Ensemble Methods for Survival Data with Time-Varying Covariates

Survival data with time-varying covariates are common in practice. If relevant, such covariates can improve on the estimation of a survival function. However, the traditional survival forests - conditional inference forest, relative risk forest and random survival forest - have accommodated only time-invariant covariates. We generalize the conditional inference and relative risk forests to allow time-varying covariates. We compare their performance with that of the extended Cox model, a commonly used method, and the transformation forest method, designed to detect non-proportional hazards deviations and adapted here to accommodate time-varying covariates, through a comprehensive simulation study in which the Kaplan-Meier estimate serves as a benchmark and the integrated L2 difference between the true and estimated survival functions is used for evaluation. In general, the performance of the two proposed forests substantially improves over the Kaplan-Meier estimate. Under the proportional-hazard setting, the best method is always one of the two proposed forests, while under the non-proportional hazards setting, it is the adapted transformation forest. We use K-fold cross-validation to choose between the methods, which is shown to be an effective tool to provide guidance in practice. The performance of the proposed forest methods for time-invariant covariate data is broadly similar to that found for time-varying covariate data.