The m×m correlation matrix ρ between the p-values is not observed in practice. Following Li et al [15], we used simulation to show that matrix ρ can be accurately approximated through the m×m correlation matrix r between the phenotypes. We simulated 55 continuous standard normally distributed phenotypes whose intercorrelations ranged between −.90 and .90, and a GV (MAF = .5) that was simulated to be unrelated to the 55 phenotypes. The association between the GV and all phenotypes was tested, yielding 55 p-values, and this simulation was run 10,000 times. We then calculated, across the 10,000 simulations, the mean pair-wise correlations between the 55 phenotypes (i.e., (55*55−55)/2 = 1485 correlations), and the mean pair-wise correlations between the p-values. Regressing the vector of correlations between the p-values on the vector of correlations between the phenotypes, we obtain the 6th order polynomial ρ = −0.0008−0.0023r+0.6226r 2+0.0149r 3+0.1095r 4−0.0219r 5+0.2179r 6 (coefficient of determination R2 = .992; see Figure S1), allowing accurate approximation of the correlations between the p-values from the observed correlations between the phenotypes. The thus obtained matrix ρ is subjected to the eigenvalue decomposition in Eq. 2.
