A Statistical Learning Take on the Concordance Index for Survival Analysis

TL;DR

This paper analyzes the statistical properties of the concordance index in survival analysis, providing consistency results for various cost functions and introducing a new method aligned with the C-index.

Contribution

It offers the first Fisher-consistency and excess risk bounds for C-index optimization in survival models, and proposes a new consistent, though computationally intensive, method.

Findings

Identifies conditions for consistency of common survival models

Provides theoretical bounds linking cost functions and C-index

Introduces a new method consistent with the C-index

Abstract

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.