Bounds on potential risks and causal risk differences under assumptions about confounding parameters

Nonparametric bounds on causal effects in observational studies are available under deterministic potential-outcome models. We derive narrower bounds by adding assumptions regarding bias due to confounding. This bias is defined as the difference between the expectation of potential outcomes for the exposed group and that for the unexposed group. We show that crude effect measures bound causal effects under the given assumptions. We then derive bounds for randomized studies with noncompliance, which are given by the per protocol effect. With perfect compliance in one treatment group, the direction of effect becomes identifiable under our assumptions. Although the assumptions are not themselves identifiable, they are nonetheless reasonable in some situations.