Improving the Efficiency of the Log-rank Test Using Auxiliary Covariates

Under the assumption of proportional hazards, the log-rank test is optimal for testing the null hypothesis of H0 : β = 0, where β denotes the logarithm of the hazard ratio. However, if there are additional covariates that correlate with survival times, making use of their information will increase the efficiency of the log-rank test. We apply the theory of semiparametrics to characterize a class of regular and asymptotically linear estimators for β when auxiliary covariates are incorporated into the model, and derive estimators that are more efficient. The Wald tests induced by these estimators are shown to be more powerful than the log-rank test. Simulation studies are used to illustrate the gains in efficiency. Some key words: Efficient estimator; Influence function; Log-rank test; Nuisance tangent space; Proportional hazard model; Regular and asymptotically linear estimator.