In order to test for partial genetic causality and to estimate the gcp, we exploit the fact that if trait 1 is partially genetically causal for trait 2, then most SNPs affecting trait 1 will have proportional effects on trait 2, but not vice versa (Figure 1c-e). Instead of using thresholds to select subsets of SNPs[11], we compare the mixed fourth moments E(α12α1α2) and E(α22α1α2) of marginal effect sizes for each trait. The rationale for utilizing these mixed fourth moments is that if trait 1 is causal for trait 2, then SNPs with large effects on trait 1 (large α12) will have proportional effects on trait 2 (large α1α2), so that E(α12α1α2) will be large; conversely, SNPs with large effects on trait 2 (large α22) will generally not affect trait 1 (small α1α2), so that E(α22α1α2) will be smaller. Thus, estimates of the mixed fourth moments can be used to test for partial genetic causality and to estimate the gcp. We note that LCV, unlike MR, does not distinguish between the “exposure” and the “outcome”; trait 1 and trait 2 are interchangeable labels.
