认知诊断测验最简完备Q矩阵统一设计方法

doi:10.3724/SP.J.1041.2025.1849

摘要/Abstract

摘要：

属性水平(二分属性和多分属性)和项目理想评分方式(0-1评分与多级评分)是认知诊断测验设计的两个重要维度。其中, 多分属性测验能提供更详细的诊断信息, 而多级评分测验能提高判准率, 但现有认知诊断测验缺乏对多分属性和多级评分的整合设计。借鉴二分属性多级评分结构化/非结构化最简完备Q矩阵(SSCQM/USCQM)的概念, 本文提出统一的认知诊断测验最简完备Q矩阵设计方法, 解决不同属性水平和不同项目理想评分方式的各种组合情境下的认知诊断测验设计问题, 并在长测验和短测验两种条件下, 以(拟)可达阵为参照, 通过模拟研究对各种SSCQM和USCQM准确率进行了比较。结果表明,总体而言, SSCQM和USCQM具有更高的判准率。实证研究数据进一步验证了SSCQM和USCQM测验的优势。

关键词: 认知诊断测验, 测验设计, 最简完备Q矩阵, 统一设计方法

Abstract:

The quality of cognitive diagnose test (CDT) directly influences the performance of the diagnostic results and its remedial function. Thus, CDT design plays a vital role in the cognitive diagnosis process. A key challenge is maximizing differentiation among participants while minimizing the number of items. Although CDT is commonly applied in contexts involving polytomous attributes and responses, prior research often yields fragmented results, limiting practical application and theoretical development. A complete Q matrix represents a valuable tool that can be employed as a high-quality test for various types of tests. The (quasi-) reachability matrix is widely utilized in various testing procedures. This study aims to develop a unified, simplified testing design that is simpler than (quasi-) reachability matrix and improves usability and applicability across various cognitive diagnosis scenarios.

The simplest complete Q-matrix (SCQM) is a key concept in test design, offering a straightforward yet effective way to organize test items using fewer questions. This study proposes a unified construction model of SCQM applicable to various cognitive diagnostic test types. The Unstructured SCQM (USCQM) is developed from the Structured SCQM (SSCQM) to enhance organization. The proposed unified SSCQM model is based on the item attribute total score and selecting the SSCQM in the (quasi-) reachability matrix, specifically: (1) retaining the maximum column from each group; (2) keeping columns in each group with the distinct highest average attribute scores from the maximum columns. Following Tang Xiaojuan et al. (2024), a unified USCQM model for dichotomous attributes and polytomous responses is also proposed.

A simulation study was conducted to evaluate the effects of the number of attributes, attribute hierarchy, attribute levels, and complete Q matrices on the classification capabilities of SSCQM and USCQM in both long and short tests. Their performance was compared with the (quasi-) reachability and identity matrices. Results showed that classification accuracy generally declined with increasing attribute hierarchy, number of attributes, and attribute levels, except in long tests with multilevel attributes. Accuracy improved with more complete Q matrices. For short tests, SSCQM performed better with dichotomous responses, while USCQM was superior with polytomous responses. In long tests, the trend reversed. When attribute levels were 3 or 4, tests with polytomous attributes and responses achieved the highest accuracy. In other cases, binary attribute tests performed better. Empirical findings confirmed SSCQM and USCQM outperform the (quasi-) reachability matrix.

Polytomous attribute and response tests provide richer diagnostic information and higher accuracy, making them valuable in educational assessments. However, research on their design remains limited and fragmented. This study introduces a unified design model of the Simplest Complete Q Matrix (SSCQM and USCQM) for cognitive diagnostic tests based on dichotomous attributes and polytomous responses. Classification accuracy was assessed in both long and short tests. Simulation and empirical results show SSCQM and USCQM outperform or match the (quasi-) reachability matrix, with especially strong performance in short test scenarios.

Key words: cognitive diagnose test, the simplest complete Q matrix, test design, unified design method

中图分类号:

B841

唐小娟, 毛萌萌, 李瑜, 丁树良, 彭志霞. (2025). 认知诊断测验最简完备Q矩阵统一设计方法. 心理学报, 57(10), 1849-1866.

TANG Xiaojuan, MAO Mengmeng, LI Yu, DING Shuliang, PENG Zhixia. (2025). A unified design method of the simplest complete Q matrix for cognitive diagnostic tests. Acta Psychologica Sinica, 57(10), 1849-1866.

图/表 19

图1 3个属性线型结构和分支结构

图2 认知诊断测验USCQM统一设计方法

表1 模拟条件

模拟对象	数量	模拟要求	模拟方法
被试	2000	多分属性多级评分被试服从正态分布; 多分属性0-1评分被试服从均匀分布。	所有知识状态由扩张算法和广义扩张算法(丁树良, 罗芬等, 2015)得到。
多分属性0-1评分测验	长度为拟可达阵(记为R_P)的列数(N^*)	测验分别为SSCQM(也即拟可达阵)、USCQM和UC。项目参数s_j和g_j服从U(0,0.25)。	SSCQM和USCQM由各自的构造方法得到, 非完备Q矩阵从拟可达阵中随机选取。
多分属性0-1评分测验	长度为50(N^*)	测验分别包含SSCQM、USCQM和UC。项目参数s_j和g_j服从U(0,0.25)。	SSCQM、USCQM和UC生成方法如上; 测验其他题目从非零知识状态中选取并固定。
多分属性多级评分测验	长度为R_P的列数(N^*)	测验分别为R_P、UC和包含SSCQM、USCQM各1个。项目参数s_j和g_j服从U(0,0.35)。	SSCQM、USCQM和UC生成方法如上; 测验其他题目从非零知识状态中选取并固定。
	长度为SSCQM的列数(n^*)	测验分别为SSCQM、USCQM和UC。项目参数s_j和g_j服从U(0,0.35)。	SSCQM、USCQM和UC生成方法如上。
	长度为35(N^/n^)	测验分别包含拟可达阵、SSCQM、USCQM和UC。项目参数s_j和g_j服从U(0,0.35)。	SSCQM、USCQM和UC生成方法如上; 测验其他题目从非零知识状态中选取并固定。
认知诊断模型	2	多分属性0-1评分采用RPa-DINA模型, 多分属性多级评分采用修改的GRPa-DINA模型。	见4.1和4.2部分
被试作答反应	2000	被试的观察反应模式	根据被试模拟的真值、测验Q矩阵和认知诊断模型模拟被试作答结果。
被试估计	2000	估计被试的知识状态。	由模拟作答数据和认知诊断模型, 采用最大后验估计 (Maximum A Posteriori, MAP) 方法估计。

表2 属性水平数

属性标号	1	2	3	4	5	6	7
属性水平数L	2	2	3	4	5	3	3
属性水平取值	0,1	0,1	0,1,2	0,1,2,3	0,1,2,3,4	0,1,2	0,1,2

表3 多分属性0-1评分测验在不同属性层级结构下的判准率

N^*	H	MMR			PMR
N^*	H	R_P (SSCQM)	USCQM	UC	R_P (SSCQM)	USCQM	UC
11	L	0.9313	0.9239	0.9031	0.7057	0.6769	0.5998
	C	0.9277	0.9217	0.9010	0.6925	0.6681	0.6026
	D	0.9112	0.9081	0.8791	0.6381	0.6241	0.5290
	U	0.8933	0.8947	0.8574	0.5683	0.5673	0.4631
50	L	0.9760	0.9830	0.9563	0.8924	0.9207	0.8024
	C	0.9729	0.9794	0.9543	0.8785	0.9055	0.7932
	D	0.9469	0.9561	0.9301	0.7682	0.8060	0.7001
	U	0.9162	0.9323	0.9086	0.6504	0.7129	0.6180

表4 多分属性多级评分测验在不同属性层级结构下的判准率

H	N^/n^	MMR				PMR
H	N^/n^	SSCQM	USCQM	R_P	UC	SSCQM	USCQM	R_P	UC
L	11/7	0.9639/0.9304	0.9730/0.9393	0.9494	0.9157/0.9237	0.8355/0.7096	0.8784/0.7437	0.7733	0.6713/0.6797
C	11/8	0.9477/0.9150	0.9589/0.9373	0.9467	0.9347/0.9101	0.7855/0.6580	0.8216/0.7375	0.7621	0.7371/0.6377
D	11/8	0.9399/0.9177	0.9532/0.9148	0.9355	0.9319/0.8872	0.7497/0.6557	0.8001/0.6577	0.7251	0.7079/0.5549
U	11/10	0.9130/0.8791	0.9152/0.8950	0.9072	0.8905/0.8826	0.6641/0.5334	0.6641/0.5786	0.6194	0.5742/0.5221
L	35	0.9934	0.9857	0.9791	0.9660	0.9671	0.9289	0.8954	0.8311
C		0.9871	0.9870	0.9863	0.9679	0.9361	0.9354	0.9319	0.8403
D		0.9912	0.9888	0.9842	0.9695	0.9563	0.9445	0.9212	0.8484
U		0.9939	0.9901	0.9890	0.9720	0.9710	0.9521	0.9469	0.8629

表5 多分属性0-1评分测验在不同属性个数下的判准率

N^*	K	MMR			PMR
N^*	K	R_P (SSCQM)	USCQM	UC	R_P (SSCQM)	USCQM	UC
11	4	0.9010	0.9148	0.8681	0.7002	0.7124	0.5603
	5	0.9112	0.9081	0.8791	0.6381	0.6241	0.5290
	6	0.9160	0.9044	0.8683	0.5967	0.5520	0.4277
	7	0.9124	0.9030	0.8612	0.5431	0.5041	0.3677
50	4	0.9751	0.9796	0.9306	0.9005	0.9186	0.7240
	5	0.9469	0.9561	0.9301	0.7682	0.8060	0.7001
	6	0.9186	0.9274	0.8853	0.6280	0.6663	0.5052
	7	0.9035	0.9072	0.8730	0.5088	0.5308	0.4170

表6 多分属性多级评分测验不同属性个数下的判准率

K	N^/n^	MMR				PMR
K	N^/n^	SSCQM	USCQM	R_P	UC	SSCQM	USCQM	R_P	UC
4	7/5	0.9189/0.9138	0.9465/0.9161	0.9231	0.9275/0.8900	0.7450/0.7026	0.8196/0.7199	0.7293	0.7371/0.6269
5	11/8	0.9399/0.9177	0.9532/0.9148	0.9355	0.9319/0.8872	0.7497/0.6557	0.8001/0.6577	0.7251	0.7079/0.5549
6	13/10	0.9310/0.9071	0.9430/0.9107	0.9209	0.8810/0.8626	0.6864/0.5637	0.7281/0.5867	0.6149	0.4935/0.3915
7	15/12	0.9272/0.9248	0.9364/0.9111	0.9404	0.8895/0.8814	0.6288/0.5868	0.6670/0.5341	0.6190	0.4622/0.3976
4	35	0.9903	0.9830	0.9838	0.9566	0.9610	0.9319	0.9354	0.8273
5		0.9912	0.9888	0.9842	0.9695	0.9563	0.9445	0.9212	0.8484
6		0.9787	0.9795	0.9849	0.9527	0.8766	0.8846	0.9114	0.7259
7		0.9777	0.9834	0.9851	0.9535	0.8500	0.8894	0.9001	0.6926

表7 多分属性0-1评分测验在不同属性水平数下的判准率

N^*	L	MMR			PMR
N^*	L	R_P (SSCQM)	USCQM	UC	R_P (SSCQM)	USCQM	UC
11	2	0.9360	0.9303	0.8782	0.7938	0.7784	0.5485
	3	0.8748	0.8868	0.7971	0.5911	0.6269	0.4190
	4	0.8567	0.8619	0.8166	0.5410	0.5559	0.4419
	5	0.8629	0.8462	0.8160	0.5625	0.5214	0.4610
50	2	0.9788	0.9808	0.8994	0.9152	0.9232	0.5977
	3	0.9565	0.9730	0.9113	0.8338	0.8968	0.6549
	4	0.8716	0.8918	0.8386	0.6144	0.6626	0.5182
	5	0.8438	0.8511	0.8257	0.5286	0.5457	0.4888

表8 多分属性多级评分测验在不同属性水平数下的判准率

L	N^/n^	MMR				PMR
L	N^/n^	SSCQM	USCQM	R_P	UC	SSCQM	USCQM	R_P	UC
2	4/2	0.9598/0.9138	0.9575/0.9161	0.9534	0.8569/0.8548	0.8511/0.7026	0.8431/0.7199	0.8347	0.4718/0.4773
3	8/6	0.9454/0.9088	0.9255/0.8914	0.9309	0.8817/0.8508	0.8125/0.6852	0.7640/0.6461	0.7665	0.5925/0.5100
4	12/10	0.9228/0.8781	0.9183/0.8802	0.9238	0.8940/0.8643	0.7412/0.6021	0.7328/0.6159	0.7345	0.6327/0.5611
5	16/14	0.8797/0.8525	0.9118/0.8839	0.8905	0.8779/0.8380	0.6101/0.5398	0.7101/0.6224	0.6368	0.6137/0.4972
2	35	0.9762	0.9760	0.9854	0.8795	0.9048	0.9042	0.9415	0.5179
3		0.9894	0.9785	0.9934	0.9393	0.9580	0.9140	0.9735	0.7587
4		0.9857	0.9831	0.9743	0.9499	0.9440	0.9345	0.9004	0.8046
5		0.9621	0.9674	0.9655	0.9443	0.8554	0.8770	0.8713	0.7875

表9 多分属性0-1评分测验在不同完备Q矩阵个数下的判准率

M	MMR			PMR
M	R_P (SSCQM)	USCQM	UC	R_P (SSCQM)	USCQM	UC
1	0.9469	0.9561	0.9300	0.7682	0.8060	0.7539
2	0.9825	0.9787	0.9688	0.9155	0.8992	0.8896
3	0.9926	0.9875	0.9870	0.9639	0.9399	0.9375

表10 多分属性多级评分测验在不同完备Q矩阵个数下的判准率

M	MMR				PMR
M	SSCQM	USCQM	R_P	UC	SSCQM	USCQM	R_P	UC
1	0.9910	0.9899	0.9876	0.9697	0.9553	0.9501	0.9382	0.8487
2	0.9912	0.9935	0.9903	0.9673	0.9563	0.9678	0.9521	0.8385
3	0.9962	0.9951	0.9925	0.9683	0.9813	0.9757	0.9632	0.8432

表11 测验项目

测验项目	多分属性			二分属性
测验项目	A₁^’	A₂^’	A₃^’	A₁	A₂	A₃	A₄	A₅
1	1	1	0	1	0	1	0	0
2	1	0	0	1	0	0	0	0
3	1	0	0	1	0	0	0	0
4	1	0	0	1	0	0	0	0
5	1	2	0	1	1	0	0	0
6	1	1	0	1	0	1	0	0
7	1	2	0	1	1	0	0	0
8	1	1	0	1	0	1	0	0
9	1	2	0	1	1	0	0	0
10	1	0	1	1	0	0	1	0
11	1	0	2	1	0	0	0	1
12	1	0	1	1	0	0	1	0
13	1	0	2	1	0	0	0	1
14	1	1	0	1	0	1	0	0

表12 二分属性测验识别被试属性判准率/属性模式判准率

判准率	属性或可达阵	二分属性0-1评分				二分属性多级评分
判准率	属性或可达阵	R₁	R₂	USCQM	UC	R₂	SSCQM	USCQM	UC
AR	A₁	1	0.9348	0.9858	0.9674	0.9035	0.9092	0.9092	0.9589
	A₂	1	0.6667	0.6851	0.8596	0.7222	0.7504	0.7504	0.7447
	A₃	1	0.9390	0.9546	0.9759	0.9050	0.9319	0.9206	0.9348
	A₄	1	0.9943	0.9972	0.8355	0.8482	0.8809	0.9163	0.5206
	A₅	1	0.9972	0.9972	0.5504	0.8326	0.8837	0.8738	0.5418
	平均	1	0.9064	0.9240	0.8377	0.8423	0.8712	0.8740	0.7401
PR	R₁	1	0.6170	0.6511	0.3844	0.5220	0.5730	0.5674	0.2057

表13 多分属性测验识别被试属性判准率/属性模式判准率

判准率	属性或拟可达阵	多分属性0-1评分				多分属性多级评分
判准率	属性或拟可达阵	R_P₁	R_P₂	USCQM	UC	R_P₂	SSCQM	USCQM	UC
AR	A₁^’	1	0.9489	0.9972	0.9362	0.9390	0.9319	0.9730	0.6454
	A₂^’	1	0.7674	0.9716	0.4511	0.8170	0.8326	0.9645	0.4383
	A₃^’	1	0.9999	1	0.7929	0.9404	0.9234	0.8865	0.2979
	平均	1	0.9054	0.9896	0.7267	0.8988	0.8960	0.9414	0.4605
PR	R_P₁	1	0.7348	0.9716	0.3461	0.7915	0.8000	0.8426	0.2270

附表1 3个属性分支结构SSCQM/USCQM运用于RPa-DINA模型的S(α)

KS	SSCQM $R P$ / USCQM $R P$
KS	$q 1$ (100)/(100)	$q 2$ (110)/(010)	$q 3$ (120)/(020)	$q 4$ (101)/(101)	$q 5$ (102)/(002)	$q 6$ (103)/(003)
$α$	$S 1 α$	$S 2 α$	$S 3 α$	$S 4 α$	$S 5 α$	$S 6 α$
(000)	$g 1$	$g 2$	$g 3$	$g 4$	$g 5$	$g 6$
(100)	$1 − s 1$	$g 2$	$g 3$	$g 4$	$g 5$	$g 6$
(110)	$1 − s 1$	$1 − s 2$	$g 3$	$g 4$	$g 5$	$g 6$
(120)	$1 − s 1$	$1 − s 2$	$1 − s 3$	$g 4$	$g 5$	$g 6$
(101)	$1 − s 1$	$g 2$	$g 3$	$1 − s 4$	$g 5$	$g 6$
(102)	$1 − s 1$	$g 2$	$g 3$	$1 − s 4$	$1 − s 5$	$g 6$
(103)	$1 − s 1$	$g 2$	$g 3$	$1 − s 4$	$1 − s 5$	$1 − s 6$
(111)	$1 − s 1$	$1 − s 2$	$g 3$	$1 − s 4$	$g 5$	$g 6$
(112)	$1 − s 1$	$1 − s 2$	$g 3$	$1 − s 4$	$1 − s 5$	$g 6$
(113)	$1 − s 1$	$1 − s 2$	$g 3$	$1 − s 4$	$1 − s 5$	$1 − s 6$
(121)	$1 − s 1$	$1 − s 2$	$1 − s 3$	$1 − s 4$	$g 5$	$g 6$
(122)	$1 − s 1$	$1 − s 2$	$1 − s 3$	$1 − s 4$	$1 − s 1$	$g 6$
(123)	$1 − s 1$	$1 − s 2$	$1 − s 3$	$1 − s 4$	$1 − s 1$	$1 − s 6$

附表1 3个属性分支结构SSCQM/USCQM运用于RPa-DINA模型的S(α)

KS	SSCQM $R P$ / USCQM $R P$
KS	$q 1$ (100)/(100)	$q 2$ (110)/(010)	$q 3$ (120)/(020)	$q 4$ (101)/(101)	$q 5$ (102)/(002)	$q 6$ (103)/(003)
$α$	$S 1 α$	$S 2 α$	$S 3 α$	$S 4 α$	$S 5 α$	$S 6 α$
(000)	$g 1$	$g 2$	$g 3$	$g 4$	$g 5$	$g 6$
(100)	$1 − s 1$	$g 2$	$g 3$	$g 4$	$g 5$	$g 6$
(110)	$1 − s 1$	$1 − s 2$	$g 3$	$g 4$	$g 5$	$g 6$
(120)	$1 − s 1$	$1 − s 2$	$1 − s 3$	$g 4$	$g 5$	$g 6$
(101)	$1 − s 1$	$g 2$	$g 3$	$1 − s 4$	$g 5$	$g 6$
(102)	$1 − s 1$	$g 2$	$g 3$	$1 − s 4$	$1 − s 5$	$g 6$
(103)	$1 − s 1$	$g 2$	$g 3$	$1 − s 4$	$1 − s 5$	$1 − s 6$
(111)	$1 − s 1$	$1 − s 2$	$g 3$	$1 − s 4$	$g 5$	$g 6$
(112)	$1 − s 1$	$1 − s 2$	$g 3$	$1 − s 4$	$1 − s 5$	$g 6$
(113)	$1 − s 1$	$1 − s 2$	$g 3$	$1 − s 4$	$1 − s 5$	$1 − s 6$
(121)	$1 − s 1$	$1 − s 2$	$1 − s 3$	$1 − s 4$	$g 5$	$g 6$
(122)	$1 − s 1$	$1 − s 2$	$1 − s 3$	$1 − s 4$	$1 − s 1$	$g 6$
(123)	$1 − s 1$	$1 − s 2$	$1 − s 3$	$1 − s 4$	$1 − s 1$	$1 − s 6$

附表2 3个属性分支结构SSCQM运用于修改GRPa-DINA模型的S(α)

KS	SSCQM $Q R P S$
KS	$q 1$ $q 1$	$q 2$ $q 2$	$q 3$ $q 3$	$q 4$ $q 4$	$q 5$ $q 5$
$α$	$S 1 α$	$S 2 α$	$S 3 α$	$S 4 α$	$S 5 α$
(000)	$1 × g 11 − g 12$ $1 × g 11 − g 12$	$1 × g 21 − g 23$ $1 × g 21 − g 23$	$1 × g 31 − g 32$ $1 × g 31 − g 32$	$1 × g 41 − g 43$ $1 × g 41 − g 43$	$1 × g 51 − g 54$ $1 × g 51 − g 54$
(100)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × 1 − s 31 − g 32$ $1 × 1 − s 31 − g 32$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
KS	SSCQM $Q R P S$
KS	$q 1$ $q 1$	$q 2$ $q 2$	$q 3$ $q 3$	$q 4$ $q 4$	$q 5$ $q 5$
$α$	$S 1 α$	$S 2 α$	$S 3 α$	$S 4 α$	$S 5 α$
(110)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × 1 − s 31 − g 32$ $1 × 1 − s 31 − g 32$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(120)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × s 23 − s 21$ $1 × s 23 − s 21$	$1 × 1 − s 31 − g 32$ $1 × 1 − s 31 − g 32$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(101)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(102)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(103)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × s 54 − s 51$ $1 × s 54 − s 51$
(111)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(112)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(113)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × s 54 − s 51$ $1 × s 54 − s 51$
(121)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × s 23 − s 21$ $1 × s 23 − s 21$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(122)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × s 23 − s 21$ $1 × s 23 − s 21$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(123)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × s 23 − s 21$ $1 × s 23 − s 21$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × s 54 − s 51$ $1 × s 54 − s 51$

附表2 3个属性分支结构SSCQM运用于修改GRPa-DINA模型的S(α)

KS	SSCQM $Q R P S$
KS	$q 1$ $q 1$	$q 2$ $q 2$	$q 3$ $q 3$	$q 4$ $q 4$	$q 5$ $q 5$
$α$	$S 1 α$	$S 2 α$	$S 3 α$	$S 4 α$	$S 5 α$
(000)	$1 × g 11 − g 12$ $1 × g 11 − g 12$	$1 × g 21 − g 23$ $1 × g 21 − g 23$	$1 × g 31 − g 32$ $1 × g 31 − g 32$	$1 × g 41 − g 43$ $1 × g 41 − g 43$	$1 × g 51 − g 54$ $1 × g 51 − g 54$
(100)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × 1 − s 31 − g 32$ $1 × 1 − s 31 − g 32$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
KS	SSCQM $Q R P S$
KS	$q 1$ $q 1$	$q 2$ $q 2$	$q 3$ $q 3$	$q 4$ $q 4$	$q 5$ $q 5$
$α$	$S 1 α$	$S 2 α$	$S 3 α$	$S 4 α$	$S 5 α$
(110)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × 1 − s 31 − g 32$ $1 × 1 − s 31 − g 32$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(120)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × s 23 − s 21$ $1 × s 23 − s 21$	$1 × 1 − s 31 − g 32$ $1 × 1 − s 31 − g 32$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(101)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(102)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(103)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × s 54 − s 51$ $1 × s 54 − s 51$
(111)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(112)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(113)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × 1 − s 21 − g 23$ $1 × 1 − s 21 − g 23$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × s 54 − s 51$ $1 × s 54 − s 51$
(121)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × s 23 − s 21$ $1 × s 23 − s 21$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × 1 − s 41 − g 43$ $1 × 1 − s 41 − g 43$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(122)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × s 23 − s 21$ $1 × s 23 − s 21$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × 1 − s 51 − g 54$ $1 × 1 − s 51 − g 54$
(123)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$1 × s 23 − s 21$ $1 × s 23 − s 21$	$1 × s 32 − s 31$ $1 × s 32 − s 31$	$1 × s 43 − s 41$ $1 × s 43 − s 41$	$1 × s 54 − s 51$ $1 × s 54 − s 51$

附表3 3个属性分支结构USCQM运用于修改GRPa-DINA模型的 S α

KS	USCQM $Q R P U 2$
KS	$q 1$ $q 1$	$q 2$ $q 2$	$q 3$ $q 3$	$q 4$ $q 4$	$q 5$ $q 5$
$α$	$S 1 α$	$S 2 α$	$S 3 α$	$S 4 α$	$S 5 α$
(000)	$1 × g 11 − g 12$ $1 × g 11 − g 12$	$2 × g 22$	$1 × g 31$	$2 × g 42$	$3 × g 53$
(100)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$2 × g 22$	$1 × g 31$	$2 × g 42$	$3 × g 53$
(110)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × g 22$	$1 × g 31$	$2 × g 42$	$3 × g 53$
(120)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × 1 − s 22$	$1 × g 31$	$2 × g 42$	$3 × g 53$
(101)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$2 × g 22$	$1 × 1 − s 31$	$2 × g 42$	$3 × g 53$
(102)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$2 × g 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × g 53$
(103)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$2 × g 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × 1 − s 53$
(111)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × g 22$	$1 × 1 − s 31$	$2 × g 42$	$3 × g 53$
(112)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × g 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × g 53$
(113)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × g 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × 1 − s 53$
(121)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × 1 − s 22$	$1 × 1 − s 31$	$2 × g 42$	$3 × g 53$
(122)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × 1 − s 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × g 53$
(123)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × 1 − s 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × 1 − s 53$

附表3 3个属性分支结构USCQM运用于修改GRPa-DINA模型的 S α

KS	USCQM $Q R P U 2$
KS	$q 1$ $q 1$	$q 2$ $q 2$	$q 3$ $q 3$	$q 4$ $q 4$	$q 5$ $q 5$
$α$	$S 1 α$	$S 2 α$	$S 3 α$	$S 4 α$	$S 5 α$
(000)	$1 × g 11 − g 12$ $1 × g 11 − g 12$	$2 × g 22$	$1 × g 31$	$2 × g 42$	$3 × g 53$
(100)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$2 × g 22$	$1 × g 31$	$2 × g 42$	$3 × g 53$
(110)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × g 22$	$1 × g 31$	$2 × g 42$	$3 × g 53$
(120)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × 1 − s 22$	$1 × g 31$	$2 × g 42$	$3 × g 53$
(101)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$2 × g 22$	$1 × 1 − s 31$	$2 × g 42$	$3 × g 53$
(102)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$2 × g 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × g 53$
(103)	$1 × 1 − s 11 − g 12$ $1 × 1 − s 11 − g 12$	$2 × g 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × 1 − s 53$
(111)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × g 22$	$1 × 1 − s 31$	$2 × g 42$	$3 × g 53$
(112)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × g 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × g 53$
(113)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × g 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × 1 − s 53$
(121)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × 1 − s 22$	$1 × 1 − s 31$	$2 × g 42$	$3 × g 53$
(122)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × 1 − s 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × g 53$
(123)	$1 × s 12 − s 11$ $1 × s 12 − s 11$	$2 × 1 − s 22$	$1 × 1 − s 31$	$2 × 1 − s 42$	$3 × 1 − s 53$

附图1 倒金字塔结构

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