Ronald Christensen. University of New Mexico. Albuquerque, New Mexico. Wesley Johnson. University of California, Irvine. Irvine, California
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1 Texts in Statistical Science Bayesian Ideas and Data Analysis An Introduction for Scientists and Statisticians Ronald Christensen University of New Mexico Albuquerque, New Mexico Wesley Johnson University of California, Irvine Irvine, California Adam Branscum Oregon State University Corvallis, Oregon Timothy E. Hanson University of South Carolina Columbia, South Carolina & CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor St Francis Group an informa business A CHAPMAN & HALL BOOK
2 Contents Preface XV 1 Prologue Probability of a Defective: Binomial Data Brass Alloy Zinc Content: Normal Data Armadillo Hunting: Poisson Data Abortion in Dairy Cattle: Survival Data Ache Hunting with Age Trends Lung Cancer Treatment: Log-Normal Regression Survival with Random Effects: Ache Hunting 8 2 Fundamental Ideas I Simple Probability Computations Science, Priors, and Prediction Statistical Models Posterior Analysis Commonly Used Distributions 33 3 Integration Versus Simulation Introduction WinBUGS I: Getting Started Method of Composition Monte Carlo Integration* Posterior Computations in R 51 4 Fundamental Ideas II Statistical Testing Checking Bayesian Models Predictive P-Values Lindley-Jeffreys Paradox Exchangeability Likelihood Functions Sufficient Statistics Analysis Using Predictive Distributions Flat Priors Data Translated Likelihoods Jeffreys' Priors Multiple Parameter Jeffreys' Prior* Bayes Factors* General Parametric Testing Nested Models Simulating Bayes Factors 75 ix
3 CONTENTS 4.9 Other Model Selection Criteria Bayesian Information Criterion LPML Deviance Information Criterion Final Comments Normal Approximations to Posteriors* Bayesian Consistency and Inconsistency Hierarchical Models Some Final Comments on Likelihoods* Identiflability and Noninformative Data 94 Comparing Populations Inference for Proportions Prior Distributions Reference Priors Informative Beta Priors Rare Events Non-Beta Priors Effect Measures Independent Binomials Case-Control Sampling Inference for Normal Populations lit Reference Priors Conjugate Priors Independence Priors Some Curious Distributional Results* Two-Sample Normal Model Inference for Rates One-Sample Poisson Data Informative Priors Reference Priors Two-Sample Poisson Data Sample Size Determination* 136 Simulations Generating Random Samples Traditional Monte Carlo Methods Acceptance-Rejection Sampling Importance Sampling Markov Chain Monte Carlo Markov Chains Gibbs Sampling Proof that p(6) is the Stationary Distribution in the Two-Block Case* Metropolis Algorithm Proof that p(6) is the Stationary Distribution* Slice Sampling Checking MCMC Samples 159
4 CONTENTS xi 7 Basic Concepts of Regression Introduction Data Notation and Format Predictive Models: An Overview Modeling with Linear Structures Continuous Predictors Binary Predictors Multi-Category Predictors Predictor Selection Several Categorical Covariates Confounding Effect Modification/Interaction Two Categorical Predictors One Continuous and One Categorical Predictor Two Continuous Predictors Illustration: FEV Data Binomial Regression The Sampling Model Binomial Regression Analysis Predictive Probabilities Inference for Regression Coefficients Inference for LDa Model Checking Box's Method Link Selection Prior Distributions Simple Regression General Regression Prior Elicitation Data Augmentation Priors Standardized Variables Reference Priors Partial Prior Information Partial Priors: Theoretical Considerations* Mixed Models Prior Elicitation Mixed Model Likelihood Gibbs Sampling and Centering* Linear Regression The Sampling Model Reference Priors Least Squares Estimation Posterior Analysis A Proper Reference Prior Conjugate Priors Independence Priors Prior on j Prior on t Partial Prior Information Inference and Displays 238
5 xii CONTENTS Gibbs Sampling* WinBUGS and R Code ANOVA Independence Prior Allocation and Diagnosis Hierarchical Priors and Models Model Diagnostics Model Selection Nonlinear Regression* Correlated Data Introduction Mixed Models Random Intercept Model Random Slopes and Random Intercepts Multivariate Normal Models Parameterized Covariance Matrices Analytic Formulas for CS and AR(1) Precision Matrices Multivariate Normal Regression Posterior Sampling and Missing Data Count Data Poisson Regression Poisson Regression for Rates Over-Dispersion and Mixtures of Poissons Zero-Inflated Poisson Data SAS Analysis of Foot-and-Mouth Disease Data Longitudinal Data Time to Event Data Introduction Survival and Hazard Functions Censoring The Likelihood One-Sample Models Distributional Models Posterior Analysis Log-Normal Data Exponential Data WinBUGS for Censored Data WeibullData Prediction Interval Censoring Two-Sample Data Two-Sample Exponential Model Two-Sample Weibull Model Two-Sample Log-Normal Model Plotting Survival and Hazard Functions' 322
6 CONTENTS xiii 13 Time to Event Regression Accelerated Failure Time Models Abortion Data Prior Elicitation for AFTs Specifying the Marginal Prior for j Partial Prior Information for/ Uncertainty About t Case Deletion Diagnostics for AFT Models Predictive Influence Bayes Factor Model Selection Sensitivity Analysis Final Comments Proportional Hazards Modeling The Proportional Hazards (PH) Model A Baseline Hazard Model The Likelihood NoninformativeData* Priors for J Priors for X Our Data Model WinBUGSCode Posterior Analysis for Leukemia Data SAS Analysis of Leukemia Data Another Example Survival with Random Effects Binary Diagnostic Tests Basic Ideas One Test, One Population Gold-Standard Data No Gold-Standard Data Two Tests, Two Populations Methods for Conditionally Independent Tests Prevalence Distributions Nonparametric Models Flexible Density Shapes Finite Mixtures Identifiability Issues* Dirichlet Process Mixtures: Infinite Mixtures Mixtures of Polya Trees Flexible Regression Functions Proportional Hazards Modeling 414 Appendix A: Matrices and Vectors 419 A.l Matrix Addition and Subtraction 420 A.2 Scalar Multiplication 420 A.3 Matrix Multiplication 420 A.4 Special Matrices 422 A.5 Linear Dependence and Rank 423 A.6 Inverse Matrices 424 A.7 A List of Useful Properties 426
7 xiv CONTENTS A.8 Eigenvalues and Eigenvectors 426 A.9 Properties of Determinants 428 A. 10 Calculus and Taylor's Theorem 428 A. 11 Partitioned Matrices 428 Appendix B: Probability 431 B. 1 Univariate Probability 431 B.2 Multivariate Probability 432 B.2.1 Joint Distribution of Two Vectors 434 B.2.2 Conditional Distributions 434 B.2.3 Independence 436 B.2.4 Moment Generating Functions 437 B.2.5 Change of Variables 437 B. 3 Models and Conditional Independence 438 Appendix C: Getting Started in R 443 C. l Getting R 443 C.2 Some R Basics 443 C.3 User-Contributed Packages 446 C.4 Reading Data 447 C.5 Graphing 447 C.6 Interface Between R and WinBUGS 456 C.7 Writing New R Functions 456 References 459 Author Index 467 Subject Index 473
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