The sign of the unmeasured confounding bias under various standard populations

Unmeasured confounders are a common problem in drawing causal inferences in observational studies. VanderWeele (Biometrics 2008, 64, 702-706) presented a theorem that allows researchers to determine the sign of the unmeasured confounding bias when monotonic relationships hold between the unmeasured confounder and the treatment, and between the unmeasured confounder and the outcome. He showed that his theorem can be applied to causal effects with the total group as the standard population, but he did not mention the causal effects with treated and untreated groups as the standard population. Here, we extend his results to these causal effects, and apply our theorems to an observational study. When researchers have a sense of what the unmeasured confounder may be, conclusions can be drawn about the sign of the bias.