Copula-Based Survival Models for Dependent Competing Risks

Abstract

A survival dataset describes a collection of instances, such as patients, and associates each instance with either the time until an event (such as death), or the censoring time (eg, when the instance is lost to follow-up), which is a lower bound on the time until the event. While there are several approaches to survival prediction, this thesis focuses on models that produce an individual survival curve (providing P(death ≥ t|x) for each t > 0) for each individual patient, x – here based on a “Deep Weibull” model. Most survival prediction methods assume that the event and censoring distributions are independent given the instance’s covariates. This assumption is challenging to verify since we only observe a single outcome (event xor censor time) for each instance. Moreover, models that assume this independence can be substantially biased when this independence does not hold. Moreover, the standard methods to evaluate survival models do not provide meaningful values here. In this study, we present a way to relax the assumption of conditional independence, using a parametric model of survival that incorporates Archimedean copulas to address residual dependency that cannot be explained by the co-variates in the dataset. Additionally, we show how to extend this to a broader range of dependencies by using a convex combination of members from the Archimedean copula family, rather than relying on a specific member. Our empirical studies, conducted on synthetic and semi-synthetic data, demonstrated that our approach significantly improves the estimation of survival distributions in terms of log-likelihood (which is a proper scoring for the survival analysis task) and L1 survival distance (which we proposed), compared to the standard approach that assumes conditional independence.