An Analysis of (Dis)Ordered Categories, Thresholds, and Crossings in Difference and Divide-by-Total IRT Models for Ordered Responses.

Threshold parameters have distinct referents across models for ordered responses. In difference models, thresholds are trait levels at which responding beyond category k is as likely as responding at or below it; in divide-by-total models, thresholds are trait levels at which responding in category k is as likely as responding in category k - 1. Thus, thresholds in divide-by-total models (but not in difference models) are the crossings of the option response functions for consecutive categories. Thresholds in difference models are always ordered but they may inconsequentially yield ordered or disordered crossings. In contrast, assimilation of thresholds and crossings in divide-by-total models questions category order when crossings are disordered. We analyze these aspects of difference and divide-by-total models, their relation to the order of response categories, and the consequences of collapsing categories to instate ordered crossings under divide-by-total models. We also show that item parameters in models for ordered responses can never contradict the pre-assumed order of categories and that the empirical order can only be established using a polytomous model that does not assume ordered categories, although this often gives rise to spurious outcomes. Practical implications for scale development are discussed.