We first consider methods in which the contribution of some genetic variants (for example, heterogeneous or outlying variants) to the analysis is downweighted (or penalized). If the causal conclusion from a Mendelian randomization investigation depends only on a single genetic variant (particularly if the estimate from this variant is heterogeneous with those from other variants) then the result may be driven by a pleiotropic effect of that particular variant and not by the causal effect of the risk factor.
The simplest way of performing a penalization method is to omit some of the variants from the analysis. This could be done systematically. For example, with a small number of genetic variants, the causal estimates omitting one variant at a time could be considered. Alternatively, it could be done stochastically. For example, we could consider estimates omitting (say) 30% of the genetic variants at a time by selecting the 30% of variants at random a large number of times, and calculating the causal estimate in each case. This sensitivity analysis has been undertaken for the effect of LDL-c on aortic stenosis seeeFigure in ref.49 (link). If the spread of results includes only (say) positive effect estimates, then we can be confident that the overall finding does not depend only on the influence of a few variants. However, even if only a small proportion of the estimates are discordant, these cases should be investigated and the omitted variants leading to the discordant estimates should be carefully investigated for potential violations of the instrumental variable assumptions. The causal estimates for the example of CRP on CAD risk based on the genome-wide significant variants using the inverse-variance weighted method are displayed in Figure 6 . Two of the 17 variants are omitted from the analysis in turn in a systematic way, and then the 136 resulting estimates are arranged in order of magnitude. The overall estimate excluding the two strongest variants with negative causal estimates is positive, indicating that the overall negative finding based on all the variants seems to be driven by these two variants, and is not supported by the majority of variants.
A more focused approach to omitting genetic variants is to omit genetic variants from the analysis with heterogeneous instrumental variable estimates. This could be done by calculating the contribution to Cochran’s Q statistic for each genetic variant, and omitting any variant whose contribution to the statistic is greater than the upper 95th percentile of a chi-squared distribution on one degree of freedom (3.84). This approach has been applied for investigating the causal effect of lipid fractions on CAD risk.50 (link) More formal penalization methods have been proposed using L1-penalization to downweight the contribution of outlying variants to the analysis in a continuous way.51 , 52 (link) These methods have desirable theoretical properties, giving consistent estimates of the causal effect even if up to half of the genetic variants are not valid instrumental variables. However, they require individual-level data and a one-sample setting (genetic variants, risk factor and outcome measurements are available for the same individuals).
The simplest way of performing a penalization method is to omit some of the variants from the analysis. This could be done systematically. For example, with a small number of genetic variants, the causal estimates omitting one variant at a time could be considered. Alternatively, it could be done stochastically. For example, we could consider estimates omitting (say) 30% of the genetic variants at a time by selecting the 30% of variants at random a large number of times, and calculating the causal estimate in each case. This sensitivity analysis has been undertaken for the effect of LDL-c on aortic stenosis see
A more focused approach to omitting genetic variants is to omit genetic variants from the analysis with heterogeneous instrumental variable estimates. This could be done by calculating the contribution to Cochran’s Q statistic for each genetic variant, and omitting any variant whose contribution to the statistic is greater than the upper 95th percentile of a chi-squared distribution on one degree of freedom (3.84). This approach has been applied for investigating the causal effect of lipid fractions on CAD risk.50 (link) More formal penalization methods have been proposed using L1-penalization to downweight the contribution of outlying variants to the analysis in a continuous way.51 , 52 (link) These methods have desirable theoretical properties, giving consistent estimates of the causal effect even if up to half of the genetic variants are not valid instrumental variables. However, they require individual-level data and a one-sample setting (genetic variants, risk factor and outcome measurements are available for the same individuals).
