Piecewise-linear criterion functions in oblique survival tree induction

Recursive partitioning is a common, assumption-free method of survival data analysis. It focuses mainly on univariate trees, which use splits based on a single variable in each internal node. In this paper, I provide an extension of an oblique survival tree induction technique, in which axis-parallel splits are replaced by hyperplanes, dividing the feature space into areas with a homogeneous survival experience.

The proposed tree induction algorithm consists of two steps. The first covers the induction of a large tree with internal nodes represented by hyperplanes, whose positions are calculated by the minimization of a piecewise-linear criterion function, the dipolar criterion. The other phase uses a split-complexity algorithm to prune unnecessary tree branches and a 10-fold cross-validation technique to choose the best tree. The terminal nodes of the final tree are characterised by Kaplan-Meier survival functions. A synthetic data set was used to test the performance, while seven real data sets were exploited to validate the proposed method.

The evaluation of the method was focused on two features: predictive ability and tree size. These were compared with two univariate tree models: the conditional inference tree and recursive partitioning for survival trees, respectively. The comparison of the predictive ability, expressed as an integrated Brier score, showed no statistically significant differences (p=0.486) among the three methods. Similar results were obtained for the tree size (p=0.11), which was calculated as a median value over 20 runs of a 10-fold cross-validation.

The predictive ability of trees generated using piecewise-linear criterion functions is comparable to that of univariate tree-based models. Although a similar conclusion may be drawn from the analysis of the tree size, in the majority of the studied cases, the number of nodes of the dipolar tree is one of the smallest among all the methods.