Adjusting for observational secondary treatments in estimating the effects of randomized treatments

In randomized clinical trials, for example, on cancer patients, it is not uncommon that patients may voluntarily initiate a secondary treatment postrandomization, which needs to be properly adjusted for in estimating the "true" effects of randomized treatments. As an alternative to the approach based on a marginal structural Cox model (MSCM) in Zhang and Wang [(2012). Estimating treatment effects from a randomized trial in the presence of a secondary treatment. Biostatistics 13, 625-636], we propose methods that treat the time to start a secondary treatment as a dependent censoring process, which is handled separately from the usual censoring such as the loss to follow-up. Two estimators are proposed, both based on the idea of inversely weighting by the probability of having not started a secondary treatment yet. The second estimator focuses on improving efficiency of inference by a robust covariate-adjustment that does not require any additional assumptions. The proposed methods are evaluated and compared with the MSCM-based method in terms of bias and variance tradeoff using simulations and application to a cancer clinical trial.