on study controls and for the test statistics from stage 2 analyses of the replication-based two-stage study designs (because cases and controls would be genotyped at the same time in stage 2). The mean systematic inflation of the test statistics, μ − 1, is equal to λ ~ Np(1−p)Δ2 (i.e., the non-centrality parameter of a chi-square test with 1 df), where N is the total sample size, p is the proportion of cases in the sample and Δ is the metric reflecting the difference in genotype frequencies between cases and controls due to batch effects. Consequently, the magnitude of the systematic inflation of the test statistics does not impact all study designs equally. Hence, we recalculated the mean inflation of the test statistics for the one-stage design with both study and public controls and for stage 1 of each replication-based two-stage study design (based on the number of cases included in stage 1). For the one-stage studies that include public 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
