Bayesian Hierarchical Models: With Applications Using R, Second Edition
Peter D. Congdon
Résumé
An intermediate-level treatment of Bayesian hierarchical models and their applications, this book demonstrates the advantages of a Bayesian approach to data sets involving inferences for collections of related units or variables, and in methods where parameters can be treated as random collections. Through illustrative data analysis and attention to statistical computing, this book facilitates practical implementation of Bayesian hierarchical methods.
The new edition is a revision of the book Applied Bayesian Hierarchical Methods . It maintains a focus on applied modelling and data analysis, but now using entirely R-based Bayesian computing options. It has been updated with a new chapter on regression for causal effects, and one on computing options and strategies. This latter chapter is particularly important, due to recent advances in Bayesian computing and estimation, including the development of rjags and rstan. It also features updates throughout with new examples.
The examples exploit and illustrate the broader advantages of the R computing environment, while allowing readers to explore alternative likelihood assumptions, regression structures, and assumptions on prior densities.
Features:
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- Provides a comprehensive and accessible overview of applied Bayesian hierarchical modelling
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- Includes many real data examples to illustrate different modelling topics
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- R code (based on rjags, jagsUI, R2OpenBUGS, and rstan) is integrated into the book, emphasizing implementation
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- Software options and coding principles are introduced in new chapter on computing
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- Programs and data sets available on the book's website
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Contents
Preface...............................................................................................................................................xi
1. Bayesian Methods for Complex Data: Estimation and Inference ..................................1
1.1 Introduction....................................................................................................................1
1.2 Posterior Inference from Bayes Formula....................................................................2
1.3 MCMC Sampling in Relation to Monte Carlo Methods; Obtaining
Posterior Inferences.......................................................................................................3
1.4 Hierarchical Bayes Applications..................................................................................5
1.5 Metropolis Sampling.....................................................................................................8
1.6 Choice of Proposal Density..........................................................................................9
1.7 Obtaining Full Conditional Densities....................................................................... 10
1.8 Metropolis-Hastings Sampling................................................................................. 14
1.9 Gibbs Sampling............................................................................................................ 17
1.10 Hamiltonian Monte Carlo........................................................................................... 18
1.11 Latent Gaussian Models.............................................................................................. 19
1.12 Assessing Efficiency and Convergence; Ways of Improving Convergence.........20
1.12.1 Hierarchical Model Parameterisation to Improve Convergence.............22
1.12.2 Multiple Chain Methods................................................................................ 24
1.13 Choice of Prior Density...............................................................................................25
1.13.1 Including Evidence.........................................................................................26
1.13.2 Assessing Posterior Sensitivity; Robust Priors...........................................27
1.13.3 Problems in Prior Selection in Hierarchical Bayes Models......................29
1.14 Computational Notes.................................................................................................. 31
References................................................................................................................................ 37
2. Bayesian Analysis Options in R, and Coding for BUGS, JAGS, and Stan ................45
2.1 Introduction..................................................................................................................45
2.2 Coding in BUGS and for R Libraries Calling on BUGS .........................................46
2.3 Coding in JAGS and for R Libraries Calling on JAGS............................................47
2.4 Coding for rstan ..........................................................................................................49
2.4.1 Hamiltonian Monte Carlo.............................................................................49
2.4.2 Stan Program Syntax......................................................................................49
2.4.3 The Target + Representation......................................................................... 51
2.4.4 Custom Distributions through a Functions Block.....................................53
2.5 Miscellaneous Differences between Generic
Packages (BUGS, JAGS, and Stan)..............................................................................55
References................................................................................................................................56
3. Model Fit, Comparison, and Checking .............................................................................59
3.1 Introduction.................................................................................................................. 59
3.2 Formal Model Selection..............................................................................................59
3.2.1 Formal Methods: Approximating Marginal Likelihoods......................... 62
3.2.2 Importance and Bridge Sampling Estimates..............................................63
3.2.3 Path Sampling.................................................................................................65
3.2.4 Marginal Likelihood for Hierarchical Models........................................... 67
3.3 Effective Model Dimension and Penalised Fit Measures......................................71
3.3.1 Deviance Information Criterion (DIC).........................................................72
3.3.2 Alternative Complexity Measures................................................................73
3.3.3 WAIC and LOO-IC.........................................................................................75
3.3.4 The WBIC.........................................................................................................77
3.4 Variance Component Choice and Model Averaging..............................................80
3.4.1 Random Effects Selection..............................................................................80
3.5 Predictive Methods for Model Choice and Checking............................................ 87
3.5.1 Predictive Model Checking and Choice...................................................... 87
3.5.2 Posterior Predictive Model Checks..............................................................89
3.5.3 Mixed Predictive Checks............................................................................... 91
3.6 Computational Notes..................................................................................................95
References................................................................................................................................98
4. Borrowing Strength via Hierarchical Estimation ......................................................... 103
4.1 Introduction................................................................................................................ 103
4.2 Hierarchical Priors for Borrowing Strength Using Continuous Mixtures........ 105
4.3 The Normal-Normal Hierarchical Model and Its Applications.......................... 106
4.3.1 Meta-Regression............................................................................................ 110
4.4 Prior for Second Stage Variance............................................................................... 111
4.4.1 Non-Conjugate Priors................................................................................... 113
4.5 Multivariate Meta-Analysis...................................................................................... 116
4.6 Heterogeneity in Count Data: Hierarchical Poisson Models............................... 121
4.6.1 Non-Conjugate Poisson Mixing.................................................................. 124
4.7 Binomial and Multinomial Heterogeneity............................................................. 126
4.7.1 Non-Conjugate Priors for Binomial Mixing............................................. 128
4.7.2 Multinomial Mixtures.................................................................................. 130
4.7.3 Ecological Inference Using Mixture Models............................................ 131
4.8 Discrete Mixtures and Semiparametric Smoothing Methods............................ 134
4.8.1 Finite Mixtures of Parametric Densities.................................................... 135
4.8.2 Finite Mixtures of Standard Densities....................................................... 136
4.8.3 Inference in Mixture Models...................................................................... 137
4.8.4 Particular Types of Discrete Mixture Model............................................ 141
4.8.5 The Logistic-Normal Alternative to the Dirichlet Prior.......................... 142
4.9 Semiparametric Modelling via Dirichlet Process and Polya Tree Priors.......... 144
4.9.1 Specifying the Baseline Density................................................................. 146
4.9.2 Truncated Dirichlet Processes and Stick-Breaking Priors...................... 148
4.9.3 Polya Tree Priors........................................................................................... 149
4.10 Computational Notes................................................................................................ 154
References.............................................................................................................................. 156
5. Time Structured Priors ....................................................................................................... 165
5.1 Introduction................................................................................................................ 165
5.2 Modelling Temporal Structure: Autoregressive Models...................................... 166
5.2.1 Random Coefficient Autoregressive Models............................................ 168
5.2.2 Low Order Autoregressive Models............................................................ 169
5.2.3 Antedependence Models............................................................................. 170
5.3 State-Space Priors for Metric Data........................................................................... 172
5.3.1 Simple Signal Models................................................................................... 175
5.3.2 Sampling Schemes........................................................................................ 176
5.3.3 Basic Structural Model................................................................................. 178
5.3.4 Identification Questions............................................................................... 179
5.3.5 Nonlinear State-Space Models for Continuous Data............................... 184
5.4 Time Series for Discrete Responses; State-Space Priors and Alternatives......... 186
5.4.1 Other Approaches......................................................................................... 188
5.5 Stochastic Variances.................................................................................................. 193
5.6 Modelling Discontinuities in Time......................................................................... 197
5.7 Computational Notes................................................................................................ 202
References..............................................................................................................................206
6. Representing Spatial Dependence ................................................................................... 213
6.1 Introduction................................................................................................................ 213
6.2 Spatial Smoothing and Prediction for Area Data.................................................. 214
6.2.1 SAR Schemes................................................................................................. 216
6.3 Conditional Autoregressive Priors.......................................................................... 221
6.3.1 Linking Conditional and Joint Specifications...........................................222
6.3.2 Alternative Conditional Priors....................................................................223
6.3.3 ICAR(1) and Convolution Priors.................................................................226
6.4 Priors on Variances in Conditional Spatial Models..............................................227
6.4 Spatial Discontinuity and Robust Smoothing.......................................................229
6.5 Models for Point Processes.......................................................................................234
6.5.1 Covariance Functions................................................................................... 237
6.5.2 Sparse and Low Rank Approaches............................................................238
6.6 Discrete Convolution Models................................................................................... 241
6.7 Computational Notes................................................................................................ 245
References.............................................................................................................................. 246
7. Regression Techniques Using Hierarchical Priors .......................................................253
7.1 Introduction................................................................................................................253
7.2 Predictor Selection.....................................................................................................253
7.2.1 Predictor Selection........................................................................................254
7.2.2 Shrinkage Priors...........................................................................................256
7.3 Categorical Predictors and the Analysis of Variance........................................... 259
7.3.1 Testing Variance Components....................................................................260
7.4 Regression for Overdispersed Data........................................................................264
7.4.1 Overdispersed Poisson Regression............................................................264
7.4.2 Overdispersed Binomial and Multinomial Regression.......................... 267
7.5 Latent Scales for Binary and Categorical Data...................................................... 270
7.5.1 Augmentation for Ordinal Responses....................................................... 273
7.6 Heteroscedasticity and Regression Heterogeneity............................................... 276
7.6.1 Nonconstant Error Variances...................................................................... 276
7.6.2 Varying Regression Effects via Discrete Mixtures..................................277
7.6.3 Other Applications of Discrete Mixtures.................................................. 278
7.7 Time Series Regression: Correlated Errors and Time-Varying
Regression Effects...................................................................................................... 282
7.7.1 Time-Varying Regression Effects...............................................................283
7.8 Spatial Regression......................................................................................................288
7.8.1 Spatial Lag and Spatial Error Models........................................................288
7.8.2 Simultaneous Autoregressive Models.......................................................288
7.8.3 Conditional Autoregression........................................................................290
7.8.4 Spatially Varying Regression Effects: GWR and Bayesian SVC
Models............................................................................................................ 291
7.8.5 Bayesian Spatially Varying Coefficients.................................................... 292
7.8.6 Bayesian Spatial Predictor Selection Models............................................ 293
7.9 Adjusting for Selection Bias and Estimating Causal Effects............................... 296
7.9.1 Propensity Score Adjustment...................................................................... 296
7.9.2 Establishing Causal Effects: Mediation and Marginal Models..............299
7.9.3 Causal Path Sequences.................................................................................299
7.9.4 Marginal Structural Models........................................................................306
References..............................................................................................................................308
8. Bayesian Multilevel Models .............................................................................................. 317
8.1 Introduction................................................................................................................ 317
8.2 The Normal Linear Mixed Model for Hierarchical Data..................................... 318
8.2.1 The Lindley-Smith Model Format............................................................. 320
8.3 Discrete Responses: GLMM, Conjugate, and Augmented Data Models........... 322
8.3.1 Augmented Data Multilevel Models.......................................................... 324
8.3.2 Conjugate Cluster Effects............................................................................. 325
8.4 Crossed and Multiple Membership Random Effects........................................... 328
8.5 Robust Multilevel Models......................................................................................... 331
References..............................................................................................................................336
9. Factor Analysis, Structural Equation Models, and Multivariate Priors ................... 339
9.1 Introduction................................................................................................................ 339
9.2 Normal Linear Structural Equation and Factor Models......................................340
9.2.1 Forms of Model.............................................................................................342
9.2.2 Model Definition...........................................................................................343
9.2.3 Marginal and Complete Data Likelihoods, and MCMC Sampling.......345
9.3 Identifiability and Priors on Loadings....................................................................346
9.3.1 An Illustration of Identifiability Issues......................................................348
9.4 Multivariate Exponential Family Outcomes and Generalised Linear
Factor Models.............................................................................................................354
9.4.1 Multivariate Count Data..............................................................................355
9.4.2 Multivariate Binary Data and Item Response Models............................ 357
9.4.3 Latent Scale IRT Models............................................................................... 359
9.4.4 Categorical Data............................................................................................360
9.5 Robust Density Assumptions in Factor Models.................................................... 370
9.6 Multivariate Spatial Priors for Discrete Area Frameworks................................. 373
9.7 Spatial Factor Models................................................................................................ 379
9.8 Multivariate Time Series........................................................................................... 381
9.8.1 Multivariate Dynamic Linear Models....................................................... 381
9.8.2 Dynamic Factor Analysis............................................................................386
9.8.3 Multivariate Stochastic Volatility...............................................................388
9.9 Computational Notes................................................................................................ 396
References.............................................................................................................................. 397
10. Hierarchical Models for Longitudinal Data ..................................................................405
10.1 Introduction................................................................................................................405
10.2 General Linear Mixed Models for Longitudinal Data.........................................406
10.2.1 Centred or Non-Centred Priors..................................................................408
10.2.2 Priors on Unit Level Random Effects.........................................................409
10.2.3 Priors for Random Covariance Matrix and
Random Effect Selection.............................................................................. 411
10.2.4 Priors for Multiple Sources of Error Variation.......................................... 415
10.3 Temporal Correlation and Autocorrelated Residuals........................................... 418
10.3.1 Explicit Temporal Schemes for Errors....................................................... 419
10.4 Longitudinal Categorical Choice Data....................................................................423
10.5 Observation Driven Autocorrelation: Dynamic Longitudinal Models.............427
10.5.1 Dynamic Models for Discrete Data............................................................429
10.6 Robust Longitudinal Models: Heteroscedasticity, Generalised Error
Densities, and Discrete Mixtures............................................................................433
10.6.1 Robust Longitudinal Data Models: Discrete Mixture Models...............436
10.7 Multilevel, Multivariate, and Multiple Time Scale Longitudinal Data..............443
10.7.1 Latent Trait Longitudinal Models..............................................................445
10.7.2 Multiple Scale Longitudinal Data..............................................................446
10.8 Missing Data in Longitudinal Models.................................................................... 452
10.8.1 Forms of Missingness Regression (Selection Approach)........................454
10.8.2 Common Factor Models...............................................................................455
10.8.3 Missing Predictor Data................................................................................ 457
10.8.4 Pattern Mixture Models............................................................................... 459
References.............................................................................................................................. 462
11. Survival and Event History Models ................................................................................ 471
11.1 Introduction................................................................................................................ 471
11.2 Survival Analysis in Continuous Time..................................................................472
11.2.1 Counting Process Functions....................................................................... 474
11.2.2 Parametric Hazards...................................................................................... 475
11.2.3 Accelerated Hazards.................................................................................... 478
11.3 Semiparametric Hazards.......................................................................................... 481
11.3.1 Piecewise Exponential Priors......................................................................482
11.3.2 Cumulative Hazard Specifications.............................................................484
11.4 Including Frailty........................................................................................................488
11.4.1 Cure Rate Models..........................................................................................490
11.5 Discrete Time Hazard Models................................................................................. 494
11.5.1 Life Tables...................................................................................................... 496
11.6 Dependent Survival Times: Multivariate and Nested Survival Times..............502
11.7 Competing Risks........................................................................................................507
11.7.1 Modelling Frailty..........................................................................................509
11.8 Computational Notes................................................................................................ 514
References.............................................................................................................................. 519
12. Hierarchical Methods for Nonlinear and Quantile Regression ................................ 525
12.1 Introduction................................................................................................................ 525
12.2 Non-Parametric Basis Function Models for the Regression Mean..................... 526
12.2.1 Mixed Model Splines.................................................................................... 527
12.2.2 Basis Functions Other Than Truncated Polynomials.............................. 529
12.2.3 Model Selection............................................................................................. 532
12.3 Multivariate Basis Function Regression.................................................................536
12.4 Heteroscedasticity via Adaptive Non-Parametric Regression............................ 541
12.5 General Additive Methods.......................................................................................543
12.6 Non-Parametric Regression Methods for Longitudinal Analysis......................546
12.7 Quantile Regression.................................................................................................. 552
12.7.1 Non-Metric Responses.................................................................................554
12.8 Computational Notes................................................................................................560
References..............................................................................................................................560
Index ..............................................................................................................................................565
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Taylor&francis |
| Auteur(s) | Peter D. Congdon |
| Parution | 01/10/2019 |
| Nb. de pages | 580 |
| EAN13 | 9781498785754 |
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