Stochastic Ordering Using the Latent Trait and the Sum Score in Polytomous IRT Models

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Abstract
In a restricted class of item response theory (IRT) models for polytomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait ϑ. MLR implies two stochastic ordering (SO) properties, denoted SOM and SOL, which are both weaker than MLR, but very useful for measurement with IRT models. Therefore, these SO properties are investigated for a broader class of IRT models for which the MLR property does not hold.In this study, first a taxonomy is given for nonparametric and parametric models for polytomous items based on the hierarchical relationship between the models. Next, it is investigated which models have the MLR property and which have the SO properties. It is shown that all models in the taxonomy possess the SOM property. However, counterexamples illustrate that many models do not, in general, possess the even more useful SOL property.