Stochastic ordering of the latent trait by the sum score under various polytomous IRT models

The sum score is often used to order respondents on the latent trait measured by the test. Therefore, it is desirable that under the chosen model the sum score stochastically orders the latent trait. It is known that unlike dichotomous item response theory (IRT) models, most polytomous IRT models do not imply stochastic ordering. It is unknown, however, (1) whether stochastic ordering is often or rarely violated and (2) whether violations yield a serious problem for practical data analysis. These are the central issues of this paper. First, some unanswered questions that pertain to polytomous IRT models implying stochastic ordering were investigated. Second, simulation studies were conducted to evaluate stochastic ordering in practical situations. It was found that for most polytomous IRT models that do not imply stochastic ordering, the sum score can be used safely to order respondents on the latent trait.