A General Approach to Sensitivity Analysis for Mendelian Randomization

Abstract

Mendelian Randomization (MR) represents a class of instrumental variable methods using genetic variants. It has become popular in epidemiological studies to account for the unmeasured confounders when estimating the effect of exposure on outcome. The success of Mendelian Randomization depends on three critical assumptions, which are difficult to verify. Therefore, sensitivity analysis methods are needed for evaluating results and making plausible conclusions. We propose a general and easy to apply approach to conduct sensitivity analysis for Mendelian Randomization studies. Bound et al. (J Am Stat Assoc 90:443–450. 10.2307/2291055, 1995) derived a formula for the asymptotic bias of the instrumental variable estimator. Based on their work, we derive a new sensitivity analysis formula. The parameters in the formula include sensitivity parameters such as the correlation between instruments and unmeasured confounder, the direct effect of instruments on outcome and the strength of instruments. In our simulation studies, we examined our approach in various scenarios using either individual SNPs or unweighted allele score as instruments. By using a previously published dataset from researchers involving a bone mineral density study, we demonstrate that our proposed method is a useful tool for MR studies, and that investigators can combine their domain knowledge with our method to obtain bias-corrected results and make informed conclusions on the scientific plausibility of their findings.

Appendix 1

We assume independence of the p genetic variants. We first expand the ${\sigma }_{\widehat{x},\varepsilon },$ which is the covariance between the fitted exposure and the error term in Eq. (5).

$$\begin{aligned}{\sigma }_{\widehat{x},\varepsilon} & =cov\left(\widehat{x},\varepsilon \right) \\ & =cov\left({\alpha }_{0}+{\alpha }_{u2}{\gamma }_{0}+Z(\alpha_{z}+{\alpha }_{u2}{\gamma }_{z}), {\beta }_{u1}{U}_{1}+{\beta }_{u2}{U}_{2}+Z{\beta }_{z}+{e}_{y}\right) \\ & = cov\left(Z({\alpha }_{z }+{\alpha }_{u2}{\gamma }_{z}), {\beta }_{u2}{U}_{2}+Z{\beta }_{z}\right)\\ &= cov\left(Z({\alpha}_{z }+{\alpha }_{u2}{\gamma }_{z}), {\beta }_{u2}{U}_{2}\right) \\ & \quad +cov\left(Z(\alpha_{z }+{\alpha}_{u2}{\gamma}_{z}), Z{\beta }_{z}\right) \\ & =cov\left(Z(\alpha_{z}+{\alpha}_{u2}{\gamma }_{z}), {\beta }_{u2}({\gamma }_{0}+{Z\gamma }_{z }+{e}_{u2})\right)+ (\alpha_{z}+{\alpha}_{u2}{\gamma}_{z})^{\prime} var(Z){\beta }_{z} \\ & =(\alpha_{z}+{\alpha }_{u2}{\gamma }_{z})^{\prime}var(Z){\beta }_{u2}{\gamma }_{z }+ (\alpha_{z }+{\alpha }_{u2}{\gamma }_{z})^{\prime}var(Z){\beta }_{z} \\ & =(\alpha_{z }+{\alpha }_{u2}{\gamma }_{z})^{\prime}var(Z)({\beta }_{u2}{\gamma }_{z }+{\beta }_{z})={\sum }_{i=1}^{p}(\alpha_{zi }+{\alpha }_{u2}{\gamma }_{zi})({\beta }_{u2}{\gamma }_{zi }+{\beta}_{zi})var\left({Z}_{i}\right)\end{aligned}$$

$${\sigma }_{\widehat{x}}^{2}=var\left({\alpha }_{0}+{\alpha }_{u2}{\gamma }_{0}+{Z(\alpha }_{z }+{\alpha }_{u2}{\gamma }_{z})\right)={(\alpha }_{z }+{\alpha }_{u2}{\gamma }_{z})^{\prime}var(Z){(\alpha }_{z }+{\alpha }_{u2}{\gamma }_{z})={\sum }_{i=1}^{p}{{(\alpha }_{zi }+{\alpha }_{u2}{\gamma }_{zi})}^{2}var({Z}_{i}).$$

Hence,

$$\frac{{\sigma }_{\widehat{x},\varepsilon }}{{\sigma }_{\widehat{x}}^{2}}=\frac{{\sum }_{i=1}^{p}{(\alpha }_{zi }+{\alpha }_{u2}{\gamma }_{zi})({\beta }_{u2}{\gamma }_{zi }+{\beta }_{zi})var\left({Z}_{i}\right)}{{\sum }_{i=1}^{p}{{(\alpha }_{zi }+{\alpha }_{u2}{\gamma }_{zi})}^{2}var({Z}_{i})}.$$

When there is a single SNPs (p = 1), the bias simplifies to

$${\sigma }_{\widehat{x},\varepsilon }/{\sigma }_{\widehat{x}}^{2}={\beta }_{u2}{\gamma }_{z}/{(\alpha }_{z }+{\alpha }_{u2}{\gamma }_{z})+{\beta }_{z}/{(\alpha }_{z }+{\alpha }_{u2}{\gamma }_{z}).$$

When multiple SNPs are used in MR, it was suggested that a summary score should be used in place of the multiple SNPs to reduce the finite-sample bias. This simplified equation may be used in such situation by treating the summary score as the single instrument.