We use statistical mechanics techniques, viz. the replica method, to model the effect of censoring on overfitting in Cox's proportional hazards model, the dominant regression method for time-to-event data. In the overfitting regime, Maximum Likelihood parameter estimators are known to be biased already for small values of the ratio of the number of covariates over the number of samples. The inclusion of censoring was avoided in previous overfitting analyses for mathematical convenience, but is vital to make any theory applicable to real-world medical data, where censoring is ubiquitous. Upon constructing efficient algorithms for solving the new (and more complex) RS equations and comparing the solutions with numerical simulation data, we find excellent agreement, even for large censoring rates. We then address the practical problem of using the theory to correct the biased ML estimators {without} knowledge of the data-generating distribution. This is achieved via a novel numerical algorithm that self-consistently approximates all relevant parameters of the data generating distribution while simultaneously solving the RS equations. We investigate numerically the statistics of the corrected estimators, and show that the proposed new algorithm indeed succeeds in removing the bias of the ML estimators, for both the association parameters and for the cumulative hazard.

翻译：我们运用统计力学技术（即副本方法）对右删失在Cox比例风险模型（时间至事件数据的主要回归方法）过拟合中的效应进行建模。在过拟合区域，当协变量数量与样本数量之比很小时，最大似然参数估计量已知存在偏差。此前为数学便利，过拟合分析中均规避了右删失的纳入，但这对于使理论适用于右删失普遍存在的真实医学数据至关重要。通过构建求解新型（更复杂）RS方程的高效算法，并将数值解与模拟数据对比，我们发现即便在右删失率极高的情况下仍具有极佳吻合性。随后我们攻克了实际难题：在未知数据生成分布的情况下，利用理论修正有偏的ML估计量。这通过新型数值算法实现，该算法在同时求解RS方程时能自洽近似数据生成分布的所有相关参数。我们通过数值方法研究修正估计量的统计特性，证明所提出的新算法确实能成功消除ML估计量在关联参数和累积风险函数上的偏差。