Selection of the number of components for finite mixtures of linear mixed models

Over the last decades, linear models have been studied by the scientific community as an important tool of statistical modelling in a great variety of phenomena. However, in many situations the data are grouped according to factors, so the introduction of random effects is required in order to consider the correlation between observations from the same individual, in which case linear mixed models are used. In addition, it is often observed that the data comes from a heterogeneous population, giving rise to situations where the estimation of a single linear model is not sufficient. Therefore, it is necessary to use models that incorporate this unobserved heterogeneity, as is the case of mixture models. Thus, mixtures of linear mixed models allow modelling the heterogeneity among the individuals and, at the same time, to account for correlations between observations from the same individual. Choosing the number of components for mixture models has long been considered as an important but difficult research problem. There is wide variety of literature available on the performance of model selection statistics for determining the number of components in mixture models. In this article, we study the problem of determining the number of components in mixtures of linear mixed models, investigating the performance of various model selection methods. In order to evaluate the methodologies developed, we carry out a simulation study and we illustrate these methodologies using a real data set.