A Derivation of the Optimal Answer-Copying Index and Some Applications

Abstract

Multiple choice exams are frequently used as an efficient and objective way of evaluating knowledge. Nevertheless, they are more vulnerable to answer-copying than tests based on open questions. Several statistical tests (known as indices) have been proposed to detect cheating but all of them are ad-hoc lacking a derivation that guarantees optimatility in any sense. We derived the uniform most powerful (UMP) test under certain assumptions. Given that we do not know the correct behavioral model for item answering, necessary to derive the UMP, we calculate the empirical type-I and type-II error rates for several indices using Monte Carlo simulations and data from twelve multiple choice exams taken by 5th and 9th graders in Colombia. According to the results, the index with the highest power among those that preserve the type-I error rate, uses a nominal response model for item answering, conditions on the responses of the individual that is suspected of being the sources of answers and uses a normal approximation to calculate the critical values. This index was first studied by Wollack (1997) and is superior to the index studied by Wesolowsky (2000) and used in his software. Furthermore, we compare the performance of the indexes on examination rooms with different levels of proctoring and find that increasing the level of proctoring can reduce copying by as much as 30% and that simple strategies such as having different students answer different portion of the test at different times can also reduce cheating by over 30%. Finally, a Bonferroni type false discovery rate procedure is used to detect massive cheating. Although the application is straightforward it not currently as a tool for making entire examination rooms retake an exam under stricter surveillance conditions when massive cheating is detected.