Effect modification in a randomized trial under non‐ignorable non‐compliance: an application to the alpha‐tocopherol beta‐carotene study

We propose an inference procedure that allows an estimation of the effects of treatment modifying factors on a survival end point in a placebo‐controlled randomized trial with non‐ignorable non‐compliance. Against prior expectation the alpha‐tocopherol beta‐carotene (ATBC) study showed an increase in the incidence of lung cancer and total mortality following supplementation with beta‐carotene. Any attempt to assess interactions between beta‐carotene supplementation and base‐line covariates needs to address the issue of non‐compliance. In the ATBC study about 25% of the participants dropped out from the active follow‐up prematurely for reasons other than death and they stopped receiving the study supplementation. Survival status was observed regardless of drop‐out for all participants. Drop‐out is related to base‐line covariates, to the treatment arm and presumably also to the health status of the individuals through both measured and unmeasured factors. Non‐ignorable non‐compliance is induced by unmeasured covariates related both to drop‐out from active participation in the study and to health status and death. Instead of comparing survival between the randomized groups we focus on an estimation of the effect of beta‐carotene actually received and its interaction with base‐line covariates such as age, smoking and alcohol consumption. We use an accelerated failure time model to link a potential treatment‐free survival time to the actual observed survival time and to the treatment received and its interactions with base‐line covariates. We refer to this as the structural model. The estimation of the parameters in the structural model is based on minimizing a score test statistic obtained from the partial likelihood specified for the treatment‐free survival time under the structural model. We compare the method proposed with Robins's g‐estimation based on a linear rank test statistic. If the structural model holds, then the method proposed can substantially increase efficiency. We propose diagnostic techniques for evaluating the adequacy of the structural model and discuss how a misspecification of the model may induce bias into the score equations.