Suppose that a study or meta-analysis reports the results of regression analyses for each of m genetic variants, *G*_{j}, that are associated with their trait, *X*. These results might be in the form of the estimated regression coefficients, *a*_{X j}, and their variances, *V*_{X j}, or other statistics from which these quantities can be derived. It is important that the selection of the genetic variants is not based on the same data that is used to calculate *a*_{X j} for otherwise the estimated coefficients will be biased away from zero due to the Winner’s curse [10], so *a*_{X j} and *V*_{X j} might be taken from a replication study. These estimated regression coefficients will be modelled as,
${a}_{X\phantom{\rule{thinmathspace}{0ex}}j}\sim N\left({\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j},\phantom{\rule{thinmathspace}{0ex}}{V}_{X\phantom{\rule{thinmathspace}{0ex}}j}\right)\phantom{\rule{1em}{0ex}}j=1,\phantom{\rule{thinmathspace}{0ex}}\dots ,\phantom{\rule{thinmathspace}{0ex}}m$where ${\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j}$ is the true regression coefficient and the *V*_{X j} will be treated as known.

A second study or meta-analysis publishes similar data for the same variants but a different outcome, *Y*. The variants are unlikely to be top hits for Y so now we will need access to the full set of results in order to look-up the required estimates. The Winner’s curse is no longer a concern because the variants were not chosen for their effect in the second study. Suppose that the regression coefficients and variances from the look-up are *b*_{Y j} and the *V*_{Y j}, we can model them as,
${b}_{Y\phantom{\rule{thinmathspace}{0ex}}j}\sim N\left({\mathrm{\beta}}_{Y\phantom{\rule{thinmathspace}{0ex}}j},\phantom{\rule{thinmathspace}{0ex}}{V}_{Y\phantom{\rule{thinmathspace}{0ex}}j}\right)\phantom{\rule{1em}{0ex}}j=1,\phantom{\rule{thinmathspace}{0ex}}\dots ,\phantom{\rule{thinmathspace}{0ex}}m$where the ${\mathrm{\beta}}_{Y\phantom{\rule{thinmathspace}{0ex}}j}$ represent the true coefficients and the *V*_{Y j} are treated as known.

A Mendelian randomization for a continuous outcome targets the unconfounded regression coefficient, $\mathrm{\varphi},$ of *X* on *Y*. Provided that the assumptions for Mendelian randomization hold for every genetic variant, there are m relationships ${\mathrm{\beta}}_{Y\phantom{\rule{thinmathspace}{0ex}}j}=\mathrm{\varphi}{\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j},$ each of which creates a ratio, or Wald, estimator ${\stackrel{\u02c6}{\mathrm{\varphi}}}_{j}={b}_{Y\phantom{\rule{thinmathspace}{0ex}}j}/{a}_{X\phantom{\rule{thinmathspace}{0ex}}j}$ [11, 12]. These can be averaged with weights inversely proportional to their variances in order to create an overall Mendelian randomization estimate $\stackrel{\u02c6}{\mathrm{\varphi}}.$

When the selected genetic variants are independent, we can estimate the variance of $\stackrel{\u02c6}{\mathrm{\varphi}}$ without needing to make assumptions about the unknown pattern of linkage disequilibrium and the coefficients ${\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j}$ that apply to each variant separately will also be the coefficients in the joint regression of *X* on the genetic risk score *S*_{Xi}. That is,
$\begin{array}{c}E\{{X}_{i}|{S}_{Xi}\}=\mathrm{\mu}+{S}_{Xi}=\mathrm{\mu}+\sum _{j=1}^{m}{\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j}{g}_{ij}\\ E\{{Y}_{i}|{S}_{Xi}\}=\mathrm{\nu}+\mathrm{\varphi}E\{{X}_{i}|{S}_{Xi}\}=\mathrm{\nu}+\mathrm{\varphi}\mathrm{\mu}+\mathrm{\varphi}\sum _{j=1}^{m}{\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j}{g}_{ij}\end{array}$where the confounder is omitted because it is assumed independent of each *G*_{j} and *g*_{i j} represents the measured genotype of the *i*^{th} subject for variant *G*_{j} coded as the number of effect alleles, 0,1 or 2. This genetic risk score *S*_{Xi}, assumes a per allele effect of each variant and ignores any interactions. Dominant or recessive genetic effects could be created but the necessary estimates of the coefficients are rarely published. In this model *S*_{Xi} represents the ideally weighted combination of the variants for use as a combined instrument in a Mendelian randomization.

We could estimate the variances of the ratio estimates of each variant using a Taylor series [13],
$Var\{{\stackrel{\u02c6}{\mathrm{\varphi}}}_{j}\}\approx [{\mathrm{\beta}}_{Y\phantom{\rule{thinmathspace}{0ex}}j}/{\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j}{]}^{2}[{V}_{X\phantom{\rule{thinmathspace}{0ex}}j}/{\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j}^{2}+{V}_{Y\phantom{\rule{thinmathspace}{0ex}}j}/{\mathrm{\beta}}_{Y\phantom{\rule{thinmathspace}{0ex}}j}^{2})]$where the covariance term is omitted because *X* and *Y* come from different studies and their regression coefficients are independent. To estimate this variance we could just replace ${\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j}$ and ${\mathrm{\beta}}_{Y\phantom{\rule{thinmathspace}{0ex}}j}$ by *a*_{X j} and *b*_{Y j}. The weights, *w*_{j}, needed for averaging the ${\mathrm{\varphi}}_{j}$ would then be the inverse of these variances and the resulting Mendelian randomization estimate of $\mathrm{\varphi}$ would have variance,
$Var\{\stackrel{\u02c6}{\mathrm{\varphi}}\}=\sum {w}_{j}^{2}Var\{{\stackrel{\u02c6}{\mathrm{\varphi}}}_{j}\}/{\left[\sum {w}_{j}\right]}^{2}=1/\sum {w}_{j}$

However, as we will see in the simulations, inverse variance weighting does not work well in this context.

Should we want to test the hypothesis that $\mathrm{\varphi}=0,$ we would need the variance when this hypothesis is true, that is when ${\mathrm{\beta}}_{Y\phantom{\rule{thinmathspace}{0ex}}j}=0.$ In that case the variance reduces to ${V}_{Y\phantom{\rule{thinmathspace}{0ex}}j}/{\mathrm{\alpha}}_{X\phantom{\rule{thinmathspace}{0ex}}j}^{2}.$

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