study as a whole. We simulated sets of 100,000 X 1 values from a Normal distribution with mean = √η 1 and variance = 1, where η 1 is the presumed non-centrality parameter from the GWAS study (n = 1734, Π = 0.506), and an additional set of 100,000 X 2 values from a Normal distribution with mean = √η 2 and variance = 1, where η 2 is the presumed non-centrality parameter from the first replication study (n = 1011, Π = 0.295). To score a “hit”, we required both that X 1 exceeded the upper critical value for a two-tailed test at α = 0.0003 (to mimic being passed to the 1st replication stage), and that both X 1 and X 2 had the same sign and had a joint P<1.6×10−7 when combined using Stouffer's weighted-Z method [60] (to mimic achieving a Bonferroni-corrected genome-wide significance level after both stages). Power was defined as the number of hits divided by 10,000. Solutions to θ based on fixed values of the other parameters were found by an iterative root-finding procedure (function “uniroot” in the R statistical package, http://www.r-project.org/). The Total Lambda-s expected based on k independent SNPs each with a
