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Acta Psychologica Sinica    2020, Vol. 52 Issue (1) : 93-106     DOI: 10.3724/SP.J.1041.2020.00093
Reports of Empirical Studies |
A method of Q-matrix validation for polytomous response cognitive diagnosis model based on relative fit statistics
WANG Daxun1,GAO Xuliang2,CAI Yan1,TU Dongbo1()
1 School of Psychology, Jiangxi Normal University, Nanchang 330022, China
2 School of Psychology, Guizhou Normal University, Guiyang 550000, China
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Abstract  

Cognitive diagnostic assessments (CDAs) can provide fine-grained diagnostic information about students' knowledge states, so as to help to teach in accordance with the students’ aptitude. The development of cognitive diagnosis model for polytomous response data expands the application scope of cognitive diagnostic assessment. As the basis of CDAs, Q-matrix has aroused more and more attention for the subjective tendency in Q-matrix construction that is typically performed by domain experts. Due to the subjective process of Q-matrix construction, there inevitably have some misspecifications in the Q-matrix, if left unchecked, can result in a serious negative impact on CDAs. To avoid the subjective tendency from experts and to improve the correctness of the Q-matrix, several objective Q-matrix validation methods have been proposed. Many Q-matrix validation methods have been proposed in dichotomous CDMs, however, the research of the Q-matrix validation method under polytomous CDMs is stalling lacking. To address this concern, several relative fit statistics (i.e., -2LL, AIC, BIC) were applied to the Q-matrix validation for polytomous cognitive diagnosis model in this research. The process of Q-matrix validation is as follows:
First, the reduced Q-matrix is represented by${{Q}_{r}}$, which represents a set of potential q-vectors and contains ${{2}^{K}}-1$ possible q-vectors when attributes are independent. When validating the q-vector of the first category of item j, all possible q-vectors in${{Q}_{r}}$can be used as the q-vector of the first category of item j, and the Q-matrix of remaining items remains intact. From this, the item parameters and the attribute patterns of students can be estimated, and the -2LL, AIC, and BIC can be calculated accordingly. The q-vector with the largest likelihood (or smallest AIC/BIC) is regarded as the q-vector of the first category of item j. The q-vector of the next category of the item j can also be obtained in the same way. The algorithm stops when the validated Q-matrix is same as the previous Q-matrix, or every item has been reached. In order to improve the efficiency of the method, a sequential search algorithm was proposed.
Several simulation studies were conducted to evaluate the effectiveness and practicality of these methods, and the performance of the methods in this paper was compared with the stepwise method ( Ma & de la Torre, 2019). Three experimental factors were considered in simulation studies, including sample size, Q-matrix error types and CDMs. The results show that (1) BIC method can be used for Q-matrix validation under polytomous response CDMs, and the performance of the BIC method is better than the stepwise method. (2) In general, the performance of the three methods from good to bad is the BIC method, AIC method, and -2LL method. (3) The performance of Q-matrix validation methods is affected by the sample size, and increasing the number of sample size can improve the accuracy of the Q-matrix validation.
In this study, Q-matrix validation methods for polytomous response CDMs were studied. It was found that the BIC method can be used for the Q-matrix validation under polytomous response CDMs. The method proposed in this paper can not only improve the accuracy of Q-matrix specification but also increase the model-data fit level. Besides, the data-based Q-matrix validation method can also reduce the workload of experts in Q-matrix construction and improve the classification accuracy of cognitive diagnosis.

Keywords cognitive diagnostic assessment      Q-matrix      seq-GDINA      BIC     
ZTFLH:  B841  
Corresponding Authors: Dongbo TU     E-mail: tudongbo@aliyun.com
Issue Date: 21 November 2019
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Daxun WANG
Xuliang GAO
Yan CAI
Dongbo TU
Cite this article:   
Daxun WANG,Xuliang GAO,Yan CAI, et al. A method of Q-matrix validation for polytomous response cognitive diagnosis model based on relative fit statistics[J]. Acta Psychologica Sinica, 2020, 52(1): 93-106.
URL:  
http://journal.psych.ac.cn/xlxb/EN/10.3724/SP.J.1041.2020.00093     OR     http://journal.psych.ac.cn/xlxb/EN/Y2020/V52/I1/93
题目 类别 A1 A2 A3 A4 A5 题目 类别 A1 A2 A3 A4 A5
1 1 1 0 0 0 0 11 1 1 1 0 0 0
1 2 0 1 0 0 0 11 2 0 0 0 0 1
2 1 0 0 1 0 0 12 1 1 1 1 0 0
2 2 0 0 0 1 0 12 2 0 0 0 1 1
3 1 0 0 0 0 1 13 1 1 1 0 0 0
3 2 1 0 0 0 0 13 2 0 0 1 1 1
4 1 0 0 0 0 1 14 1 1 0 1 0 0
4 2 0 0 0 1 0 14 2 0 0 0 1 0
5 1 0 0 1 0 0 14 3 0 0 0 0 1
5 2 0 1 0 0 0 15 1 0 0 0 0 1
6 1 1 0 0 0 0 15 2 0 0 1 1 0
6 2 0 1 1 0 0 15 3 0 1 0 0 0
7 1 0 0 1 0 0 16 1 1 0 0 0 0
7 2 0 0 0 1 1 16 2 0 1 0 0 0
8 1 0 0 0 0 1 16 3 0 0 1 1 0
8 2 1 1 0 0 0 17 1 1 0 0 0 0
9 1 0 0 0 1 1 18 1 0 1 0 0 0
9 2 0 0 1 0 0 19 1 0 0 1 0 0
10 1 0 1 0 1 0 20 1 0 0 0 1 0
10 2 1 0 0 0 0 21 1 0 0 0 0 1
  
Q Q矩阵错误模拟规则 调整的类别 调整的属性个数 备注
Q1 ${{q}_{jk}}=0\to {{q}_{jk}}=1$ $K_{jh}^{*}=1$的类别 5 属性冗余
Q2 ${{q}_{jk}}=1\to {{q}_{jk}}=0$ $K_{jh}^{*}>2$的类别 5 属性缺失
Q3 ${{q}_{jk}}=0\to {{q}_{jk}}=1,{{q}_{j{k}'}}=1\to {{q}_{j{k}'}}=0$ $K_{jh}^{*}>2$的类别 10 属性既冗余又缺失
Q4 ${{q}_{jk}}=0\to {{q}_{jk}}=1$
${{q}_{jk}}=1\to {{q}_{jk}}=0$${{q}_{jk}}=0\to {{q}_{jk}}=1\text{ }{{q}_{j{k}'}}=1\to {{q}_{j{k}'}}=0$
分别为Q1、Q2和Q3的类别 20 Q1、Q2和Q3的组合
Q5 10%随机调整 随机 20 调整后$1<K_{jh}^{*}<3$
Q6 20%随机调整 随机 40 调整后$1<K_{jh}^{*}<3$
  
Q-matrix N PMR AMR FPR TPR RMSEA
Stepwise BIC Stepwise BIC Stepwise BIC Stepwise BIC QW QStepwise QBIC
Q1 500 0.795 0.788 0.957 0.963 0.118 0.157 0.958 0.965 0.017 0.015 0.007
1000 0.879 0.863 0.975 0.977 0.065 0.074 0.975 0.978 0.018 0.009 0.005
2000 0.918 0.911 0.984 0.986 0.048 0.049 0.985 0.986 0.019 0.005 0.003
Q2 500 0.763 0.790 0.953 0.962 0.367 0.021 0.958 0.962 0.017 0.016 0.007
1000 0.826 0.856 0.967 0.975 0.257 0.004 0.971 0.975 0.016 0.011 0.005
2000 0.865 0.903 0.976 0.984 0.219 0.002 0.980 0.984 0.017 0.008 0.003
Q3 500 0.758 0.786 0.952 0.962 0.339 0.126 0.963 0.966 0.033 0.016 0.006
1000 0.815 0.861 0.964 0.976 0.251 0.089 0.972 0.979 0.034 0.010 0.005
2000 0.856 0.910 0.974 0.985 0.180 0.065 0.980 0.987 0.035 0.009 0.004
Q4 500 0.680 0.776 0.938 0.961 0.363 0.110 0.962 0.966 0.041 0.020 0.007
1000 0.721 0.853 0.950 0.975 0.288 0.064 0.968 0.978 0.041 0.015 0.005
2000 0.745 0.905 0.956 0.984 0.251 0.040 0.972 0.986 0.042 0.013 0.003
Q5 500 0.760 0.777 0.951 0.959 0.112 0.068 0.956 0.961 0.075 0.020 0.008
1000 0.835 0.851 0.968 0.975 0.082 0.041 0.972 0.976 0.073 0.011 0.004
2000 0.874 0.903 0.975 0.984 0.065 0.022 0.978 0.984 0.076 0.013 0.004
Q6 500 0.629 0.687 0.924 0.933 0.184 0.105 0.943 0.940 0.100 0.035 0.015
1000 0.656 0.744 0.931 0.942 0.173 0.097 0.949 0.949 0.102 0.038 0.017
2000 0.687 0.793 0.935 0.953 0.163 0.081 0.951 0.959 0.102 0.037 0.010
  
Q-matrix N PMR AMR FPR TPR RMSEA
Stepwise BIC Stepwise BIC Stepwise BIC Stepwise BIC QW QStepwise QBIC
Q1 500 0.750 0.841 0.952 0.975 0.083 0.022 0.952 0.975 0.006 0.007 0.006
1000 0.823 0.884 0.968 0.982 0.037 0.041 0.968 0.983 0.005 0.005 0.005
2000 0.864 0.915 0.976 0.987 0.029 0.020 0.977 0.987 0.004 0.005 0.004
Q2 500 0.746 0.839 0.953 0.975 0.332 0.199 0.958 0.978 0.027 0.008 0.007
1000 0.819 0.890 0.968 0.983 0.264 0.153 0.972 0.986 0.026 0.006 0.005
2000 0.843 0.919 0.974 0.988 0.252 0.117 0.978 0.990 0.026 0.005 0.003
Q3 500 0.734 0.847 0.949 0.976 0.300 0.121 0.959 0.980 0.022 0.008 0.006
1000 0.794 0.877 0.963 0.981 0.241 0.086 0.971 0.983 0.023 0.006 0.005
2000 0.843 0.914 0.973 0.987 0.171 0.057 0.979 0.989 0.023 0.005 0.003
Q4 500 0.714 0.832 0.946 0.974 0.275 0.123 0.963 0.981 0.030 0.008 0.007
1000 0.770 0.881 0.959 0.982 0.215 0.085 0.973 0.987 0.031 0.006 0.005
2000 0.796 0.917 0.966 0.987 0.195 0.058 0.978 0.991 0.032 0.005 0.003
Q5 500 0.751 0.841 0.952 0.975 0.098 0.047 0.956 0.976 0.039 0.008 0.006
1000 0.807 0.880 0.965 0.982 0.073 0.038 0.968 0.983 0.035 0.005 0.005
2000 0.849 0.914 0.974 0.987 0.053 0.021 0.976 0.987 0.039 0.005 0.004
Q6 500 0.686 0.817 0.941 0.968 0.134 0.063 0.953 0.973 0.063 0.014 0.008
1000 0.726 0.848 0.948 0.973 0.127 0.058 0.960 0.978 0.070 0.012 0.007
2000 0.748 0.896 0.953 0.982 0.120 0.032 0.966 0.984 0.063 0.009 0.004
  
Q-matrix N PMR AMR FPR TPR RMSEA
Stepwise BIC Stepwise BIC Stepwise BIC Stepwise BIC QW QStepwise QBIC
Q1 500 0.795 0.861 0.961 0.979 0.075 0.006 0.962 0.979 0.007 0.007 0.007
1000 0.875 0.913 0.978 0.987 0.032 0.004 0.978 0.987 0.005 0.006 0.005
2000 0.919 0.950 0.986 0.993 0.020 0.001 0.986 0.993 0.004 0.005 0.004
Q2 500 0.799 0.864 0.964 0.980 0.209 0.211 0.967 0.983 0.029 0.008 0.009
1000 0.877 0.916 0.979 0.988 0.108 0.110 0.980 0.990 0.030 0.006 0.006
2000 0.915 0.948 0.986 0.993 0.093 0.067 0.987 0.994 0.030 0.004 0.004
Q3 500 0.794 0.867 0.961 0.980 0.236 0.125 0.969 0.984 0.022 0.007 0.008
1000 0.854 0.910 0.974 0.987 0.170 0.069 0.979 0.989 0.025 0.006 0.006
2000 0.904 0.949 0.984 0.993 0.106 0.038 0.988 0.994 0.025 0.005 0.003
Q4 500 0.775 0.863 0.958 0.979 0.193 0.106 0.970 0.985 0.033 0.009 0.008
1000 0.840 0.912 0.972 0.987 0.136 0.062 0.981 0.991 0.033 0.007 0.006
2000 0.884 0.945 0.981 0.992 0.112 0.040 0.989 0.994 0.034 0.005 0.004
Q5 500 0.786 0.866 0.959 0.980 0.086 0.040 0.962 0.981 0.034 0.009 0.009
1000 0.859 0.911 0.975 0.987 0.053 0.023 0.977 0.988 0.036 0.006 0.006
2000 0.912 0.949 0.985 0.993 0.031 0.011 0.987 0.993 0.041 0.005 0.004
Q6 500 0.731 0.838 0.948 0.973 0.122 0.059 0.960 0.978 0.061 0.015 0.009
1000 0.784 0.885 0.959 0.980 0.104 0.039 0.970 0.983 0.066 0.010 0.007
2000 0.913 0.948 0.985 0.992 0.037 0.015 0.987 0.993 0.041 0.004 0.004
  
Item Code 类别 A1 A2 A3 A4 A5 A6 A7
1 M042041 1 0 1 0 0 0 0 0
2 M042024 1 0 1 0 0 0 0 0
3 M042016 1 1 0 0 0 0 0 1*#
4 M042002 1 1 0 0 0 0 0 0
5 M042198A 1 0 0 1 0 0 0 0*#
6 M042198B 1 0 0 1 0 0 0 0
7 M042198C 1 0 0 1 0 0* 0 0
8 M042077 1 1 0 0 1 0 0 0
9 M042235 1 0 0 0 1 0* 0 0
10 M042150 1 0 0 0 0 1 0 0
11 M042300Z 1 0 0 0 0 0 1 1
11 M042300Z 2 0 0 0 0 1 0 0
12 M042169A 1 0* 0 0 0 0 0 1
13 M042169B 1 0 0 0 0 0 0 1
14 M042169C 1 0* 0 0 0 0 0 1
15 M032352 1 1 0 1*# 0 0 0 1*
16 M032725 1 0 1* 0 0 0 0*# 0
17 M032738 1 0 0 0 1 0 0 0
18 M032295 1 0 0 0 1 0 0 0
19 M032331 1 0 0 0 0 1 1 0
20 M032679 1 0 0 0 0 1 1* 0
21 M032047 1 1 0 0 1*# 0 0 0
22 M032398 1 0* 0 0 0 1 0 0
23 M032424 1 0 0*# 0 1 0 0 0
  
Q Qoriginal QBIC QStepwise
Qoriginal 1
QBIC 0.92 1
QStepwise 0.96 0.95 1
  
Q 相对拟合指标 绝对拟合指标
-2*LL AIC BIC M2检验 RMSEA SRMSR
M2 df p
Qoriginal 18888.23 19274.23 20165.39 123.51 83 0.003 0.026 0.059
QBIC 18624.73 19014.73 19915.13 89.02 81 0.254 0.012 0.044
QStepwise 18757.88 19139.88 20021.88 89.90 85 0.337 0.009 0.050
  
Item Code 类别 A1 A2 A3 A4 A5 A6 A7 A8
1 M041052 1 1 1 0 0 0 0 0 0
2 M041281 1 0 1 1* 0 1* 0 0 0
3 M041275 1 1 0 0 0 0 1 0 1*
3 M041275 2 1* 0 0 0 0 1 0 1*
4 M031303 1 0 1 1 0 0 0 0 0
5 M031309 1 0 1 1 0 0 0 0 0
6 M031245 1 0 1 0 1 0 0 0 0
7 M031242A 1 0 1 1 0 1 0 0 0
7 M031242B 2 0 0 0 0 0 0 1 0
8 M031242C 1 0 1* 1* 0 1 0 1* 0
9 M031247 1 0 1* 1 1 0 0 0 0
9 M031247 2 0 1 1 1 0 0 0 0
10 M031173 1 0* 1* 1 0 0 0 0 0
11 M031172 1 1* 1* 0 0 0 1* 0 1
  
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