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基于选项层面的认知诊断非参数方法

本站小编 Free考研考试/2022-01-01
郭磊1,2(), 周文杰1
1西南大学心理学部, 重庆 400715
2中国基础教育质量监测协同创新中心西南大学分中心, 重庆 400715
收稿日期:2020-11-02出版日期:2021-09-25发布日期:2021-07-22
通讯作者:郭磊E-mail:happygl1229@swu.edu.cn

基金资助:国家自然科学基金青年项目(31900793);北京师范大学中国基础教育质量监测协同创新中心重大成果培育性项目(2019-06-023- BZPK01);中央高校基本科研业务费专项资金(SWU2109222)

Nonparametric methods for cognitive diagnosis to multiple-choice test items

GUO Lei1,2(), ZHOU Wenjie1
1Faculty of Psychology, Southwest University, Chongqing 400715, China
2Southwest University Branch, Collaborative Innovation Center of Assessment toward Basic Education Quality, Chongqing 400715, China
Received:2020-11-02Online:2021-09-25Published:2021-07-22
Contact:GUO Lei E-mail:happygl1229@swu.edu.cn






摘要/Abstract


摘要： 充分挖掘选择题(Multiple-Choice, MC)的诊断信息受到了较多关注, 将干扰项信息考虑在内可以提升诊断精度。为了弥补参数模型基于大样本才能获得可靠估计的不足, 以及适用于班级水平的小样本诊断测验情境, 本研究提出了非参数的多选题诊断方法。模拟和实证研结果表明：(1)当MC测验中题目参数不存在较大差异时, ${{d}_{\text{ph}-\text{MC}}}$法在多数情况下表现优于参数类诊断模型。(2)当MC测验中题目参数存在较大差异时, ${{d}_{ph-MC}}$法的表现最优。(3)实证研究中非参数方法和参数类模型的分类一致性程度较高, ${{d}_{\text{ph}-\text{MC}}}$距离法估计得到的考生属性总体掌握程度与总分相关最高。最后, 基于MC诊断测验的特点提出了若干研究方向。


表1选项编码的分数减法示例
$2\frac{4}{12}-\frac{7}{12}$ 属性
S1 S2 S3
A $2\frac{3}{12}$ √
B $2\frac{1}{4}$ √ √
C $1\frac{9}{12}$ √ √
D $1\frac{3}{4}$ √ √ √

表1选项编码的分数减法示例
$2\frac{4}{12}-\frac{7}{12}$ 属性
S1 S2 S3
A $2\frac{3}{12}$ √
B $2\frac{1}{4}$ √ √
C $1\frac{9}{12}$ √ √
D $1\frac{3}{4}$ √ √ √


表2MC题目中干扰项已编码的Q矩阵
属性 题目
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
A1 1 0 0 0 0 1 0 0 0 0 2 2 1 2 0 0 0 0 0 0 2 1 3 1 2 2 0 0 0 0
A2 0 1 0 0 0 0 1 0 0 0 1 0 0 0 2 1 1 0 0 0 2 2 1 0 0 0 2 2 2 0
A3 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 2 2 0 2 2 0 2
A4 0 0 0 1 0 0 0 0 1 0 0 0 2 0 0 2 0 2 0 1 0 2 0 2 0 2 2 0 2 2
A5 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 2 0 2 2 0 0 1 0 2 2 0 2 2 2

表2MC题目中干扰项已编码的Q矩阵
属性 题目
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
A1 1 0 0 0 0 1 0 0 0 0 2 2 1 2 0 0 0 0 0 0 2 1 3 1 2 2 0 0 0 0
A2 0 1 0 0 0 0 1 0 0 0 1 0 0 0 2 1 1 0 0 0 2 2 1 0 0 0 2 2 2 0
A3 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 2 2 0 2 2 0 2
A4 0 0 0 1 0 0 0 0 1 0 0 0 2 0 0 2 0 2 0 1 0 2 0 2 0 2 2 0 2 2
A5 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 2 0 2 2 0 0 1 0 2 2 0 2 2 2


表3两类诊断方法的模式判准率和属性判准率(真模型为MC1)
题目质量 题目数量 样本量 PCCR AACCR
${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ MC1 MC2
高质量 10 30 0.784 0.710 0.763 0.703 0.918 0.884 0.906 0.896
50 0.783 0.701 0.749 0.690 0.916 0.883 0.900 0.889
100 0.789 0.703 0.757 0.704 0.922 0.888 0.902 0.896
20 30 0.911 0.893 0.896 0.888 0.968 0.962 0.930 0.928
50 0.911 0.895 0.879 0.863 0.976 0.962 0.918 0.970
100 0.912 0.895 0.905 0.896 0.973 0.963 0.921 0.968
30 30 0.957 0.947 0.979 0.964 0.987 0.984 0.992 0.991
50 0.951 0.934 0.973 0.966 0.986 0.980 0.992 0.989
100 0.954 0.940 0.976 0.970 0.986 0.982 0.993 0.983
低质量 10 30 0.575 0.495 0.498 0.450 0.843 0.798 0.814 0.799
50 0.588 0.501 0.505 0.428 0.843 0.801 0.820 0.788
100 0.590 0.501 0.518 0.420 0.849 0.806 0.828 0.784
20 30 0.802 0.768 0.742 0.655 0.933 0.919 0.917 0.888
50 0.798 0.762 0.742 0.651 0.935 0.921 0.919 0.889
100 0.793 0.760 0.752 0.671 0.930 0.917 0.922 0.892
30 30 0.865 0.849 0.820 0.757 0.964 0.959 0.952 0.935
50 0.868 0.845 0.837 0.777 0.965 0.957 0.957 0.940
100 0.874 0.853 0.848 0.801 0.967 0.959 0.961 0.947

表3两类诊断方法的模式判准率和属性判准率(真模型为MC1)
题目质量 题目数量 样本量 PCCR AACCR
${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ MC1 MC2
高质量 10 30 0.784 0.710 0.763 0.703 0.918 0.884 0.906 0.896
50 0.783 0.701 0.749 0.690 0.916 0.883 0.900 0.889
100 0.789 0.703 0.757 0.704 0.922 0.888 0.902 0.896
20 30 0.911 0.893 0.896 0.888 0.968 0.962 0.930 0.928
50 0.911 0.895 0.879 0.863 0.976 0.962 0.918 0.970
100 0.912 0.895 0.905 0.896 0.973 0.963 0.921 0.968
30 30 0.957 0.947 0.979 0.964 0.987 0.984 0.992 0.991
50 0.951 0.934 0.973 0.966 0.986 0.980 0.992 0.989
100 0.954 0.940 0.976 0.970 0.986 0.982 0.993 0.983
低质量 10 30 0.575 0.495 0.498 0.450 0.843 0.798 0.814 0.799
50 0.588 0.501 0.505 0.428 0.843 0.801 0.820 0.788
100 0.590 0.501 0.518 0.420 0.849 0.806 0.828 0.784
20 30 0.802 0.768 0.742 0.655 0.933 0.919 0.917 0.888
50 0.798 0.762 0.742 0.651 0.935 0.921 0.919 0.889
100 0.793 0.760 0.752 0.671 0.930 0.917 0.922 0.892
30 30 0.865 0.849 0.820 0.757 0.964 0.959 0.952 0.935
50 0.868 0.845 0.837 0.777 0.965 0.957 0.957 0.940
100 0.874 0.853 0.848 0.801 0.967 0.959 0.961 0.947


表4两类诊断方法的模式判准率和属性判准率(真模型为MC2)
题目质量 题目数量 样本量 PCCR AACCR
${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ MC1 MC2.
高质量 10 30 0.772 0.700 0.746 0.697 0.915 0.884 0.904 0.896
50 0.781 0.700 0.747 0.701 0.917 0.880 0.900 0.893
100 0.788 0.705 0.753 0.705 0.921 0.889 0.903 0.897
20 30 0.907 0.888 0.887 0.888 0.966 0.961 0.935 0.967
50 0.909 0.892 0.884 0.905 0.965 0.959 0.923 0.972
100 0.911 0.896 0.886 0.916 0.967 0.961 0.923 0.971
30 30 0.953 0.938 0.960 0.976 0.985 0.980 0.991 0.991
50 0.949 0.938 0.966 0.973 0.985 0.981 0.989 0.992
100 0.952 0.936 0.972 0.973 0.986 0.981 0.987 0.993
低质量 10 30 0.566 0.501 0.490 0.424 0.835 0.798 0.807 0.787
50 0.580 0.493 0.497 0.424 0.841 0.797 0.815 0.786
100 0.593 0.501 0.516 0.422 0.847 0.803 0.823 0.786
20 30 0.787 0.752 0.723 0.642 0.931 0.917 0.915 0.886
50 0.793 0.761 0.744 0.656 0.930 0.917 0.917 0.889
100 0.792 0.762 0.754 0.666 0.931 0.918 0.921 0.892
30 30 0.872 0.849 0.830 0.759 0.964 0.957 0.954 0.935
50 0.873 0.846 0.844 0.777 0.965 0.956 0.959 0.940
100 0.873 0.848 0.849 0.797 0.965 0.956 0.959 0.945

表4两类诊断方法的模式判准率和属性判准率(真模型为MC2)
题目质量 题目数量 样本量 PCCR AACCR
${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ MC1 MC2.
高质量 10 30 0.772 0.700 0.746 0.697 0.915 0.884 0.904 0.896
50 0.781 0.700 0.747 0.701 0.917 0.880 0.900 0.893
100 0.788 0.705 0.753 0.705 0.921 0.889 0.903 0.897
20 30 0.907 0.888 0.887 0.888 0.966 0.961 0.935 0.967
50 0.909 0.892 0.884 0.905 0.965 0.959 0.923 0.972
100 0.911 0.896 0.886 0.916 0.967 0.961 0.923 0.971
30 30 0.953 0.938 0.960 0.976 0.985 0.980 0.991 0.991
50 0.949 0.938 0.966 0.973 0.985 0.981 0.989 0.992
100 0.952 0.936 0.972 0.973 0.986 0.981 0.987 0.993
低质量 10 30 0.566 0.501 0.490 0.424 0.835 0.798 0.807 0.787
50 0.580 0.493 0.497 0.424 0.841 0.797 0.815 0.786
100 0.593 0.501 0.516 0.422 0.847 0.803 0.823 0.786
20 30 0.787 0.752 0.723 0.642 0.931 0.917 0.915 0.886
50 0.793 0.761 0.744 0.656 0.930 0.917 0.917 0.889
100 0.792 0.762 0.754 0.666 0.931 0.918 0.921 0.892
30 30 0.872 0.849 0.830 0.759 0.964 0.957 0.954 0.935
50 0.873 0.846 0.844 0.777 0.965 0.956 0.959 0.940
100 0.873 0.848 0.849 0.797 0.965 0.956 0.959 0.945


表5题目质量存在较大差异时各方法的模式判准率和属性判准率
真模型 题目数量 样本量 PCCR AACCR
${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2
MC1 10 30 0.631 0.547 0.669 0.596 0.523 0.865 0.820 0.877 0.858 0.835
50 0.644 0.549 0.675 0.605 0.518 0.866 0.825 0.877 0.856 0.822
100 0.645 0.543 0.678 0.623 0.523 0.869 0.825 0.880 0.866 0.826
20 30 0.839 0.812 0.888 0.857 0.796 0.945 0.935 0.964 0.958 0.937
50 0.840 0.817 0.882 0.859 0.800 0.948 0.939 0.964 0.960 0.938
100 0.844 0.819 0.894 0.877 0.829 0.947 0.937 0.967 0.964 0.946
30 30 0.904 0.878 0.938 0.930 0.906 0.975 0.968 0.986 0.984 0.978
50 0.904 0.883 0.943 0.933 0.916 0.974 0.968 0.987 0.984 0.981
100 0.908 0.891 0.942 0.939 0.925 0.976 0.970 0.986 0.986 0.983
MC2 10 30 0.623 0.546 0.647 0.578 0.512 0.866 0.820 0.868 0.847 0.825
50 0.638 0.548 0.672 0.601 0.521 0.866 0.824 0.876 0.858 0.827
100 0.643 0.548 0.676 0.621 0.519 0.870 0.824 0.879 0.865 0.825
20 30 0.834 0.803 0.886 0.853 0.801 0.944 0.933 0.967 0.957 0.939
50 0.836 0.808 0.897 0.862 0.817 0.942 0.931 0.969 0.959 0.944
100 0.838 0.808 0.892 0.868 0.828 0.944 0.932 0.966 0.960 0.948
30 30 0.905 0.879 0.942 0.925 0.900 0.973 0.966 0.986 0.982 0.976
50 0.906 0.884 0.942 0.928 0.909 0.974 0.968 0.986 0.984 0.979
100 0.905 0.884 0.937 0.933 0.924 0.974 0.968 0.985 0.984 0.982

表5题目质量存在较大差异时各方法的模式判准率和属性判准率
真模型 题目数量 样本量 PCCR AACCR
${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{wh}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2
MC1 10 30 0.631 0.547 0.669 0.596 0.523 0.865 0.820 0.877 0.858 0.835
50 0.644 0.549 0.675 0.605 0.518 0.866 0.825 0.877 0.856 0.822
100 0.645 0.543 0.678 0.623 0.523 0.869 0.825 0.880 0.866 0.826
20 30 0.839 0.812 0.888 0.857 0.796 0.945 0.935 0.964 0.958 0.937
50 0.840 0.817 0.882 0.859 0.800 0.948 0.939 0.964 0.960 0.938
100 0.844 0.819 0.894 0.877 0.829 0.947 0.937 0.967 0.964 0.946
30 30 0.904 0.878 0.938 0.930 0.906 0.975 0.968 0.986 0.984 0.978
50 0.904 0.883 0.943 0.933 0.916 0.974 0.968 0.987 0.984 0.981
100 0.908 0.891 0.942 0.939 0.925 0.976 0.970 0.986 0.986 0.983
MC2 10 30 0.623 0.546 0.647 0.578 0.512 0.866 0.820 0.868 0.847 0.825
50 0.638 0.548 0.672 0.601 0.521 0.866 0.824 0.876 0.858 0.827
100 0.643 0.548 0.676 0.621 0.519 0.870 0.824 0.879 0.865 0.825
20 30 0.834 0.803 0.886 0.853 0.801 0.944 0.933 0.967 0.957 0.939
50 0.836 0.808 0.897 0.862 0.817 0.942 0.931 0.969 0.959 0.944
100 0.838 0.808 0.892 0.868 0.828 0.944 0.932 0.966 0.960 0.948
30 30 0.905 0.879 0.942 0.925 0.900 0.973 0.966 0.986 0.982 0.976
50 0.906 0.884 0.942 0.928 0.909 0.974 0.968 0.986 0.984 0.979
100 0.905 0.884 0.937 0.933 0.924 0.974 0.968 0.985 0.984 0.982


表6包含干扰项信息的大学英语高级英语阅读测验Q矩阵
题目1 题目2 题目3 题目4
A 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
B 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
C 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0
D 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0
题目5 题目6 题目7 题目8
A 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0
B 1 1 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
C 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0
D 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
题目9 题目10 题目11 题目12
A 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
B 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0
D 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
题目13 题目14 题目15
A 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0
B 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0
C 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
D 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0

表6包含干扰项信息的大学英语高级英语阅读测验Q矩阵
题目1 题目2 题目3 题目4
A 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
B 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
C 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0
D 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0
题目5 题目6 题目7 题目8
A 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0
B 1 1 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
C 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0
D 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
题目9 题目10 题目11 题目12
A 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
B 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0
D 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
题目13 题目14 题目15
A 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0
B 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0
C 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
D 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0


表7各模型间的分类一致性程度
指标 平均属性分类一致性指标
(AAR)		 模式分类一致性指标1
(PAR(K = 6))		 模式分类一致性指标2
(PAR(K ≥ 5))
${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2
${{d}_{\text{h}-\text{MC}}}$ 1 1 1
${{d}_{\text{ph}-\text{MC}}}$ 0.88 1 0.61 1 0.92 1
MC1 0.85 0.86 1 0.55 0.59 1 0.88 0.89 1
MC2 0.84 0.85 0.92 1 0.51 0.57 0.71 1 0.87 0.88 0.94 1

表7各模型间的分类一致性程度
指标 平均属性分类一致性指标
(AAR)		 模式分类一致性指标1
(PAR(K = 6))		 模式分类一致性指标2
(PAR(K ≥ 5))
${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2 ${{d}_{\text{h}-\text{MC}}}$ ${{d}_{\text{ph}-\text{MC}}}$ MC1 MC2
${{d}_{\text{h}-\text{MC}}}$ 1 1 1
${{d}_{\text{ph}-\text{MC}}}$ 0.88 1 0.61 1 0.92 1
MC1 0.85 0.86 1 0.55 0.59 1 0.88 0.89 1
MC2 0.84 0.85 0.92 1 0.51 0.57 0.71 1 0.87 0.88 0.94 1







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[1]汪文义;丁树良;宋丽红. 认知诊断中基于条件期望的距离判别方法[J]. 心理学报, 2015, 47(12): 1499-1510.





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