Initially, we consider the causal effect of a risk factor X on an outcome Y using genetic variants G j(j=1,…,J) that are assumed to be uncorrelated (not in linkage disequilibrium). Then, we expand to consider multiple risk factors X 1,X 2,…,X K. Increasingly, MR investigations are implemented using summarized data from consortia to leverage their large sample sizes, thereby improving the precision of causal estimates.11 We therefore assume that summarized data are available on the associations of each genetic variant with the risk factor (or with each risk factor for the multivariable setting) and with the outcome: the beta‐coefficients ( β^Xj,β^Yj) and their standard errors ( se(β^Xj),se(β^Yj)) from univariable regression on each variant G j in turn. We additionally assume that the associations of genetic variants with the risk factor and the outcome, and the causal effect of the risk factor on the outcome, are linear and homogeneous across the population; these assumptions are discussed in detail elsewhere.12 To distinguish between the parameters from the different methods considered, we use the following subscript notation: UI (“univariable inverse variance weighted (IVW)”); UE (“univariable MR‐Egger”); MI (“multivariable IVW”); and ME (“multivariable MR‐Egger”).
