Adjusting for unnecessary covariates incurs the risk of decreasing power [31]. We have presented a simulation to demonstrate that the unnecessary adjustment of population substructure (even one PC) could lead to a significant loss in power (Text S1, Table S1). A permutation procedure is proposed to identify the minimal number of PCs while allowing an effective correction of the confounding effect. By applying this new procedure to the two original GWAS with internal controls and two reconstructed studies with external controls, we documented its advantage over other commonly used PC selection strategies. At the expense of computing time, the new procedure is able to pick fewer PCs while reducing the over-dispersion factor to a similar or even lower level.
