Bounds on causal effects in randomized trials with noncompliance under monotonicity assumptions about covariates

In randomized trials with nonrandom noncompliance, the causal effects of a treatment among the entire population cannot be estimated in an unbiased manner. Therefore, several authors have considered the bounds on the causal effects. Here, we propose bounds by applying an idea of VanderWeele (Biometrics 2008; 64:702-706), who showed that the sign of the unmeasured confounding bias can be determined under monotonicity assumptions about covariates in the framework of observational studies. In randomized trials with noncompliance by switching the treatment, we show that the lower or upper bound on the expectation of the potential outcome becomes the expectation from the per-protocol analysis under monotonicity assumptions similar to those of VanderWeele. In particular, the monotonicity assumptions can yield both the lower and the upper bounds on causal effects when the monotonic relationship between the covariates and the treatment actually received depends on the treatment assigned. The results are extended to cases of noncompliance by subjects not receiving any treatment. Although the monotonicity assumptions are not themselves identifiable, they are nonetheless reasonable in some situations.