三参数Normal-Ogive模型参数估计的SAEM算法

A Stochastic Approximation EM Algorithm for the Estimation of Item Parameters in the 3-Parameter Normal-Ogive Model

Normal-Ogive模型是项目反应理论的代表性模型之一,其参数估计主要是基于MCMC抽样实现的,但MCMC抽样的不足是计算效率较低。针对这一问题,本文以混合模型(mixture model)的视角,通过变量扩充,提出三参数normalogive(3PNO)模型题目参数估计的随机逼近EM(stochastic approximation EM,简称SAEM)算法,并通过Monte Carlo模拟对SAEM算法的主要影响因素、计算效率、估计的返真性进行验证。模拟研究的结果表明:SAEM算法能够准确实现3PNO模型题目参数估计的计算,并且具有较高的计算效率,表现出优良的计算性质。

The normal ogive(NO)model is the first item response theory(IRT)model,which was developed by Lord(1953).However,the NO model has not been widely used in psychological and educational measurement since the estimation of parameters is of great low efficiency.The NO model derives from the assumption of normally distributed measurement error and is theoretically appealing on that basis.Recently a lot of the frontier IRT models were developed based on the NO model,for instance,the multilevel IRT model and the response time models.Therefore,to make sure the NO model can be applied in practice,a more efficient estimation approach must be developed for the NO model,which is the main work of our study.In this study,the 3-parameter NO(3PNO)model was revised to be a mixture model,and a stochastic approximation EM algorithm was developed for calculating the marginalized maximum a posteriori estimation(MMAP)of the 3PNO model.As an extension of the EM method,the SAEM algorithm must be more efficient than the MCMC sampler which is commonly used for estimating the NO model.Furthermore,the 3PNO model under the mixture modeling framework is the exponential distribution family.Sufficient statistics exist for the item parameters,which also greatly simplified the SAEM algorithm.To investigate the computation efficiency and the impact factors of the SAEM algorithm,two Monte Carlo simulation studies were constructed.Finally,an empirical example was analyzed to display the practical application value of the 3PNO model with the SAEMalgorithm.The results from the first simulation study demonstrated that the step size is very important for the performance of SAEM iteration.To ensure the SAEM algorithm is used accurately,we proposed some valuable suggestions for implementing the SAEM for the 3PNO model according to the results of the simulation study.In the second simulation study,the MMAPISAEM estimates displayed excellent accuracy,and it is greatly faster than the Gibbs sampler.Finally,the results of the empirical study are that the values of MMAPISAEM estimates were highly correlated with the same item characteristic values from classical test theory,furthermore,they were stronger positively correlated with the EAP estimates obtained by the MCMC samplers.Therefore,it can be concluded that the MMAPISAEM estimates are accurate and have high reliability.Furthermore,the fit of the 3PNO model is better than that of the 2PNO model for this real data.According to the results from both the simulation and the empirical studies,it can be concluded that the SAEM algorithm given by us is an accurate and efficient estimation method for the 3PNO model,which is superior than the 2PNO model.But,some important issues should be further clarified.First,an SAEM algorithm should be proposed for estimating the multidimensional NO model,because the multidimensional test is commonly used in psychological and educational measurement.Second,in recent years the four-parameter IRT model is receiving increasing attention and some studies have shown that the four-parameter model is valuable for testing design.Therefore,it may be interesting to propose an SAEM algorithm to estimate the 4PNO model.Finally,cognitive diagnostic modeling(CDM)in educational measurement has attracted much attention from researchers nowadays.However,its applications have been lagged by the computational complexity of model estimation.So,it is greatly valuable to give an SAEM algorithmforcalculatingtheCDM estimation.