A simple descriptive method for multidimensional item response theory based on stochastic dominance

In this paper we develop a descriptive concept of a (partially) ordinal joint scaling of items and persons in the context of (dichotomous) item response analysis. The developed method has to be understood as a purely descriptive method describing relations among the data observed in a given item response data set, it is not intended to directly measure some presumed underlying latent traits. We establish a hierarchy of pairs of item di culty and person ability orderings that empirically support each other. The ordering principles we use for the construction are essentially related to the concept of rst order stochastic dominance. Our method is able to avoid a paradoxical result of multidimensional item response theory models described in Hooker et al. [2009]. We introduce our concepts in the language of formal concept analysis. This is due to the fact that our method has some similarities with formal concept analysis and knowledge space theory: Both our methods as well as descriptive techniques used in knowledge space theory (concretely, item tree analysis) could be seen as two di erent stochastic generalizations of formal implications from formal concept analysis.