认知诊断模型中项目参数的方差-协方差矩阵估计方法比较:Bootstrap与解析法

A Comparison of Item Parameter Variance-Covariance Matrix Estimation Procedures in Cognitive Diagnostic Models: Analytical and Bootstrap Methods

认知诊断模型中,项目参数的方差-协方差矩阵具有很重要的作用。作为一种非参数化的方差-协方差矩阵估计方法,Bootstrap法的一个主要优势在于它不需要解析推导。比较认知诊断模型中基于解析法的经验交叉相乘信息矩阵、观察信息矩阵和三明治协方差矩阵法,与Bootstrap法在估计项目参数标准误时的表现,模拟结果显示,认知诊断模型及Q矩阵正确设定或是模型中错误设定较少时,解析法的表现优于Bootstrap法,只有在样本量N=5000的条件下,Bootstrap法的表现才基本与解析法接近;当模型中错误设定较多时,Bootstrap法也没有表现出明显的稳健性。因此,在认知诊断模型中,推荐使用基于解析法的方差-协方差矩阵估计方法,尤其是三明治协方差矩阵法;当没有现成的基于解析法的方差-协方差矩阵估计方法可用时,Bootstrap法可以作为一种粗略的估计方法使用,尤其是在样本量较小的情况下。

The item parameter variance-covariance matrix plays an important role in cognitive diagnostic models. As a non-parametric variance-covariance matrix estimation method,one of the main advantages of the Bootstrap method is that it does not require analytical derivation. The performance of the empirical cross-product information matrix,observed information matrix and sandwich covariance matrix based on the analytical method,and the bootstrap method with respect to the item parameter standard error calculation were compared. The simulation results indicated that the analytical method is better than the Bootstrap method when the cognitive diagnosis model and Q matrix are correctly set or the error setting is small in the model;only under the condition of sample size N=5000,the performance of Bootstrap method is basically close to the analytical method. The Bootstrap method also showed no obvious robustness when there were more errors in the model.Therefore,in the cognitive diagnosis model,it is recommended to use the analytic-based variance-covariance matrix estimation method,especially the sandwich covariance matrix method;when there is no ready-made analytic-based variance-covariance matrix estimation method available,the Bootstrap method can be used as a rough estimation method,especially if the sample size is small.