Discrete-time survival forests with Hellinger distance decision trees

Abstract

Random survival forests (RSF) are a powerful nonparametric method for building prediction models with a time-to-event outcome. RSF do not rely on the proportional hazards assumption and can be readily applied to both low- and higher-dimensional data. A remaining limitation of RSF, however, arises from the fact that the method is almost entirely focussed on continuously measured event times. This issue may become problematic in studies where time is measured on a discrete scale t = 1, 2, ..., referring to time intervals [0,a_1), [a_1,a_2), ldots . In this situation, the application of methods designed for continuous time-to-event data may lead to biased estimators and inaccurate predictions if discreteness is ignored. To address this issue, we develop a RSF algorithm that is specifically designed for the analysis of (possibly right-censored) discrete event times. The algorithm is based on an ensemble of discrete-time survival trees that operate on transformed versions of the original time-to-event data using tree methods for binary classification. As the outcome variable in these trees is typically highly imbalanced, our algorithm implements a node splitting strategy based on Hellinger’s distance, which is a skew-insensitive alternative to classical split criteria such as the Gini impurity. The new algorithm thus provides flexible nonparametric predictions of individual-specific discrete hazard and survival functions. Our numerical results suggest that node splitting by Hellinger’s distance improves predictive performance when compared to the Gini impurity. Furthermore, discrete-time RSF improve prediction accuracy when compared to RSF approaches treating discrete event times as continuous in situations where the number of time intervals is small.