Parameter Estimation for Mixtures of Generalized Linear Mixed - effects Models

Finite mixtures of simpler component models such as mixtures of normals and mixtures of generalized linear models (GLM) have proven useful for modelling data arising from a heterogeneous population, typically under an independence assumption. Mixed-effects models are often used to handle correlation as arises in longitudinal or other clustered data. In Chapter 3 of this dissertation, we present a more general class of models consisting of finite mixtures of generalized linear mixed effect models to handle correlation and heterogeneity simultaneously. For this class of models, we consider maximum likelihood (ML) as our main approach to estimation. Due to the complexity of the marginal loglikelihood of this model, the EM algorithm is employed to facilitate computation. To evaluate the integral in the Estep, when assuming normally distributed random effects, we consider numerical integration methods such as ordinary Gaussian quadrature (OGQ) and adaptive Gaussian quadrature (AGQ). We discuss nonparametric ML estimation (Aitkin, 1999) when we relax the normal assumption on the random effects. We also present the methods for computing the information matrix. In Chapter 4, restricted maximum likelihood method (REML) for Zero-Inflated (ZI) mixed effect models are developed. Zero-Inflated mixed effect models are submodels of two-component mixtures of GLMMs with one component degenerate to zero. For this type of models, we adapt an estimator of variance components proposed by Liao and Lipsitz (2002) and think this method is more in the spirit of REML estimation in linear mixed effect models. This estimator is obtained based upon correcting the bias in the profile score function of the variance components. The idea is from McCullagh and Tibshirani (1990). The estimating procedure involves Monte Carlo EM algorithm which uses important sampling to generate random variates to construct Monte Carlo approximations at E-step. Simulation results show that the estimates of variance component parameters obtained from the REML method have significantly less bias than corresponding estimates from ML estimation method. In Chapter 5, we discuss some issues we encountered in the research and point out the potential topics for future research. Index words: Mixture models, Generalized linear mixed effect models, Maximum likelihood estimation, Zero-inflated models, Restricted maximum likelihood estimation Parameter Estimation for Mixtures of Generalized Linear Mixed-effects Models