Estimating treatment effect in a proportional hazards model in randomized clinical trials with all‐or‐nothing compliance

Summary We consider methods for estimating the treatment effect and/or the covariate by treatment interaction effect in a randomized clinical trial under noncompliance with time‐to‐event outcome. As in Cuzick et al. (2007), assuming that the patient population consists of three (possibly latent) subgroups based on treatment preference: the ambivalent group, the insisters , and the refusers , we estimate the effects among the ambivalent group. The parameters have causal interpretations under standard assumptions. The article contains two main contributions. First, we propose a weighted per‐protocol (Wtd PP) estimator through incorporating time‐varying weights in a proportional hazards model. In the second part of the article, under the model considered in Cuzick et al. (2007), we propose an EM algorithm to maximize a full likelihood (FL) as well as the pseudo likelihood (PL) considered in Cuzick et al. (2007). The E step of the algorithm involves computing the conditional expectation of a linear function of the latent membership, and the main advantage of the EM algorithm is that the risk parameters can be updated by fitting a weighted Cox model using standard software and the baseline hazard can be updated using closed‐form solutions. Simulations show that the EM algorithm is computationally much more efficient than directly maximizing the observed likelihood. The main advantage of the Wtd PP approach is that it is more robust to model misspecifications among the insisters and refusers since the outcome model does not impose distributional assumptions among these two groups.