Unlü Ali
Department of Computer-Oriented Statistics and Data Analysis, Institute of Mathematics, University of Augsburg, Germany.
Br J Math Stat Psychol. 2008 May;61(Pt 1):179-87. doi: 10.1348/000711007X173391. Epub 2007 Jan 4.
This note provides a direct, elementary proof of the fundamental result on monotone likelihood ratio of the total score variable in unidimensional item response theory (IRT). This result is very important for practical measurement in IRT, because it justifies the use of the total score variable to order participants on the latent trait. The proof relies on a basic inequality for elementary symmetric functions which is proved by means of few purely algebraic, straightforward transformations. In particular, flaws in a proof of this result by Huynh [(1994). A new proof for monotone likelihood ratio for the sum of independent Bernoulli random variables. Psychometrika, 59, 77-79] are pointed out and corrected, and a natural generalization of the fundamental result to non-linear (quasi-ordered) latent trait spaces is presented. This may be useful for multidimensional IRT or knowledge space theory, in which the latent 'ability' spaces are partially ordered with respect to, for instance, coordinate-wise vector-ordering or set-inclusion, respectively.
本笔记给出了一维项目反应理论（IRT）中总分变量单调似然比这一基本结果的直接、初等证明。该结果在IRT的实际测量中非常重要，因为它证明了使用总分变量对参与者在潜在特质上进行排序的合理性。证明依赖于一个关于初等对称函数的基本不等式，该不等式通过一些纯代数的、直接的变换得以证明。特别地，指出并纠正了Huynh（1994年）[《独立伯努利随机变量之和的单调似然比的新证明》。《心理测量学》，59，77 - 79]对该结果证明中的缺陷，并给出了该基本结果到非线性（拟序）潜在特质空间的自然推广。这对于多维IRT或知识空间理论可能是有用的，在这些理论中，潜在的“能力”空间分别相对于例如坐标方向的向量排序或集合包含是部分有序的。
