Comparison of median survival times with adjustment for covariates

Brookmeyer and Crowley derived a non-parametric confidence interval for the median survival time of a homogeneous population by inverting a generalization of the sign test for censored data. The 1 - alpha confidence interval for the median is essentially the set of all values t such that the Kaplan-Meier estimate of the survival curve at time t does not differ significantly from one-half at the two-sided alpha level. Su and Wei extended this approach to the two-sample problem and derived a confidence interval for the difference in median survival times based on the Kaplan-Meier estimates of the individual survival curves and a 'minimum dispersion' test statistic. Here, I incorporate covariates into the analysis by assuming a proportional hazards model for the covariate effects, while leaving the two underlying survival curves virtually unconstrained. I generate a simultaneous confidence region for the two median survival times, adjusted to any selected value, z, of the covariate vector using a test-based approach analogous to Brookmeyer and Crowley's for the one-sample case. This region is, in turn, used to derive a confidence interval for the difference in median survival times between the two treatment groups at the selected value of z. Employment of a procedure suggested by Aitchison sets the level of the simultaneous region to a value that should yield, at least approximately, the desired confidence coefficient for the difference in medians. Simulation studies indicate that the method provides reasonably accurate coverage probabilities.