多维二参数Logistic项目反应模型的Gibbs抽样法

项目反应理论主要有3个基本假设：单维性,局部独立性和单调性。但是这3个假设存在一些弊端亟待解决。一些科学研究表明,用单维模型来模拟多维测量数据往往会增大测量误差,导致对学生的能力做出不正确的推论。因此,研究者基于各种不同的测验背景,将单维项目反应模型推广到多维项目反应模型。多维项目反应模型涉及到的参数较多,如果采用传统的估计方法,如边际最大似然法和贝叶斯众数估计法处理起来比较困难。然而,在项目反应理论中,Gibbs抽样法可以作为一种高效灵活的估计方法加以应用。基于Gibbs抽样的增加数据的技巧,给出了多维二参数Logistic项目反应模型的Bayes估计方法。随着潜在变量的引入,每个参数的满条件分布都很容易得到,并且不受先验分布选取的限制,这样Gibbs抽样的方法就很容易实施。

Gibbs sampling method in multidimensional two parameter logistic item response model

Item response theory（IRT）is based on three assumptions：unidimensionality,local independence and monotonicity.However these assumption has some defects need to be improved.Research shows that use unidimensional model to fit multidimensional data will increase measurement error and make wrong inference to students＇ability.Just because of this researchers extend the unidimensional IRT to the multidimensional IRT from different perspectives.Since the multidimensional model has more parameters need to be estimated,traditional methods such as marginal maximum likelihood and Bayes modal estimation procedures are not suitable.However,Gibbs sampler has a great potential to be an efficient and versatile estimation procedure in item response theory.In this article,based on a data augmentation scheme using the Gibbs sampler,we propose a Bayesian procedure to estimate the multidimensional two parameter logistic model（2PLM）.With the introduction of latent variable,the full conditional distributions are tractable,and consequently the Gibbs sampling is easy to implement for any prior assumptions.