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We use statistical mechanics techniques, viz. the replica method, to modelthe effect of censoring on overfitting in Cox's proportional hazards model, thedominant regression method for time-to-event data. In the overfitting regime,Maximum Likelihood parameter estimators are known to be biased already forsmall values of the ratio of the number of covariates over the number ofsamples. The inclusion of censoring was avoided in previous overfittinganalyses for mathematical convenience, but is vital to make any theoryapplicable to real-world medical data, where censoring is ubiquitous. Uponconstructing efficient algorithms for solving the new (and more complex) RSequations and comparing the solutions with numerical simulation data, we findexcellent agreement, even for large censoring rates. We then address thepractical problem of using the theory to correct the biased ML estimators{without} knowledge of the data-generating distribution. This is achieved via anovel numerical algorithm that self-consistently approximates all relevantparameters of the data generating distribution while simultaneously solving theRS equations. We investigate numerically the statistics of the correctedestimators, and show that the proposed new algorithm indeed succeeds inremoving the bias of the ML estimators, for both the association parameters andfor the cumulative hazard.

Replica analysis of overfitting in regression models for time to event data: the impact of censoring