Nonparametric covariate adjustment is considered for log-rank type tests of
treatment effect with right-censored time-to-event data from clinical trials
applying covariate-adaptive randomization. Our proposed covariate-adjusted
log-rank test has a simple explicit formula and a guaranteed efficiency gain
over the unadjusted test. We also show that our proposed test achieves
universal applicability in the sense that the same formula of test can be
universally applied to simple randomization and all commonly used
covariate-adaptive randomization schemes such as the stratified permuted block
and Pocock and Simon's minimization, which is not a property enjoyed by the
unadjusted log-rank test. Our method is supported by novel asymptotic theory
and empirical results for type I error and power of tests