Wasserstein-Kaplan-Meier Survival Regression

Abstract

Survival analysis plays a pivotal role in medical research, offering valuable insights into the timing of events such as survival time. One common challenge in survival analysis is the necessity to adjust the survival function to account for additional factors, such as age, gender, and ethnicity. We propose an innovative regression model for right-censored survival data across heterogeneous populations, leveraging the Wasserstein space of probability measures. Our approach models the probability measure of survival time and the corresponding non-parametric Kaplan-Meier estimator for each subgroup as elements of the Wasserstein space. The Wasserstein space provides a flexible framework for modeling heterogeneous populations, allowing us to capture complex relationships between covariates and survival times. We address an underexplored aspect by deriving the non-asymptotic convergence rate of the Kaplan-Meier estimator to the underlying probability measure in terms of the Wasserstein metric. The proposed model is supported with a solid theoretical foundation including pointwise and uniform convergence rates, along with an efficient algorithm for model fitting. The proposed model effectively accommodates random variation that may exist in the probability measures across different subgroups, demonstrating superior performance in both simulations and two case studies compared to the Cox proportional hazards model and other alternative models.