  We propose a structural equation model, which reduces to a multidimensional
latent class item response theory model, for the analysis of binary item
responses with non-ignorable missingness. The missingness mechanism is driven
by two sets of latent variables: one describing the propensity to respond and
the other referred to the abilities measured by the test items. These latent
variables are assumed to have a discrete distribution, so as to reduce the
number of parametric assumptions regarding the latent structure of the model.
Individual covariates may also be included through a multinomial logistic
parametrization of the probabilities of each support point of the distribution
of the latent variables. Given the discrete nature of this distribution, the
proposed model is efficiently estimated by the Expectation-Maximization
algorithm. A simulation study is performed to evaluate the finite sample
properties of the parameter estimates. Moreover, an application is illustrated
to data coming from a Students' Entry Test for the admission to some university
courses.
