Survival Analysis via Ordinary Differential Equations

TL;DR

This paper introduces an ODE-based framework for survival analysis that unifies existing models, enabling scalable estimation and inference with strong theoretical guarantees and practical advantages.

Contribution

It develops a general ODE modeling framework for survival analysis, unifying many existing models and providing a scalable, stable estimation procedure with theoretical validation.

Findings

The proposed estimator is consistent and asymptotically normal.

It achieves the semi-parametric efficiency bound.

Finite sample performance is validated through simulations and real data.

Abstract

This paper introduces an Ordinary Differential Equation (ODE) notion for survival analysis. The ODE notion not only provides a unified modeling framework, but more importantly, also enables the development of a widely applicable, scalable, and easy-to-implement procedure for estimation and inference. Specifically, the ODE modeling framework unifies many existing survival models, such as the proportional hazards model, the linear transformation model, the accelerated failure time model, and the time-varying coefficient model as special cases. The generality of the proposed framework serves as the foundation of a widely applicable estimation procedure. As an illustrative example, we develop a sieve maximum likelihood estimator for a general semi-parametric class of ODE models. In comparison to existing estimation methods, the proposed procedure has advantages in terms of computational scalability and numerical stability. Moreover, to address unique theoretical challenges induced by the ODE notion, we establish a new general sieve M-theorem for bundled parameters and show that the proposed sieve estimator is consistent and asymptotically normal, and achieves the semi-parametric efficiency bound. The finite sample performance of the proposed estimator is examined in simulation studies and a real-world data example.