Deep Neural Networks for Survival Analysis Using Pseudo Values

Abstract

There has been increasing interest in modelling survival data using deep learning methods in medical research. Current approaches have focused on designing special cost functions to handle censored survival data. We propose a very different method with two simple steps. In the first step, we transform each subject's survival time into a series of jackknife pseudo conditional survival probabilities and then use these pseudo probabilities as a quantitative response variable in the deep neural network model. By using the pseudo values, we reduce a complex survival analysis to a standard regression problem, which greatly simplifies the neural network construction. Our two-step approach is simple, yet very flexible in making risk predictions for survival data, which is very appealing from the practice point of view. The source code is freely available at http://github.com/lilizhaoUM/DNNSurv.

Fig. 1.

DNNSurv Architecture with two fully connected hidden layers.

Fig. 2.

Boxplots of c-index (a) and Brier score (b) from DNNSurv, DeepSurv, Cox-nnet, and nnet-survival over 100 simulated datasets generated from the AFT model with Friedman’s random function generator with 40% censoring. The c-index and Brier scores were evaluated at six time points, which were determined from the six percentiles of the empirical survival distribution.

Fig. 3.

Boxplots of c-index (a) and Brier score (b) for the 100 simulated datasets generated from the simple Cox model in the case of covariate-dependent censoring.