Robust Mendelian Randomization with MR-Egger Regression

An alternative robust method for Mendelian randomization with summary data has been recently proposed by Bowden et al. [2015], referred to as “MR‐Egger regression.” This approach was motivated from a method in the meta‐analysis literature for the assessment of small‐study bias (often called “publication bias”) (Egger et al., 1997). This performs a weighted linear regression of the gene‐outcome coefficients Γ^j on the gene‐exposure coefficients γ^j:
${\widehat{\Gamma }}_j={\beta }_{0E}+{\beta }_E{\widehat{\gamma }}_j$ in which all the γ^j associations are orientated to be positive (the orientation of the Γ^j associations should be altered if necessary to match the orientation of the γ^j parameters), and the weights in the regression are the inverse variances of the gene‐outcome associations (σYj−2). Reorientation of the variants is performed as the orientation of genetic variants is arbitrary (i.e., estimates can be presented with reference to either the major or minor allele), and different orientations of genetic variants change the estimate of the intercept, as well as the sign and magnitude of the pleiotropic effect of the genetic variant. If there is no intercept term in the regression model, then the MR‐Egger slope estimate β^E will equal the IVW estimate (Burgess et al., 2015a).
The value of the intercept term β^0E can be interpreted as an estimate of the average pleiotropic effect across the genetic variants (Bowden et al., 2015). The pleiotropic effect is the effect of the genetic variant on the outcome that is not mediated via the exposure. An intercept term that differs from zero is indicative of overall directional pleiotropy; that is, pleiotropic effects do not cancel out and the IVW estimate is biased.
MR‐Egger regression additionally provides an estimate for the true causal effect β^E that is consistent even if all genetic variants are invalid due to violation of IV3, but under a weaker assumption known as the InSIDE (instrument strength independent of direct effect) assumption. If the association of the jth genetic variant with the outcome Γj=βγj+αj, where αj is the pleiotropic (direct) effect of the variant, then the InSIDE assumption states that the pleiotropic effects αj must be distributed independently of the instrument strength parameters γj (Kolesár et al., 2014). (Formally, the consistency property holds both as the sample size and the number of instruments increases. For a fixed number of instruments, consistency only holds asymptotically if the correlation between the αj and γj parameters is zero.) The InSIDE assumption is likely to be satisfied if pleiotropic effects on the outcome are direct (i.e., not via a confounder). There is some empirical evidence supporting the proposition that genetic effects on separate exposures are independent (Pickrell, 2015). However, if the pleiotropic effects of genetic variants are all via a single confounder, then they will be correlated with instrument strength, and the InSIDE assumption will be violated.