Analysis of overfitting in the regularized Cox model

TL;DR

This paper investigates overfitting in regularized Cox models using statistical physics methods, revealing how to optimally choose regularization to correct biases in high-dimensional survival analysis.

Contribution

It extends previous work by analyzing the relationship between true and inferred parameters in regularized Cox models with large p and N, providing a way to select optimal regularization.

Findings

Derived a relationship between regularization parameter and p/N

Generalized maximum likelihood to maximum a posteriori inference

Provided a method for overfitting correction in high-dimensional data

Abstract

The Cox proportional hazards model is ubiquitous in the analysis of time-to-event data. However, when the data dimension p is comparable to the sample size $N$, maximum likelihood estimates for its regression parameters are known to be biased or break down entirely due to overfitting. This prompted the introduction of the so-called regularized Cox model. In this paper we use the replica method from statistical physics to investigate the relationship between the true and inferred regression parameters in regularized multivariate Cox regression with L2 regularization, in the regime where both p and N are large but with p/N ~ O(1). We thereby generalize a recent study from maximum likelihood to maximum a posteriori inference. We also establish a relationship between the optimal regularization parameter and p/N, allowing for straightforward overfitting corrections in time-to-event analysis.