Motivated by an empirical analysis of ecological momentary assessment data (EMA) collected in a smoking cessation study, we propose a joint modeling technique for estimating the time-varying association between two intensively measured longitudinal responses: a continuous one and a binary one. A major challenge in joint modeling these responses is the lack of a multivariate distribution. We suggest introducing a normal latent variable underlying the binary response and factorizing the model into two components: a marginal model for the continuous response, and a conditional model for the binary response given the continuous response. We develop a two-stage estimation procedure and establish the asymptotic normality of the resulting estimators. We also derived the standard error formulas for estimated coefficients. We conduct a Monte Carlo simulation study to assess the finite sample performance of our procedure. The proposed method is illustrated by an empirical analysis of smoking cessation data, in which the question of interest is to investigate the association between urge to smoke, continuous response, and the status of alcohol use, the binary response, and how this association varies over time.
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