  The area under the curve (AUC) of the receiver operating characteristics
curve (ROC) evaluates the separation between patients and nonpatients or
discrimination. For risk prediction models these risk distributions can be
derived from the population risk distribution so are not independent as in
diagnosis. A ROC curve AUC formula based on the underlying population risk
distribution clarifies how discrimination is defined mathematically and that
generation of the equivalent c-statistic effects a Monte Carlo integration of
the formula. For a selection of continuous risk distributions, exact analytic
formulas or numerical results for the ROC curve AUC and overlap measure are
presented and demonstrate a linear or near-linear dependence on their standard
deviation. The ROC curve AUC is also shown to be highly dependent on the mean
population risk, a distinction from the independence from disease prevalence
for diagnostic tests. The converse of discrimination, overlap, has been
quantified by the overlap measure, which appears to provide equivalent
information. As achieving wider population risk distributions is the goal of
risk prediction modeling for clinical risk stratification, interpreting the ROC
curve AUC as a measure of dispersion, rather than discrimination, when
comparing risk prediction models may be more relevant.
