Latent variable model for joint analysis of multiple repeated measures and bivariate event times

This article presents a novel approach to analyzing a complex dataset from a prevention trial, where outcomes comprise multiple repeated mental health items and times to initiation of alcohol and tobacco use. The dataset has a nonnegligible portion of missing values and interval or left censored events. The substantive interest of the trial suggests a psychiatric distress latent variable that is reflected in the mental health items and potentially affects initiation of alcohol and tobacco use. We describe the data with a combination of three types of component model: a marginal model for the longitudinal latent process for psychiatric distress given study interventions and covariates; logistic regression models for the repeated mental health items given the latent process; and hazard models for times to initiation of alcohol and tobacco use given the latent process, study interventions, and covariates. To aid in fitting these models simultaneously, we use automatic differentiation to find the first two derivatives of the total log-likelihood function, thus speeding up convergence relative to the regular expectation-maximization algorithm with a direct calculation of valid variance estimates.