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

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 c...

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 infer...

Nearly all statistical inference methods were developed for the regime where the number $N$ of data samples is much larger than the data dimension $p$. Inference protocols such as maximum likelihood (ML) or maximum a posteriori probability (MAP) are unreliable if $p=O(N)$, due to overfitting. This limitation has for many disciplines with increasingly high-dimensional data become a serious bottleneck. We recently showed that in Cox regression for time-to-event data the overfit...

We investigate analytically the behaviour of the penalized maximum partial likelihood estimator (PMPLE). Our results are derived for a generic separable regularization, but we focus on the elastic net. This penalization is routinely adopted for survival analysis in the high dimensional regime, where the Maximum Partial Likelihood estimator (no regularization) might not even exist. Previous theoretical results require that the number $s$ of non-zero association coefficients is...

Regression analysis based on many covariates is becoming increasingly common. However, when the number of covariates $p$ is of the same order as the number of observations $n$, maximum likelihood regression becomes unreliable due to overfitting. This typically leads to systematic estimation biases and increased estimator variances. It is crucial for inference and prediction to quantify these effects correctly. Several methods have been proposed in literature to overcome overf...

Recently, the increasing availability of repeated measurements in biomedical studies has motivated the development of several statistical methods for the dynamic prediction of survival in settings where a large (potentially high-dimensional) number of longitudinal covariates is available. These methods differ in both how they model the longitudinal covariates trajectories, and how they specify the relationship between the longitudinal covariates and the survival outcome. Beca...

In the era of precision medicine, time-to-event outcomes such as time to death or progression are routinely collected, along with high-throughput covariates. These high-dimensional data defy classical survival regression models, which are either infeasible to fit or likely to incur low predictability due to over-fitting. To overcome this, recent emphasis has been placed on developing novel approaches for feature selection and survival prognostication. We will review various c...

The proportional hazards model has been extensively used in many fields such as biomedicine to estimate and perform statistical significance testing on the effects of covariates influencing the survival time of patients. The classical theory of maximum partial-likelihood estimation (MPLE) is used by most software packages to produce inference, e.g., the coxph function in R and the PHREG procedure in SAS. In this paper, we investigate the asymptotic behavior of the MPLE in the...

Many researchers have investigated first hitting times as models for survival data. First hitting times arise naturally in many types of stochastic processes, ranging from Wiener processes to Markov chains. In a survival context, the state of the underlying process represents the strength of an item or the health of an individual. The item fails or the individual experiences a clinical endpoint when the process reaches an adverse threshold state for the first time. The time s...

In the realm of contemporary data analysis, the use of massive datasets has taken on heightened significance, albeit often entailing considerable demands on computational time and memory. While a multitude of existing works offer optimal subsampling methods for conducting analyses on subsamples with minimized efficiency loss, they notably lack tools for judiciously selecting the optimal subsample size. To bridge this gap, our work introduces tools designed for choosing the op...