Bespoke instrumental variables with nonideal reference populations

Recently, a bespoke instrumental variable method was proposed, which, under certain assumptions, can eliminate bias due to unmeasured confounding when estimating the causal exposure effect among the exposed. This method uses data from both the study population of interest and a reference population in which the exposure is completely absent. In this article, we extend the bespoke instrumental variable method to allow for a nonideal reference population that may include exposed individuals. Such an extension is particularly important in randomized trials with nonadherence, where even individuals in the control arm may have access to the treatment under investigation. We further scrutinize the assumptions underlying the bespoke instrumental method and caution the reader about the potential nonrobustness of the method to these assumptions.