Consistency of Nonparametric Classification in Cognitive Diagnosis

Abstract

Latent class models for cognitive diagnosis have been developed to classify examinees into one of the 2^K attribute profiles arising from a K-dimensional vector of binary skill indicators. These models recognize that response patterns tend to deviate from the ideal responses that would arise if skills and items generated item responses through a purely deterministic conjunctive process. An alternative to employing these latent class models is to minimize the distance between observed item response patterns and ideal response patterns, in a nonparametric fashion that utilizes no stochastic terms for these deviations. Theorems are presented that show the consistency of this approach, when the true model is one of several common latent class models for cognitive diagnosis. Consistency of classification is independent of sample size, because no model parameters need to be estimated. Simultaneous consistency for a large group of subjects can also be shown given some conditions on how sample size and test length grow with one another.