Following Li et al [15], we obtain an estimate of the effective number of p-values m ej through a correction based on eigenvalue decomposition of the m×m correlation matrix ρ between the p-values associated with the m phenotypes. The effective number of p-values m ej for the top j p-values is calculated as:(2)where j is the number of top j p-values, λi denotes the i th eigenvalue, and I( λi−1) is an indicator function taking on value 0 if λi≤1 and 1 if λi>1. That is, the effective number of p-values m ej is calculated as the observed number of p-values j minus the sum of the difference between the eigenvalues λi and 1 for those eigenvalues λi>1. If the j phenotypes are all uncorrelated, then all j eigenvalues equal 1, and m ej = j−0 = j. In contrast, if the j phenotypes are perfectly correlated, then the first eigenvalue equals j, and the other eigenvalues equal 0, rendering m ej = j−(j−1) = 1 (i.e., j perfectly correlated phenotypes represent only 1 unique unit of information). In practice,
