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首页> 外文期刊>Educational and Psychological Measurement >Numerical Differentiation Methods for Computing Error Covariance Matrices in Item Response Theory Modeling: An Evaluation and a New Proposal
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Numerical Differentiation Methods for Computing Error Covariance Matrices in Item Response Theory Modeling: An Evaluation and a New Proposal

机译：项目响应理论建模中计算误差协方差矩阵的数值微分方法：一项评估和一项新建议

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摘要

In item response theory (IRT) modeling, the item parameter error covariance matrix plays a critical role define? in statistical inference procedures. When item parameters are estimated using the EM algorithm, the parameter error covariance matrix is not an automatic by-product of item calibration. Cai proposed the use of Supplemented EM algorithm for computing the item parameter error covariance matrix. This method has been subsequently implemented in commercial IRT software programs such as IRTPRO and flexMIRT. Jamshidian and Jennrich noted that Supplemented EM is among a class of methods based on numerically differentiating the EM map, and they proposed noniterative alternatives, such as forward difference and Richardson extrapolation, that are mathematically simpler and may lead to a reduction in computational burden when compared with Supplemented EM. However, the relative merits of the various numerical differentiation methods have not been evaluated in the context of IRT modeling. We perform such an evaluation, using both simulated and empirical data. It is found that the accuracy of the simpler noniterative alternatives is heavily dependent on the choice of the numerical differentiation perturbation constants. On the other hand, Supplemented EM consistently maintains accuracy and does not require the selection of perturbation constants. Furthermore, when implemented with an adaptive iteration scheme, an updated Supplemented EM algorithm can be as computationally efficient as the alternatives. The expected (Fisher) information matrix, while accurate, requires too heavy computation for realistic test lengths. Therefore, we recommend the routine use of the updated Supplemented EM algorithm in IRT applications.
机译：在项目响应理论（IRT）建模中，项目参数误差协方差矩阵在定义？在统计推断程序中。使用EM算法估算商品参数时，参数误差协方差矩阵不是商品校准的自动副产品。 Cai提出使用补充EM算法来计算项目参数误差协方差矩阵。此方法随后已在商业IRT软件程序（例如IRTPRO和flexMIRT）中实现。 Jamshidian和Jennrich指出，补充EM是基于对EM映射进行数值微分的一类方法，他们提出了非迭代替代方案，例如前向差分和Richardson外推法，它们在数学上更简单，并且在进行比较时可以减少计算负担与补充EM。但是，尚未在IRT建模中评估各种数值微分方法的相对优点。我们使用模拟和经验数据进行这种评估。发现更简单的非迭代替代方法的精度在很大程度上取决于数值微分扰动常数的选择。另一方面，增补EM始终保持精度，并且不需要选择扰动常数。此外，当采用自适应迭代方案实施时，更新的补充EM算法的计算效率与替代方法一样。预期的（Fisher）信息矩阵虽然准确，但对于实际的测试长度来说，计算量太大。因此，我们建议在IRT应用程序中例行使用更新后的补充EM算法。

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