Instruments and Bounds for Causal Effects under the Monotonic Selection Assumption

Noncompliance with assigned treatment is an important problem of randomized clinical trials. In this situation, the structural mean model (SMM) approach focuses on the average treatment effect among patients actually treated (ATT). In contrast, the principal stratification (PS) approach addresses the effect on a certain subgroup defined by latent compliance behavior. While these approaches target different causal effects, the estimators have the same form as the classical instrumental variable estimator, under the assumption of no effect modification (NEM) and monotonic selection. In this article, we clarify the relation between SMM and PS under the monotonic selection assumption. Specifically, we translate the NEM assumption for the SMM estimator into the words of the PS approach. Then, we propose a new bound for the ATT by making a possibly more plausible assumption than the NEM assumption based on the PS approach. Furthermore, we extend these results to the average treatment effect for the entire population. The proposed bounds are illustrated with applications to a real clinical trial data. Although our assumption cannot be empirically verified, the proposed bounds can be considerably tighter than those previously proposed.