Higher criticism for rare and weak non-proportional hazard deviations in survival analysis

Abstract We propose a method for comparing survival data based on higher criticism of $ p $-values obtained from many exact hypergeometric tests. The method accommodates noninformative right-censorship and is sensitive to hazard differences in unknown and relatively rare time intervals. It attains much better power against such differences than the log-rank test and its variants. We demonstrate the usefulness of our method in detecting rare and weak non-proportional hazard differences compared to existing tests using simulations and gene expression data. Additionally, we analyse the asymptotic power of our method and other tests under a theoretical framework describing two groups experiencing failure rates that are usually identical over time, except in a few unknown instances where one group’s failure rate is higher. Our test’s power experiences a phase transition across the plane of rarity and intensity parameters that mirrors the phase transition of higher criticism in two-sample rare and weak normal and Poisson means settings. The region of the plane in which our method has asymptotically full power is larger than the corresponding region for the log-rank test.