Bayesian Modeling Using WinBUGS

CONTENTS ... 7
	1  Introduction to Bayesian Inference ... 7
	2  Markov Chain Monte Carlo Algorithms in Bayesian Inference ... 8
	3  WinBUGS Software:  Introduction, Setup, and Basic Analysis ... 8
	4  Win BUGS Software: Illustration,  Results, and  Further Analysis ... 8
PREFACE ... 16
ACKNOWLEDGMENTS ... 18
ACRONYMS ... 19
Chapter 1 INTRODUCTION TO BAYESIAN INFERENCE ... 22
	1.1 INTRODUCTION: BAYESIAN MODELING IN THE 21ST CENTURY ... 22
	1.2 DEFINITION OF STATISTICAL MODELS ... 24
	1.3 BAYES THEOREM ... 24
	1.4 MODEL-BASED BAYESIAN INFERENCE ... 25
	1.5 INFERENCE USING CONJUGATE PRIOR DISTRIBUTIONS ... 28
		1.5.1 Inference for the Poisson rate of  count data ... 28
		1.5.2 Inference for the success probability of binomial data ... 29
		1.5.3 Inference for the mean of normal data with known variance ... 30
		1.5.4 Inference for the mean and variance of normal data ... 32
		1.5.5 Inference for normal regression models ... 33
		1.5.6 Other conjugate prior distributions ... 35
		1.5.7 Illustrative examples ... 35
	1.6 NONCONJUGATE ANALYSIS ... 45
	Problems ... 48
Chapter 2 MARKOV CHAIN MONTE CARLO ALGORITHMS IN BAYESIAN INFERENCE ... 51
	2.1 SIMULATION, MONTE  CARLO INTEGRATION, AND THEIR IMPLEMENTATION IN BAYESIAN INFERENCE ... 51
	2.2 MARKOV CHAIN MONTE CARLO METHODS ... 55
		2.2.1 The algorithm ... 56
		2.2.2 Terminology and implementation details ... 57
			2.2.2.1 Definitions and initial terminology ... 57
			2.2.2.2 Describing the target distribution using MCMC output. ... 58
			2.2.2.3 Monte Carlo error ... 59
			2.2.2.4 Convergence of the algorithm ... 61
	2.3 POPULAR MCMC ALGORITHMS ... 62
		2.3.1 The Metropolis-Hastings algorithm ... 62
			2.3.1.1 Random-walk Metropolis ... 63
			2.3.1.2 The independence  sampler. ... 64
		2.3.2 Componentwise Metropolis-Hastings ... 65
			2.3.2.1 Simple examples ... 66
		2.3.3 The Gibbs sampler ... 91
			2.3.3.1 A simple  example using  the Gibbs sampler ... 92
		2.3.4 Metropolis within Gibbs ... 96
		2.3.5 The slice  Gibbs sampler ... 96
		2.3.6 A simple example using the slice sampler ... 97
	2.4 SUMMARY AND CLOSING REMARKS ... 101
	Problems ... 101
Chapter 3 WinBUGS SOFTWARE: INTRODUCTION,SETUP, AND BASIC ANALYSIS ... 103
	3.1 INTRODUCTION AND HISTORICAL BACKGROUND ... 103
	3.2 THE WinBUGS ENVIRONMENT ... 104
		3.2.1 Downloading and installing WinBUGS ... 104
		3.2.2 A short description of the menus ... 105
	3.3 PRELIMINARIES ON USING WinBUGS ... 108
		3.3.1 Code structure and type of parameters/nodes ... 108
		3.3.2 Scalar, vector, matrix, and array nodes ... 109
	3.4 BUILDING BAYESIAN MODELS IN WinBUGS ... 113
		3.4.1 Function description ... 113
		3.4.2 Using the for syntax and array, matrix,and vector calculations ... 117
		3.4.3 Use of parentheses, brackets and curly braces in WinBUGS ... 118
		3.4.4 Differences between WinBUGS and R/Splus syntax ... 118
		3.4.5 Model specification in WinBUGS ... 119
		3.4.6 Data and initial value specification ... 120
			3.4.6.1 Rectangular data format ... 120
			3.4.6.2 List data format ... 121
			3.4.6.3 Importing data from R/Splus ... 122
			3.4.6.4 A simple example of data specification. ... 124
			3.4.6.5 A simple example using arrays ... 125
			3.4.6.6 Mixed and  multiple data definition ... 125
			3.4.6.7 Initial values ... 127
			3.4.6.8 Other details. ... 127
		3.4.7 An example of a complete model specification ... 127
		3.4.8 Data transformations ... 128
	3.5 COMPILING THE MODEL AND SIMULATING VALUES ... 128
	3.6 BASIC OUTPUT ANALYSIS USING  THE SAMPLE  MONITOR TOOL ... 137
	3.7 SUMMARIZING THE PROCEDURE ... 140
	3.8 CHAPTER SUMMARY AND CONCLUDING COMMENTS ... 141
	Problems ... 141
Chapter 4 Win BUGS SOFTWARE: ILLUSTRATION,RESULTS,AND FURTHER ANALYSIS ... 144
	4.1 A COMPLETE  EXAMPLE OF RUNNING MCMC IN WinBUGS FOR A SIMPLE MODEL ... 144
		4.1.1 The model ... 144
		4.1.2 Data and initial values ... 146
		4.1.3 Compiling and running the model ... 146
		4.1.4 MCMC output analysis and results ... 148
			4.1.4.1 Checking convergence ... 148
			4.1.4.2 Calculation of posterior summaries ... 150
	4.2 FURTHER OUTPUT ANALYSIS USING THE INFERENCE MENU ... 151
		4.2.1 Comparison of nodes ... 152
		4.2.2 Calculation of correlations ... 155
		4.2.3 Using the summary tool ... 156
		4.2.4 Evaluation and ranking of individuals ... 157
		4.2.5 Calculation of deviance information criterion ... 159
	4.3 MULTIPLE CHAINS ... 160
		4.3.1 Generation of multiple chains ... 160
		4.3.2 Output analysis ... 161
		4.3.3 The Gelman-Rubin convergence diagnostic ... 162
	4.4 CHANGING THE PROPERTIES OF A FIGURE ... 164
		4.4.1 General graphical options ... 164
		4.4.2 Special graphical options ... 164
	4.5 OTHER TOOLS AND MENUS ... 167
		4.5.1 The node info tool ... 167
		4.5.2 Monitoring the acceptance rate of the Metropolis-Hastings algorithm ... 167
		4.5.3 Saving the current state of the chain ... 168
		4.5.4 Setting the starting seed number ... 168
		4.5.5 Running the model as a script ... 168
	4.6 SUMMARY AND CONCLUDING REMARKS ... 168
	Problems ... 169
Chapter 5 INTRODUCTION TO BAYESIAN MODELS: NORMAL MODELS ... 170
	5.1 GENERAL MODELING PRINCIPLES ... 170
	5.2 MODEL SPECIFICATION IN NORMAL REGRESSION MODELS ... 171
		5.2.1 Specifying the likelihood ... 172
		5.2.2 Specifying a simple independent prior distribution ... 173
		5.2.3 Interpretation of the regression coeff icients ... 173
		5.2.4 A regression example using WinBUGS ... 176
	5.3 USING VECTORS AND MULTIVARIATE PRIORS IN NORMAL REGRESSION MODELS ... 180
		5.3.1 Defining the model using matrices ... 180
		5.3.2 Prior distributions for normal regression models ... 181
		5.3.3 Multivariate normal priors in WinBUGS ... 182
		5.3.4 Continuation of Example 5.1 ... 183
	5.4 ANALYSIS OF VARIANCE MODELS ... 186
		5.4.1 The one-way ANOVA model ... 186
		5.4.2 Parametrization and parameter interpretation ... 187
			5.4.2.1 Corner constraints ... 187
			5.4.2.2 Sum-to-zero constraints. ... 188
		5.4.3 One-way ANOVA model in WinBUGS ... 188
		5.4.4 A one-way ANOVA example using WinBUGS ... 190
		5.4.5 Two-way ANOVA models ... 192
			5.4.5.1 The main effects model. ... 192
			5.4.5.2 Parametrization and parameter interpretation ... 193
			5.4.5.3 The two-way interaction  model. ... 193
			5.4.5.4 Data in tabular format (equal observations per cell). ... 195
			5.4.5.5 A two-way ANOVA example. ... 197
		5.4.6 Multifactor analysis of variance ... 203
	Problems ... 203
Chapter 6 INCORPORATING CATEGORICAL VARIABLES IN NORMAL MODELS  AND FURTHER MODELING ISSUES ... 207
	6.1 ANALYSIS OF VARIANCE MODELS USING DUMMY VARIABLES ... 209
	6.2 ANALYSIS OF COVARIANCE MODELS ... 213
		6.2.1 Models using one quantitative variable and one qualitative variable ... 215
		6.2.2 The parallel lines model ... 215
		6.2.3 The separate lines  model ... 219
	6.3 A BIOASSAY EXAMPLE ... 221
		6.3.1 Parallel lines analysis ... 222
		6.3.2 Slope ratio analysis: Models with common intercept and different slope ... 230
		6.3.3 Comparison of the two approaches ... 235
	6.4 FURTHER MODELING ISSUES ... 236
		6.4.1 Extending the simple ANCOVA model ... 236
		6.4.2 Using binary indicators to specify models in multiple regression ... 237
		6.4.3 Selection of variables using the deviance information criterion (DIC) ... 237
			6.4.3.1 A stepwise method for DIC based  variable selection  in WinBUGS ... 238
	6.5 CLOSING REMARKS ... 244
	Problems ... 244
Chapter 7 INTRODUCTION TO GENERALIZED LINEAR MODELS: BINOMIAL AND POISSON DATA ... 246
	7.1 INTRODUCTION ... 246
		7.1.1 The exponential family ... 247
		7.1.2 Common distributions as members of the exponential family ... 248
		7.1.3 Link functions ... 251
			7.1.3.1 Common link functions ... 251
			7.1.3.2 More complicated link functions for binomial data. ... 252
		7.1.4 Common generalized linear models ... 253
		7.1.5 Interpretation of GLM coefficients ... 255
	7.2 PRIOR DISTRIBUTIONS ... 256
	7.3 POSTERIOR INFERENCE ... 258
		7.3.1 The posterior distribution of a generalized linear model ... 258
		7.3.2 GLM specification in WinBUGS ... 259
	7.4 POISSON REGRESSION MODELS ... 259
		7.4.1 Interpretation of Poisson log-linear parameters ... 259
		7.4.2 A simple Poisson regression example ... 262
			7.4.2.1 Model specification in WinBUGS ... 262
			7.4.2.2 Results ... 263
			7.4.2.3 Interpretation of the model parameters. ... 263
			7.4.2.4 Estimating specific profiles ... 264
			7.4.2.5 Selection of variables using DIC ... 265
		7.4.3 A Poisson regression model for modeling football data ... 266
			7.4.3.1 Background information and the model ... 266
			7.4.3.2 Model specification in WinBUGS ... 267
			7.4.3.3 Results. ... 267
			7.4.3.4 Prediction of future games ... 268
			7.4.3.5 Regeneration of the full  league ... 270
	7.5 BINOMIAL RESPONSE MODELS ... 272
		7.5.1 Interpretation of model parameters in binomial response models ... 274
			7.5.1.1 Odds and odds ratios. ... 274
			7.5.1.2 Logistic regression parameters and odds ratios ... 276
			7.5.1.3 Parameter interpretation in probit models. ... 276
			7.5.1.4 Relationship between log it and probit parameters. ... 278
			7.5.1.5 Parameter interpretation in log-log and clog-log models. ... 279
		7.5.2 A simple example ... 280
			7.5.2.1 Model specification in WinBUGS. ... 280
			7.5.2.2 Results and parameter interpretation. ... 284
	7.6 MODELS FOR CONTINGENCY TABLES ... 286
	Problems ... 287
Chapter 8 MODELS FOR  POSITIVE CONTINUOUS DATA, COUNT DATA,AND OTHER GLM-BASED EXTENSIONS ... 292
	8.1 MODELS WITH NONSTANDARD  DISTRIBUTIONS ... 292
		8.1.1 Specification of arbitrary likelihood using the zeros-ones trick ... 293
		8.1.2 The inverse Gaussian model ... 294
	8.2 MODELS FOR POSITIVE CONTINUOUS RESPONSE VARIABLES ... 296
		8.2.1 The gamma model ... 296
		8.2.2 Other models ... 297
		8.2.3 An example ... 298
	8.3 ADDITIONAL MODELS FOR COUNT DATA ... 299
		8.3.1 The negative binomial model ... 300
		8.3.2 The generalized Poisson model ... 303
		8.3.3 Zero inflated models ... 305
		8.3.4 The bivariate Poisson  model ... 308
		8.3.5 The Poisson difference model ... 310
	8.4 FURTHER GLM·BASED MODELS AND EXTENSIONS ... 313
		8.4.1 Survival analysis models ... 314
		8.4.2 Multinomial models ... 315
		8.4.3 Additional models and further reading ... 317
	Problems ... 318
Chapter 9 BAYESIAN HIERARCHICAL MODELS ... 322
	9.1 INTRODUCTION ... 322
		9.1.1 A simple motivating example ... 323
		9.1.2 Why use a hierarchical model? ... 324
		9.1.3 Other advantages and characteristics ... 325
	9.2 SOME SIMPLE EXAMPLES ... 325
		9.2.1 Repeated measures data ... 325
			9.2.1.1 Model formulation. ... 325
			9.2.1.2 Win BUGS code. ... 327
			9.2.1.3 Results. ... 327
			9.2.1.4 Handling missing data. ... 327
		9.2.2 Introducing random effects in performance parameters ... 330
			9.2.2.1 State space model. ... 330
		9.2.3 Poisson mixture models for count data ... 332
			9.2.3.1 The Poisson-gamma model. ... 332
			9.2.3.2 The Poisson-log-normal model. ... 333
		9.2.4 The use of hierarchical models in meta-analysis ... 335
	9.3 THE GENERALIZED LINEAR MIXED MODEL FORMULATION ... 337
		9.3.1 A hierarchical normal  model: A simple crossover  trial ... 338
		9.3.2 Logit GLMM for correlated binary responses ... 342
			9.3.2.1 The logit  model in 2x2 tables of dependent binary  data. ... 343
		9.3.3 Poisson log-linear GLMMs for correlated count data ... 350
	9.4 DISCUSSION, CLOSING  REMARKS,AND FURTHER READING ... 355
	Problems ... 357
Chapter 10 THE PREDICTIVE DISTRIBUTION AND MODEL CHECKING ... 358
	10.1 INTRODUCTION ... 358
		10.1.1 Prediction within  Bayesian framework ... 358
		10.1.2 Using posterior  predictive densities for model evaluation and checking ... 359
		10.1.3 Cross-validation predictive densities ... 361
	10.2 ESTIMATING THE PREDICTIVE DISTRIBUTION  FOR FUTURE OR MISSING OBSERVATIONS USING MCMC ... 361
		10.2.1 A simple example: Estimating missing observations ... 362
		10.2.2 An example of Bayesian prediction using a simple model ... 364
			10.2.2.1 Model formulation. ... 366
	10.3 USING THE PREDICTIVE DISTRIBUTION FOR MODEL CHECKING ... 371
		10.3.1 Comparison of actual and predictive frequencies for discrete data ... 371
		10.3.2 Comparison of cumulative frequencies for predictive and actual values for continuous data ... 374
		10.3.3 Comparison of ordered predictive and actual values for continuous data ... 375
		10.3.4 Estimation of the posterior  predictive ordinate ... 376
		10.3.5 Checking individual observations using residuals ... 379
		10.3.6 Checking structural  assumptions of the model ... 382
		10.3.7 Checking the goodness-of-fit of a model ... 385
	10.4 USING CROSS-VALIDATION PREDICTIVE DENSITIES FOR MODEL CHECKING, EVALUATION, AND COMPARISON ... 392
		10.4.1 Estimating the conditional predictive ordinate ... 392
		10.4.2 Generating values from the leave-one-out cross-validatory predictive distributions ... 394
	10.5 ILLUSTRATION OF A COMPLETE  PREDICTIVE ANALYSIS:  NORMAL REGRESSION MODELS ... 395
		10.5.1 Checking structural  assumptions of the  model ... 395
		10.5.2 Detailed checks based on residual analysis ... 396
		10.5.3 Overall goodness-of-fit of the  model ... 397
		10.5.4 Implementation using WinBUGS ... 397
		10.5.5 An Illustrative example ... 400
		10.5.6 Summary of the model checking procedure ... 403
	10.6 DISCUSSION ... 404
	Problems ... 405
Chapter 11 BAYESIAN MODEL AND VARIABLE EVALUATION ... 406
	11.1 PRIOR PREDICTIVE DISTRIBUTIONS  AS MEASURES OF MODEL COMPARISON: POSTERIOR MODEL ODDS AND BAYES FACTORS ... 406
	11.2 SENSITIVITY OF THE POSTERIOR MODEL PROBABILITIES: THE LINDLEY-BARTLETT PARADOX ... 408
	11.3 COMPUTATION OF THE MARGINAL LIKELIHOOD ... 409
		11.3.1 Approximations based on  the normal distribution ... 409
		11.3.2 Sampling from the prior: A naive Monte Carlo  estimator ... 409
		11.3.3 Sampling from the posterior: The harmonic mean estimator ... 410
		11.3.4 Importance sampling estimators ... 411
		11.3.5 Bridge sampling estimators ... 411
		11.3.6 Chib's marginal likelihood estimator ... 412
		11.3.7 Additional details and further reading ... 414
	11.4 COMPUTATION OF THE MARGINAL LIKELIHOOD USING WinBUGS ... 414
		11.4.1 A beta-binomial example ... 416
		11.4.2 A normal regression example with conjugate normal-inverse gamma prior ... 420
	11.5 BAYESIAN VARIABLE SELECTION USING GIBBS-BASED METHODS ... 422
		11.5.1 Prior distributions for variable selection in GLM ... 423
		11.5.2 Gibbs variable selection ... 426
		11.5.3 Other Gibbs-based methods for variable selection ... 427
	11.6 POSTERIOR INFERENCE USING THE OUTPUT OF BAYESIAN VARIABLE SELECTION SAMPLERS ... 429
	11.7 IMPLEMENTATION OF GIBBS VARIABLE SELECTION IN WinBUGS USING AN ILLUSTRATIVE EXAMPLE ... 431
	11.8 THE CARLIN-CHIB METHOD ... 436
	11.9 REVERSIBLE JUMP MCMC(RJMCMC) ... 437
	11.10 USING POSTERIOR PREDICTIVE DENSITIES  FOR MODEL EVALUATION ... 438
		11.10.1 Estimation from an MCMC output ... 440
		11.10.2 A simple example in WinBUGS ... 441
	11.11 INFORMATION CRITERIA ... 441
		11.11.1 The Bayes information criterion (BIC) ... 442
		11.11.2 The Akaike information criterion (AIC) ... 443
		11.11.3 Other criteria ... 444
		11.11.4 Calculation of penalized deviance measures from the MCMC output ... 445
		11.11.5 Implementation in WinBUGS ... 445
		11.11.6 A simple example in WinBUGS ... 446
	11.12 DISCUSSION AND FURTHER READING ... 449
	Problems ... 449
APPENDIX A MODEL SPECIFICATION VIA DIRECTED ACYCLIC GRAPHS: THE DOODLE MENU ... 451
	A.1 INTRODUCTION: STARTING WITH DOODLE ... 451
	A.2 NODES ... 452
	A.3 EDGES ... 454
	A.4 PANELS ... 454
	A.5 A SIMPLE EXAMPLE ... 455
APPENDIX B THE BATCH  MODE: RUNNING A MODEL IN THE BACKGROUND USING SCRIPTS ... 458
	B.1 INTRODUCTION ... 458
	B.2 BASIC COMMANDS:COMPILING AND RUNNING THE MODEL ... 459
APPENDIX C CHECKING CONVERGENCE USING CODA/BOA ... 461
	C.1 INTRODUCTION ... 461
	C.2 A SHORT HISTORICAL REVIEW ... 462
	C.3 DIAGNOSTICS IMPLEMENTED BY CODA/BOA ... 462
		C.3.1 The Geweke diagnostic ... 462
		C.3.2 The Gelman-Rubin diagnostic ... 463
		C.3.3 The Raftery-Lewis diagnostic ... 463
		C.3.4 The Heidelberger-Welch diagnostic ... 463
		C.3.5 Final remarks ... 464
	C.4 A FIRST LOOK AT CODA/BOA ... 464
		C.4.1 CODA ... 464
		C.4.2 BOA ... 465
	C.5 A SIMPLE EXAMPLE ... 467
		C.5.1 Illustration in  CODA ... 467
		C.5.2 Illustration in BOA ... 471
APPENDIX D NOTATION SUMMARY ... 475
REFERENCES ... 482
INDEX ... 498