and Koots and Gibson (1996) proposed a modified version as [25], where is the phenotypic correlation between the traits. However, both approximations have an unsatisfying property that will approach 0 if or is close to 1. We derived an approximation, which does not have this problem (Text S2), i.e.(7)As described above, for a GRM estimated from common SNPs in unrelated individuals in human populations, therefore . When , i.e. the traits are completely independent, . We tested equation (7) by simulations based on real genotype data (Text S1). The simulation results suggest that the approximation is reasonably accurate (Table 1). For real data analysis, we previously estimated the genetic correlation between intelligence at age 11 years and in old age of 0.62 with a SE of 0.23 using 1729 samples [5], consistent with the predicted SE of 0.22 from the approximation theory.
