A Taxonomy of IRT Models for Ordering Persons and Items Using Simple Sum Scores

Abstract
The stochastic ordering of the latent trait by means of the unweighted total score is considered for 10 dichotomous IRT models and 10 polytomous IRT models. The conclusion is that the stochastic ordering property holds for all dichotomous IRT models and for two polytomous IRT models. Also, the invariant item ordering property is considered for the same 20 IRT models. It is concluded that invariant item ordering holds for three dichotomous IRT models and three polytomous IRT models. The person and item ordering results are summarized in a taxonomy of IRT models. Some consequences far practical test construction are briefly discussed.