On Joint Estimation of Regression and Overdispersion Parameters in Generalized Linear Models for Longitudinal Data

Liang and Zeger introduced a class of estimating equations that gives consistent estimates of regression parameters and of their variances in the class of generalized linear models for longitudinal data. When the response variable in such models is subject to overdispersion, the oerdispersion parameter does not only influence the marginal variance, it may also influence the mean of the response variable. In such cases, the overdispersion parameter plays a significant role in the estimation of the regression parameters. This raises the necessity for a joint estimation of the regression, as well as overdispersion parameters, in order to describe the marginal expectation of the outcome variable as a function of the covariates. To correct for the effect of overdispersion, we, therefore, exploit a general class of joint estimating equations for the regression and overdispersion parameters. This is done, first, under the working assumption that the observations for a subject are independent and then under the general condition that the observations are correlated. In the former case, both score and quasi-score estimating equations are developed. The score equations are obtained from the marginal likelihood of the data, and the quasi-score equations are derived by exploiting the first two moments of the marginal distribution. This quasi-score equations approach requires a weight matrix, usually referred to as the pseudo-covariance weight matrix, which we construct under the assumption that the observations for a subject (or in a cluster) are independent. In the later case when observations are correlated, quasi-score estimating equations are developed in the manner similar to that of the independence case but the pseudo-covariance weight matrix is constructed from a suitable working covariance matrix of the longitudinal observations, the joint distribution of the observations being unknown. Asymptotic theory is provided for the general class of joint estimators for the regression and overdispersion parameters. The asymptotic distributional results are also applied to develop suitable chi-square test for testing for the regression of the overdispersed data.