  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.
