Mendelian Randomization Egger Method

We assume that all relationships between variables (in particular, the genetic associations with the risk factor and with the outcome, and the causal effect of the risk factor on the outcome) are linear with no effect modification. We also assume that all genetic variants are uncorrelated (that is, not in linkage disequilibrium), although conventional instrumental variable methods for analysing summarized data from correlated variants have been developed [14 (link)], and similar extensions to the MR-Egger method are discussed later in this paper. The association between genetic variant $G_j$ ( $j=1,2,\dots ,J$ ) and the outcome is denoted ${\mathit{\beta }}_{Yj}$ , and the association between genetic variant $G_j$ and the risk factor is denoted ${\mathit{\beta }}_{Xj}$ .
The genetic association with the outcome can be decomposed into the sum of a direct (pleiotropic) effect and an indirect (causal) effect: $$\begin{array}{c}\hfill {\mathit{\beta }}_{Yj}={\mathit{\alpha }}_j+\mathit{\theta }{\mathit{\beta }}_{Xj}\end{array}$$ where ${\mathit{\alpha }}_j$ is the effect of the genetic variant on the outcome that is not mediated via the risk factor of interest, and $\mathit{\theta }$ is the causal effect of the risk factor on the outcome [15 ]; see Fig. 1. A genetic variant is referred to as pleiotropic if it has associations with more than one risk factor on different causal pathways [16 (link)]. Any such effect is included in the parameter ${\mathit{\alpha }}_j$ ; a genetic variant is pleiotropic if ${\mathit{\alpha }}_j\ne 0$ . A pleiotropic genetic variant is not a valid instrumental variable.Fig. 1

Decomposing association for genetic variant \documentclass[12pt]{minimal}
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\begin{document}$$G_j$$\end{document}Gj with the outcome into a indirect (causal) effect via the risk factor and an direct (pleiotropic) effect (see Eq. 1)