Replica analysis of overfitting in regression models for time-to-event data
Replica analysis of overfitting in regression models for time-to-event data
复制标题
DOI：
10.1088/1751-8121/aa812f
复制
发表时间：
2017-09-15
影响因子：
2.1
通讯作者：
Perez-Vicente, C. J.
中科院分区：
文献类型：
--
作者：
Coolen, A. C. C.;Barrett, J. E.;Perez-Vicente, C. J.
过度拟合是生存分析中的一个严重且日益严重的问题，当模型中的参数数量与可用于确定这些参数的数据点数量相比过大时会发生过度拟合。虽然现代医学为我们提供了前所未有的多维数据，但这些数据还不能有效地用于临床结果预测。最大似然回归中的标准误差测量，如p值和z分数，对过拟合是盲目的，即使对于考克斯的比例风险模型（医学统计学家的主要工具），人们在文献中也只能找到避免过拟合所需样本数量的经验法则。在本文中，我们提出了一个数学理论的过拟合的回归模型的时间到事件的数据，其目的是增加我们的定量理解的问题，并提供实用的工具，以纠正回归结果的影响过拟合。它是基于复制方法，一种用于分析异质多变量系统的统计力学技术，已在物理学，生物学和计算机科学中成功使用了几十年，但尚未在医学统计学中使用。我们开发的理论最初为任意的回归模型的时间到事件的数据，并验证其预测的详细流行的考克斯模型。
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.