To Adjust or Not to Adjust? Sensitivity Analysis of M -Bias and Butterfly-Bias

Abstract “ M -Bias,” as it is called in the epidemiologic literature, is the bias introduced by conditioning on a pretreatment covariate due to a particular “ M -Structure” between two latent factors, an observed treatment, an outcome, and a “collider.” This potential source of bias, which can occur even when the treatment and the outcome are not confounded, has been a source of considerable controversy. We here present formulae for identifying under which circumstances biases are inflated or reduced. In particular, we show that the magnitude of M -Bias in linear structural equation models tends to be relatively small compared to confounding bias, suggesting that it is generally not a serious concern in many applied settings. These theoretical results are consistent with recent empirical findings from simulation studies. We also generalize the M -Bias setting (1) to allow for the correlation between the latent factors to be nonzero and (2) to allow for the collider to be a confounder between the treatment and the outcome. These results demonstrate that mild deviations from the M -Structure tend to increase confounding bias more rapidly than M -Bias, suggesting that choosing to condition on any given covariate is generally the superior choice. As an application, we re-examine a controversial example between Professors Donald Rubin and Judea Pearl.