Hazard-based distributional regression via ordinary differential equations

TL;DR

This paper introduces a flexible survival regression framework modeling hazard functions through ODEs, allowing for complex hazard shapes and covariate effects, with efficient Bayesian inference and practical case studies.

Contribution

It proposes a novel hazard-based regression model using ODEs, enhancing flexibility and interpretability over traditional parametric models, with new Bayesian computational methods.

Findings

Successfully modeled crossing survival curves in clinical trial data.

Revealed covariate effects on hazard shapes in cancer recurrence study.

Provided efficient Bayesian inference tools for the proposed models.

Abstract

The hazard function is central to the formulation of commonly used survival regression models such as the proportional hazards and accelerated failure time models. However, these models rely on a shared baseline hazard, which, when specified parametrically, can only capture limited shapes. To overcome this limitation, we propose a general class of parametric survival regression models obtained by modelling the hazard function using autonomous systems of ordinary differential equations (ODEs). Covariate information is incorporated via transformed linear predictors on the parameters of the ODE system. Our framework capitalises on the interpretability of parameters in common ODE systems, enabling the identification of covariate values that produce qualitatively distinct hazard shapes associated with different attractors of the system of ODEs. This provides deeper insights into how covariates influence survival dynamics. We develop efficient Bayesian computational tools, including parallelised evaluation of the log-posterior, which facilitates integration with general-purpose Markov Chain Monte Carlo samplers. We also derive conditions for posterior asymptotic normality, enabling fast approximations of the posterior. A central contribution of our work lies in the case studies. We demonstrate the methodology using clinical trial data with crossing survival curves, and a study of cancer recurrence times where our approach reveals how the efficacy of interventions (treatments) on hazard and survival are influenced by patient characteristics.