Two-Part Hidden Semi-Markov Mixed Effects Models for Semi-Continuous Longitudinal Data

Modeling dynamic heterogeneity is essential for revealing the distinct longitudinal trajectories of individual change. Dynamic heterogeneity analysis of semi-continuous longitudinal data is commonly difficult due to the semi-continuity of longitudinal responses. The hidden semi-Markov model is a powerful tool that can reveal the longitudinal dependency structure and the dynamic heterogeneity of the observation process by introducing the sojourn time distribution. To address the challenge of modeling dynamic heterogeneity in semi-continuous longitudinal data, this study develops a two-part hidden semi-Markov mixed-effects model. The proposed model mainly consists of two parts: a discrete binary indicator model to estimate the probability of a zero outcome for the semi-continuous longitudinal response, and a continuous hidden semi-Markov model to fit the positive values of semi-continuous longitudinal responses. In order to accurately obtain the state of each individual at different observation times, a set of likelihood ratio test state iteration algorithms is developed. Bayesian methods are used to estimate the regression coefficients and state parameters of the proposed model. The proposed methodology is applied to analyze the dataset of the Health and Retirement Study conducted by the University of Michigan. Simulation studies are conducted to assess the flexibility of the proposed model under various scenarios.