Optimal model-based clustering with multilevel data

In many contexts, sample units are clustered in groups according to a certain criterion, for instance employees in firms, students in classes, or patients in hospitals. These data are analyzed by multilevel models (Goldstein, 2011) and have important applications in the evaluation of public services, particularly in education and health. For instance, it may be of interest to make comparisons between schools or classes at national and international level on the basis of the students’acquired knowledge. Accountability systems in education have been promoted in the statistical literature mainly since the 90’s by Goldstein and Spiegelhalter (1996), who supported the idea that the performance monitoring approach may improve efficiency. In this work, we focus on models in which the multilevel structure is accounted for by a hierarchical set of discrete latent variables, even in the presence of multivariate responses; these latent variables are used to represent the unobserved heterogeneity between clusters (i.e., groups) of units and between units in each cluster, extending the Latent Class (LC) approach (Lazarsfeld and Henry, 1968) to the multilevel setting. In particular, two cases are of interest. The first is when the observed outcomes are polytomous, as they correspond to item responses, and data are collected at the same time occasion. This approach has been applied by many authors in the educational context, see among others Vermunt (2008) and Gnaldi et al. (2016). The second case of interest is when the data have a longitudinal dimension and heterogeneity between units is represented in a dynamic fashion by a Latent Markov (LM) chain, as proposed in Bartolucci et al. (2011); see also Bartolucci et al. (2013). While maximum likelihood estimation through the Expectation-Maximization algorithm (Dempster et al. 1977) of the models mentioned above is already well established, an issue that still deserves attention is that of predicting the latent variables at cluster