SUFFICIENT AND NECESSARY CONDITIONS FOR THE IDENTIFIABILITY OF THE Q-MATRIX

Restricted latent class models (RLCMs) have recently gained prominence in educational assessment, psychiatric evaluation, and medical diagnosis. In contrast to conventional latent class models, the restrictions on RLCM parameters are imposed using a design matrix, in order to respect practitioners’ scientific assumptions. The design matrix, called the Q-matrix in the cognitive diagnosis literature, is usually constructed by practitioners and domain experts; however, it remains subjective and can be misspecified. To address this problem, researchers have proposed estimating the Q-matrix from sample data. However, the fundamental learnability of the Q-matrix and the model parameters remains underexplored. As a result, studies often impose stronger than needed (or even impractical) conditions. Here, we propose sufficient and necessary conditions for the joint identifiability of the Q-matrix and the RLCM parameters under different types of RLCMs. The proposed identifiability conditions depend only on the design matrix, and are easy to verify in practice.