2PL模型的两种马尔可夫蒙特卡洛缺失数据处理方法比较

马尔科夫蒙特卡洛（MCMC）是项目反应理论中处理缺失数据的一种典型方法。文章通过模拟研究比较了在不同被试人数，项目数，缺失比例下两种MCMC方法（M-H within Gibbs和DA-T Gibbs）参数估计的精确性，并结合了实证研究。研究结果表明，两种方法是有差异的，项目参数估计均受被试人数影响很大，受缺失比例影响相对更小。在样本较大缺失比例较小时，M-H within Gibbs参数估计的均方误差（RMSE）相对略小，随着样本数的减少或缺失比例的增加，DA-T Gibbs方法逐渐优于M-H within Gibbs方法

Comparison of Two MCMC Approaches to Missing Response Data in 2PL Model

Missing response data is common in educational assessment surveys. MCMC (Markov chain Monte Carlo) is a method of simulating random samples from any theoretical multivariate distribution-in particular, from the multivariate posterior distribution that is the focus of Bayesian inference. So it can obtain Bayes parameter estimation using the simulated sample, such as the mean of the simulated sample of posterior distribution can be used to estimate the EAP (expected a posteriori) of parameters. This algorithm is easy to implement when the IRT model is complex and the most important is that it can deal with missing data easily. In the past studies involving MCMC algorithm in IRT, researchers just compared MCMC method with other algorithms, no one compared different MCMC approaches. But in fact, there are vary MCMC algorithms, and some of them have been implemented in IRT. So, if the different MCMC approaches in IRT perform same is needed to be explored. This paper explored the relative performance of two different MCMC approaches: MH within Gibbs and DA-T Gibbs Sampler in the estimation of the two-parameter logistic (2PL) item parameters. Simulation studies and real data examples were used in the comparison. Within the simulation, the factor effects of sample size, test length and missing rate were investigated. We considered three different sample size (100, 500, 1000), two different test length (15, 40) and three different missing rate (0.05, 0.1, 0.25). So there are 18 combinatorial situations. In each situations, we generated 20 subject response matrix, used the two methods to estimate the item parameters and then used the index RMSE (root mean square error) to compare the two approaches. The simulation study results showed that the two MCMC approaches were indeed different in item parameter estimation. The parameter estimations of the two methods were both affected by sample size significantly, while the effect of the missing rate was relatively small. When the sample size is large and the missing rate is small, the RMSE of parameter estimation in MH within Gibbs is relatively small, and as the decreasing of sample size or the increasing of missing rate, DA-T Gibbs Sampler became better than MH within Gibbs. The results of real data example are consistent with the simulation results.