Causal progress with imperfect placebo treatments and outcomes

Abstract: In the quest to make defensible causal claims from observational data, it is sometimes possible to leverage information from "placebo treatments" and "placebo outcomes". Existing approaches employing such information focus largely on point identification and assume (i) "perfect placebos", meaning placebo treatments have precisely zero effect on the outcome and the real treatment has precisely zero effect on a placebo outcome; and (ii) "equiconfounding", meaning that the treatment-outcome relationship where one is a placebo suffers the same amount of confounding as does the real treatment-outcome relationship, on some scale. We instead consider an omitted variable bias framework, in which users can postulate ranges of values for the degree of unequal confounding and the degree of placebo imperfection. Once postulated, these assumptions identify or bound the linear estimates of treatment effects. Our approach also does not require using both a placebo treatment and placebo outcome, as some others do. While applicable in many settings, one ubiquitous use-case for this approach is to employ pre-treatment outcomes as (perfect) placebo outcomes, as in difference-in-difference. The parallel trends assumption in this setting is identical to the equiconfounding assumption, on a particular scale, which our framework allows the user to relax. Finally, we demonstrate the use of our framework with two applications and a simulation, employing an R package that implements these approaches.