A Constrained Metropolis-Hastings Robbins-Monro Algorithm for Q Matrix Estimation in DINA Models

In diagnostic classification models (DCMs), theQmatrix encodes in which attributes are required for each item. TheQmatrix is usually predetermined by the researcher but may in practice be misspecified which yields incorrect statistical inference. Instead of using a predeterminedQmatrix, it is possible to estimate it simultaneously with the item and structural parameters of the DCM. Unfortunately, current methods are computationally intensive when there are many attributes and items. In addition, the identification constraints necessary for DCMs are not always enforced in the estimation algorithms which can lead to non-identified models being considered. We address these problems by simultaneously estimating the item, structural andQmatrix parameters of the Deterministic Input Noisy And gate model using a constrained Metropolis-Hastings Robbins-Monro algorithm. Simulations show that the new method is computationally efficient and can outperform previously proposed Bayesian Markov chain Monte-Carlo algorithms in terms ofQmatrix recovery, and item and structural parameter estimation. We also illustrate our approach using Tatsuoka's fraction-subtraction data and Certificate of Proficiency in English data.