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

  • Ding P
  • Miratrix L
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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.

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Ding, P., & Miratrix, L. W. (2014). To Adjust or Not to Adjust? Sensitivity Analysis of M-Bias and Butterfly-Bias. Journal of Causal Inference, 3(1), 41–57. https://doi.org/10.1515/jci-2013-0021

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