Sharp bounds for causal effects based on Ding and VanderWeele's sensitivity parameters

In a seminal article, Ding and VanderWeele proposed a method of constructing bounds for causal effects that has become widely recognized in causal inference. This method requires the analyst to provide guesses of certain “sensitivity parameters,” loosely defined as the maximal strength of association that an unmeasured confounder may have with the exposure and with the outcome. Ding and VanderWeele stated that their bounds are sharp, but without defining this term. Using a common definition of sharpness, Sjölander A. A note on a sensitivity analysis for unmeasured confounding, and the related E-value. J Causal Inference. 2020;8(1):229–48 showed that Ding and VanderWeele’s bounds are sharp in some regions of the sensitivity parameters, but are non-sharp in other regions. In this note, we follow up the work by Sjölander A. A note on a sensitivity analysis for unmeasured confounding, and the related E-value. J Causal Inference. 2020;8(1):229–48, by deriving bounds that are guaranteed to be sharp in all regions of Ding and VanderWeele’s sensitivity parameters. We illustrate the discrepancy between Ding and VanderWeele’s bounds and the sharp bounds with a real data example on vitamin D insufficiency and urine incontinence in pregnant women.