Subgroup Discovery with the Cox Model

We study the problem of subgroup discovery for survival analysis, where the goal is to find an interpretable subset of the data on which a Cox model is highly accurate. Our work is the first to study this particular subgroup problem, for which we make several contributions. Subgroup discovery methods generally require a "quality function" in order to sift through and select the most advantageous subgroups. We first examine why existing natural choices for quality functions are insufficient to solve the subgroup discovery problem for the Cox model. To address the shortcomings of existing metrics, we introduce two technical innovations: the expected prediction entropy (EPE), a novel metric for evaluating survival models which predict a hazard function; and the conditional rank statistics (CRS), a statistical object which quantifies the deviation of an individual point to the distribution of survival times in an existing subgroup. We study the EPE and CRS theoretically and show that they can solve many of the problems with existing metrics. Having established the fundamentals of the problem, we then turn to methodology for solving it. To this end, we introduce a total of eight algorithms for the Cox subgroup discovery problem. The main algorithm, which is based on the DDGroup framework of [22] , is able to take advantage of both the EPE and the CRS, allowing us to give theoretical correctness results for this algorithm in a well-specified setting. We evaluate all of the proposed methods empirically on both synthetic and real data. The experiments confirm our theory, showing that our contributions allow for the recovery of a ground-truth subgroup in well-specified cases, as well as leading to better model fit compared to naively fitting the Cox model to the whole dataset in practical settings. Lastly, to showcase the utility of the subgroups themselves beyond improving model predictions, we conduct a case study on jet engine simulation data from NASA. The discovered subgroups uncover known nonlinearities/homogeneity in the data, and which suggest design choices which have been mirrored in practice.