Survival analysis is a type of semi-supervised ranking task where the target output (the survival time) is often right-censored. Utilizing this information is a challenge because it is not obvious how to correctly incorporate these censored examples into a model. We study how three categories of loss functions, namely partial likelihood methods, rank methods, and our classification method based on a Wasserstein metric (WM) and the non-parametric Kaplan Meier estimate of the probability density to impute the labels of censored examples, can take advantage of this information. The proposed method allows us to have a model that predict the probability distribution of an event. If a clinician had access to the detailed probability of an event over time this would help in treatment planning. For example, determining if the risk of kidney graft rejection is constant or peaked after some time. Also, we demonstrate that this approach directly optimizes the expected C-index which is the most common evaluation metric for ranking survival models.

An accurate model of patient-specific kidney graft survival distributions can help to improve shared-decision making in the treatment and care of patients. In this paper, we propose a deep learning method that directly models the survival function instead of estimating the hazard function to predict survival times for graft patients based on the principle of multi-task learning. By learning to jointly predict the time of the event, and its rank in the cox partial log likelihood framework, our deep learning approach outperforms, in terms of survival time prediction quality and concordance index, other common methods for survival analysis, including the Cox Proportional Hazards model and a network trained on the cox partial log-likelihood.