Eggen Theo J H M, Verhelst Norman D
Cito, The Netherlands.
Cito, PO Box 1034, 6801 MG, Arnhem, The Netherlands.
Psychometrika. 2006 Jun;71(2):303-322. doi: 10.1007/s11336-004-1205-6. Epub 2017 Feb 11.
In this paper, the efficiency of conditional maximum likelihood (CML) and marginal maximum likelihood (MML) estimation of the item parameters of the Rasch model in incomplete designs is investigated. The use of the concept of F-information (Eggen, 2000) is generalized to incomplete testing designs. The scaled determinant of the F-information matrix is used as a scalar measure of information contained in a set of item parameters. In this paper, the relation between the normalization of the Rasch model and this determinant is clarified. It is shown that comparing estimation methods with the defined information efficiency is independent of the chosen normalization. The generalization of the method to other models than the Rasch model is discussed.In examples, information comparisons are conducted. It is found that for both CML and MML some information is lost in all incomplete designs compared to complete designs. A general result is that with increasing test booklet length the efficiency of an incomplete design, compared to a complete design, is increasing, as is the efficiency of CML compared to MML. The main difference between CML and MML is seen in the effect of the length of the test booklet. It will be demonstrated that with very small booklets, there is a substantial loss in information (about 35%) with CML estimation, while this loss is only about 10% in MML estimation. However, with increasing test length, the differences between CML and MML quickly disappear.
本文研究了在不完全设计中，Rasch模型项目参数的条件最大似然估计（CML）和边际最大似然估计（MML）的效率。F-信息（Eggen，2000）概念的应用被推广到不完全测试设计中。F-信息矩阵的缩放行列式被用作一组项目参数中所含信息的标量度量。本文阐明了Rasch模型的归一化与该行列式之间的关系。结果表明，用定义的信息效率比较估计方法与所选的归一化无关。文中还讨论了将该方法推广到Rasch模型以外的其他模型的情况。在实例中进行了信息比较。结果发现，与完全设计相比，在所有不完全设计中，CML和MML都会损失一些信息。一个普遍的结果是，随着测试手册长度的增加，与完全设计相比，不完全设计的效率会提高，CML相对于MML的效率也会提高。CML和MML的主要区别体现在测试手册长度的影响上。结果表明，对于非常小的手册，CML估计会有大量信息损失（约35%），而MML估计的信息损失仅约为10%。然而，随着测试长度的增加，CML和MML之间的差异会迅速消失。
