Generalized linear mixed hidden semi‐Markov models in longitudinal settings: A Bayesian approach

Hidden Markov and semi‐Markov models (H(S)MMs) constitute useful tools for modeling observations subject to certain dependency structures. The hidden states render these models very flexible and allow them to capture many different types of latent patterns and dynamics present in the data. This has led to the increased popularity of these models, which have been applied to a variety of problems in various domains and settings, including longitudinal data. In many longitudinal studies, the response variable is categorical or count‐type. Generalized linear mixed models (GLMMs) can be used to analyze a wide range of variables, including categorical and count. The present study proposes a model that combines HSMMs with GLMMs, leading to generalized linear mixed hidden semi‐Markov models (GLM‐HSMMs). These models can account for time‐varying unobserved heterogeneity and handle different response types. Parameter estimation is achieved using a Monte Carlo Newton‐Raphson (MCNR)‐like algorithm. In our proposed model, the distribution of the random effects depends on hidden states. We illustrate the applicability of GLM‐HSMMs with an example in the field of occupational health, where the response variable consists of count values. Furthermore, we assess the performance of our MCNR‐like algorithm through a simulation study.