2.2. Revisit Egger Regression and MR-Egger

Let Γ~j and γ~j denote the coefficient estimates of the simple linear regression of the outcome Y and the exposure X on the genotype G.j=(G1j,G2j,…,Gnj)T at variant j, respectively, and SE(Γ~j) denote the standard error of Γ~j,1≤j≤m. An adaption of Egger regression was proposed (Bowden et al., 2015) as follows to estimate the causal effect,

where Γ~=(Γ~1,Γ~2,…,Γ~m)T, γ~=(γ~1,γ~2,…,γ~m)T.

Imposing the constraint of β_0E = 0 on the above regression model yields the inverse-variance weighted (IVW) estimate of the causal effect (Burgess et al., 2013), which is also commonly used in the meta-analysis. Notice that both MR-Egger and IVW are applicable to the summary data that are accessible in most GWASs.