A Statistical Learning Take on the Concordance Index for Survival Analysis
This work addresses a foundational issue for researchers in survival analysis by clarifying when ML models align with C-index evaluation, though it is incremental with limited experimental validation.
The paper analyzes the relationship between machine learning cost functions and the concordance index (C-index) in survival analysis, providing Fisher-consistency results and excess risk bounds for common cost functions under specific model conditions, and introduces a new consistent method that is computationally expensive.
The introduction of machine learning (ML) techniques to the field of survival analysis has increased the flexibility of modeling approaches, and ML based models have become state-of-the-art. These models optimize their own cost functions, and their performance is often evaluated using the concordance index (C-index). From a statistical learning perspective, it is therefore an important problem to analyze the relationship between the optimizers of the C-index and those of the ML cost functions. We address this issue by providing C-index Fisher-consistency results and excess risk bounds for several of the commonly used cost functions in survival analysis. We identify conditions under which they are consistent, under the form of three nested families of survival models. We also study the general case where no model assumption is made and present a new, off-the-shelf method that is shown to be consistent with the C-index, although computationally expensive at inference. Finally, we perform limited numerical experiments with simulated data to illustrate our theoretical findings.