中小学生几何类比推理能力诊断评价中的理论与技术研究
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摘要
认知诊断(cognitive diagnosis)研究有助于人们更好地了解人类内部心理活动规律及其加工机制,实现对个体认知强项和弱项的诊断评估。在Embretson (1999)、Stout(2002)和Leighton & Gierl(2007)等研究者看来,认知诊断研究是一项新的课题,在心理测量实践中具有重要价值,必将成为21世纪新的测量范式。几何类比推理是以几何图形为材料的类比推理。它是测量非言语智力中经常使用的项目之一。利用认知诊断测量范式研究几何类比推理,有助于更好地理解几何类比推理的实质,有助于更好地了解学生几何类比推理能力的强项和弱项,以提升学生的推理能力。
本研究是使用HO-sRUM和HO-DINA两模型对中小学生几何类比推理能力进行认知诊断时的几个重要的理论和技术问题的探讨。首先,在文献调查基础上提出了“几何类比推理问题解决认知模型”,然后分别采用口语报告法和LLTM模型实测数据验证法对所得模型进行了验证。其次,在采用MCMC方法对HO-sRUM和HO-DINA模型进行参数估计的基础上,分析了两种模型的性能。具体包括三个子研究,一是在属性独立的条件下探讨HO-sRUM和HO-DINA模型在不同实验条件下参数估计精度;二是考察属性独立条件下HO-sRUM和HO-DINA在不同实验条件下的稳健性;三是考察属性层次关系下HO-sRUM模型和HO-DINA模型参数估计的返真性,进一步评价两个模型的统计性能。最后,采用HO-sRUM和HO-DINA模型对学生几何类比推理能力进行认知诊断,同时对四种诊断结果进行比较分析,并从模型数据拟合检验和诊断结果质量两个方面对四种诊断结果进行评价。研究表明:
(1)口语报告法和LLTM模型分析结果表明,学生对几何类比推理问题的解决主要涉及七个认知属性,即变换的知识。变换包括两大类变化,即空间移置和空间变形。移置具体包括位置移置、旋转、翻转,变形具体包括数量、大小、颜色、形状的变化。LLTM模型分析结果还表明,翻转变量对项目难度贡献率最大,其次是旋转,最小是数量。
(2)在各种实验条件下,HO-DINA、HO-sRUM两个模型的返真性都比较理想。在高项目—属性关联度和低水平失误、猜测参数下,模型参数估计精度更高;当更多的项目测量一个属性,属性分类准确性也更高。在使用HO-sRUM模型模拟数据、使用HO-DINA模型进行参数估计时,模型的返真性良好,表明HO-DINA模型具有一定的稳健性;当使用HO-DINA模型模拟数据而使用HO-sRUM模型估计参数时,在低复杂Q矩阵情形下,HO-sRUM模型具有一定的稳健性,但在高复杂Q矩阵下,HO-sRUM模型的稳健性比较差。对于四种属性层次关系,处于下层属性的边际判准率逐步下降;无结构型、线型、收敛型和发散型的边际判判准率都比较高,相对而言,发散型和无结构型的模式判准率比较高,但线型和收敛型的模式判准率比较低。
(3)结合数据拟合检验和认知诊断结果的有效性看,HO-sRUM模型诊断结果的质量最好。与HO-sRUM模型相比,HO-DINA模型估计的学生属性掌握比例更高。4-8年级的学生对几何类别推理问题的掌握情况比较良好,其中对属性A1和A4掌握的比较理想,对属性A2掌握的比较差。同时七个认知属性存在年级差异,六年级是学生几何类别推理能力发展的最快时期。学生所犯的认知错误主要有三类(0000000,1001000,1011111),而这些错误均与属性A2有关。
Cognitive diagnosis can help people better understand human internal psychological rules and its processing mechanism, and diagnose the strengths and weaknesses of individual cognition, and promote development of the individuals. Embretson(1999),Stout(2002) and Leighton & Gierl(2007) all think , cognitive diagnosis is new research topic, the actual value of cognitive diagnosis is also unceasingly apparent in practice,and cognitive diagnosis is the new testing paradigm in the 21st century. Geometric analogies are analogical reasoning based on geometric figure. The ability that Geometric analogy measured is at the core of general intelligence. Using cognitive diagnosis to research geometric analogical reasoning, it help to better understand the essence of geometric analogical reasoning, and help better understand students’analogical reasoning ability's strengths and weaknesses, and improve students' analogical reasoning ability.
This study discussed three important theoretical and technical problems during using HO-sRUM and HO– DINA to diagnose students’geometric analogical reasoning ability. First of all, we put forward cognitive model involved in solving geometric analogies based on the literature survey, and validate cognitive model by means of oral report method and LLTM respectively. Second, this study first estimate parameters of the two models using the higher-order approach given by de la Torre & Douglas(2004). In order to explore estimated precision and properties of HO-DINA and HO-sRUM, we use the Monte Carlo simulation method to simulate various situations to evaluate models performance. It include three researches, the first is to explore parameter estimated precision of HO-DINA and HO-sRUM in different experimental conditions while the attributes are independent. The second is to examine model robustness of HO-DINA and HO-sRUM in different experimental conditions while the attributes are independent. Third, we use Monte Carlo simulation method to investigate the influence of diagnosis accuracy based on different levels of item parameters under four attribute hierarchical structures. Finally, this study adopts HO-DINA and HO-sRUM model to evaluate the students’geometric analogical reasoning ability, and evaluated diagnostic result from model data fitting inspection and diagnosis of quality. Research shows that:
(1) Oral report method and LLTM model results show that cognitive attributes involved in solving geometric analogical reasoning test are seven types of transformations. Transformations include two kinds change, namely spatial displacement and spatial distortions. Spatial displacements include rotation, reflection, and exchange. Spatial distortions include size, shading, shape and number. LLTM model result also shows that cognitive component for the greatest impact on item difficulty is reflection, followed by rotation, the minimum is number.
(2) In all cases, the precision of item and person parameters is preferably great for HO-sRUM and HO-DINA. In high cognitive structure and low level slip and guess parameters condition, the precision of item and person parameters is higher. The results also show that, when the more items measure an attribute, the attribute classification accuracy is higher. At the same time, the result also shows, HO-DINA model has certain robustness, however, HO-sRUM model has certain robustness in low complex Q-matrix circumstances, but in highly complex Q-matrix circumstance, the precision of item and person parameters of HO-sRUM model is relatively poor, but it does not result in serious mistake. For four hierarchical structure, attribute marginal match ratio gradually decreases in lower strata. Attribute marginal match ratios between four hierarchical structure are high, convergence and linear’s pattern match ratio is lower than unstructure and divergence’s.
(3) From the data fitting inspection and quality of diagnostic results, the diagnostic result of HO-sRUM model is the best. The results of this study indicate that students’attribute master proportion that HO-DINA estimates than HO-sRUM’s . 4-8 grade students’mastery of 7 key attributes on geometric analogical reason is goodish in total. But there exists significant different effect between diffirent grades. The result also indicated that the sixth grade is the fastest development period for geometric analogical reasoning abilities. The cognitive error of students can be categorized into three types, all of them are related to attribute 2.
本研究是使用HO-sRUM和HO-DINA两模型对中小学生几何类比推理能力进行认知诊断时的几个重要的理论和技术问题的探讨。首先,在文献调查基础上提出了“几何类比推理问题解决认知模型”,然后分别采用口语报告法和LLTM模型实测数据验证法对所得模型进行了验证。其次,在采用MCMC方法对HO-sRUM和HO-DINA模型进行参数估计的基础上,分析了两种模型的性能。具体包括三个子研究,一是在属性独立的条件下探讨HO-sRUM和HO-DINA模型在不同实验条件下参数估计精度;二是考察属性独立条件下HO-sRUM和HO-DINA在不同实验条件下的稳健性;三是考察属性层次关系下HO-sRUM模型和HO-DINA模型参数估计的返真性,进一步评价两个模型的统计性能。最后,采用HO-sRUM和HO-DINA模型对学生几何类比推理能力进行认知诊断,同时对四种诊断结果进行比较分析,并从模型数据拟合检验和诊断结果质量两个方面对四种诊断结果进行评价。研究表明:
(1)口语报告法和LLTM模型分析结果表明,学生对几何类比推理问题的解决主要涉及七个认知属性,即变换的知识。变换包括两大类变化,即空间移置和空间变形。移置具体包括位置移置、旋转、翻转,变形具体包括数量、大小、颜色、形状的变化。LLTM模型分析结果还表明,翻转变量对项目难度贡献率最大,其次是旋转,最小是数量。
(2)在各种实验条件下,HO-DINA、HO-sRUM两个模型的返真性都比较理想。在高项目—属性关联度和低水平失误、猜测参数下,模型参数估计精度更高;当更多的项目测量一个属性,属性分类准确性也更高。在使用HO-sRUM模型模拟数据、使用HO-DINA模型进行参数估计时,模型的返真性良好,表明HO-DINA模型具有一定的稳健性;当使用HO-DINA模型模拟数据而使用HO-sRUM模型估计参数时,在低复杂Q矩阵情形下,HO-sRUM模型具有一定的稳健性,但在高复杂Q矩阵下,HO-sRUM模型的稳健性比较差。对于四种属性层次关系,处于下层属性的边际判准率逐步下降;无结构型、线型、收敛型和发散型的边际判判准率都比较高,相对而言,发散型和无结构型的模式判准率比较高,但线型和收敛型的模式判准率比较低。
(3)结合数据拟合检验和认知诊断结果的有效性看,HO-sRUM模型诊断结果的质量最好。与HO-sRUM模型相比,HO-DINA模型估计的学生属性掌握比例更高。4-8年级的学生对几何类别推理问题的掌握情况比较良好,其中对属性A1和A4掌握的比较理想,对属性A2掌握的比较差。同时七个认知属性存在年级差异,六年级是学生几何类别推理能力发展的最快时期。学生所犯的认知错误主要有三类(0000000,1001000,1011111),而这些错误均与属性A2有关。
Cognitive diagnosis can help people better understand human internal psychological rules and its processing mechanism, and diagnose the strengths and weaknesses of individual cognition, and promote development of the individuals. Embretson(1999),Stout(2002) and Leighton & Gierl(2007) all think , cognitive diagnosis is new research topic, the actual value of cognitive diagnosis is also unceasingly apparent in practice,and cognitive diagnosis is the new testing paradigm in the 21st century. Geometric analogies are analogical reasoning based on geometric figure. The ability that Geometric analogy measured is at the core of general intelligence. Using cognitive diagnosis to research geometric analogical reasoning, it help to better understand the essence of geometric analogical reasoning, and help better understand students’analogical reasoning ability's strengths and weaknesses, and improve students' analogical reasoning ability.
This study discussed three important theoretical and technical problems during using HO-sRUM and HO– DINA to diagnose students’geometric analogical reasoning ability. First of all, we put forward cognitive model involved in solving geometric analogies based on the literature survey, and validate cognitive model by means of oral report method and LLTM respectively. Second, this study first estimate parameters of the two models using the higher-order approach given by de la Torre & Douglas(2004). In order to explore estimated precision and properties of HO-DINA and HO-sRUM, we use the Monte Carlo simulation method to simulate various situations to evaluate models performance. It include three researches, the first is to explore parameter estimated precision of HO-DINA and HO-sRUM in different experimental conditions while the attributes are independent. The second is to examine model robustness of HO-DINA and HO-sRUM in different experimental conditions while the attributes are independent. Third, we use Monte Carlo simulation method to investigate the influence of diagnosis accuracy based on different levels of item parameters under four attribute hierarchical structures. Finally, this study adopts HO-DINA and HO-sRUM model to evaluate the students’geometric analogical reasoning ability, and evaluated diagnostic result from model data fitting inspection and diagnosis of quality. Research shows that:
(1) Oral report method and LLTM model results show that cognitive attributes involved in solving geometric analogical reasoning test are seven types of transformations. Transformations include two kinds change, namely spatial displacement and spatial distortions. Spatial displacements include rotation, reflection, and exchange. Spatial distortions include size, shading, shape and number. LLTM model result also shows that cognitive component for the greatest impact on item difficulty is reflection, followed by rotation, the minimum is number.
(2) In all cases, the precision of item and person parameters is preferably great for HO-sRUM and HO-DINA. In high cognitive structure and low level slip and guess parameters condition, the precision of item and person parameters is higher. The results also show that, when the more items measure an attribute, the attribute classification accuracy is higher. At the same time, the result also shows, HO-DINA model has certain robustness, however, HO-sRUM model has certain robustness in low complex Q-matrix circumstances, but in highly complex Q-matrix circumstance, the precision of item and person parameters of HO-sRUM model is relatively poor, but it does not result in serious mistake. For four hierarchical structure, attribute marginal match ratio gradually decreases in lower strata. Attribute marginal match ratios between four hierarchical structure are high, convergence and linear’s pattern match ratio is lower than unstructure and divergence’s.
(3) From the data fitting inspection and quality of diagnostic results, the diagnostic result of HO-sRUM model is the best. The results of this study indicate that students’attribute master proportion that HO-DINA estimates than HO-sRUM’s . 4-8 grade students’mastery of 7 key attributes on geometric analogical reason is goodish in total. But there exists significant different effect between diffirent grades. The result also indicated that the sixth grade is the fastest development period for geometric analogical reasoning abilities. The cognitive error of students can be categorized into three types, all of them are related to attribute 2.
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Jang, E. E. (2005). A validity narrative: effects of reading skills diagnosis on teaching and learning in the context of NG TOEFL. Unpublished doctoral dissertation, University of Illinois.
Jiang, H. (2002). Applications of computational statistics in congnitive diagnosis and IRT modeling. Unpublished doctoral dissertation, University of Illinois at Urbana-Champaign.
Johnson, J. F. (2006). Diagnosing skill mastery in the national assessment of educational progress: Applicaion of the fusion model. Unpublished doctoral dissertation, University of Alabama.
Junker, B. W. & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied psychological measurement, Vol. 25, No. 3, pp. 258-272.
Kairong, W.(2008). Investigating the domain of geometric inductive reasoning problems: A structural equation modeling analysis. Unpublished doctoraldissertation, Brigham Young University.
Kato, K. (2008). Improving efficiency of cognitive diagnosis by using diagnostic items and adaptive testing. Unpublished doctoral dissertation, University of Minnesota.
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Laura, R. N. & Barbara T.(1987).Cognitive constraints on ordering operations: the case of geometric analogical. Joural of experimental psychology: General. 1987, Vol. 116, No. 1, 50-67.
Leighton, J. P., Gierl, M & Hunka, S.E.(2004). The attribute hierarchy method for cognitive assessment: A variation on Tatsuoka’s rule-space approach. Journal of educational measurement, 41(3), 205-236.
Leighton, J. P., & Gierl M. J.(2007). Cognitive Diagnostic Assessment for Education: Theory and Applications. Cambridge University Press .The Edinburgh Building, Cambridge CB2 8RU, UK.
Maris, E. (1995). Psychometric latent response models. Psychometrika, 60(4), 523-547.
Maris, E.(1999). Estimating multiple classification latent class models. Psychometrika. Vol. 64, No. 2, 187-212.
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Mislevy, R.J.(1996). Test theory reconceived. Journal of Educational Measurement, 33,379-416.
Mislevy, R.J., Steinberg, L.S. & Almond, R. G.(2003). On the role of task model variables in assessment design. In S. Irvine & P. Kyullonene(Eds.), Generating items for cognitive tests: Theory and practice. Hillsdale, NJ: Erlbaum.
Mulholland ,T,Pellegrino, J. W. & Glaser, R. Components of geometric analogy solution. Cognitive Psychology, 1980.
Nichols, S. F. Chipman & R. L. Brennan (Eds.)(1995). Cognitively diagnostic assessment (pp. 1-18). Hillsdale, NJ, England: Lawrence Erlbaum Associates, Inc.
Patricia, A. A. & Victor L. W(.1987). Analogical reasoning in young children. Journal of educational psychology. Vol. 79, No.4, 401-408.
Patz, R. & Junker, B. (1999a), Application and extensions of MCMC in IRT: multiple item types, missing data, and rated responses. Journal of educational and behavioral statistics24, 342-366.
Patz, R. & Junker, B.(1999b). A straightforward approach to Markov Chain Monte Carlo methods for item response models. Journal of educational and behavioral statistics, 24, 146-178.
Rochelles, S. M. (2006). The development of a cognitive model to provide psychometrically sound and useful diagnostic information for a quantitative measure. Unpublished doctoral dissertation, Fordham University.
Sarah von Schrader (2006). On the feasibility of applying skills assessment models to achievement test data. Unpublished doctoral dissertation, University Of Iowa.
Seongah, Im (2007). Statistical consequeces of attribute misspecification in the rule space model. Unpublished doctoral dissertation, the executive committee of the graduate school of arts and sciences.
Sinharay, S.(2006). Model diagnostics for Bayesian networks. Jounal of educational and behavioral statistics. Vol. 31, No. 1, pp. 1-33.
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Sternberg , R J(1977).Component processes in analogical reasoning.Psychological Review.31:356—378.
Sternberg, R J(2000.).Practical intelligence in everyday 1ife.Cambridge university press.
Stout, W. (2002). Psychometrics: form practice to theory and back. Psychometric. Vol. 67, No.4, 485-518.
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Templin, J.L., (2004). Genatalized linear mixed proficiency models for cognitive diagnosis. Unpublished doctoral dissertation, University of Illinois.
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Won-Chan, L., Bradley A. H. & Robert L. B.(2002). Estimating consistency and accuracy indices for multiple classifications. Applied psychological measurement, Vol.26 No. 4,pp. 412-432.
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Yi-Hsin, Ch. (2006). Cognitively diagnostic examination of Taiwanese mathematics achievement on TIMSS-1999. Unpublished doctoral dissertation, Arizona state university.
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Jianbin, F. & Yanmei L. (2008). Cognitively diagnostic psychometric models: An integrative review. ETS research report.
Jang, E. E. (2005). A validity narrative: effects of reading skills diagnosis on teaching and learning in the context of NG TOEFL. Unpublished doctoral dissertation, University of Illinois.
Jiang, H. (2002). Applications of computational statistics in congnitive diagnosis and IRT modeling. Unpublished doctoral dissertation, University of Illinois at Urbana-Champaign.
Johnson, J. F. (2006). Diagnosing skill mastery in the national assessment of educational progress: Applicaion of the fusion model. Unpublished doctoral dissertation, University of Alabama.
Junker, B. W. & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied psychological measurement, Vol. 25, No. 3, pp. 258-272.
Kairong, W.(2008). Investigating the domain of geometric inductive reasoning problems: A structural equation modeling analysis. Unpublished doctoraldissertation, Brigham Young University.
Kato, K. (2008). Improving efficiency of cognitive diagnosis by using diagnostic items and adaptive testing. Unpublished doctoral dissertation, University of Minnesota.
Kuhn, D. (2001). Why development does (and does not) occur: Evidence from the domain of inductive reasoning. In J. L. McClelland & R. Siegler (Eds.), Mechanisms of cognitive development: Behavioral and neural perspectives (pp. 221–249). Hillsdale, NJ: Erlbaum.
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Leighton, J. P., Gierl, M & Hunka, S.E.(2004). The attribute hierarchy method for cognitive assessment: A variation on Tatsuoka’s rule-space approach. Journal of educational measurement, 41(3), 205-236.
Leighton, J. P., & Gierl M. J.(2007). Cognitive Diagnostic Assessment for Education: Theory and Applications. Cambridge University Press .The Edinburgh Building, Cambridge CB2 8RU, UK.
Maris, E. (1995). Psychometric latent response models. Psychometrika, 60(4), 523-547.
Maris, E.(1999). Estimating multiple classification latent class models. Psychometrika. Vol. 64, No. 2, 187-212.
Mislevy, R. J. (1993) Introduction. In N. Frederiksen, R.J. Mislevy, & I.I.Bejar. (Eds), Test theory for a new generation of tests (pp.ix-xii).Hillsdale, NJ: Lawrence Erlbaum.
Mislevy, R.J.(1996). Test theory reconceived. Journal of Educational Measurement, 33,379-416.
Mislevy, R.J., Steinberg, L.S. & Almond, R. G.(2003). On the role of task model variables in assessment design. In S. Irvine & P. Kyullonene(Eds.), Generating items for cognitive tests: Theory and practice. Hillsdale, NJ: Erlbaum.
Mulholland ,T,Pellegrino, J. W. & Glaser, R. Components of geometric analogy solution. Cognitive Psychology, 1980.
Nichols, S. F. Chipman & R. L. Brennan (Eds.)(1995). Cognitively diagnostic assessment (pp. 1-18). Hillsdale, NJ, England: Lawrence Erlbaum Associates, Inc.
Patricia, A. A. & Victor L. W(.1987). Analogical reasoning in young children. Journal of educational psychology. Vol. 79, No.4, 401-408.
Patz, R. & Junker, B. (1999a), Application and extensions of MCMC in IRT: multiple item types, missing data, and rated responses. Journal of educational and behavioral statistics24, 342-366.
Patz, R. & Junker, B.(1999b). A straightforward approach to Markov Chain Monte Carlo methods for item response models. Journal of educational and behavioral statistics, 24, 146-178.
Rochelles, S. M. (2006). The development of a cognitive model to provide psychometrically sound and useful diagnostic information for a quantitative measure. Unpublished doctoral dissertation, Fordham University.
Sarah von Schrader (2006). On the feasibility of applying skills assessment models to achievement test data. Unpublished doctoral dissertation, University Of Iowa.
Seongah, Im (2007). Statistical consequeces of attribute misspecification in the rule space model. Unpublished doctoral dissertation, the executive committee of the graduate school of arts and sciences.
Sinharay, S.(2006). Model diagnostics for Bayesian networks. Jounal of educational and behavioral statistics. Vol. 31, No. 1, pp. 1-33.
Snow,R.E.(1984).The topography of ability and learning correlations.In R.J.Sternberg(Eds.),Advances in the psychology of human intelligence(PP. 47—103).Hillsdale,NJ:Erlbaum.
Snow, R.J. & Lohman, D.F. (1989). Implications of cognitive psychology for educational measurement. InR. Linn(Eds).Educational measurement (pp.263-331).New York: Macmillan.
Spearman, C.(1923). The nature of intelligence and the principles of cognition. London: McMillan and Company.
Stella, V.(1995). Analogical reasoning in cognitive development. Metaphor and symbolic activity 10(4), 297-308.
Sternberg , R J(1977).Component processes in analogical reasoning.Psychological Review.31:356—378.
Sternberg, R J(2000.).Practical intelligence in everyday 1ife.Cambridge university press.
Stout, W. (2002). Psychometrics: form practice to theory and back. Psychometric. Vol. 67, No.4, 485-518.
Tatsuoka, K.K.(1990). Toward an integration of item response theory and cognitive error diagnosis. In N. Frederiksen. R. Glaser. A. Lesgold. & M. G. Shafto(Eds.). Dignostic monitoring of skills and knowledge acquistition(pp.453-488). Hillsdale, NJ: Erlbaum.
Tatsuoka, K. K. (1995). Architecture of knowledge structures and cognitive diagnosis: A statistical pattern recognition and classification approach. In P. D. Nichols, S. F.
Tatsuoka, K. K. (2009). Cognitive assessment: an introduction to the rule space method. Routledge, Taylor & Francis group, New York London.
Templin, J., Roussos, L., & Stout, W. (2003). An extension of the current fusion model to treat polytomous attributes (Draft Technical Report). Urbana-Champaign, IL: University of Illinois at Urbana-Champaign.
Templin, J.L., (2004). Genatalized linear mixed proficiency models for cognitive diagnosis. Unpublished doctoral dissertation, University of Illinois.
Templin, J. L. Robert A. Henson & Sara E. T. (2008). Robustness of hierarchical modeling of skill association in cognitive diagnosis model. Applied psychological measurement, Vol. 32, No. 7, 559-574.
Thurston, L.L.(1938). Primary mental abilities. Chicago: University of Chocago press.
Tomoko, Y.(2008). Comparison of cognitively diagnostic adaptive testing algorithms. Unpublished doctoral dissertation, Columbia University.
Whitely, S. E. & Schneider L., M.(1981).Information structure for geometric analogies: A Test Theory Approach. Applied psychological measurement Vol. 5, No.3, pp.383-397.
Won-Chan, L., Bradley A. H. & Robert L. B.(2002). Estimating consistency and accuracy indices for multiple classifications. Applied psychological measurement, Vol.26 No. 4,pp. 412-432.
Xiangdong, Y. (2003). Inferring diagnostic information from abstract reasoning test item. Unpublished doctoral dissertation, University of Kansas.
Yi-Hsin, Ch. (2006). Cognitively diagnostic examination of Taiwanese mathematics achievement on TIMSS-1999. Unpublished doctoral dissertation, Arizona state university.
Ying, L., Douglas, J. A. & Henson, R. A. (2009). Testing person fit in cognitive diagnosis. Applied psychological measurement.