The causal effect of the exposure on the outcome can be estimated using the jth variant as the ratio of the gene‐outcome association and the gene‐exposure association estimates (Lawlor et al., 2008): β^j=Γ^jγ^j.If the IV assumptions are satisfied for genetic variant j, then Γj=βγj and the ratio estimate is consistent asymptotically. Furthermore, if the genetic variants are uncorrelated (not in linkage disequilibrium) then the ratio estimates from each genetic variant can be combined into an overall estimate using a formula from the meta‐analysis literature (Johnson, 2013): β^IVW=∑jγ^j2σYj−2β^j∑jγ^j2σYj−2,where σYj is the standard error of the gene‐outcome association estimate for variant j. This is referred to as the inverse‐variance weighted (IVW) estimator (Burgess et al., 2013). Provided that the genetic variants are uncorrelated, the IVW estimate is asymptotically equal to the two‐stage least squares estimate commonly used with individual‐level data. If all genetic variants satisfy the IV assumptions, then the IVW estimate is a consistent estimate of the causal effect (i.e., it converges to the true value as the sample size increases), as it is a weighted mean of the individual ratio estimates.
