Confounding by PS could lead to an inflated type I error rate for the association test. For each of the four combinations of cases and controls, assuming an additive genetic model we performed a 1-df Wald test (without adjusting for PS) using the standard logistic regression for each testing autosomal SNP. Table 4 shows the over-dispersion factor and empirical type I error (assuming the vast majority of testing SNPs are disease unrelated) under various significance levels. All estimates presented in Table 4 are based on tests performed with the same set of 241,238 genomic control SNPs. They are close to estimates obtained when all 475,116 testing SNPs were used (results not shown). As expected, the two reconstructed studies with external controls have larger over-dispersion factors and higher empirical type I errors compared with the original studies with internal controls. Similar conclusion can be reached based on the Q-Q plot (on the log scale) comparison in Figure 3. From Table 4, the over-dispersion factor is positively correlated with empirical type I errors under various considered significant levels. The over-dispersion factor is
