The Risk Distribution Curve and its Derivatives

Risk stratification is most directly and informatively summarized as a risk distribution curve. From this curve the ROC curve, predictiveness curve, and other curves depicting risk stratification can be derived, demonstrating that they present similar information. A mathematical expression for the ROC curve AUC is derived which clarifies how this measure of discrimination quantifies the overlap between patients who have and don’t have events. This expression is used to define the positive correlation between the dispersion of the risk distribution curve and the ROC curve AUC. As more disperse risk distributions and greater separation between patients with and without events characterize superior risk stratification, the ROC curve AUC provides useful information. Many statistical methods of risk stratification have been developed for clinical risk assessment. Evaluation of the clinical utility of these methods requires risk estimation for the members of a patient population. The most direct and informative way to present these results and understand the achieved risk stratification is a risk distribution curve.[1] Previously Huang et al. [2] had proposed use of the predictiveness curve which also presents risk distribution. Neither approach is being used. Instead the ROC (receiver operating characteristics) curve is usually presented and the area under this curve evaluated. However the value of this approach has been questioned.[3,4] In this paper, the relationship between the multiple graphical presentations of risk stratification used in the literature is outlined, providing a better understanding of the value of ROC curve analysis. Lognormal example of a risk distribution curve The risk distribution curve of cardiovascular risk in adults has not been presented. The expected distribution when risk factors interact multiplicatively is lognormal.[5] For cardiovascular risk, this is especially so for two reasons. First, levels of individual risk factors, such as blood pressure and cholesterol, are often lognormal. This reflects the fact that their levels in turn are determined by the action of multiple other influences.[6] Second, an exponential increase in risk with increasing levels of a risk factor will also produce lognormal risk distribution curves. A lognormal curve is one whose natural logarithm is normally distributed with mean μ and variance σ. Lognormal variates are greater than zero and lognormal curves are variably skewed to right. The equation for a lognormal risk distribution curve where risk, r, varies from >0 to 1 is: