Improved Non-parametric Penalized Maximum Likelihood Estimation for Arbitrarily Censored Survival Data

TL;DR

This paper introduces a kernel smoothing enhancement to non-parametric maximum likelihood estimators for censored survival data, significantly reducing overfitting and improving accuracy in survival function estimation and prediction.

Contribution

It proposes a novel smoothing approach with a BIC-type criterion for censored data, along with an optimization algorithm, improving existing survival analysis methods.

Findings

Reduces discrepancy between estimated and true survival functions by up to 49%.

Improves within-sample and out-of-sample prediction accuracy by up to 41% and 23%.

Demonstrates effectiveness on real breast cancer censored data.

Abstract

Non-parametric maximum likelihood estimation encompasses a group of classic methods to estimate distribution-associated functions from potentially censored and truncated data, with extensive applications in survival analysis. These methods, including the Kaplan-Meier estimator and Turnbull's method, often result in overfitting, especially when the sample size is small. We propose an improvement to these methods by applying kernel smoothing to their raw estimates, based on a BIC-type loss function that balances the trade-off between optimizing model fit and controlling model complexity. In the context of a longitudinal study with repeated observations, we detail our proposed smoothing procedure and optimization algorithm. With extensive simulation studies over multiple realistic scenarios, we demonstrate that our smoothing-based procedure provides better overall accuracy in both survival function estimation and individual-level time-to-event prediction by reducing overfitting. Our smoothing procedure decreases the discrepancy between the estimated and true simulated survival function using interval-censored data by up to 49% compared to the raw un-smoothed estimate, with similar improvements of up to 41% and 23% in within-sample and out-of-sample prediction, respectively. Finally, we apply our method to real data on censored breast cancer diagnosis, which similarly shows improvement when compared to empirical survival estimates from uncensored data. We provide an R package, SISE, for implementing our penalized likelihood method.