# Replica analysis of overfitting in regression models for time-to-event data

## TL;DR

This paper develops a mathematical theory using the replica method to analyze and correct overfitting in regression models for time-to-event data, addressing a critical challenge in high-dimensional survival analysis.

## Contribution

It introduces a novel application of the replica method to quantify and mitigate overfitting in survival regression models, including Cox's proportional hazards model.

## Key findings

- The theory accurately predicts overfitting effects in Cox models.
- Provides practical tools for correcting overfitting in survival analysis.
- Enhances understanding of overfitting in high-dimensional clinical data.

## Abstract

Overfitting, which happens when the number of parameters in a model is too large compared to the number of data points available for determining these parameters, is a serious and growing problem in survival analysis. While modern medicine presents us with data of unprecedented dimensionality, these data cannot yet be used effectively for clinical outcome prediction. Standard error measures in maximum likelihood regression, such as p-values and z-scores, are blind to overfitting, and even for Cox's proportional hazards model (the main tool of medical statisticians), one finds in literature only rules of thumb on the number of samples required to avoid overfitting. In this paper we present a mathematical theory of overfitting in regression models for time-to-event data, which aims to increase our quantitative understanding of the problem and provide practical tools with which to correct regression outcomes for the impact of overfitting. It is based on the replica method, a statistical mechanical technique for the analysis of heterogeneous many-variable systems that has been used successfully for several decades in physics, biology, and computer science, but not yet in medical statistics. We develop the theory initially for arbitrary regression models for time-to-event data, and verify its predictions in detail for the popular Cox model.