Concavity of ROC curve under a very general condition
Honghong Liu, Hongyue Wang, Changyong Feng · Communication in Statistics- Theory and Methods · 2025
The ROC curve analysis serves as a valuable tool for evaluating the performance of binary classification methods, providing insights into their sensitivity, specificity, and overall effectiveness across various threshold settings. When the ROC curve is concave, we prove that there is a unique point on the curve that is closest to the perfect point, where both sensitivity and specificity are 1. This article introduces a comprehensive condition ensuring the concavity of the ROC curve and establishes the criteria for a unique Youden’s index point. Additionally, for binormal data, the ROC curve is concave across its entire range if and only if the data sets from two groups have the same standard deviation.