Survival Analysis with Time-Varying Regression Effects Using a TreeBased Approach

Nonproportional hazards often arise in survival analysis, as is evident in the data from the International Non-Hodgkinʹs Lymphoma Prognostic Factors Project. A tree-based method to handle such survival data is developed for the assessment and estimation of timedependent regression effects under a Cox-type model. The tree method approximates the time-varying regression effects as piecewise constants and is designed to estimate change points in the regression parameters. A fast algorithm that relies on maximized score statistics is used in recursive segmentation of the time axis. Following the segmentation, a pruning algorithm with optimal properties similar to those of classification and regression trees (CART) is used to determine a sparse segmentation. Bootstrap resampling is used in correcting for overoptimism due to split point optimization. The piecewise constant model is often more suitable for clinical interpretation of the regression parameters than the more flexible spline models. The utility of the algorithm is shown on the lymphoma data, where we further develop the published International Risk Index into a time-varying risk index for non-Hodgkinʹs lymphoma.