Applied Bayesian Hierarchical Methods

The use of Markov chain Monte Carlo (MCMC) methods for estimating hierarchical models involves complex data structures and is often described as a revolutionary development. An intermediate-level treatment of Bayesian hierarchical models and their applications, Applied Bayesian Hierarchical Methods 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.

Emphasizing computational issues, the book provides examples of the following application settings: meta-analysis, data structured in space or time, multilevel and longitudinal data, multivariate data, nonlinear regression, and survival time data. For the worked examples, the text mainly employs the WinBUGS package, allowing readers to explore alternative likelihood assumptions, regression structures, and assumptions on prior densities. It also incorporates BayesX code, which is particularly useful in nonlinear regression. To demonstrate MCMC sampling from first principles, the author includes worked examples using the R package.

Through illustrative data analysis and attention to statistical computing, this book focuses on the practical implementation of Bayesian hierarchical methods. It also discusses several issues that arise when applying Bayesian techniques in hierarchical and random effects models.

Bayesian Methods for Complex Data: Estimation and Inference

Introduction

Posterior Inference from Bayes Formula

Markov Chain Sampling in Relation to Monte Carlo Methods: Obtaining Posterior Inferences

Hierarchical Bayes Applications

Metropolis Sampling

Choice of Proposal Density

Obtaining Full Conditional Densities

Metropolis–Hastings Sampling

Gibbs Sampling

Assessing Efficiency and Convergence: Ways of Improving Convergence

Choice of Prior Density

Model Fit, Comparison, and Checking

Introduction

Formal Methods: Approximating Marginal Likelihoods

Effective Model Dimension and Deviance Information Criterion

Variance Component Choice and Model Averaging

Predictive Methods for Model Choice and Checking

Estimating Posterior Model Probabilities

Hierarchical Estimation for Exchangeable Units: Continuous and Discrete Mixture Approaches

Introduction

Hierarchical Priors for Ensemble Estimation using Continuous Mixtures

The Normal-Normal Hierarchical Model and Its Applications

Priors for Second Stage Variance Parameters

Multivariate Meta-Analysis

Heterogeneity in Count Data: Hierarchical Poisson Models

Binomial and Multinomial Heterogeneity

Discrete Mixtures and Nonparametric Smoothing Methods

Nonparametric Mixing via Dirichlet Process and Polya Tree Priors

Structured Priors Recognizing Similarity over Time and Space

Introduction

Modeling Temporal Structure: Autoregressive Models

State Space Priors for Metric Data

Time Series for Discrete Responses: State Space Priors and Alternatives

Stochastic Variances

Modeling Discontinuities in Time

Spatial Smoothing and Prediction for Area Data

Conditional Autoregressive Priors

Priors on Variances in Conditional Spatial Models

Spatial Discontinuity and Robust Smoothing

Models for Point Processes

Regression Techniques using Hierarchical Priors

Introduction

Regression for Overdispersed Discrete Data

Latent Scales for Binary and Categorical Data

Nonconstant Regression Relationships and Variance Heterogeneity

Heterogeneous Regression and Discrete Mixture Regressions

Time Series Regression: Correlated Errors and Time-Varying Regression Effects

Spatial Correlation in Regression Residuals

Spatially Varying Regression Effects: Geographically Weighted Linear Regression and Bayesian Spatially Varying Coefficient Models

Bayesian Multilevel Models

Introduction

The Normal Linear Mixed Model for Hierarchical Data

Discrete Responses: General Linear Mixed Model, Conjugate, and Augmented Data Models

Crossed and Multiple Membership Random Effects

Robust Multilevel Models

Multivariate Priors, with a Focus on Factor and Structural Equation Models

Introduction

The Normal Linear SEM and Factor Models

Identifiability and Priors on Loadings

Multivariate Exponential Family Outcomes and General Linear Factor Models

Robust Options in Multivariate and Factor Analysis

Multivariate Spatial Priors for Discrete Area Frameworks

Spatial Factor Models

Multivariate Time Series

Hierarchical Models for Panel Data

Introduction

General Linear Mixed Models for Panel Data

Temporal Correlation and Autocorrelated Residuals

Categorical Choice Panel Data

Observation-Driven Autocorrelation: Dynamic Panel Models

Robust Panel Models: Heteroscedasticity, Generalized Error Densities, and Discrete Mixtures

Multilevel, Multivariate, and Multiple Time Scale Longitudinal Data

Missing Data in Panel Models

Survival and Event History Models

Introduction

Survival Analysis in Continuous Time

Semiparametric Hazards

Including Frailty

Discrete Time Hazard Models

Dependent Survival Times: Multivariate and Nested Survival Times

Competing Risks

Hierarchical Methods for Nonlinear Regression

Introduction

Nonparametric Basis Function Models for the Regression Mean

Multivariate Basis Function Regression

Heteroscedasticity via Adaptive Nonparametric Regression

General Additive Methods

Nonparametric Regression Methods for Longitudinal Analysis

Appendix: Using WinBUGS and BayesX

References

Index

Peter D. Congdon is a research professor of quantitative geography and health statistics in the Centre for Statistics and Department of Geography at the University of London, UK.

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