The Brier Score under Administrative Censoring: Problems and Solutions

The Brier score is commonly used for evaluating probability predictions. In survival analysis, with right-censored observations of the event times, this score can be weighted by the inverse probability of censoring (IPCW) to retain its original interpretation. It is common practice to estimate the censoring distribution with the Kaplan-Meier estimator, even though it assumes that the censoring distribution is independent of the covariates. This paper discusses the general impact of the censoring estimates on the Brier score and shows that the estimation of the censoring distribution can be problematic. In particular, when the censoring times can be identified from the covariates, the IPCW score is no longer valid. For administratively censored data, where the potential censoring times are known for all individuals, we propose an alternative version of the Brier score. This administrative Brier score does not require estimation of the censoring distribution and is valid even if the censoring times can be identified from the covariates.

Continuous and Discrete-Time Survival Prediction with Neural Networks

Application of discrete-time survival methods for continuous-time survival prediction is considered. For this purpose, a scheme for discretization of continuous-time data is proposed by considering the quantiles of the estimated event-time distribution, and, for smaller data sets, it is found to be preferable over the commonly used equidistant scheme. Furthermore, two interpolation schemes for continuous-time survival estimates are explored, both of which are shown to yield improved performance compared to the discrete-time estimates. The survival methods considered are based on the likelihood for right-censored survival data, and parameterize either the probability mass function (PMF) or the discrete-time hazard rate, both with neural networks. Through simulations and study of real-world data, the hazard rate parametrization is found to perform slightly better than the parametrization of the PMF. Inspired by these investigations, a continuous-time method is proposed by assuming that the continuous-time hazard rate is piecewise constant. The method, named PC-Hazard, is found to be highly competitive with the aforementioned methods in addition to other methods for survival prediction found in the literature.

Time-to-Event Prediction with Neural Networks and Cox Regression

New methods for time-to-event prediction are proposed by extending the Cox proportional hazards model with neural networks. Building on methodology from nested case-control studies, we propose a loss function that scales well to large data sets, and enables fitting of both proportional and non-proportional extensions of the Cox model. Through simulation studies, the proposed loss function is verified to be a good approximation for the Cox partial log-likelihood. The proposed methodology is compared to existing methodologies on real-world data sets, and is found to be highly competitive, typically yielding the best performance in terms of Brier score and binomial log-likelihood. A python package for the proposed methods is available at https://github.com/havakv/pycox.