Abstract: In this paper, the concept of Q matrix and its application in cognitive diagnosis are summarized, analyzed and discussed. This paper mainly includes five parts. The first part introduces the Q matrix and its related concepts. The Q matrix can represent the relationship between the items and the attributes, it can also represent the relationship between the examinees' knowledge states and the attributes, the construction of the Q matrix also has a significant impact on the performance of the cognitive diagnostic model. Considering the hierarchical relationship between attributes, there exists some transformation relations among the adjacency matrix A, the reachability matrix R and the Q matrix: A can be obtained from the cognitive model, we can get R by exponentiation of A or Wallshell algorithm, the potential Q matrix can be obtained by using the augment algorithm on R. If the potential Q matrix is the necessary Q matrix, we can implement a cleaning algorithm on it to get R in turn. The test Q matrix, obtained by picking several columns from the potential Q matrix, is a blueprint of a cognitive diagnostic test that can be used to guide the repertoire. Given the hierarchy of attributes, the more rows of R have, the more columns of potential Q matrices obtained based on R using the augment algorithm would have. The second part introduces the important role of Q matrix in cognitive diagnosis: (1) The important role of Q matrix in cognitive model and CDM: From the cognitive model point of view, R is a special Q matrix, also the presentation of cognitive model, and the attribute hierarchy that is mined from the "retrofitting" data does not necessarily coincide with the theoretical hierarchy structure; From the CDM point of view, Q matrices represent cognitive blueprints, the item attribute vector and the examinees’ KS (vector) are under the same "gauge", which making it possible to "compare" them under a partial order relationship, under certain conditions we can use Qs and the necessary Qt to calculate the ideal response pattern (IRP), we can define operators of union and intersection on the Qs column set, and thus construct a lattice. (2) The role of Qp in cognitive diagnosis and the union representation of its column: Firstly, R can generate Qp, so R is a good "representative" of Qp, which has an important influence on the design of cognitive diagnostic test; Secondly, each column of the Qp matrix can be represented as a union of the R column, and the representation is not unique, which allows us to define a redundant expression and a concise expression for Qp columns, and the redundant expression has its special effects. The third part introduces several recent advances related to the Q matrix theory: To start with, we point out that Leighton, Gierl, and Hunka (2004) have some problems in the discussion of convergent structure paths. With the augment algorithm, we reveal that there is only one top-down path in the convergent structure. Next, we construct a test containing three independent attributes and three items, when a different column is selected from the set of columns augmented by R and added to the given test Q matrix, the theoretic construct validity ( TCV ) of resulting Q would vary accordingly, thus we found that there is a discrepancy between Q's columns which came from the augment of R. Finally, Lee, Park, and Taylan (2011) conducted a cognitive diagnosis of TIMSS2007 in Massachusetts, Minnesota and the United States. We further analyzed the results and found that there exists big difference between the knowledge states derived from R and the knowledge states based on DINA model used in the article. The fourth part points out some possible research directions related to the Q matrix, including the problem of Q matrix calibration, the measurement of the goodness and the unreasonable of test Q matrix's design, the properties and applications of polytomous Q matrices. The fifth part summarizes this paper and discusses some extra problems to be studied relate to the necessary Qt matrix, polytomous Q matrix and item attribute calibration. We expect that this paper can provide reference and help for people to use the Q matrix more flexibly.

Key words: Cogonive diagnosis, Q matrix, Attribute structures, Reachability Matrix

摘要： Q矩阵作为连接认知和测量的桥梁，在认知诊断中起重要作用。本文梳理了应用Q矩阵解决认知诊断相关问题的理论与方法。首先整理Q矩阵的相关概念、算法、性质及其在认知诊断中的作用；并根据Q矩阵可计算理论构念效度、可以构成格等，指出Q矩阵是特殊的关联矩阵；接着介绍Q矩阵理论研究方面的几个近期发展；并对Q矩阵未来的应用研究作出展望。期望本文能为测量工作者更灵活地利用Q矩阵提供参考和帮助。

关键词: 认知诊断, Q矩阵, 属性层级结构, 可达阵