Abstract
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.
Highlights
The hallmark of an observational study is selection bias (Heckman, 1979, Copas and Li, 1997, Hernan, Hernandez-Diaz, and Robins, 2004)
Rubin and Rosenbaum suggest balancing all the pretreatment covariate in observational studies to parallel with the design of randomized experiments (Rubin, 2007, 2008, 2009, Rosenbaum, 2002), which is called the “pretreatment criterion” (VanderWeele and Shpitser, 2011)
Pearl and other researchers (Pearl, 2009b,c, Shrier, 2008, 2009, Sjolander, 2009) criticize the “pretreatment criterion” by pointing out that this criterion may lead to biased inference in presence of a possible M-Structure even if the treatment assignment is unconfounded
Summary
The hallmark of an observational study is selection bias (Heckman, 1979, Copas and Li, 1997, Hernan, Hernandez-Diaz, and Robins, 2004). Many statisticians believe that “there is no reason to avoid adjustment for a variable describing subjects before treatment” in observational studies (Rosenbaum, 2002, pp 76), because “typically, the more conditional an assumption, the more generally acceptable it is” (Rubin, 2009). Sprites (2002) uses linear models to illustrate M-Bias in observational studies, and Pearl (2013) utilize the transparency of such linear models to examine various types of causal phenomena, biases, and paradoxes. We here extend these works and provide exact formulae for biases, allowing for a more detailed quantitative analysis of M-bias. We conclude with a brief discussion and present all technical details in the Appendix
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