Here me is the effective number of independent phenotypes, mej is the number of independent phenotypes among top j phenotypes (after ordering by p-value). To estimate me and mej, the TATES test uses phenotypic information and the argument that p-value correlations and phenotype correlations are related. van der Sluis et al. (2013) used a 6 degree polynomial to approximate the correlations between the p-values and the phenotypes (i.e., the relationship between phenotypic correlation (x) and the p-value correlation (y)), as follows: y=0.2179x6−0.0219x5+0.1095x4+0.0149x3+0.6226x2−0.0023x−0.008, R2=0.992.When phenotypes are independent, the TATES test is the same as the Simes test. The estimated number of independent phenotypes/p-values among top j phenotypes is defined as: mej=j−∑i=1j(λi−1)I(λi−1),where I is an indicator function and λi is ith eigenvalue of the approximated p-value correlation matrix based on top j phenotypes. We note that this formula corresponds to formula (2) from van der Sluis and et al. (2013) and mem=m−∑i=1m(λi−1)I(λi−1)=me.
