The Robustness of the "Binormal" Assumptions Used in Fitting ROC Curves

The binormal form is the most common model used to formally fit ROC curves to the data from signal detection studies that employ the "rating" method. The author lists a number of justifications that have been offered for this choice, ranging from theoretical considerations of probability laws and signal detection theory, to mathematical tractability and convenience, to empirical results showing that "it fits!" To these justifications is added another, namely that even if an alternative formulation based on another underlying form (e.g., power law) or model (e.g., binomial, Poisson, or gamma type distributions) were in fact correct, the binormal fit differs so little from the true form as to be of no practical consequence. Moreover, the small lack of fit is unlikely to be demonstrated in practice: it is obscured by the much larger variation that can be attributed to sampling of cases. In addition, even if a very large sample of cases could be studied, the small number of rating categories used does not permit seemingly very different models to be distinguished from one another.