Generalized quasi-likelihood versus hierarchical likelihood inferences in generalized linear mixed models for count data

Abstract

Inferences in generalized linear mixed models (GLMMs) which includes count and binary data as special cases are extremely important. As it is proven to be difficult to obtain consistent and efficient estimates of the parameters (regression effects and variance of the random effects) of such models, there is a vast growing literature dealing with this important estimation problem. Among them, the method of moments (MM), Penalized quasilikelihood (PQL) and Hierarchical likelihood (HL) approaches are more familiar. It is however known that the MM approach always produces consistent estimates, whereas the PQL approach may not provide consistent estimates for all the parameters of the model. A recently proposed generalized quasilikelihood (GQL) approach has proven to be better in the sense of consistency and efficiency as compared to the MM and other improved MM (IMM) procedures. There does not, however, exist any comparative study between the GQL and the HL approaches. In this thesis, we have made a comparison between these two approaches mainly through an extensive simulation study involving the Poisson-normal mixed model. It is found that the HL approach may not produce consistent estimates for the regression effects specially when the variance of the random effects is large. In contrast, the GQL approach is found to always produce consistent estimates for all parameters of the model. These two estimation methodologies are also illustrated by analyzing a data set on the health care utilization in St. John's, Canada.