Direct Estimation of Diagnostic Classification Model Attribute Mastery Profiles via a Collapsed Gibbs Sampling Algorithm

Abstract

This paper proposes a novel collapsed Gibbs sampling algorithm that marginalizes model parameters and directly samples latent attribute mastery patterns in diagnostic classification models. This estimation method makes it possible to avoid boundary problems in the estimation of model item parameters by eliminating the need to estimate such parameters. A simulation study showed the collapsed Gibbs sampling algorithm can accurately recover the true attribute mastery status in various conditions. A second simulation showed the collapsed Gibbs sampling algorithm was computationally more efficient than another MCMC sampling algorithm, implemented by JAGS. In an analysis of real data, the collapsed Gibbs sampling algorithm indicated good classification agreement with results from a previous study.

Bayesian Analysis Methods for Two-Level Diagnosis Classification Models

“…Swapping of attribute mastery and switching of latent class labels may occur if identification conditions are not satisfied. For identification of diagnostic model parts, we need monotonicity constraints on the correct item response probabilities (e.g., Henson et al, 2009; Yamaguchi & Templin, 2022a). In our model formulation, monotonicity constraints were not employed.…”

“…For the diagnostic measurement, we employed the latent class formulation and notations utilized in Yamaguchi and Okada (2020a) and Yamaguchi and Templin (2022a, 2022b). Attribute mastery patterns α = [ α 1 , … , α k , … , α K ] ⊤ are defined as permutations of the mastery status of each attribute.…”

“…Note that θ j h can be sampled to satisfy the monotonicity constraints using the sampling procedure of Yamaguchi and Templin (2022a). The other full conditional posteriors are listed hereafter.…”

“…The true Θ was generated using the method employed by Yamaguchi and Templin (2022a), in which the correct item response probabilities of all nonmastering item-specific patterns and mastering patterns are represented as g and 1 − s . In addition, the number of mastering attributes in the h th item-specific pattern is defined as # j h , and the true correct item response probability is given by θ j h = g + # j h ( 1 − s − g ) / ( false ∑ k q j k ) .…”

On the boundary problems in diagnostic classification models

“…In this case, we do not interpret the parameter estimates. As mentioned above, full Bayesian estimations, not discussed in this study, are also a solution (e.g., Yamaguchi & Templin, 2022b). One reason we employed the MAP estimation is that can be tractable in mathematical analysis.…”