Conjunctive and Disjunctive Extensions of the Least Squares Distance Model of Cognitive Diagnosis

Abstract

Many models of cognitive diagnosis, including the least squares distance model (LSDM), work under the conjunctive assumption that a correct item response occurs when all latent attributes required by the item are correctly performed. This article proposes a disjunctive version of the LSDM under which the correct item response occurs when at least one attribute is correctly applied. Also, under both the conjunctive and disjunctive versions of the LSDM, this article demonstrates an approach to estimating the conditional probability that (a) a specific pattern of p attributes, (b) exactly p attributes, and (c) at least p attributes will be correctly performed across locations on the logit scale in the item response theory under the one-, two-, or three-parameter logistic model. Such information can be useful for interpretations and decisions based on a person’s performance on attributes that govern the correct responses on binary items under unidimensional item response theory calibrations for assessment in education, psychology, and other fields.