A generalized linear mixed model for longitudinal binary data with a marginal logit link function

TL;DR

This paper introduces a flexible generalized linear mixed model for longitudinal binary data that maintains a logistic form at the marginal level, accommodating multiple correlated random intercepts over time.

Contribution

It extends existing models by allowing separate, correlated random intercepts at each measurement occasion, enabling more realistic correlation structures in longitudinal binary data.

Findings

Model retains logistic form at the marginal level.

Allows flexible correlation structures among random intercepts.

Applied to longitudinal study of cardiac abnormalities in children.

Abstract

Longitudinal studies of a binary outcome are common in the health, social, and behavioral sciences. In general, a feature of random effects logistic regression models for longitudinal binary data is that the marginal functional form, when integrated over the distribution of the random effects, is no longer of logistic form. Recently, Wang and Louis [Biometrika 90 (2003) 765--775] proposed a random intercept model in the clustered binary data setting where the marginal model has a logistic form. An acknowledged limitation of their model is that it allows only a single random effect that varies from cluster to cluster. In this paper we propose a modification of their model to handle longitudinal data, allowing separate, but correlated, random intercepts at each measurement occasion. The proposed model allows for a flexible correlation structure among the random intercepts, where the correlations can be interpreted in terms of Kendall's $\tau$. For example, the marginal correlations among the repeated binary outcomes can decline with increasing time separation, while the model retains the property of having matching conditional and marginal logit link functions. Finally, the proposed method is used to analyze data from a longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women.