The motivation for utilizing the normalization by A1(x)2+A2(x)2 is that the magnitude of A1(x) and A2(x) tend to be highly correlated, leading to greatly increased standard errors if we only use the numerator of S. However, the denominator tends to zero when the genetic correlation is zero, leading to instability in the test statistic and false positives. The use of the threshold leads to conservative, rather than inflated, standard errors when the genetic correlation is zero or nearly zero. We recommend only analyzing trait pairs with a significant genetic correlation, and this threshold usually has no effect on the results. Another reason not to analyze trait pairs whose genetic correlation is non-significant is that for positive LCV results, the genetic correlation provides critical information about the causal effect size and direction.
