Improving latent class analysis through outlier detection– an example from criminal careers research

We propose a comparison between two alternative approaches of modeling longitudinal data, mainly focusing on the case of univariate ordinal responses when time fixed and time-varying covariates are available: latent Markov model and growth mixture model. Both approaches try to model patterns of changes through time and are built on the finite mixture model. They are commonly applied in many fields and explain the dynamic change from a different perspective. In the latent Markov model the latent process fully explains the observable behavior of the subjects. This process is made by time-varying latent variables having a discrete distribution which are assumed to follow a first-order Markov chain. The available covariates may affect the conditional distribution of the response variables given the latent process (measurement model) or the distribution of the latent process (latent model). The model estimation is performed by the maximum likelihood procedure through the EM algorithm. The growth mixture model is tailored to study the evolution of a latent characteristic of the subjects with the idea of estimating trajectories across all time points and it is designed to account for individual variability about a mean population trend. It aims at estimating class specific variance components (intercept variance, slope variance). A latent categorical variable is used to account for the heterogeneity in observed development trajectories and the population variability in growth is modeled through a mixture of different subpopulations. The comparison we propose is focused on the basic assumptions, the inferential properties, the computational aspects and the predictive features of each methodology. Strengths and weaknesses of the latent Markov and of the growth mixture model are illustrated having in mind a specific research question related to an applicative example concerning the perception of the health status of elderly people.