On Weighted Log-Rank Combination Tests and Companion Cox Model Estimators

In randomized clinical trials, the log-rank test and Cox proportional hazards model are the gold standard in survival data analyses. While the log-rank test is generally valid, in the presence of non-proportional hazards, the power can be substantially decreased relative to the proportional hazards assumptions under which studies are usually designed. In contrast, weighted log-rank tests can be more powerful for specific treatment differences under non-proportional hazards scenarios. However, a poor choice of the weighting form can be detrimental. Recent work on combining various weighted log-rank tests allows for tests that are capable of detecting treatment effects across a broad range of non-proportional hazards scenarios. In this paper, we expand on these ideas with a framework based on a flexible resampling approach [ 5 ] which allows for the combination of various testing procedures in addition to weighted log-rank tests. In particular, we describe how tests based on restricted mean survival time (RMST) comparisons can be included within combinations of weighted log-rank tests as well as other test statistics such as Tarone-Ware and Renyi-type supremum families. For estimation, we propose companion weighted Cox model estimators [ 14 , 21 , 22 ] which utilize the weighting form that is “selected” through the combination test and provide simultaneous confidence intervals. The performance of various combinations and their companion Cox estimators as well as RMST are evaluated in simulation studies under null, proportional hazards, late-separation, and early-separation scenarios. We find the combination tests perform quite well in controlling type-1 error rates and in achieving higher power than individual tests across the scenarios considered here. We suggest the companion Cox estimators are a natural link to the testing procedures and can be a useful complementary summary of treatment effects with careful interpretation. For illustration, we apply the proposals to a randomized clinical trial study of the PD-L1-targeted therapy atezolizumab in comparison with docetaxel in previously treated non-small-cell lung cancer patients. R code can be found at https://github.com/larryleon/combination-tests-and-estimators .