Theoretical evaluation of partial credit scoring of the multiple-choice test item

Abstract

In multiple-choice tests, guessing is a source of test error which can be suppressed if its expected score is made negative by either penalizing wrong answers or rewarding expressions of partial knowledge. Starting from the most general formulation of the necessary and sufficient scoring conditions for guessing to lead to an expected loss beyond the test-taker’s knowledge, we formulate a class of optimal scoring functions, including the proposal by Zapechelnyuk (Econ. Lett. 132, 24–27 (2015)) as a special case. We then consider an arbitrary multiple-choice test taken by a rational test-taker whose knowledge of a test item is defined by the fraction of the answer options which can be ruled out. For this model, we study the statistical properties of the obtained score for both standard marking (where guessing is not penalized), and marking where guessing is suppressed either by expensive score penalties for incorrect answers or by different marking schemes that reward partial knowledge.