Statistical relationships between traffic crashes and traffic flows at roadway intersections have been extensively modeled and evaluated in recent years. The underlying assumptions adopted in the popular models for intersections are challenged. First, the assumption that the dispersion parameter is a fixed parameter across sites and time periods is challenged. Second, the mathematical limitations of some functional forms used in these models, particularly their properties at the boundaries, are examined. It is also demonstrated that, for a given data set, a large number of plausible functional forms with almost the same overall statistical goodness of fit (GOF) is possible, and an alternative class of logical formulations that may enable a richer interpretation of the data is introduced. A comparison of site estimates from the empirical Bayes and full Bayes methods is also presented. All discussions and comparisons are illustrated with a set of data collected for an urban four-legged signalized intersection in Toronto, Ontario, Canada, from 1990 to 1995. In discussing functional forms, the need for some goodness-of-logic measures, in addition to the GOF measure, is emphasized and demonstrated. Finally, analysts are advised to be mindful of the underlying assumptions adopted in the popular models, especially the assumption that the dispersion parameter is a fixed parameter, and the limitations of the functional forms used. Promising directions in which this study may be extended are also discussed.