Proper and Useful Distractors in Multiple-Choice Diagnostic Classification Models

The multiple-choice (MC) item format has been implemented in educational assessments that are used across diverse content domains. MC items comprise two components: the stem that provides the context with a motivating narrative, and the collection of response options consisting of the correct answer, called the "key," and several incorrect alternatives, the "distractors." The MC-DINA model was the first diagnostic classification model for MC items that used distractors explicitly as potential sources of diagnostic information. However, the MC-DINA model requires that the q-vectors of the distractors are nested within each other and that of the key, which poses a serious constraint on item development. Consequently, later adaptations of the MC item format to cognitive diagnosis dropped the nestedness condition. The relaxation of the nestedness-condition, however, comes at a price: distractors may become redundant (i.e., they do not contribute to any further diagnostic differentiation between examinees), and they may induce undesirable diagnostic ambiguity (i.e., they are equally likely to be chosen by an examinee, but their q-vectors point at different diagnostic classifications). In this article, two criteria, useful and proper, are proposed to identify redundant and diagnostically ambiguous distractors.