Logistic regression when covariates are random effects from a non-linear mixed model

In many studies, the association of longitudinal measurements of a continuous response and a binary outcome are often of interest. A convenient framework for this type of problems is the joint model, which is formulated to investigate the association between a binary outcome and features of longitudinal measurements through a common set of latent random effects. The joint model, which is the focus of this article, is a logistic regression model with covariates defined as the individual-specific random effects in a non-linear mixed-effects model (NLMEM) for the longitudinal measurements. We discuss different estimation procedures, which include two-stage, best linear unbiased predictors, and various numerical integration techniques. The proposed methods are illustrated using a real data set where the objective is to study the association between longitudinal hormone levels and the pregnancy outcome in a group of young women. The numerical performance of the estimating methods is also evaluated by means of simulation.