Unobserved covariates in the two-sample comparison of survival times: a maximum efficiency robust test

In analysing a clinical trial with the logrank test, the hazards between the two groups are usually assumed to be proportional. Nevertheless, this hypothesis is no longer valid with unobserved covariates. As a consequence, there is a loss of power of the logrank test for testing the null hypothesis H(0) of no treatment effect. We propose a test suited for taking into account unobserved covariates. The proposed approach is based on a proportional hazard frailty model whereby the omitted covariates are considered as an unobserved frailty variable. The procedure is as follows. In a first step, the weighted logrank test optimal for testing H(0) against a general proportional hazard frailty model is obtained and its specialization for a gamma frailty variable is derived. In a second step, the proposed test is obtained by combining the maximin efficiency robustness principle and the gamma frailty distribution properties. Simulation studies investigate the power properties of the test for different frailty distributions. A breast cancer clinical trial is analysed as an example. The proposed test might be recommended rather than the logrank for practical situations in which one expects heterogeneity related to omitted covariates.