A unified design method of the simplest complete Q matrix for cognitive diagnostic tests

doi:10.3724/SP.J.1041.2025.1849

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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.

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URL: https://journal.psych.ac.cn/acps/EN/10.3724/SP.J.1041.2025.1849

https://journal.psych.ac.cn/acps/EN/Y2025/V57/I10/1849

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Table 1 The accuracy rate of polytomous attributes and dichotomous responses tests under different attribute hierarchical structures

${{N}^{*}}$	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

Table 1 The accuracy rate of polytomous attributes and dichotomous responses tests under different attribute hierarchical structures

${{N}^{*}}$	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

Table 2 The accuracy rate of polytomous attributes and polytomous responses tests under different attribute hierarchical structures

H	*${{N}^{}}$*/${{n}^{\text{}}}$	MMR				PMR
H	*${{N}^{}}$*/${{n}^{\text{}}}$	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

Table 2 The accuracy rate of polytomous attributes and polytomous responses tests under different attribute hierarchical structures

H	*${{N}^{}}$*/${{n}^{\text{}}}$	MMR				PMR
H	*${{N}^{}}$*/${{n}^{\text{}}}$	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

Table 3 The accuracy rate of polytomous attributes and dichotomous responses tests under thenumber of different attributes

${{N}^{*}}$	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

Table 3 The accuracy rate of polytomous attributes and dichotomous responses tests under thenumber of different attributes

${{N}^{*}}$	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

Table 4 The accuracy rate of polytomous attributes and polytomous responses tests under the numberof different attributes

K	${{N}^{}}$/${{n}^{\text{}}}$	MMR				PMR
K	${{N}^{}}$/${{n}^{\text{}}}$	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

Table 4 The accuracy rate of polytomous attributes and polytomous responses tests under the numberof different attributes

K	${{N}^{}}$/${{n}^{\text{}}}$	MMR				PMR
K	${{N}^{}}$/${{n}^{\text{}}}$	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

Table 5 The accuracy rate of polytomous attributes and dichotomous responses tests under the number of different attribute levels

${{N}^{*}}$	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

Table 5 The accuracy rate of polytomous attributes and dichotomous responses tests under the number of different attribute levels

${{N}^{*}}$	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

Table 6 The accuracy rate of polytomous attributes and polytomous responses tests under thenumber of different attribute levels

L	${{N}^{}}$/${{n}^{\text{}}}$	MMR				PMR
L	${{N}^{}}$/${{n}^{\text{}}}$	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

Table 6 The accuracy rate of polytomous attributes and polytomous responses tests under thenumber of different attribute levels

L	${{N}^{}}$/${{n}^{\text{}}}$	MMR				PMR
L	${{N}^{}}$/${{n}^{\text{}}}$	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

Table 7 The accuracy rate of polytomous attributes and dichotomous responses tests under the number of complete Q matrix

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

Table 7 The accuracy rate of polytomous attributes and dichotomous responses tests under the number of complete Q matrix

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

Table 8 The accuracy rate of polytomous attributes and polytomous responses tests under the number of complete Q matrix

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

Table 8 The accuracy rate of polytomous attributes and polytomous responses tests under the number of complete Q matrix

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

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