Adaptive test for testing the difference in survival distributions

An adaptive test is proposed for the problem of testing the difference in survival distributions when the shape of the hazard ratio is unknown, hence the efficient test is unknown. The proposed adaptive test selects a test statistic from a finite set of the weighted logrank statistics T on the basis of the estimates of the efficiencies of the tests in T for given data. The efficiency estimator uses the length of the test based nonparametric confidence interval for the shift in a time transformed shift model. The suggested adaptive test is shown to be asymptotically efficient among the tests in T under the time transformed shift model and conditions commonly used in survival analysis. Simulations demonstrate that the adaptive test enjoys good small sample properties and in most situations is more powerful than the test using the maximum of the tests in T.