Abstract: The Q-matrix helps bridge the gap between cognitive psychology and psychometrics, and thus it plays a very important role in cognitive diagnostic assessment. Significant progress has been made in the Q-matrix theory and its applications in recent years. Numerous researchers have made significant contributions to the Q-matrix theory from structured to unstructured matrices, binary to polytomous attributes, simple to complex models, independent to general structures, and dichotomous to polytomous item responses.
The studies on the Q-matrix theory mainly contain four aspects of contents. The first is related research on the Q-matrix theory under ideal item response patterns. The representative study was the sufficient Q-matrix proposed by Tatsuoka (1995, 2009), which is used for representing the prerequisite relationships among the attributes and improving the construct validity of a test. Under the hierarchical relationships among the attributes, Ding et al. (2009) and their subsequent research found that a sufficient Q-matrix cannot completely distinguish the ideal item response patterns of different knowledge states. Therefore, they proposed a sufficient and necessary Q-matrix that must include a reachability matrix. Due to the rich information contained in polytomous scores, Ding et al. (2014) explored a method for constructing a complete Q-matrix under different attribute hierarchies for a certain polytomous scoring rule. As students’ cognitive levels at the same attribute may change over time in their learning progression, the polytomous sufficient Q-matrix was introduced and can be used to guide test design and item construction, and establish a one-to-one correspondence between knowledge states and ideal item response patterns (Ding et al., 2015; Sun et al., 2013).
The second is the study on complete Q-matrix theory under the nonparametric cognitive diagnostic framework. Under independent structure, Chiu (2009) proposed a complete Q-matrix containing an identity matrix, which can be used to distinguish different knowledge states from ideal item response patterns, and established the asymptotic classification theory of cluster analysis for cognitive diagnosis. Chiu and Köhn (2015) provided a general definition of a complete Q matrix based on expected item response patterns, and delved into the sufficient conditions for the Q matrix to be complete in general cognitive diagnostic models (such as the generalized deterministic inputs, noisy and gate model, the log-linear cognitive diagnosis model, the general diagnostic model) or simplified cognitive diagnostic models with two item parameters (such as the deterministic inputs, noisy and gate model, or the deterministic inputs, noisy or gate model, referred as the DINA or DINO model). The purpose of the design of the compete Q-matrix mainly focused on the identifiability of the knowledge state and derived the asymptotic classification theory under the general cognitive diagnostic model. At the same time, they also explored the completeness conditions of the structured and unstructured Q-matrices of the DINA model with attribute hierarchy structures.
The third kind of the complete Q-matrix was proposed for the general attribute structure under the framework of knowledge space theory. Heller (2022) summarizes and sorts out the relevant conclusions on complete Q-matrices under knowledge space theory. Unlike the independent structure and attribute hierarchy discussed earlier, the complete Q-matrix in conjunctive models under knowledge space theory is suitable for more general attribute structures. The fourth category is the complete Q-matrix under the framework of the model identifiability (Gu & Xu, 2021, 2023), which mainly includes the necessary and sufficient conditions for the strict identifiability of the DINA and DINO model parameters, the sufficient conditions for partial identifiability of cognitive diagnostic models with two-parameters or multi-parameters, and the model identifiability conditions for both item parameters and attribute hierarchy structures.
Following the introduction of the Q-matrix theory, four examples were presented to illustrate its applications in the theoretical validity criterion of diagnostic tests, the design of item selection methods in computerized adaptive test, the methods for Q-matrix learning and specification, and test construction for cognitive diagnosis. Model-free or model-based Q-matrix theory, and the applications of the latest Q-matrix theory needs to be further investigated.

Key words: cognitive diagnosis, Q-matrix, attribute structure, complete, polytomous attributes