Negative-Control Exposures: Adjusting for Unmeasured and Measured Confounders With Bounds for Remaining Bias

Negative-control exposures can be used to detect and even adjust for confounding that remains after control of measured confounders. A newly described method allows the analyst to reduce residual confounding by unmeasured confounders U by using negative-control exposures to define and select a subcohort wherein the U-distribution among the exposed is similar to that among the unexposed. Here, we show that conventional methods can be used to control for measured confounders in conjunction with the new method to control for unmeasured ones. We also derive an expression for bias that remains after applying this approach. We express remaining bias in terms of a "balancing" parameter and show that this parameter is bounded by a summary variational distance between the U-distribution in the exposed and the unexposed. These measures describe and bound the extent of remaining confounding after using negative controls to adjust for unmeasured confounders with conventional control of measured confounders.