认知诊断评估中Q矩阵理论及应用

doi:10.3724/SP.J.1042.2024.01010

摘要/Abstract

摘要：

Q矩阵是认知心理学与心理计量学结合的重要载体, Q矩阵在认知诊断中发挥着十分重要的作用。Q矩阵理论和应用研究近年来取得了重要进展。众多研究者从结构化到非结构化、属性二值到多值、简单到复杂模型、独立到一般结构、0-1到多级评分方面不断深入和拓展Q矩阵理论。Q矩阵理论也广泛应用于测验构念效度评价、计算机化自适应测验选题策略设计、Q矩阵学习和标定、认知诊断测验组卷等。与模型无关的Q矩阵理论和适合特定认知诊断模型下Q矩阵理论, 以及最新Q矩阵理论的应用都值得深入研究。

关键词: 认知诊断, Q矩阵, 属性结构, 完备性, 多值属性

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

中图分类号:

B841

宋丽红, 汪文义, 丁树良. (2024). 认知诊断评估中Q矩阵理论及应用. 心理科学进展 , 32(6), 1010-1033.

SONG Lihong, WANG Wenyi, DING Shuliang. (2024). Q-matrix theory and its applications in cognitive diagnostic assessment. Advances in Psychological Science, 32(6), 1010-1033.

图/表 3

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知识状态	$\varnothing $	$\text{ }\!\!\{\!\!\text{ 1,2 }\!\!\}\!\!\text{ }$	$\text{ }\!\!\{\!\!\text{ 2,3 }\!\!\}\!\!\text{ }$	$\text{ }\!\!\{\!\!\text{ 3,4 }\!\!\}\!\!\text{ }$	$\text{ }\!\!\{\!\!\text{ 1,2,3 }\!\!\}\!\!\text{ }$	$\text{ }\!\!\{\!\!\text{ 2,3,4 }\!\!\}\!\!\text{ }$	$\text{S}$
$\varnothing $	-	$a$	$b$	$b$	$a/b/c$	$a/b/d$	$a/b/c/d$
$\text{ }\!\!\{\!\!\text{ 1,2 }\!\!\}\!\!\text{ }$		-	$b$	$b$	$c$	$b/d$	$b/c/d$
$\text{ }\!\!\{\!\!\text{ 2,3 }\!\!\}\!\!\text{ }$			-	$a$	$c$	$d$	$c/d$
$\text{ }\!\!\{\!\!\text{ 3,4 }\!\!\}\!\!\text{ }$				-	$c$	$d$	$c/d$
$\text{ }\!\!\{\!\!\text{ 1,2,3 }\!\!\}\!\!\text{ }$					-	$c/d$	$d$
$\text{ }\!\!\{\!\!\text{ 2,3,4 }\!\!\}\!\!\text{ }$						-	$c$
$\text{S}$							-

知识状态	$\varnothing $	$\text{ }\!\!\{\!\!\text{ 1,2 }\!\!\}\!\!\text{ }$	$\text{ }\!\!\{\!\!\text{ 2,3 }\!\!\}\!\!\text{ }$	$\text{ }\!\!\{\!\!\text{ 3,4 }\!\!\}\!\!\text{ }$	$\text{ }\!\!\{\!\!\text{ 1,2,3 }\!\!\}\!\!\text{ }$	$\text{ }\!\!\{\!\!\text{ 2,3,4 }\!\!\}\!\!\text{ }$	$\text{S}$
$\varnothing $	-	$a$	$b$	$b$	$a/b/c$	$a/b/d$	$a/b/c/d$
$\text{ }\!\!\{\!\!\text{ 1,2 }\!\!\}\!\!\text{ }$		-	$b$	$b$	$c$	$b/d$	$b/c/d$
$\text{ }\!\!\{\!\!\text{ 2,3 }\!\!\}\!\!\text{ }$			-	$a$	$c$	$d$	$c/d$
$\text{ }\!\!\{\!\!\text{ 3,4 }\!\!\}\!\!\text{ }$				-	$c$	$d$	$c/d$
$\text{ }\!\!\{\!\!\text{ 1,2,3 }\!\!\}\!\!\text{ }$					-	$c/d$	$d$
$\text{ }\!\!\{\!\!\text{ 2,3,4 }\!\!\}\!\!\text{ }$						-	$c$
$\text{S}$							-

结构	水平	机制	Q矩阵	理论基础	样本量	诊断方法
层级	二值	连接	Q矩阵包含单位矩阵 Q矩阵可达矩阵R (定理1) Q矩阵包含介于两者之间的E* (定理4)	Chiu (2009) 丁树良等(2010) Köhn和Chiu (2019, 2021) Heller (2022)	小	NPC
层级	二值	连接	定理10或定理11的条件	Gu和Xu (2021a, 2023)	中	DINA-AHM
		非连接	Q矩阵包含单位矩阵(充分条件)	Chiu和Köhn (2015b)	小	NPC
			定理A6	Gu和Xu (2021a, 2023)	中	DINO-AHM
			原文中定理3(充分条件)	Gu和Xu (2021a, 2023)	中	ACDM/LLTM
独立	二值	连接	Q矩阵包含单位矩阵定理2或定理3的条件	Chiu (2009) Heller (2022)	小	NPC
			定理8的条件	Gu和Xu (2019b)	中	DINA
		非连接	Q矩阵包含单位矩阵	Chiu和Köhn (2015b)	小	NPC
			定理8的条件	Gu和Xu (2019b)	中	DINO
			定理9的条件	Gu和Xu (2020, 2021b)	中	ACDM/LLTM
					大	GDINA LCDM GDM
层级 (含独立)	多值	非连接	Q矩阵包含拟可达矩阵	丁树良, 罗芬等(2015) Sun等人(2013) 蔡艳和涂冬波(2015)	中	GDD-P
一般	二值	连接	Q矩阵包含基本属性模式矩阵B或定理5、6、7中条件	Heller (2022)	小	NPC

结构	水平	机制	Q矩阵	理论基础	样本量	诊断方法
层级	二值	连接	Q矩阵包含单位矩阵 Q矩阵可达矩阵R (定理1) Q矩阵包含介于两者之间的E* (定理4)	Chiu (2009) 丁树良等(2010) Köhn和Chiu (2019, 2021) Heller (2022)	小	NPC
层级	二值	连接	定理10或定理11的条件	Gu和Xu (2021a, 2023)	中	DINA-AHM
		非连接	Q矩阵包含单位矩阵(充分条件)	Chiu和Köhn (2015b)	小	NPC
			定理A6	Gu和Xu (2021a, 2023)	中	DINO-AHM
			原文中定理3(充分条件)	Gu和Xu (2021a, 2023)	中	ACDM/LLTM
独立	二值	连接	Q矩阵包含单位矩阵定理2或定理3的条件	Chiu (2009) Heller (2022)	小	NPC
			定理8的条件	Gu和Xu (2019b)	中	DINA
		非连接	Q矩阵包含单位矩阵	Chiu和Köhn (2015b)	小	NPC
			定理8的条件	Gu和Xu (2019b)	中	DINO
			定理9的条件	Gu和Xu (2020, 2021b)	中	ACDM/LLTM
					大	GDINA LCDM GDM
层级 (含独立)	多值	非连接	Q矩阵包含拟可达矩阵	丁树良, 罗芬等(2015) Sun等人(2013) 蔡艳和涂冬波(2015)	中	GDD-P
一般	二值	连接	Q矩阵包含基本属性模式矩阵B或定理5、6、7中条件	Heller (2022)	小	NPC