On the Optimality of Answer-Copying Indices

Abstract

Multiple-choice exams are frequently used as an efficient and objective method to assess learning, but they are more vulnerable to answer copying than tests based on open questions. Several statistical tests (known as indices in the literature) have been proposed to detect cheating; however, to the best of our knowledge, they all lack mathematical support that guarantees optimality in any sense. We partially fill this void by deriving the uniformly most powerful (UMP) test under the assumption that the response distribution is known. In practice, however, we must estimate a behavioral model that yields a response distribution for each question. As an application, we calculate the empirical type I and type II error rates for several indices that assume different behavioral models using simulations based on real data from 12 nationwide multiple-choice exams taken by fifth and ninth graders in Colombia. We find that the most powerful index among those studied, subject to the restriction of preserving the type I error, is one based on the work of Wollack and is superior to the index developed by Wesolowsky.