Abstract
Dichotomous IRT models can be viewed as families of stochastically ordered distributions of responses to test items. This paper explores several properties of such distributions. In particular, it is examined under what conditions stochastic order in families of conditional distributions is transferred to their inverse distributions, from two families of related distributions to a third family, or from multivariate conditional distributions to a marginal distribution. The main results are formulated as a series of theorems and corollaries which apply to dichotomous IRT models. One part of the results holds for unidimensional models with fixed item parameters. The other part holds for models with random item parameters as used, for example, in adaptive testing or for tests with multidimensional abilities.
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Portions of this paper were presented at the 9th European Meeting of the Psychometric Society, Leiden, The Netherlands, July 4–7, 1995. The author is indebted to the referees for their comments on the previous version of the manuscript, as well as to the Greek fisherman who picked up the only copy of the set of handwritten notes for this paper from the harbor of Karpathos.
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van der Linden, W.J. Stochastic order in dichotomous item response models for fixed, adaptive, and multidimensional tests. Psychometrika 63, 211–226 (1998). https://doi.org/10.1007/BF02294852
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DOI: https://doi.org/10.1007/BF02294852