterior summary measures for the linear regression model 98 4.7.4 Sampling from the posterior distribution 99 4.7.5 An informative Bayesian linear regression model 101 4.8 Bayesian generalized linear models 101 4.9 More complex regression models 102 4.10 Closing remarks 102 Exercises 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 5.3.1 Univariate data distributions 106 5.3.2 Normal distribution - mean and variance unknown 109 5.3.3 Multivariate data distributions 110 5.3.4 Conditional conjugate and semiconjugate distributions 111 5.3.5 Hyperpriors 112 5.4 Noninformative prior distributions 113 5.4.1 Introduction 113 5.4.2 Expressing ignorance 114 5.4.3 General principles to choose noninformative priors 115 5.4.4 Improper prior distributions 119 5.4.5 Weak/vague priors 120 5.5 Informative prior distributions 121 5.5.1 Introduction 121 5.5.2 Data-based prior distributions 121 5.5.3 Elicitation of prior knowledge 122 5.5.4 Archetypal prior distributions 126 5.6 Prior distributions for regression models 129 5.6.1 Normal linear regression 129 5.6.2 Generalized linear models 131 5.6.3 Specification of priors in Bayesian software 134 5.7 Modeling priors 134 5.8 Other regression models 136 5.9 Closing remarks 136 Exercises 137 6 Markov chain Monte Carlo sampling 139 6.1 Introduction 139 6.2 The Gibbs sampler 140 6.2.1 The bivariate Gibbs sampler 140 6.2.2 The general Gibbs sampler 146 6.2.3 Remarks 150 6.2.4 Review of Gibbs sampling approaches 152 6.2.5 The Slice sampler 153 6.3 The Metropolis(-Hastings) algorithm 154 6.3.1 The Metropolis algorithm 155 6.3.2 The Metropolis-Hastings algorithm 157 6.3.3 Remarks 159 6.3.4 Review of Metropolis(-Hastings) approaches 161 6.4 Justification of the MCMC approaches 162 6.4.1 Properties of the MH algorithm 164 6.4.2 Properties of the Gibbs sampler 165 6.5 Choice of the sampler 165 6.6 The Reversible Jump MCMC algorithm 168 6.7 Closing remarks 172 Exercises 173 7 Assessing and improving convergence of the Markov chain 175 7.1 Introduction 175 7.2 Assessing convergence of a Markov chain 176 7.2.1 Definition of convergence for a Markov chain 176 7.2.2 Checking convergence of the Markov chain 176 7.2.3 Graphical approaches to assess convergence 177 7.2.4 Formal diagnostic tests 180 7.2.5 Computing the Monte Carlo standard error 186 7.2.6 Practical experience with the formal diagnostic procedures 188 7.3 Accelerating convergence 189 7.3.1 Introduction 189 7.3.2 Acceleration techniques 189 7.4 Practical guidelines for assessing and accelerating convergence 194 7.5 Data augmentation 195 7.6 Closing remarks 200 Exercises 201 8 Software 202 8.1 WinBUGS and related software 202 8.1.1 A first analysis 203 8.1.2 Information on samplers 206 8.1.3 Assessing and accelerating convergence 207 8.1.4 Vector and matrix manipulations 208 8.1.5 Working in batch mode 210 8.1.6 Troubleshooting 212 8.1.7 Directed acyclic graphs 212 8.1.8 Add-on modules: GeoBUGS and PKBUGS 214 8.1.9 Related software 214 8.2 Bayesian analysis using SAS 215 8.2.1 Analysis using procedure GENMOD 215 8.2.2 Analysis using procedure MCMC 217 8.2.3 Other Bayesian programs 220 8.3 Additional Bayesian software and comparisons 221 8.3.1 Additional Bayesian software 221 8.3.2 Comparison of Bayesian software 222 8.4 Closing remarks 222 Exercises 223 Part II Bayesian Tools for Statistical Modeling 9 Hierarchical models 227 9.1 Introduction 227 9.2 The Poisson-gamma hierarchical model 228 9.2.1 Introduction 228 9.2.2 Model specification 229 9.2.3 Posterior distributions 231 9.2.4 Estimating the parameters 232 9.2.5 Posterior predictive distributions 237 9.3 Full versus empirical Bayesian approach 238 9.4 Gaussian hierarchical models 240 9.4.1 Introduction 240 9.4.2 The Gaussian hierarchical model 240 9.4.3 Estimating the parameters 241 9.4.4 Posterior predictive distributions 243 9.4.5 Comparison of FB and EB approach 244 9.5 Mixed models 244 9.5.1 Introduction 244 9.5.2 The linear mixed model 244 9.5.3 The generalized linear mixed model 248 9.5.4 Nonlinear mixed models 253 9.5.5 Some further extensions 256 9.5.6 Estimation of the random effects and posterior predictive distributions 256 9.5.7 Choice of the level-2 variance prior 258 9.6 Propriety of the posterior 260 9.7 Assessing and accelerating convergence 261 9.8 Comparison of Bayesian and frequentist hierarchical models 263 9.8.1 Estimating the level-2 variance 263 9.8.2 ML and REml estimates compared with Bayesian estimates 264 9.9 Closing remarks 265 Exercises 265 10 Model building and assessment 267 10.1 Introduction 267 10.2 Measures for model selection 268 10.2.1 The Bayes factor 268 10.2.2 Information theoretic measures for model selection 274 10.2.3 Model selection based on predictive loss functions 286 10.3 Model checking 288 10.3.1 Introduction 288 10.3.2 Model-checking procedures 289 10.3.3 Sensitivity analysis 295 10.3.4 Posterior predictive checks 300 10.3.5 Model expansion 308 10.4 Closing remarks 316 Exercises 316 11 Variable selection 319 11.1 Introduction 319 11.2 Classical variable selection 320 11.2.1 Variable selection techniques 320 11.2.2 Frequentist regularization 322 11.3 Bayesian variable selection: Concepts and questions 325 11.4 Introduction to Bayesian variable selection 326 11.4.1 Variable selection for K small 326 11.4.2 Variable selection for K large 330 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 11.7.1 Stochastic Search Variable Selection 340 11.7.2 Gibbs Variable Selection 343 11.7.3 Dependent variable selection using SSVS 345 11.8 Bayesian regularization 345 11.8.1 Bayesian LASSO regression 346 11.8.2 Elastic Net and further extensions of the Bayesian LASSO 350 11.9 The many regressors case 351 11.10 Bayesian model selection 355 11.11 Bayesian model averaging 357 11.12 Closing remarks 359 Exercises 360 Part III Bayesian Methods in Practical Applications 12 Bioassay 365 12.1 Bioassay essentials 365 12.1.1 Cell assays 365 12.1.2 Animal assays 366 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 13.1.1 Measurement error in a variable 375 13.1.2 Two types of measurement error on the predictor in linear and nonlinear models 376 13.1.3 Accommodation of predictor measurement error 378 13.1.4 Nonadditive errors and other extensions 382 13.2 Discrete measurement error 382 13.2.1 Sources of misclassification 382 13.2.2 Misclassification in the binary predictor 383 13.2.3 Misclassification in a binary response 386 13.3 Closing remarks 389 14 Survival analysis 390 14.1 Basic terminology 390 14.1.1 Endpoint distributions 391 14.1.2 Censoring 392 14.1.3 Random effect specification 393 14.1.4 A general hazard model 393 14.1.5 Proportional hazards 394 14.1.6 The Cox model with random effects 394 14.2 The Bayesian model formulation 394 14.2.1 A Weibull survival model 395 14.2.2 A Bayesian AFT model 397 14.3 Examples 397 14.3.1 The gastric cancer study 397 14.3.2 Prostate cancer in Louisiana: A spatial AFT model 401 14.4 Closing remarks 406 15 Longitudinal analysis 407 15.1 Fixed time periods 407 15.1.1 Introduction 407 15.1.2 A classical growth-curve example 408 15.1.3 Alternate data models 414 15.2 Random event times 417 15.3 Dealing with missing data 420 15.3.1 Introduction 420 15.3.2 Response missingness 421 15.3.3 Missingness mechanisms 422 15.3.4 Bayesian considerations 424 15.3.5 Predictor missingness 424 15.4 Joint modeling of longitudinal and survival responses 424 15.4.1 Introduction 424 15.4.2 An example 425 15.5 Closing remarks 429 16 Spatial applications: Disease mapping and image analysis 430 16.1 Introduction 430 16.2 Disease mapping 430 16.2.1 Some general spatial epidemiological issues 431 16.2.2 Some spatial statistical issues 433 16.2.3 Count data models 433 16.2.4 A special application area: Disease mapping/risk estimation 434 16.2.5 A special application area: Disease clustering 438 16.2.6 A special application area: Ecological analysis 443 16.3 Image analysis 444 16.3.1 fMRI modeling 446 16.3.2 A note on software 455 17 Final chapter 456 17.1 What this book covered 456 17.2 Additional Bayesian developments 456 17.2.1 Medical decision making 456 17.2.2 Clinical trials 457 17.2.3 Bayesian networks 457 17.2.4 Bioinformatics 458 17.2.5 Missing data 458 17.2.6 Mixture models 458 17.2.7 Nonparametric Bayesian methods 459 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

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