With a single genetic variant
\(G_j\) that satisfies the instrumental variable assumptions, the causal effect of the risk factor on the outcome can be consistently estimated as a simple ratio of association estimates:
\({\hat{\theta }_{j}} = \frac{\hat{\beta }_{Yj}}{\hat{\beta }_{Xj}}\) [
16], where
\(\hat{\beta }_{Yj}\) is the estimated coefficient from univariable regression of the outcome on the
jth genetic variant, and likewise
\({\hat{\beta }_{Xj}}\) from univariable regression of the risk factor on the
jth genetic variant. With multiple genetic variants, the ratio estimates from each genetic variant can be averaged using an inverse-variance weighted formula taken from the meta-analysis literature to provide an overall causal estimate known as the inverse-variance weighted (IVW) estimate [
17]. This assumes that the ratio estimates all provide independent evidence on the causal effect; this occurs when the genetic variants are uncorrelated. If the variance terms are taken as
\(\frac{{{\mathrm{se}}}({\hat{\beta }_{Yj}})^2}{\hat{\beta }_{Xj}^2}\) (this is the first term from a delta method expansion for the ratio estimate [
18]), then the pooled estimate (assuming a fixed-effect model) is [
19]:
$$\begin{aligned} \hat{\theta }_{IVW} = \frac{\sum _j \hat{\beta }_{Yj} \hat{\beta }_{Xj} {{\mathrm{se}}}({\hat{\beta }}_{Yj})^{-2}}{\sum _j \hat{\beta }_{Xj}^2 {{\mathrm{se}}}(\hat{\beta }_{Yj})^{-2}}. \end{aligned}$$
(2)
This same estimate is obtained from the two-stage least squares analysis method for individual-level data when the genetic variants are uncorrelated [
20]. The same estimate can also be obtained from a weighted linear regression of the genetic associations with the outcome (
\({\hat{\beta }}_{Yj}\)) on the genetic associations with the risk factor (
\({\hat{\beta }}_{Xj}\)) using inverse-variance weights (
\({{\mathrm{se}}}(\hat{\beta }_{Yj})^{-2}\)) when there is no intercept term in the regression model [
14]:
$$\begin{aligned} \hat{\beta }_{Yj} = \theta _{IVW} \hat{\beta }_{Xj} + \epsilon _{Ij}; \quad \epsilon _{Ij} \sim \mathcal {N}(0, \sigma ^2 {{\mathrm{se}}}(\hat{\beta }_{Yj})^{2}) \end{aligned}$$
(3)
where
\(\hat{\beta }_{Yj}\) and
\(\hat{\beta }_{Xj}\) are the data in the model,
\(\theta _{IVW}\) is the parameter, and
\(\epsilon _{Ij}\) is the residual term. To obtain the same standard error for the causal estimate from the regression analysis as from the fixed-effect meta-analysis, the residual standard error in the regression (
\(\sigma \)) must be set to equal one [
21].