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

Overfitting, which happens when the number of parameters in a model is too\nlarge compared to the number of data points available for determining these\nparameters, is a serious and growing problem in survival analysis. While modern\nmedicine presents us with data of unprecedented dimensionality, these data\ncannot yet be used effectively for clinical outcome prediction. Standard error\nmeasures in maximum likelihood regression, such as p-values and z-scores, are\nblind to overfitting, and even for Cox's proportional hazards model (the main\ntool of medical statisticians), one finds in literature only rules of thumb on\nthe number of samples required to avoid overfitting. In this paper we present a\nmathematical theory of overfitting in regression models for time-to-event data,\nwhich aims to increase our quantitative understanding of the problem and\nprovide practical tools with which to correct regression outcomes for the\nimpact of overfitting. It is based on the replica method, a statistical\nmechanical technique for the analysis of heterogeneous many-variable systems\nthat has been used successfully for several decades in physics, biology, and\ncomputer science, but not yet in medical statistics. We develop the theory\ninitially for arbitrary regression models for time-to-event data, and verify\nits predictions in detail for the popular Cox model.\n