Fast Implementation of DeLong’s Algorithm for Comparing the Areas Under Correlated Receiver Operating Characteristic Curves

Among algorithms for comparing the areas under two or more correlated receiver operating characteristic (ROC) curves, DeLong's algorithm is perhaps the most widely used one due to its simplicity of implementation in practice. Unfortunately, however, the time complexity of DeLong's algorithm is of quadratic order (the product of sample sizes), thus making it time-consuming and impractical when the sample sizes are large. Based on an equivalent relationship between the Heaviside function and mid-ranks of samples, we improve DeLong's algorithm by reducing the order of time complexity from quadratic down to linearithmic (the product of sample size and its logarithm). Monte Carlo simulations verify the computational efficiency of our algorithmic findings in this work.