Under the assumption of proportional hazards, the log-rank test is optimal for testing the null hypothesis <inline-formula><inline-graphic xlink:href="asn003ilm1.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula>, where <inline-formula><inline-graphic xlink:href="asn003ilm2.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula> 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...