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A multilevel finite mixture item response model to cluster examinees and schools

Within the educational context, a key goal is to assess students acquired skills and to cluster students according to their ability level. In this regard, a relevant element to be accounted for is the possible effect of the school students come from. For this aim, we provide a methodological tool which takes into account the multilevel structure of the data (i.e., students in schools) in a suitable way. This approach allows us to cluster both students and schools into homogeneous classes of ability and effectiveness, and to assess the effect of certain students and school characteristics on the probability to belong to such classes. The approach relies on an extended class of multidimensional latent class IRT models characterized by: (i) latent traits defined at student level and at school level, (ii) latent traits represented through random vectors with a discrete distribution, (iii) the inclusion of covariates at student level and at school level, and (iv) a two-parameter logistic parametrization for the conditional probability of a correct response given the ability. The approach is applied for the analysis of data collected by two national tests administered in Italy to middle school students in June 2009: the INVALSI Italian Test and Mathematics Test. Results allow us to study the relationships between observed characteristics and latent trait standing within each latent class at the different levels of the hierarchy. They show that examinees and school expected observed scores, at a given latent trait level, are dependent on both unobserved (latent class) group membership and observed first and second level covariates.