Local dependence (LD) refers to the violation of the local independence assumption of most item response models. Statistics that indicate LD between a pair of items on a test or questionnaire that is being fitted with an item response model can play a useful diagnostic role in applications of item response theory. In this article, a new score test statistic, S[subscript b], based on the bifactor logistic model is described. To compare the performance of S[subscript b] with the score test statistic (S[subscript t]) based on the threshold shift model, and the LD X[squared] statistic, data were simulated under locally independent, bifactor, and threshold shift conditions. The results summarize the null distributions of all three diagnostic statistics, and their power for various degrees of bifactor and threshold shift LD. Future research directions are discussed, including the straightforward generalization of S[subscript b] for polytomous item response models, and the challenges involved in the corresponding generalizations of S[subscript t] and LD X[squared]. (Contains 8 notes, 3 tables and 9 figures.)