Boosted nonparametric hazards with time-dependent covariates
This work addresses survival analysis for researchers dealing with time-dependent covariates, offering a nonparametric method with theoretical guarantees, though it appears incremental as it builds on existing boosting techniques.
The authors tackled the problem of estimating hazard functions with time-dependent covariates in survival analysis by deriving a smooth convex representation for the nonparametric log-likelihood and developing a gradient boosting procedure, achieving consistency under correct model specification and an oracle inequality for tree-based models.
Given functional data from a survival process with time-dependent covariates, we derive a smooth convex representation for its nonparametric log-likelihood functional and obtain its functional gradient. From this, we devise a generic gradient boosting procedure for estimating the hazard function nonparametrically. An illustrative implementation of the procedure using regression trees is described to show how to recover the unknown hazard. The generic estimator is consistent if the model is correctly specified; alternatively, an oracle inequality can be demonstrated for tree-based models. To avoid overfitting, boosting employs several regularization devices. One of them is step-size restriction, but the rationale for this is somewhat mysterious from the viewpoint of consistency. Our work brings some clarity to this issue by revealing that step-size restriction is a mechanism for preventing the curvature of the risk from derailing convergence.