Statistical power is calculated from the population value of the parameter and its sampling variance, which was derived above. If the parameter is θ, where θ is either the proportion of phenotypic variance captured by SNPs () in the univariate case or the genetic correlation () in the bivariate case, then is asymptotically distributed as a non-central χ 2 with 1 degree of freedom and non-centrality parameter (NCP) of . Given λ and the type-I error rate of α, statistical power is the probability that a non-central χ 2 variable is larger than the central χ 2 threshold that is determined by α. We show in Figure 3 the statistical power based on the sampling variance from our approximation theories to detect in a univariate case and in a bivariate case under a range of scenarios. For example, for a quantitative trait, approximately 8900, 4500, 3000 and 2300 independent individuals are required to detect of 0.1, 0.2, 0.3 and 0.4 with >80% power at a type-I error rate of 0.05, respectively. For two quantitative traits measured on the same sample,
