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

Author

Listed:
  • Ding Peng

    (Department of Statistics, Harvard University, Cambridge, MA 02138, USA)

  • Miratrix Luke W.

    (Department of Statistics, Harvard University, Cambridge, MA 02138, USA)

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.

Suggested Citation

  • Ding Peng & Miratrix Luke W., 2015. "To Adjust or Not to Adjust? Sensitivity Analysis of M-Bias and Butterfly-Bias," Journal of Causal Inference, De Gruyter, vol. 3(1), pages 41-57.
  • Handle: RePEc:bpj:causin:v:3:y:2015:i:1:p:41-57:n:1003
    DOI: 10.1515/jci-2013-0021
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