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Evaluation of student proficiency through a multidimensional finite mixture IRT model

In certain academic systems, a student can enroll for an exam immediately after the end of the teaching period or can postpone it to any later examination session, so that the grade is missing until the exam is not attempted. We propose an approach for the evaluation in itinere of a student's proficiency accounting also for non-attempted exams. The approach is based on considering each exam as an item, so that responding to the item amounts to attempting the exam, and on an Item Response Theory model that includes two latent variables corresponding to the student's ability and the propensity to attempt the exam. In this way, we explicitly account for non-ignorable missing observations as the indicators of item response also contribute to measure the ability. The two latent variables are assumed to have a discrete distribution defining latent classes of students that are homogeneous in terms of ability and priority assigned to exams. The model, which also allows for individual covariates in its structural part, is fitted by the Expectation-Maximization algorithm. The approach is illustrated through the analysis of data about the first-year exams of freshmen of the School of Economics at the University of Florence (Italy).