Nonparametric item response theory and Mokken scale analysis, with relations to latent class models and cognitive diagnostic models

As the focus of this chapter, we discuss nonparametric item response theory for ordinal person scales, specifically the monotone homogeneity model and Mokken scale analysis, which is the data-analysis procedure used for investigating the compliance between the monotone homogeneity model and data. Next, we discuss the unrestricted latent class model as an even more liberal model for investigating the scalability of a set of items, producing nominal scales, but we also discuss an ordered latent class model that one can use to investigate assumptions about item response functions in the monotone homogeneity model and other nonparametric item response models. Finally, we discuss cognitive diagnostic models, which are the core of this volume, and which are a further deepening of latent class models, providing diagnostic information about the people who responded to a set of items. A data analysis example, using item scores of 1210 respondents on 44 items from the Millon Clinical Multiaxial Inventory III, demonstrates how the monotone homogeneity model, the latent class model, and two cognitive diagnostic models can be used jointly to understand one’s data.