With multiple genetic variants, the causal effect of the exposure on the outcome can be estimated using the TSLS method.19 The TSLS estimate is a weighted average of the ratio estimates calculated using each genetic variant in turn.20 If the genetic variants are uncorrelated (in linkage equilibrium), then the causal effect can be estimated from summarized data on the genetic associations with the exposure and with the outcome as:21 (3)∑j=1Jγ^j2σYj−2β^j∑j=1Jγ^j2σYj−2. where β^j=Γ^jγ^j is the ratio method estimate for variant j, and σYj is the standard error in the regression of the outcome on the jth genetic variant, assumed to be known. This same weighted average formula is used in a fixed-effect meta-analysis, where the IV-specific causal estimates β^j are the study-specific estimates, and the weights are the inverse-variance weights.22 This summarized estimate, which we refer to as an inverse-variance weighted (IVW) estimate, will differ slightly from the TSLS estimate in finite samples, as the correlation between independent genetic variants will not exactly equal zero,23 but the two estimates will be equal asymptotically (that is, they both tend towards the same
