Residual-based tree-structured survival analysis

Extensions of various non-parametric regression techniques (for example, additive models, trees, MARS) have been devised for right censored survival data. These approaches directly handle the difficulties posed by censoring. However, it is possible to bypass these difficulties by utilizing standard non-parametric regression procedures applied with (say) martingale residuals as outcome. Analytic correspondences between the direct and residual-based approaches have been established for additive models while more qualitative comparisons have been provided for MARS. Here we develop such correspondences for tree-structured regression. In particular, we provide an analytic relationship between logrank and martingale residual sum-of-squares split functions that explains the widely observed similarity of the resultant trees. Further investigation is provided by simulation and an illustrative example using time to AIDS with data deriving from a Western Australian HIV cohort study.