Causal progress with imperfect placebo treatments and outcomes

In the quest to make defensible causal claims from observational data, it is sometimes possible to leverage information from "placebo treatments" and "placebo outcomes" (or "negative outcome controls"). 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 non-zero effects of placebo treatment on real outcomes or of real treatments on placebo outcomes, and the relative strengths of confounding suffered by a placebo treatment/outcome compared to the true treatment-outcome relationship. Once postulated, these assumptions identify or bound the linear estimates of treatment effects. While applicable in many settings, one ubiquitous use-case for this approach is to employ pre-treatment outcomes as (perfect) placebo outcomes. In this setting, the parallel trends assumption of difference-in-difference is in fact a strict equiconfounding assumption on a particular scale, which can be relaxed in our framework. Finally, we demonstrate the use of our framework with two applications, employing an R package that implements these approaches.