| The generalized linear mixed model (GLMM) is one of the most useful structures in modern statistics, allowing many complications to be handled within the familiar linear model framework. The key to full generality is the use of general design matrices for both the fixed and the random components, and hence our label general design GLMMs. In this dissertation we take a Bayesian inference approach and fit the general design GLMM using Markov chain Monte Carlo (MCMC) methods.; In Chapter I we explain in detail four MCMC methods for fitting general design Bayesian GLMMs: Metropolis-Hastings algorithm, adaptive rejection sampling method, auxiliary variables method, and simple slice sampling method. We also discuss the important issues of hierarchical centering and block updating schemes. A simulation study is conducted to illustrate the implementation of these four MCMC sampling schemes and to empirically compare the performance of these methods.; In Chapter 2 we analyze three spatial data sets from public health studies using general design Bayesian GLMMs. These data sets include spatial point data for female lung cancer in Upper Cape Cod, Massachusetts, as well as small area count data for female lung cancer in part of Cape Cod, Massachusetts and for visceral leishmaniasis incidence in Teresina, Piaui State, Brazil. We also illustrate the use of Deviance Information Criterion as a semi-formal criterion for model selection.; In semiparametric regression setting, we are usually interested in testing the linearity of each nonparametric function, which usually involves testing a variance component at the boundary of its parameter space. In Chapter 3 we develop a model selection method for semiparametric regression models using Bayes factors. We describe how to estimate Bayes factors using Gibbs output for Gaussian response models and using Metropolis-Hastings output for general response models. We also extend the method to other semiparametric models including varying coefficient models and geostatistic models. The methodology is illustrated using several examples which cover Gaussian additive models, generalized additive models, and varying coefficient models. |
