Finite mixtures of generalized linear mixed effect models are presented to handle situations where within-cluster correlation and heterogeneity (subpopulations) exist 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. The major obstacle in this procedure is to integrate over the random effects' distribution to evaluation the expectation of the E-step. When assuming normally distributed random effects, we consider numerical integration methods such as ordinary Gaussian quadrature (OGQ) and adaptive Gaussian quadrature (AGQ). We also discuss nonparametric ML estimation (Aitkin, 1999) under a relaxation of the normality assumption on the random effects. This method can be understood as a modification of OGQ in the usual ML estimation. In addition, Oakes' formula (Oakes, 1999), which describes a simple relationship between the observed Hessian and the second derivative of the conditional expectation of complete data loglikelihood, is used to calculate standard errors for parameter estimates. Two real data sets are analyzed to compare our proposed model with other existing models and illustrate our estimation methods.
Mixtures of generalized Linear Mixed-Effects Models for Cluster-Correlated Data