On a class of finite mixture models that includes hidden Markov models

In the context of longitudinal data, we introduce a class of finite mixture (FM) models that generalizes that of hidden Markov (HM) models, and derive conditions under which the two classes are equivalent. On the basis of this result, we develop a likelihood ratio (LR) misspecification test for assessing the latent structure of an HM model, along with a multiple version of this test that may be used in the presence of many latent states or time occasions. This testing procedure requires the maximum likelihood estimation of the two models under comparison, that is, the assumed HM model and the more general FM model, which is performed by suitable versions of the Expectation–Maximization algorithm. The approach is validated through a simulation study, aimed at assessing the performance of the proposed tests under different circumstances, and by an application using data derived from the SCImago Journal & Country Rank database.