Asymptotic Results with Generalized Estimating Equations for Longitudinal Data

We consider the marginal models of Liang and Zeger [Biometrika 73 (1986) 13–22] for the analysis of longitudinal data and we develop a theory of statistical inference for such models. We prove the existence , weak consistency and asymptotic normality of a sequence of estimators defined as roots of pseudo-likelihood equations. 1. Introduction. Longitudinal data sets arise in biostatistics and lifetime testing problems when the responses of the individuals are recorded repeatedly over a period of time. By controlling for individual differences, longitudinal studies are well-suited to measure change over time. On the other hand, they require the use of special statistical techniques because the responses on the same individual tend to be strongly correlated. In a seminal paper Liang and Zeger (1986) proposed the use of generalized linear models (GLM) for the analysis of longitudinal data. In a cross-sectional study, a GLM is used when there are reasons to believe that each response y i depends on an observable vector x i of covariates [see the monograph of McCullagh and Nelder (1989)]. Typically this dependence is specified by an unknown parameter β and a link function µ via the relationship µ i (β) = µ(x T i β), where µ i (β) is the mean of y i. For one-dimensional observations, the maximum quasi-likelihood estimatorˆβ n is defined as the solution of the equation