Applied Bayesian Modelling

Peter Congdon is Research Professor of Quantitative Geography and Health Statistics at Queen Mary University of London. He has written three earlier books on Bayesian modelling and data analysis techniques with Wiley, and has a wide range of publications in statistical methodology and in application areas. His current interests include applications to spatial and survey data relating to health status and health service research.

“A nice guidebook to intermediate and advanced Bayesian models.”   (Scientific Computing, 13 January 2015)

• Preface xi 1 Bayesian methods and Bayesian estimation 11.1 Introduction 11.1.1 Summarising existing knowledge: Prior densities for parameters 21.1.2 Updating information: Prior, likelihood and posterior densities 31.1.3 Predictions and assessment 51.1.4 Sampling parameters 61.2 MCMC techniques: The Metropolis–Hastings algorithm 71.2.1 Gibbs sampling 81.2.2 Other MCMC algorithms 91.2.3 INLA approximations 101.3 Software for MCMC: BUGS, JAGS and R-INLA 111.4 Monitoring MCMC chains and assessing convergence 191.4.1 Convergence diagnostics 201.4.2 Model identifiability 211.5 Model assessment 231.5.1 Sensitivity to priors 231.5.2 Model checks 241.5.3 Model choice 25References 282 Hierarchical models for related units 342.1 Introduction: Smoothing to the hyper population 342.2 Approaches to model assessment: Penalised fit criteria, marginal likelihood and predictive methods 352.2.1 Penalised fit criteria 362.2.2 Formal model selection using marginal likelihoods 372.2.3 Estimating model probabilities or marginal likelihoods in practice 382.2.4 Approximating the posterior density 402.2.5 Model averaging from MCMC samples 422.2.6 Predictive criteria for model checking and selection: Cross-validation 462.2.7 Predictive checks and model choice using complete data replicate sampling 502.3 Ensemble estimates: Poisson–gamma and Beta-binomial hierarchical models 532.3.1 Hierarchical mixtures for poisson and binomial data 542.4 Hierarchical smoothing methods for continuous data 612.4.1 Priors on hyperparameters 622.4.2 Relaxing normality assumptions 632.4.3 Multivariate borrowing of strength 652.5 Discrete mixtures and dirichlet processes 692.5.1 Finite mixture models 692.5.2 Dirichlet process priors 722.6 General additive and histogram smoothing priors 782.6.1 Smoothness priors 792.6.2 Histogram smoothing 80Exercises 83Notes 86References 893 Regression techniques 973.1 Introduction: Bayesian regression 973.2 Normal linear regression 983.2.1 Linear regression model checking 993.3 Simple generalized linear models: Binomial, binary and Poisson regression 1023.3.1 Binary and binomial regression 1023.3.2 Poisson regression 1053.4 Augmented data regression 1073.5 Predictor subset choice 1103.5.1 The g-prior approach 1143.5.2 Hierarchical lasso prior methods 1163.6 Multinomial, nested and ordinal regression 1263.6.1 Nested logit specification 1283.6.2 Ordinal outcomes 130Exercises 136Notes 138References 1444 More advanced regression techniques 1494.1 Introduction 1494.2 Departures from linear model assumptions and robust alternatives 1494.3 Regression for overdispersed discrete outcomes 1544.3.1 Excess zeroes 1574.4 Link selection 1604.5 Discrete mixture regressions for regression and outlier status 1614.5.1 Outlier accommodation 1634.6 Modelling non-linear regression effects 1674.6.1 Smoothness priors for non-linear regression 1674.6.2 Spline regression and other basis functions 1694.6.3 Priors on basis coefficients 1714.7 Quantile regression 175Exercises 177Notes 177References 1795 Meta-analysis and multilevel models 1835.1 Introduction 1835.2 Meta-analysis: Bayesian evidence synthesis 1845.2.1 Common forms of meta-analysis 1855.2.2 Priors for stage 2 variation in meta-analysis 1885.2.3 Multivariate meta-analysis 1935.3 Multilevel models: Univariate continuous outcomes 1955.4 Multilevel discrete responses 2015.5 Modelling heteroscedasticity 2045.6 Multilevel data on multivariate indices 206Exercises 208Notes 210References 2116 Models for time series 2156.1 Introduction 2156.2 Autoregressive and moving average models 2166.2.1 Dependent errors 2186.2.2 Bayesian priors in ARMA models 2186.2.3 Further types of time dependence 2226.3 Discrete outcomes 2296.3.1 INAR models for counts 2316.3.2 Evolution in conjugate process parameters 2326.4 Dynamic linear and general linear models 2356.4.1 Further forms of dynamic models 2386.5 Stochastic variances and stochastic volatility 2446.5.1 ARCH and GARCH models 2446.5.2 State space stochastic volatility models 2456.6 Modelling structural shifts 2486.6.1 Level, trend and variance shifts 2496.6.2 Latent state models including historic dependence 2506.6.3 Switching regressions and autoregressions 251Exercises 258Notes 261References 2657 Analysis of panel data 2737.1 Introduction 2737.2 Hierarchical longitudinal models for metric data 2747.2.1 Autoregressive errors 2757.2.2 Dynamic linear models 2767.2.3 Extended time dependence 2767.3 Normal linear panel models and normal linear growth curves 2787.3.1 Growth curves 2807.3.2 Subject level autoregressive parameters 2837.4 Longitudinal discrete data: Binary, categorical and Poisson panel data 2857.4.1 Binary panel data 2857.4.2 Ordinal panel data 2887.4.3 Panel data for counts 2927.5 Random effects selection 2957.6 Missing data in longitudinal studies 297Exercises 302Notes 303References 3068 Models for spatial outcomes and geographical association 3128.1 Introduction 3128.2 Spatial regressions and simultaneous dependence 3138.2.1 Regression with localised dependence 3168.2.2 Binary outcomes 3178.3 Conditional prior models 3218.3.1 Ecological analysis involving count data 3248.4 Spatial covariation and interpolation in continuous space 3298.4.1 Discrete convolution processes 3328.5 Spatial heterogeneity and spatially varying coefficient priors 3378.5.1 Spatial expansion and geographically weighted regression 3388.5.2 Spatially varying coefficients via multivariate priors 3398.6 Spatio-temporal models 3438.6.1 Conditional prior representations 3458.7 Clustering in relation to known centres 3488.7.1 Areas or cases as data 3508.7.2 Multiple sources 350Exercises 352Notes 354References 3559 Latent variable and structural equation models 3649.1 Introduction 3649.2 Normal linear structural equation models 3659.2.1 Cross-sectional normal SEMs 3659.2.2 Identifiability constraints 3679.3 Dynamic factor models, panel data factor models and spatial factor models 3729.3.1 Dynamic factor models 3729.3.2 Linear SEMs for panel data 3749.3.3 Spatial factor models 3789.4 Latent trait and latent class analysis for discrete outcomes 3819.4.1 Latent trait models 3819.4.2 Latent class models 3829.5 Latent trait models for multilevel data 3879.6 Structural equation models for missing data 389Exercises 392Notes 394References 39710 Survival and event history models 40210.1 Introduction 40210.2 Continuous time functions for survival 40310.2.1 Parametric hazard models 40510.2.2 Semi-parametric hazards 40810.3 Accelerated hazards 41110.4 Discrete time approximations 41310.4.1 Discrete time hazards regression 41510.5 Accounting for frailty in event history and survival models 41710.6 Further applications of frailty models 42110.7 Competing risks 423Exercises 425References 426Index 431