A Sparse Latent Class Model for Polytomous Attributes in Cognitive Diagnostic Assessments

Diagnostic models (DMs) have been widely applied to binary response data. However, in the field of educational and psychological measurement, a wealth of ordinal data are collected to measure latent structures where the traditional binary attributes may not adequately describe the complex response patterns. Considering that, we propose an extension of the sparse latent class model (SLCM) with ordinal attributes, with the purpose of fully exploring the relationships between attributes and response patterns. Furthermore, we discuss the strict and generic identifiability conditions for the ordinal SLCMs. We apply the model to the Short Dark Triad data and revisit the underlying personality structure. Evidence supports that SLCMs have better model fit to this real data than the exploratory factor models. We also confirm the efficiency of a Gibbs algorithm in recovering the empirical item parameters via a Monte Carlo simulation study. This study discusses a way of constructing DMs with ordinal attributes which helps broaden its applicability to personality assessment.