M-Bias and Butterfly-Bias in the Gaussian Linear Structural Equation Models

• Activity Number: 491
• Type: Contributed
• Date/Time: Wednesday, August 7, 2013 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistics in Epidemiology
• Author(s): Peng Ding*+ and Luke Miratrix
• Companies: Harvard University and Harvard University
• Keywords: Causality ; Collider ; Directed acyclic graph ; Stratification ; V structure

Abstract

``M-Bias'', as it is called in the epidemiological literature, is the bias introduced by conditioning on a pretreatment variable due to a particular ``M-structure'' between two unobserved variables and an observed treatment, outcome, and collider. This potential source of bias, which can occur even when the treatment and the outcome are not confounded, has been the source of considerable controversy. We show the magnitude of the M-Bias in Gaussian linear structural equation models is relatively small compared to confounding bias, suggesting that it would generally not be a serious concern in applied settings. These formulae allow for identifying under which circumstances bias is inflated or reduced. Our theoretic 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 an actual confounder between the treatment and the outcome. Deviations from the M-Bias structure change the level of bias, shedding light on whether we should condition on a given pretreatment covariate or not.