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Alternatives to the ROC Curve AUC and C-statistic for Risk Prediction Models

Ralph H. Stern
Nov 2023
Assessment of risk prediction models has primarily utilized measures of discrimination, the ROC curve AUC and C-statistic. These derive from the risk distributions of patients and nonpatients, which in turn are derived from a population risk distribution. As greater dispersion of the population risk distribution produces greater separation of patient and nonpatient risks (discrimination), its parameters can be used as alternatives to the ROC curve AUC and C-statistic. Here continuous probability distributions are employed to develop insight into the relationship between their parameters and the ROC curve AUC and C-statistic derived from them. The ROC curve AUC and C-statistic are shown to have a straight-line relationship with the SD for uniform, half-sine, and symmetric triangular probability distributions, with slight differences in the slope: AUC approx 1/2+0.28 SD/(mean(1-mean)). This also characterizes the beta distribution over the same range of SD's. But at larger beta distribution SD's the plot of AUC versus SD deviates downward from this straight-line relationship, approaching the ROC curve AUC and SD of a perfect model (AUC=1, SD= $\sqrt{\rm mean(1-mean)}$). A simpler and more intuitive discrimination metric is the coefficient of discrimination, the difference between the mean risk in patients and nonpatients. This is SD2/(mean(1-mean)), which is also the same for any distribution. Since estimating parameters or metrics discards information, the population risk distribution should always be presented. As the ROC curve AUC and C-statistic are functions of this distribution's parameters, the parameters represent simpler, intuitive alternatives to these discrimination metrics. Among discrimination metrics, the coefficient of discrimination provides a simple, intuitive alternative to the ROC curve AUC and C-statistic.
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