If the βX1 parameters are independent of the βXk parameters for all k=2,3,…,K, then the InSIDE assumption for multivariable MR‐Egger is satisfied if the direct effects of the genetic variants α′ are independent of βX1. More formally, we require: (8)βX1⊥⊥α′,ifβX1⊥⊥βX2,⋯,βXK, for the estimate of θ 1 from multivariable MR‐Egger to be consistent. If the InSIDE assumption is satisfied, then the weighted covariance of βX1 and α′( covw(α′,βX1)) will tend to zero as the number of genetic variants J tends to infinity. The estimate of θ 1 from multivariable MR‐Egger when the βX1 parameters are independent of βXk for all k=2,3,…,K is (9)θ^1ME=covw(β^Y,β^X1)varw(β^X1)→N→∞covw(βY,βX1)varw(βX1)=θ1+covw(α′,βX1)varw(βX1), which is equal to θ 1 if the InSIDE assumption is satisfied, where covw and varw represent the weighted covariance and weighted variance using the inverse‐variance weights se(β^Yj)−2: (10)covw(α′,βX1)=∑j(αj′−α′¯w)(βX1j−β¯X1w)se(β^Yj)−2∑jse(β^Yj)−2varw(βX1)=∑j(βX1j−β¯X1w)2se(β^Yj)−2∑jse(β^Yj)−2α′¯w=∑jαj′se(β^Yj)−2∑jse(β^Yj)−2β¯X1w=∑jβX1jse(β^Yj)−2∑jse(β^Yj)−2.
