Quasi likelihood on linear mixed effect of binary response in longitudinal data

Abstract

Longitudinal data is derived from repeated measurements over time, and when analyzing such data using logistic regression models, it is essential to account for the correlations induced by these repeated observations. To address this, a random component is incorporated into the linear predictor, transforming the model into a generalized linear mixed model (GLMM). In this approach, the linear predictor is no longer purely systematic due to the inclusion of the random component. Estimation of parameters in a GLMM cannot rely on the conventional maximum likelihood estimation method, as it must account for the random component in the linear predictor. A viable alternative is the quasi-likelihood method, which provides a framework for estimating these parameters. Among various quasi-likelihood-based approaches, the generalized estimating equation (GEE) is commonly employed, as it explicitly incorporates the correlation structure in the data through a specified "working correlation" matrix. In particular, an autoregressive structure is often chosen for this matrix. The estimation procedure is then carried out iteratively through the Fisher Scoring method, with convergence achieved after several iterations. To assess the performance of this methodology, simulation studies are conducted, followed by its application to real-world data for empirical validation.