Receiver Operating Characteristic (ROC) Curve: comparing parametric estimation, Monte Carlo simulation and numerical integration

A receiver operating characteristic (ROC) curve is a plot of predictive model probabilities of true positives (sensitivity) as a function of probabilities of false positives (1 – specificity) for a set of possible cutoff points. Some of the SAS/STAT procedures do not have built-in options for ROC curves and there have been a few suggestions in previous SAS forums to address the issue by using either parametric or non-parametric methods to construct those curves. This study shows how a simple concave function verifies the properties necessary to provide good fits to the ROC curve of diverse predictive models, and has also the advantage of giving an exact solution for the estimation of the area under the ROC in terms of a Beta function – defined by the parameters of the concave function. The study proceeds to discuss the implementation of the approach to a working dataset, and compares the value of the area estimated using the parametric solution to the ones obtained through three numerical methods of integration – Trapezoidal Rule, Simpson’s Rule and Gauss Quadrature, and Monte Carlo simulation. Finally the study proposes that the observed data points can be interpreted as part of an underlying data generating process (DGP) by randomly drawing subsets of the available data points – a sampling approach to the estimation of the Area Under the Curve. The SAS products used in the study are SAS BASE and SAS/STAT, in particular PROC NLIN and PROC SURVEYSELECT.