An Empirical Bayes Approach to Scoring Multiple-Choice Tests in the Misinformation Model

Abstract

This article develops multiple-choice test scoring rules, concentrating on Bayes rules and their frequency theory analogs, empirical Bayes rules. Conditions are given for empirical Bayes estimates to lie in the probability simplex. The misinformation model is considered in detail. It is shown that ranking by raw scores is equivalent to ranking by Bayes scores when the loss function increases with error and the sampling distribution has the monotone likelihood ratio property. Application of the techniques is made to data from a multiple-choice test given to students of an elementary statistics course.