Bayesian Hierarchical Models

With Applications Using R, Second Edition

Peter D. Congdon

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9781032177151
• eBook ISBN: 9780429532900
• Edition: 2
• Topic: Mathematics
• Subtopic: Computer Science General
• Index: Mathematics

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

1

Bayesian Methods for Complex Data: Estimation and Inference

1.1 Introduction

The Bayesian approach to inference focuses on updating knowledge about unknown parameters θ in a statistical model on the basis of observations y, with revised knowledge expressed in the posterior density p(θ|y). The sample of observations y being analysed provides new information about the unknowns, while the prior density p(θ) represents accumulated knowledge about them before observing or analysing the data. There is considerable flexibility with which prior evidence about parameters can be incorporated into an analysis, and use of informative priors can reduce the possibility of confounding and provides a natural basis for evidence synthesis (Shoemaker et al., 1999; Dunson, 2001; Vanpaemel, 2011; Klement et al., 2018). The Bayes approach provides uncertainty intervals on parameters that are consonant with everyday interpretations (Willink and Lira, 2005; Wetzels et al., 2014; Krypotos et al., 2017), and has no problem comparing the fit of non-nested models, such as a nonlinear model and its linearised version.

Furthermore, Bayesian estimation and inference have a number of advantages in terms of its relevance to the types of data and problems tackled by modern scientific research which are a primary focus later in the book. Bayesian estimation via repeated sampling from posterior densities facilitates modelling of complex data, with random effects treated as unknowns and not integrated out as is sometimes done in frequentist approaches (Davidian and Giltinan, 2003). For example, much of the data in social and health research has a complex structure, involving hierarchical nesting of subjects (e.g. pupils within schools), crossed classifications (e.g. patients classified by clinic and by homeplace), spatially configured data, or repeated measures on subjects (MacNab et al., 2004). The Bayesian approach naturally adapts to such hierarchically or spatio-temporally correlated effects via conditionally specified hierarchical priors under a three-stage scheme (Lindley and Smith, 1972; Clark and Gelfand, 2006; Gustafson et al., 2006; Cressie et al., 2009), with the first stage specifying the likelihood of the data, given unknown random individual or cluster effects; the second stage specifying the density of the random effects; and the third stage providing priors on parameters underlying the random effects density or densities.

The increased application of Bayesian methods has owed much to the development of Markov chain Monte Carlo (MCMC) algorithms for estimation (Gelfand and Smith, 1990; Gilks et al., 1996; Ne...

Table of contents

• Cover
• Half-Title
• Title
• Copyright
• Contents
• Preface
• 1 Bayesian Methods for Complex Data: Estimation and Inference
• 2 Bayesian Analysis Options in R, and Coding for BUGS, JAGS, and Stan
• 3 Model Fit, Comparison, and Checking
• 4 Borrowing Strength via Hierarchical Estimation
• 5 Time Structured Priors
• 6 Representing Spatial Dependence
• 7 Regression Techniques Using Hierarchical Priors
• 8 Bayesian Multilevel Models
• 9 Factor Analysis, Structural Equation Models, and Multivariate Priors
• 10 Hierarchical Models for Longitudinal Data
• 11 Survival and Event History Models
• 12 Hierarchical Methods for Nonlinear and Quantile Regression
• Index