基于贝叶斯估计的Copula方法在项目反应理论中的应用

发布时间：2018-07-03 04:39

本文选题：项目反应理论(IRT) + Copula理论　；参考：《沈阳师范大学》2017年硕士论文

【摘要】：项目反应理论(IRT)作为一种现代教育和心理测量方法,在实际测量中的应用越来越广泛。通常我们在利用项目反应模型处理问题时,为了方便,总是假设给定同一被试的情况下,项目反应是独立且无关联的,但这与很多实际测验背景并不符合,影响了测量的准确性和可信度。有效地处理局部残差相依问题是项目反应模型得以应用的前提。随着现代统计学及数学的发展,处理局部相依问题的方法也在不断发展。常见的处理方法是额外添加随机效应因子,比如有学者提出了用题组项目反应模型。但是该方法建立的联合反应分布函数,会存在一些问题,例如边际分布的不可复制性,导致原有项目反应模型中的一些参数,比如题目难度参数、区分度参数失去具体意义,难以解释。本文主要借助Copula函数来解决项目反应理论中局部残差的问题。Copula函数作为一个新兴的连接函数,在金融领域中被广泛应用。针对多个边际反应分布建立模型,求出它们的联合分布,同时考虑各个边际分布间的相关性,解决了边际分布不可复制性及参数解释问题。对应建立起来的Copula函数就是各个边际的联合分布函数。在此基础上,通过MCMC估计方法,给出项目反应模型中的项目参数及Copula连接函数中相关系数的贝叶斯后验估计。主要利用统计软件R来模拟和分析数据,由Copula模型生成数据,然后调用R2WinBUGS软件包来得出模型的后验估计结果,相关程序代码见附录。通过选取Frank copula函数和Clayton copula函数,分析实际相依反应数据,得出结论,当忽略数据的相依性,假设项目反应理论的局部独立性建模时,带来的估计偏差比较大,这对选题及被试能力评估都会有较大的影响。
[Abstract]:The project response theory (IRT), as a modern educational and psychological measurement method, is becoming more and more widely used in actual measurement. Usually, when we use the project response model to deal with the problem, we always assume that the project reaction is independent and unrelated under the assumption that the same test is given, but this is not with many actual test background. It affects the accuracy and reliability of the measurement. It is the premise for the application of the project response model to deal with the local residual dependence effectively. With the development of modern statistics and mathematics, the methods to deal with the local dependent problems are also developing. The common treatment method is to add random effect factors, such as a scholar. There are some problems in the joint reaction distribution function established by this method, such as the non reproducibility of the marginal distribution, which leads to some parameters in the original project reaction model, such as the parameter of the title difficulty, the distinction parameter loses its specific meaning, which is difficult to explain. This paper mainly uses the Copula function to solve the problem. The.Copula function of the problem of partial residual in the theory of project response is widely used in the financial field as a new connection function. A model is established for multiple marginal reaction distribution, and their joint distribution is obtained. At the same time, the correlation between the marginal distributions is considered, and the problem of non reproducible distribution of marginal distribution and the problem of parameter interpretation are solved. The corresponding Copula function is the joint distribution function of each marginal. On this basis, the project parameters in the project response model and the Bayesian posterior estimation of the correlation coefficient in the Copula connection function are given by the MCMC estimation method. The data are simulated and analyzed by the statistical software R, and the data are generated from the Copula model, and then the data are generated by the Copula model. The R2WinBUGS software package is called to get the results of the posterior estimation of the model, and the related program code is shown in the appendix. By selecting the Frank copula function and the Clayton copula function, the actual dependent response data are analyzed, and the conclusion is drawn. When the dependency of the data is ignored, the estimation deviation is larger when the Bureau independence of the project reaction theory is modeled. This will have a greater impact on the topic selection and the ability assessment of the subjects.
【学位授予单位】：沈阳师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：F224

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