Contents

Preface xi

Chapter 1 The Basis for, and Advantages of, Bayesian Model Estimation via Repeated Sampling 1

1.1 Introduction 1

1.2 Gibbs sampling 5

1.3 Simulating random variables from standard densities 12

1.4 Monitoring MCMC chains and assessing convergence 18

1.5 Model assessment and sensitivity 20

1.6 Review 27

References 28

Chapter 2 Hierarchical Mixture Models 31

2.1 Introduction: Smoothing to the Population 31

2.2 General issues of model assessment: marginal likelihood and other approaches 32

2.2.1 Bayes model selection using marginal likelihoods 33

2.2.2 Obtaining marginal likelihoods in practice 35

2.2.3 Approximating the posterior 37

2.2.4 Predictive criteria for model checking and selection 39

2.2.5 Replicate sampling 40

2.3 Ensemble estimates: pooling over similar units 41

2.3.1 Mixtures for Poisson and binomial data 43

2.3.2 Smoothing methods for continuous data 51

2.4 Discrete mixtures and Dirichlet processes 58

2.4.1 Discrete parametric mixtures 58

2.4.2 DPP priors 60

2.5 General additive and histogram smoothing priors 67

2.5.1 Smoothness priors 68

2.5.2 Histogram smoothing 69

2.6 Review 74

References 75

Chapter 3 Regression Models 79

3.1 Introduction: Bayesian regression 79

3.1.1 Specifying priors: constraints on parameters 80

3.1.2 Prior specification: adopting robust or informative priors 81

3.1.3 Regression models for overdispersed discrete outcomes 82

3.2 Choice between regression models and sets of predictors in regression 84

3.2.1 Predictor selection 85

3.2.2 Cross-validation regression model assessment 86

3.3 Polytomous and ordinal regression 98

3.3.1 Multinomial logistic choice models 99

3.3.2 Nested logit specification 100

3.3.3 Ordinal outcomes 101

3.3.4 Link functions 102

3.4 Regressions with latent mixtures 110

3.5 General additive models for nonlinear regression effects 115

3.6 Robust Regression Methods 118

3.6.1 Binary selection models for robustness 119

3.6.2 Diagnostics for discordant observations 120

3.7 Review 126

References 129

Exercises 132

Chapter 4 Analysis of Multi-Level Data 135

4.1 Introduction 135

4.2 Multi-level models: univariate continuous and discrete outcomes 137

4.2.1 Discrete outcomes 139

4.3 Modelling heteroscedasticity 145

4.4 Robustness in multi-level modelling 151

4.5 Multi-level data on multivariate indices 156

4.6 Small domain estimation 163

4.7 Review 167

References 168

Exercises 169

Chapter 5 Models for Time Series 171

5.1 Introduction 171

5.2 Autoregressive and moving average models under stationarity and non-stationarity 172

5.2.1 Specifying priors 174

5.2.2 Further types of time dependence 179

5.2.3 Formal tests of stationarity in the AR(1) model 180

5.2.4 Model assessment 182

5.3 Discrete Outcomes 191

5.3.2 INAR models for counts 193

5.3.3 Continuity parameter models 195

5.3.4 Multiple discrete outcomes 195

5.4 Error correction models 200

5.5 Dynamic linear models and time varying coefficients 203

5.5.1 State space smoothing 205

5.6 Stochastic variances and stochastic volatility 210

5.6.1 ARCH and GARCH models 210

5.6.2 Stochastic volatility models 211

5.7 Modelling structural shifts 215

5.7.1 Binary indicators for mean and variance shifts 215

5.7.2 Markov mixtures 216

5.7.3 Switching regressions 216

5.8 Review 221

References 222

Chapter 6 Analysis of Panel Data 227

6.1 Introduction 227

6.1.1 Two stage models 228

6.1.2 Fixed vs. random effects 230

6.1.3 Time dependent effects 231

6.2 Normal linear panel models and growth curves for metric outcomes 231

6.2.1 Growth Curve Variability 232

6.2.2 The linear mixed model 234

6.2.3 Variable autoregressive parameters 235

6.3 Longitudinal discrete data: binary, ordinal and multinomial and Poisson panel data 243

6.3.1 Beta-binomial mixture for panel data 244

6.4 Panels for forecasting 257

6.4.1 Demographic data by age and time period 261

6.5 Missing data in longitudinal studies 264

6.6 Review 268

References 269

Exercises 271

Chapter 7 Models for Spatial Outcomes and Geographical Association 273

7.1 Introduction 273

7.2 Spatial regressions for continuous data with fixed interaction schemes 275

7.2.1 Joint vs. conditional priors 276

7.3 Spatial effects for discrete outcomes: ecological analysis involving count data 278

7.3.1 Alternative spatial priors in disease models 279

7.3.2 Models recognising discontinuities 281

7.4 Direct modelling of spatial covariation in regression and interpolation applications 289

7.4.1 Covariance modelling in regression 290

7.4.2 Spatial interpolation 291

7.4.3 Variogram methods 292

7.4.4 Conditional specification of spatial error 293

7.5 Spatial heterogeneity: spatial expansion, geographically weighted regression, and multivariate errors 298

7.5.1 Spatial expansion model 298

7.5.2 Geographically weighted regression 299

7.5.3 Varying regressions effects via multivariate priors 300

7.6 Clust er ing in relation t o known centres 304

7.6.1 Areas vs. case events as data 306

7.6.2 Multiple sources 306

7.7 Spatio-temporal models 310

7.7.1 Space-time interaction effects 312

7.7.2 Area Level Trends 312

7.7.3 Predictor effects in spatio-temporal models 313

7.7.4 Diffusion processes 314

7.8 Review 316

References 317

Exercises 320

Chapter 8 Structural Equation and Latent Variable Models 323

8.1 Introduction 323

8.1.1 Extensions to other applications 325

8.1.2 Benefits of Bayesian approach 326

8.2 Confirmatory factor analysis with a single group 327

8.3 Latent trait and latent class analysis for discrete outcomes 334

8.3.1 Latent class models 335

8.4 Latent variables in panel and clustered data analysis 340

8.4.1 Latent trait models for continuous data 341

8.4.2 Latent class models through time 341

8.4.3 Latent trait models for time varying discrete outcomes 343

8.4.4 Latent trait models for clustered metric data 343

8.4.5 Latent trait models for mixed outcomes 344

8.5 Latent structure analysis for missing data 352

8.6 Review 357

References 358

Exercises 360

Chapter 9 Survival and Event History Models 361

9.1 Introduction 361

9.2 Continuous time functions for survival 363

9.3 Accelerated hazards 370

9.4 Discrete time approximations 372

9.4.1 Discrete time hazards regression 375

9.4.2 Gamma process priors 381

9.5 Accounting for frailty in event history and survival models 384

9.6 Counting process models 388

9.7 Review 393

References 394

Exercises 396

Chapter 10 Modelling and Establishing Causal Relations: Epidemiological Methods and Models 397

10.1 Causal processes and establishing causality 397

10.1.1 Specific methodological issues 398

10.2 Confounding between disease risk factors 399

10.2.1 Stratification vs. multivariate methods 400

10.3 Dose-response relations 413

10.3.1 Clustering effects and other methodological issues 416

10.3.2 Background mortality 427

10.4 Meta-analysis: establishing consistent associations 429

10.4.1 Priors for study variability 430

10.4.2 Heterogeneity in patient risk 436

10.4.3 Multiple treatments 439

10.4.4 Publication bias 441

10.5 Review 443

References 444

Exercises 447