Parameter estimation for mixtures of generalized linear mixed-effects models
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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 het- erogeneous population, typically under an independence assumption. Mixed-e®ects 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 ¯nite mixtures of generalized linear mixed e®ect 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 E- step, when assuming normally distributed random e®ects, 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 e®ects. We also present the methods for computing the informa- tion matrix. In Chapter 4, restricted maximum likelihood method (REML) for Zero-In°ated (ZI) mixed e®ect models are developed. Zero-In°ated mixed e®ect 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 (200 and think this method is more in the spirit of REML estimation in linear mixed e®ect mode This estimator is obtained based upon correcting the bias in the pro¯le score function of th variance components. The idea is from McCullagh and Tibshirani (1990). The estimatin procedure involves Monte Carlo EM algorithm which uses important sampling to genera random variates to construct Monte Carlo approximations at E-step. Simulation results sho that the estimates of variance component parameters obtained from the REML method ha signi¯cantly less bias than corresponding estimates from ML estimation method. In Chapt 5, we discuss some issues we encountered in the research and point out the potential topi for future research.