controls, we calculated power after correcting for the systematic batch effects across all SNPs by multiplying the critical value of the 1 df chi-square distribution corresponding to p = 1.0×10−7 (i.e. 28.373) by the mean test statistic value, μ (Reich and Goldstein, 2001). For each two-stage study, the stage 1 test-statistic critical value was multiplied by the corresponding mean test statistic value, μ, determined by the stage 1 sample composition.