  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.
