Estimation of treatment effects based on possibly misspecified Cox regression

In randomized clinical trials, a treatment effect on a time-to-event endpoint is often estimated by the Cox proportional hazards model. The maximum partial likelihood estimator does not make sense if the proportional hazard assumption is violated. Xu and O'Quigley (Biostatistics 1:423-439, 2000) proposed an estimating equation, which provides an interpretable estimator for the treatment effect under model misspecification. Namely it provides a consistent estimator for the log-hazard ratio among the treatment groups if the model is correctly specified, and it is interpreted as an average log-hazard ratio over time even if misspecified. However, the method requires the assumption that censoring is independent of treatment group, which is more restricted than that for the maximum partial likelihood estimator and is often violated in practice. In this paper, we propose an alternative estimating equation. Our method provides an estimator of the same property as that of Xu and O'Quigley under the usual assumption for the maximum partial likelihood estimation. We show that our estimator is consistent and asymptotically normal, and derive a consistent estimator of the asymptotic variance. If the proportional hazards assumption holds, the efficiency of the estimator can be improved by applying the covariate adjustment method based on the semiparametric theory proposed by Lu and Tsiatis (Biometrika 95:679-694, 2008).