Uncovering Latent Structures: A Bayesian Approach to Estimating Q-Matrix and Attribute Hierarchies in Cognitive Diagnostic Models.

Cognitive diagnostic models (CDMs) provide fine-grained diagnostic feedback by modeling the relationship between latent attributes and item responses. Two key components required for CDM implementation are the Q-matrix, which links items to attributes, and the attribute hierarchy, which defines prerequisite relationships among attributes. In many practical settings, both structures are specified by experts based on cognitive theory. In this article, we propose a novel Bayesian estimation method that simultaneously learns the Q-matrix, the attribute hierarchy, and the parameters of the deterministic inputs, noisy "and" gate (DINA) model. We develop a Metropolis-Hastings within Gibbs algorithm and integrate a mini-batch strategy to improve computational efficiency. We conducted a series of simulation studies to evaluate the performance of the proposed algorithm under varying conditions, including sample size, test length, hierarchy structure, mini-batch size, and threshold settings. The results demonstrate strong recovery rates for both latent structures and item parameters, confirming the accuracy and robustness of our method. A real data application further illustrates the utility of the proposed framework in uncovering interpretable diagnostic structures. Our findings offer practical guidance for researchers seeking to implement CDMs when both the Q-matrix and attribute hierarchy are unknown.