Stochastic order in dichotomous item response models for fixed tests, research adaptive tests, or multiple abilities

Dichotomous item response theory (IRT) models can be viewed as families of stochastically ordered distributions of responses to test items. This paper explores several properties of such distributiom. The focus is on the conditions under which 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 two theorems that apply immediately to dichotomous IRT models. One theorem holds for unidimensional models with fixed item parameters. The other theorem holds for models with multiple abilities or with random item parameters as used, for example, in adaptive testing.