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Joint Modeling of Longitudinal Count and Binary Data Using Latent Variable Approaches Open Access

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Latent variable models are used extensively to explain association or correlation between observed outcomes. In this work, two latent variable approaches are presented characterizing the joint distribution of the observed outcomes and the latent variables. The first approach models a dichotomous outcome using a binary latent variable whereby covariates model the latent variable. A random effects model introduces heterogeneity among subjects in modeling the mean value of the latent outcome. This idea is extended to the ordinal case where the latent state is composed of $K$ ordinal classes. For the fixed effects model, the Expectation Maximization algorithm is used to determine parameter estimates. For the random effects model, a Monte-Carlo Expectation Maximization alogrithm is implemented to determine parameter estimates.The second approach models a joint outcome consisting of dichotomous and count data using a latent state via a hidden Markov model. The Expectation Maximization algorithm is used to fit the model using an implementation of the forward-backward algorithm to determine posterior probabilities. This model is extended to account for heterogeneity among subjects for the count outcome and the hidden process. Estimation for this model proceeds using a unique implementation of the forward-backward algorithm with adaptive Gaussian quadrature, providing greater computational efficiency than other methods in this setting. Applications of both models is presented throughout using data from the Naturalistic Teenage Driving Study sponsored by the Eunice Kennedy Shriver National Institute of Child Health and Human Development.

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