Receiver Operating Characteristic (ROC) Curves

TL;DR

This paper explores the theoretical foundations of ROC curves, introduces a flexible beta family for fitting them, and demonstrates improved empirical performance over classical models, especially when concavity is enforced.

Contribution

It establishes an equivalence between ROC curves and CDFs, introduces a new beta family for fitting ROC curves, and provides R software for estimation and testing.

Findings

Beta family fits empirical ROC curves better than binormal models.

Enforcing concavity improves model fit.

Software for ROC analysis is provided in R.

Abstract

Receiver operating characteristic (ROC) curves are used ubiquitously to evaluate covariates, markers, or features as potential predictors in binary problems. We distinguish raw ROC diagnostics and ROC curves, elucidate the special role of concavity in interpreting and modelling ROC curves, and establish an equivalence between ROC curves and cumulative distribution functions (CDFs). These results support a subtle shift of paradigms in the statistical modelling of ROC curves, which we view as curve fitting. We introduce the flexible two-parameter beta family for fitting CDFs to empirical ROC curves, derive the large sample distribution of the minimum distance estimator and provide software in R for estimation and testing, including both asymptotic and Monte Carlo based inference. In a range of empirical examples the beta family and its three- and four-parameter ramifications that allow for straight edges fit better than the classical binormal model, particularly under the vital constraint of the fitted curve being concave.