A mixed non-homogeneous hidden Markov model for categorical data, with application to alcohol consumption

Hidden Markov models (HMMs) are frequently used to analyse longitudinal data, where the same set of subjects is repeatedly observed over time. In this context, several sources of heterogeneity may arise at individual and/or time level, which affect the hidden process, that is, the transition probabilities between the hidden states. In this paper, we propose the use of a finite mixture of non-homogeneous HMMs (NH-HMMs) to face the heterogeneity problem. The non-homogeneity of the model allows us to take into account observed sources of heterogeneity by means of a proper set of covariates, time and/or individual dependent, explaining the variations in the transition probabilities. Moreover, we handle the unobserved sources of heterogeneity at the individual level, due to, for example, omitted covariates, by introducing a random term with a discrete distribution. The resulting model is a finite mixture of NH-HMM that can be used to classify individuals according to their dynamic behaviour or to estimate a mixed NH-HMM without any assumption regarding the distribution of the random term following the non-parametric maximum likelihood approach. We test the effectiveness of the proposal through a simulation study and an application to real data on alcohol abuse.