An overview of latent Markov models for longitudinal categorical data

TL;DR

This paper provides a comprehensive overview of latent Markov models for analyzing longitudinal categorical data, detailing estimation methods, model constraints, extensions with covariates, and applications in socio-economic research.

Contribution

It offers a detailed synthesis of LM models, including estimation techniques, constrained versions, extensions with covariates, and practical applications, highlighting their flexibility and utility.

Findings

Maximum likelihood estimation via EM algorithm is efficient.

Constrained models improve parsimony and hypothesis testing.

Extensions include covariates and multilevel data handling.

Abstract

We provide a comprehensive overview of latent Markov (LM) models for the analysis of longitudinal categorical data. The main assumption behind these models is that the response variables are conditionally independent given a latent process which follows a first-order Markov chain. We first illustrate the basic LM model in which the conditional distribution of each response variable given the corresponding latent variable and the initial and transition probabilities of the latent process are unconstrained. For this model we also illustrate in detail maximum likelihood estimation through the Expectation-Maximization algorithm, which may be efficiently implemented by recursions known in the hidden Markov literature. We then illustrate several constrained versions of the basic LM model, which make the model more parsimonious and allow us to include and test hypotheses of interest. These constraints may be put on the conditional distribution of the response variables given the latent process (measurement model) or on the distribution of the latent process (latent model). We also deal with extensions of LM model for the inclusion of individual covariates and to multilevel data. Covariates may affect the measurement or the latent model; we discuss the implications of these two different approaches according to the context of application. Finally, we outline methods for obtaining standard errors for the parameter estimates, for selecting the number of states and for path prediction. Models and related inference are illustrated by the description of relevant socio-economic applications available in the literature.