A log-rank test for equivalence of two survivor functions

We consider a hypothesis testing problem in which the alternative states that the vertical distance between the underlying survivor functions nowhere exceeds some prespecified bound delta > 0. Under the assumption of proportional hazards, this hypothesis is shown to be (logically) equivalent to the statement [beta[ < log(1 + epsilon), where beta denotes the regression coefficient associated with the treatment group indicator, and epsilon is a simple strictly increasing function of delta. The testing procedure proposed consists of carrying out in terms of beta (i.e., the standard Cox likelihood estimator of beta) the uniformly most powerful level alpha test for a suitable interval hypothesis about the mean of a Gaussian distribution with fixed variance. The computation of the critical constant of this test is very easy in practice since it admits a representation as the root of the alpha th quantile of a noncentral chi-square distribution with a single degree of freedom.