Survival Regression with Proper Scoring Rules and Monotonic Neural Networks

TL;DR

This paper demonstrates that common scoring rules for survival regression are not proper, introduces SuMo-net, a neural network model that directly optimizes the proper right-censored log-likelihood, achieving state-of-the-art results and fast inference.

Contribution

The paper proves the improperness of popular survival scoring rules, and proposes SuMo-net, a neural network that directly optimizes the proper log-likelihood with monotonic constraints.

Findings

SuMo-net achieves state-of-the-art log-likelihood scores.

SuMo-net offers 20-100x faster inference than existing methods.

The paper proves the improperness of common survival scoring rules.

Abstract

We consider frequently used scoring rules for right-censored survival regression models such as time-dependent concordance, survival-CRPS, integrated Brier score and integrated binomial log-likelihood, and prove that neither of them is a proper scoring rule. This means that the true survival distribution may be scored worse than incorrect distributions, leading to inaccurate estimation. We prove that, in contrast to these scores, the right-censored log-likelihood is a proper scoring rule, i.e., the highest expected score is achieved by the true distribution. Despite this, modern feed-forward neural-network-based survival regression models are unable to train and validate directly on the right-censored log-likelihood, due to its intractability, and resort to the aforementioned alternatives, i.e., non-proper scoring rules. We therefore propose a simple novel survival regression method capable of directly optimizing log-likelihood using a monotonic restriction on the time-dependent weights, coined SurvivalMonotonic-net (SuMo-net). SuMo-net achieves state-of-the-art log-likelihood scores across several datasets with 20--100$\times$ computational speedup on inference over existing state-of-the-art neural methods, and is readily applicable to datasets with several million observations.