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

Abstract

“M-Bias”, as it is called in the epidemiological 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 Gaussian 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 to allow for the correlation between the latent factors to be nonzero, and to allow for the collider to also 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.