Polytomous IRT Models and Monotone Likelihood Ratio of the Total Score

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Abstract
In a broad class of item response theory (IRT) models for dichotomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait θ. In this study, it is shown that for polytomous items MLR holds for the partial credit model and a trivial generalization of this model. MLR does not necessarily hold if the slopes of the item step response functions vary over items, item steps, or both. MLR holds neither for Samejima's graded response model, nor for nonparametric versions of these three polytomous models. These results are surprising in the context of Grayson's and Huynh's results on MLR for nonparametric dichotomous IRT models, and suggest that establishing stochastic ordering properties for nonparametric polytomous IRT models will be much harder.