关键词: 项目反应理论, 三参数Normal-Ogive模型, SAEM算法

Abstract: 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 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 the NO model can be widely used in practice, it is necessary that a more efficient estimation approach is developed for the NO model, and this is the main work of our study. In this article, the 3-parameter NO (3PNO) model is revised to be a mixture model, and then a stochastic approximation EM algorithm is developed for calculating the marginalized maximum a posteriori estimation (MMAP) of the 3PNO model. The SAEM algorithm is an extension of the EM method, so it must be more efficient than the MCMC sampler which is commonly used for estimating NO model. Furthermore, the 3PNO model under the mixture modelling framework is the exponential distribution family, sufficient statistics exist for the item parameters, which also highly 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 is analyzed to display the practical application value of the 3PNO model with the SAEM algorithm. 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 propose some valuable suggestion for implementing the SAEM for the 3PNO model according to the results of simulation study. In the second simulation study, the MMAP\SAEM 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 MMAP\SAEM estimates were highly correlated with the same item characteristic values form 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 MMAP\SAEM estimates are accurate and highly 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 efficiency estimation method for the 3PNO model, and 3PNO model is superior to the 2PNO model. But, there are some important issues should be further studied: First, a SAEM algorithm should be proposed for estimating the multidimension NO model, because the multidimension test is commonly used in psychological and educational measurement. Second, in recent years the four-parameter IRT model is receiving more and more attentions and some studies have displayed that the four-parameter model is valuable for testing design, therefore we believe that it is very interesting to propose a SAEM algorithm for estimating the 4PNO model. Finally, the cognitive diagnostic modeling (CDM) in educational measurement has attracted much attention from researchers nowadays, but its applications have been lagged by the computational complexity of model estimation. So, it is great valuable to give a SAEM algorithm for calculating the CDM estimation.

Key words: item response theory, 3-parameter normal ogive model, SAEM algorithm