Linearithmic and unbiased implementation of DeLong’s algorithm for comparing the areas under correlated ROC curves

Abstract

Receiver operating characteristic (ROC) analysis is a widespread framework in two-group classification problems. The area under the ROC curve (AUC) has been a popular figure of merit to summarize the performance of binary classifiers. Because of its simplicity of implementation, DeLong’s algorithm (DLA) has been widely used in practice. However, DLA suffers two drawbacks of time inefficiency and biasedness. The authors’ previous work has overcome the first drawback. Specifically, the time complexity of DLA is reduced from the quadratic order down to a linearithmic one, by exploiting the relationship between the Heaviside function and the mid-ranks of samples. Unfortunately, the second drawback of DLA remains unaddressed, to the best of our knowledge. Motivated by such unsatisfactory situation, in this paper, the authors correct the bias of DLA with the assistance of an efficient implementation of Kendall’s tau (KT). The overall computational load of the improved DLA (being strictly unbiased) is in linearithmic time. Theoretical analyses and numerical results verified the proposed algorithmic findings in this work.