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 allele frequency of 0.30 has the major allele mistakenly scored the minor allele with probability = 0.15 in public controls (increasing the frequency of the minor allele in this public control sample to 0.405). In a single-stage study design with 2,000 study cases, 2,000 study controls and 5,000 public controls, this SNP would be significantly associated with the outcome with probability near 1 under the null of no true association between the SNP and outcome, hence the family-wise error rate of the experiment would also be 1. In contrast, in the proposed replication-based two-stage study the SNP would almost certainly be included in stage 2 but the overall experimental type I error would be well controlled because the allele frequencies of the SNP in study controls and remaining cases in stage 2 are unaffected by the batch effects in the public controls in stage 1.
