粗糙集理论在认知诊断中的应用
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摘要
认知诊断通过对学生的知识结构与认知加工技能进行评估，向他们提供更好的教学指导，因而，课堂测验被认为是应用认知诊断的理想场所。要真正发挥认知诊断应有的功效，现有的认知诊断模型仍然有诸多难题需要克服。首先，现有认知诊断计量模型大多采用概率统计方法，在项目参数未知的条件下对大样本的依赖，使它们必须有较高昂的费用才能融入日常教学；其次，及时反馈甚至当堂反馈是发挥认知诊断补救性教学功能的必要条件，当前有经验的教师凭借教学经验可以进行有效的课堂评估，但我国发展不平衡，师资力量不齐整，认知诊断是解决这一问题的一个办法。但基于模型的认知诊断目前主要应用于大型测验，计算复杂，测验与反馈之间的时滞较长，未能对补救性教学产生实际效应，无法真正发挥促进教学的作用，有违认知诊断的本意。在项目参数已知条件下，虽然采用计算机自适应认知诊断测验，可及时反馈结果，但建立题库费用昂贵、周期性长，且涉及到项目参数等值、题目曝光不均匀等问题，使它推广起来很不方便，甚至在有的测验中被禁用；再者，当属性个数较多时，现有的认知诊断模型计算较为复杂，现有的文献中，也很少看到处理属性数超过10个的情况，而在实际中，属性个数多的情况很常见，这为认知诊断的应用带来了一定的限制。因此，寻求新的认知诊断方法解决无项目参数、被试量少、属性个数多、无需等值、需要及时反馈等问题是非常必要的。
作为处理不确定性知识的数学工具之一，粗糙集理论可以解决认知诊断中知识粒度大小引起的不确定性。它无需先验知识，可以导出问题的决策或分类规则，对研究对象进行分类。本文旨在将粗糙集理论应用于认知诊断，克服已有认知诊断方法对样本量的依赖和诊断费时两大难题。因为将粗糙集理论应用于认知诊断是一个全新的课题，它应用于认知诊断是否有效，效果如何，能处理认知诊断中的什么问题都值得探讨，所以本文分六个部分依次展开：首先，将目前常用的DINA模型作为比较对象，在不同属性个数、不同属性层级结构和不同可达阵个数条件下，系统研究粗糙集理论应用的可行性；其次，针对属性个数较多时判准率低的情况，研究二提出属性组块方法，并在属性个数高达10和11个的情况下采用粗糙集理论进行属性组块研究，探讨属性组块及属性组块方式对判准率的影响；第三，课堂测验是认知诊断的一个理想场所，为将认知诊断融入课堂测验之中，在研究一的基础上，研究三进一步将粗糙集理论用于小样本小题量的情况，考察样本量小和题量小对诊断结果的影响；第四，项目属性标定和被试知识状态诊断是相辅相成的两个方面。到目前为止，只有计算机化认知诊断自适应测验才能处理项目属性辅助标定问题，研究四采用粗糙集理论对纸笔测验中的项目属性进行自动标定，并从被试估计准确度、被试作答失误率和属性个数三个方面考察对项目属性标定准确度的影响；第五，一题多解是问题解决中的常态，而现有的认知诊断方法基本上讨论仅仅采用一种策略的情形。研究五采用粗糙集理论进行多策略认知诊断并与已有多策略的研究结果进行比较；第六，上述五个部分皆为模拟研究，研究六旨在在此基础上，将粗糙集理论用于认知诊断的实践之中，考察粗糙集理论在实际测验中的效果。
以上研究结果表明：
（1）在无项目参数条件下，采用粗糙集理论做认知诊断与被试量无关、估计的速度非常快且诊断结果比较理想，说明粗糙集理论能够应用于课堂测验。当属性数少于6个时，粗糙集理论的模式判准率结果比DINA好，当属性数为6个或以上时，有些情况下，粗糙集理论的模式判准率低一些。
（2）在属性个数比较多时，采用组块方法能够提高模式判准率，所需题量也大大减少，同时对属性超过10个的情况也有相同的结论。
（3）采用粗糙集理论做认知诊断时，不需项目参数，样本的大小对被试知识状态判准率的影响并不明显，且估计结果稳定。
（4）粗糙集理论能有效地进行项目属性自动标定。当作答失误较低、考察属性数较少、被试知识状态估计较准时，采用粗糙集理论进行纸笔测验的项目属性自动标定，项目属性模式判准率和属性边际判准率较高；当被试估计准确度低、或作答的失误率高或属性个数多时，项目属性模式判准率和属性边际判准率会降低。
（5）采用粗糙集理论进行多策略研究，研究结果与已有的多策略认知诊断结果基本一致。
（6）根据粗糙集理论进行诊断原理，该方法完全可以用于建立认知诊断题库，且所有项目只需提供项目属性，无需项目参数，故有关模型的拟合和等值等问题完全可以避免，可大量节省建立题库的成本。
以上研究，均在粗糙集软件环境下进行，无论被试量和题量为多大，估计速度非常快，大概10秒之内便可出结果，显示出粗糙集理论做认知诊断的强大优势。
Cognitive Diagnosis (CD) is designed to measure specific knowledge structuresand processing skills in students so as to provide information about their cognitivestrengths and weaknesses. In this way, classroom testing is regarded as an ideal areato develop CDs. To cognitive diagnosis really play a role, cognitive diagnosis modelsexisting still have many difficulties to overcome.First, most cognitive diagnosismodels are based on probability models which need a big sample in estimating itemparameters, so it is difficult to integrate into daily teaching for them. Second, timelyfeedback is a necessary condition to play CD remedial teaching function,so far, CDare mainly used in lager-scale examination. In these examinations, more sophisticatedpsychometric procedures are developed and require more complex estimationmethods (e.g., Markov Chain Monte Carlo estimation), thus affecting data processingand, ultimately, reporting times. Many large-scale assessment results are releasedmonths after the tests are administered, and are not useful for promoting teaching.Under the item parameters known conditions, although Cognitive DiagnosticComputerized Adaptive Testing(CD-CAT) can timely feedback results, theestablishment of item bank is expensive, long periodicity, and item exposure andother issues, it is not convenient to popularize.Therefore,it is very necessary to find anew method when item parameters are unknown, examinees are less and feedbacksare timely.
We apply a new method–Rough Set Theory(RST) in cognitive diagnosis. It cansolve the size of knowledge granularity caused uncertainty in CD. It requires no apriori knowledge, through the knowledge reduction, induce decision or classificationrules, the object can be classify.This paper aims at the application of rough set theoryin CD. This paper is divided into six parts.Research1is effective research with CDbased on RST, we will verificate the application of RST in CD by designing differentsituations such as different number of reachability matrix, different kinds ofhierarchical structures, different number of cognitive attribute. Research2is themethod of attribute-chunk,it discuss whether attribute-chunk will affect the correctrate. Research3is concerned on RST will be used to study the small sample. Research4is item attribute identification for paper and pen testing, Research5isabout multiple-strategies CD and compared with the existing method. Research6isempirical research.
The results show that:
(1)In the absence of item parameters, the rough set theory of CD has no business withsample size, fast diagnostic speed and good results.It shows that RST can be appliedto classroom tests.When the number of attributes is less than6,the diagnostic resultsare satisfied,on the other hand,is not.
(2)by the method of attribute-chunk, we will premote the Pattern Match Ration(PMR)under the condition of large number of attributes, the number of required items isgreatly reduced.
(3) In the item with unknown parameters, using of cognitive diagnosis of rough settheory, the sample size has no significant effect to PMR and the estimated results arestable.
(4) Application RST to item attribute identification,when examinee’s PMR is lower,the failure rate is high and the number of attributes is much, item attributeidentification’s PMR is low.
(5) RST is used for multiple strategies, results are agree with the existing research.
(6) Based on rough set theory, This method can be used to establish the cognitivediagnostic item bank, in which all items only need to provide item attributes and noitem parameters, so the model fitting and equating can be avoided completely,and thecost of the establishment can be significantly saved.
The above research,both results are estimated by rough set software,regardless ofsample size and item number,estimated speed is very fast,about10seconds.It showthe powerful advantage of RST CD.
作为处理不确定性知识的数学工具之一，粗糙集理论可以解决认知诊断中知识粒度大小引起的不确定性。它无需先验知识，可以导出问题的决策或分类规则，对研究对象进行分类。本文旨在将粗糙集理论应用于认知诊断，克服已有认知诊断方法对样本量的依赖和诊断费时两大难题。因为将粗糙集理论应用于认知诊断是一个全新的课题，它应用于认知诊断是否有效，效果如何，能处理认知诊断中的什么问题都值得探讨，所以本文分六个部分依次展开：首先，将目前常用的DINA模型作为比较对象，在不同属性个数、不同属性层级结构和不同可达阵个数条件下，系统研究粗糙集理论应用的可行性；其次，针对属性个数较多时判准率低的情况，研究二提出属性组块方法，并在属性个数高达10和11个的情况下采用粗糙集理论进行属性组块研究，探讨属性组块及属性组块方式对判准率的影响；第三，课堂测验是认知诊断的一个理想场所，为将认知诊断融入课堂测验之中，在研究一的基础上，研究三进一步将粗糙集理论用于小样本小题量的情况，考察样本量小和题量小对诊断结果的影响；第四，项目属性标定和被试知识状态诊断是相辅相成的两个方面。到目前为止，只有计算机化认知诊断自适应测验才能处理项目属性辅助标定问题，研究四采用粗糙集理论对纸笔测验中的项目属性进行自动标定，并从被试估计准确度、被试作答失误率和属性个数三个方面考察对项目属性标定准确度的影响；第五，一题多解是问题解决中的常态，而现有的认知诊断方法基本上讨论仅仅采用一种策略的情形。研究五采用粗糙集理论进行多策略认知诊断并与已有多策略的研究结果进行比较；第六，上述五个部分皆为模拟研究，研究六旨在在此基础上，将粗糙集理论用于认知诊断的实践之中，考察粗糙集理论在实际测验中的效果。
以上研究结果表明：
（1）在无项目参数条件下，采用粗糙集理论做认知诊断与被试量无关、估计的速度非常快且诊断结果比较理想，说明粗糙集理论能够应用于课堂测验。当属性数少于6个时，粗糙集理论的模式判准率结果比DINA好，当属性数为6个或以上时，有些情况下，粗糙集理论的模式判准率低一些。
（2）在属性个数比较多时，采用组块方法能够提高模式判准率，所需题量也大大减少，同时对属性超过10个的情况也有相同的结论。
（3）采用粗糙集理论做认知诊断时，不需项目参数，样本的大小对被试知识状态判准率的影响并不明显，且估计结果稳定。
（4）粗糙集理论能有效地进行项目属性自动标定。当作答失误较低、考察属性数较少、被试知识状态估计较准时，采用粗糙集理论进行纸笔测验的项目属性自动标定，项目属性模式判准率和属性边际判准率较高；当被试估计准确度低、或作答的失误率高或属性个数多时，项目属性模式判准率和属性边际判准率会降低。
（5）采用粗糙集理论进行多策略研究，研究结果与已有的多策略认知诊断结果基本一致。
（6）根据粗糙集理论进行诊断原理，该方法完全可以用于建立认知诊断题库，且所有项目只需提供项目属性，无需项目参数，故有关模型的拟合和等值等问题完全可以避免，可大量节省建立题库的成本。
以上研究，均在粗糙集软件环境下进行，无论被试量和题量为多大，估计速度非常快，大概10秒之内便可出结果，显示出粗糙集理论做认知诊断的强大优势。
Cognitive Diagnosis (CD) is designed to measure specific knowledge structuresand processing skills in students so as to provide information about their cognitivestrengths and weaknesses. In this way, classroom testing is regarded as an ideal areato develop CDs. To cognitive diagnosis really play a role, cognitive diagnosis modelsexisting still have many difficulties to overcome.First, most cognitive diagnosismodels are based on probability models which need a big sample in estimating itemparameters, so it is difficult to integrate into daily teaching for them. Second, timelyfeedback is a necessary condition to play CD remedial teaching function,so far, CDare mainly used in lager-scale examination. In these examinations, more sophisticatedpsychometric procedures are developed and require more complex estimationmethods (e.g., Markov Chain Monte Carlo estimation), thus affecting data processingand, ultimately, reporting times. Many large-scale assessment results are releasedmonths after the tests are administered, and are not useful for promoting teaching.Under the item parameters known conditions, although Cognitive DiagnosticComputerized Adaptive Testing(CD-CAT) can timely feedback results, theestablishment of item bank is expensive, long periodicity, and item exposure andother issues, it is not convenient to popularize.Therefore,it is very necessary to find anew method when item parameters are unknown, examinees are less and feedbacksare timely.
We apply a new method–Rough Set Theory(RST) in cognitive diagnosis. It cansolve the size of knowledge granularity caused uncertainty in CD. It requires no apriori knowledge, through the knowledge reduction, induce decision or classificationrules, the object can be classify.This paper aims at the application of rough set theoryin CD. This paper is divided into six parts.Research1is effective research with CDbased on RST, we will verificate the application of RST in CD by designing differentsituations such as different number of reachability matrix, different kinds ofhierarchical structures, different number of cognitive attribute. Research2is themethod of attribute-chunk,it discuss whether attribute-chunk will affect the correctrate. Research3is concerned on RST will be used to study the small sample. Research4is item attribute identification for paper and pen testing, Research5isabout multiple-strategies CD and compared with the existing method. Research6isempirical research.
The results show that:
(1)In the absence of item parameters, the rough set theory of CD has no business withsample size, fast diagnostic speed and good results.It shows that RST can be appliedto classroom tests.When the number of attributes is less than6,the diagnostic resultsare satisfied,on the other hand,is not.
(2)by the method of attribute-chunk, we will premote the Pattern Match Ration(PMR)under the condition of large number of attributes, the number of required items isgreatly reduced.
(3) In the item with unknown parameters, using of cognitive diagnosis of rough settheory, the sample size has no significant effect to PMR and the estimated results arestable.
(4) Application RST to item attribute identification,when examinee’s PMR is lower,the failure rate is high and the number of attributes is much, item attributeidentification’s PMR is low.
(5) RST is used for multiple strategies, results are agree with the existing research.
(6) Based on rough set theory, This method can be used to establish the cognitivediagnostic item bank, in which all items only need to provide item attributes and noitem parameters, so the model fitting and equating can be avoided completely,and thecost of the establishment can be significantly saved.
The above research,both results are estimated by rough set software,regardless ofsample size and item number,estimated speed is very fast,about10seconds.It showthe powerful advantage of RST CD.
引文
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Cheng, Y.(2010). Improving Cognitive Diagnostic Computerized AdaptiveTesting by Balancing Attribute Coverage: The Modified MaximumGlobal Discrimination Index Method. Educational and PsychologicalMeasurement,70(6),902–913.
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Cui, Y., Gierl, M. J.,&Chang, H.-H.(2012). Estimating ClassificationConsistency and Accuracy for Cognitive Diagnostic Assessment.Journal of Educational Measurement,49(1),19–38.
Cui, Y.,&Leighton, J. P.(2009). The hierarchy consistency index:Evaluating person fit for cognitive diagnostic assessment. Journalof Educational Measurement,46(4),429–449.
de la Torre, J.(2009). DINA Model and Parameter Estimation:A Didactic.Journal of Educational and Behavioral Statistics,34(1),115-130.
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de la Torre, J.,&Douglas, J. A.(2004). Higher-order latent trait modelsfor cognitive diagnosis Psychometrika,69,333-353.
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DeCarlo, L. T.(2011). On the Analysis of Fraction Subtraction Data: TheDINA Model, Classification, Latent Class Sizes, and the Q-Matrix.Applied Psychological Measurement,35(1),8–26.
DeCarlo, L. T.(2012). Recognizing Uncertainty in the Q-Matrix via aBayesian Extension of the DINA Model. Applied PsychologicalMeasurement,36(6),447–468.
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