title: A note on monotone likelihood ratio of the total score variable in unidimensional item response theory.
abstract: 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.
authors.affiliation: Department of Computer-Oriented Statistics and Data Analysis, Institute of Mathematics, University of Augsburg, Germany. ali.uenlue@math.uni-augsburg.de
authors.forename: Ali
authors.initials: A
authors.lastname: Unlü
mesh_terms: D003625:Data Collection / Q000706:statistics & numerical data*; D006801:Humans; D016013:Likelihood Functions; D017711:Nonlinear Dynamics; D011581:Psychological Tests / Q000706:statistics & numerical data*; D011594:Psychometrics / Q000706:statistics & numerical data; D015203:Reproducibility of Results; D013269:Stochastic Processes