Using the mean-to-sigma ratio as a measure of the improperness of binormal ROC curves

A basic assumption for a meaningful diagnostic decision variable is that there is a monotone relationship between the decision variable and the likelihood of disease. This relationship, however, generally does not hold for the binormal model. As a result, receiver operating characteristic (ROC)-curve estimation based on the binormal model produces improper ROC curves that are not concave over the entire domain and cross the chance line. Although in practice the "improperness" is typically not noticeable, there are situations where the improperness is evident. Presently, standard statistical software does not provide diagnostics for assessing the magnitude of the improperness.

We show how the mean-to-sigma ratio can be a useful, easy-to-understand and easy-to-use measure for assessing the magnitude of the improperness of a binormal ROC curve by showing how it is related to the chance-line crossing. We suggest an improperness criterion based on the mean-to-sigma ratio.

Using a real-data example, we illustrate how the mean-to-sigma ratio can be used to assess the improperness of binormal ROC curves, compare the binormal method with an alternative proper method, and describe uncertainty in a fitted ROC curve with respect to improperness.

By providing a quantitative and easily computable improperness measure, the mean-to-sigma ratio provides an easy way to identify improper binormal ROC curves and facilitates comparison of analysis strategies according to improperness categories in simulation and real-data studies.