It is important to note that our examples did not include a small number of SNPs under strong selective pressure or SNPs with extreme batch effects. The impact of including these types of SNPs on overall statistical power would be minimal because, a priori, the probability that a SNP under the alternative model is under such pressure is likely small. However, a small number of SNPs under selective pressure or with extreme batch effects would substantially inflate the family-wise error rate for the GWA study and this inflation likely cannot be controlled by analytic methods. While a single-stage or a joint-analysis two-stage study that includes both public and study controls provides the greatest power, there is an increased possibility that any given significant result could be due to population stratification or extreme batch effects. In contrast, the proposed replication-based two-stage study maintains similar control of the overall type I error rate compared to a study based only on study controls, making any single significant result more reliable. For example, suppose that a single SNP, under the null hypothesis, with minor
