In 2008, Penfield showed that measurement invariance across all response options of a multiple-choice item (correct option and the J distractors) can be modeled using a nominal response model that included a differential distractor functioning (DDF) effect for each of the J distractors. This article extends this concept to consider how the differential item functioning (DIF) effect (i.e., the conditional between-group differences in the probability of correct response) is determined by the J DDF effects. In particular, this article shows how the DIF effect can be modeled as a function of the J DDF effects and thus reveals the conditions that must hold for uniform DIF, nonuniform DIF, and crossing DIF to exist. The results provide insight into the potential item-level properties that lead to uniform, nonuniform, and crossing DIF. The findings may shed light on the etiology of different forms of DIF, which may help analysts target the particular causes of the DIF effect.
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