Exploratory General-Response Cognitive Diagnostic Models with Higher-Order Structures

Cognitive Diagnostic Models (CDMs) are popular discrete latent variable models in educational and psychological measurement. While existing CDMs mainly focus on binary or categorical responses, there is a growing need to extend them to model a wider range of response types, including but not limited to continuous and count-valued responses. Meanwhile, incorporating higher-order latent structures has become crucial for gaining deeper insights into cognitive processes. We propose a general modeling framework for higher-order CDMs for rich types of responses. Our framework features a highly flexible data layer that is adaptive to various response types and measurement models for CDMs. Importantly, we address a challenging exploratory estimation scenario where the item-attribute relationship, specified by the Q-matrix, is unknown and needs to be estimated along with other parameters. In the higher-order layer, we employ a probit-link with continuous latent traits to model the binary latent attributes, highlighting its benefits in terms of identifiability and computational efficiency. Theoretically, we propose transparent identifiability conditions for the exploratory setting. Computationally, we develop an efficient Monte Carlo Expectation-Maximization algorithm, which incorporates an efficient direct sampling scheme and requires significantly reduced simulated samples. Extensive simulation studies and a real data example demonstrate the effectiveness of our methodology.