Analysis of overfitting in the regularized Cox model
This work addresses overfitting issues in survival analysis for researchers and practitioners, but it is incremental as it generalizes a prior study from maximum likelihood to maximum a posteriori inference.
The paper tackles overfitting in the regularized Cox model for time-to-event data by using the replica method to analyze the relationship between true and inferred parameters under L2 regularization in high-dimensional settings, establishing a link between optimal regularization and the p/N ratio for overfitting corrections.
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.