On estimating equations for parameters in generalized linear mixed models with application to binary data

Abstract

In view of the cumbersome and often intractable numerical integrations required for a full likelihood analysis, several suggestions have been made recently for approximate inference in generalized linear mixed models and other nonlinear variance component models. For example, we refer to the penalized quasi-likelihood (PQL) approach of Breslow and Clayton (1993), the Stein-type estimating function based approach of Waclawiw and Liang (1993), and the corrected PQL approach of Breslow and Lin (1995). Recently, Sutradhar and Godambe (1998) provided a semiparametric solution to the estimation problem dealt with by Waclawiw and Liang. In the present article, we provide a semiparametric solution to the estimation problem investigated by Breslow and Lin, and Breslow and Clayton. More specifically, we propose a two-step joint estimating equations approach to estimate the model parameters. In the first step, we use an estimating function based approach to obtain the estimates of the random effects. In the second step, following Prentice and Zhao (1991), we construct the first two moment based joint estimating equations for the regression parameters and the variance component of the random effects. As the exact first and second order moments of the generalized linear mixed models are not available, these moments are obtained first by expanding the conditional moments for given random effects about their estimating function based estimates and then taking the expectation of the conditional moments over the true distribution of the random effects. Since binary mixed models arise in many biomedical application areas, for example, computations are provided in detail for this special case. We also discuss a second approach, namely, a three-step ad hoc estimating equations approach, in the context of the special binary mixed models. The performance of the proposed estimators are examined through a simulation study. The estimation procedure is also illustrated through an analysis of the child health and development study (CHDS) data from Yerushalmy (1970).