One way of minimizing this problem is downweighting the contribution to the analysis of genetic variants with heterogeneous ratio estimates. Heterogeneity between estimates can be quantified by Cochran's Q statistic: Q=∑jQj=∑jwj′(β^j−β^)2,where we take β^ to be the IVW estimate (Greco et al., 2015). The Q statistic has a chi‐squared distribution on J−1 degrees of freedom under the null hypothesis that all genetic variants are valid IVs and the same causal effect is identified by all variants. Under this null hypothesis, the components of the Q statistic corresponding to the individual genetic variants (Qj) approximately have chi‐squared distributions with 1 degree of freedom. So as not to distort the weightings of the majority of variants, we propose penalization using the one‐sided upper P‐value (denoted qj) on a chi‐squared 1 distribution corresponding to Qj, by multiplying the weight by the P‐value multiplied by 20 (or by 1 if the P‐value is greater than 0.05). The (unstandardized) penalized weights (wj*) are therefore wj*=wj′×min(1,20qj).This means that most variants will be unaffected by the penalization, but outlying variants will be severely downweighted.
