Table 5. Over-dispersion factor (and the empirical type I error under the significant level of 0.05) for association tests with adjustment for various numbers of PCs.

PCs chosen for the adjustment	PLOCca-PLCOcoa	PLCOca-NHScob	NHSca-NHScoc	NHSca-PLCOcod
0 PC	1.025 (0.053)	1.090 (0.059)	1.005 (0.051)	1.062 (0.057)
1st PC	1.020 (0.052)	1.055 (0.055)	1.006 (0.050)	1.040 (0.055)
1–2 PCs	1.022 (0.052)	1.040 (0.053)	1.004 (0.050)	1.013 (0.052)
1–3 PCs	1.021 (0.052)	1.040 (0.053)	1.005 (0.050)	1.006 (0.052)
1–4 PCs	1.021 (0.052)	1.032 (0.053)	1.005 (0.050)	1.007 (0.052)
1–5 PCs	1.023 (0.052)	1.032 (0.053)	1.006 (0.050)	1.008 (0.052)
1–6 PCs	1.024 (0.052)	1.032 (0.053)	1.007 (0.050)	1.010 (0.052)
1–7 PCs	1.025 (0.052)	1.033 (0.053)	1.008 (0.050)	1.010 (0.051)
1–8 PCs	1.025 (0.052)	1.036 (0.053)	1.007 (0.051)	1.011 (0.052)
1–9 PCs	1.025 (0.053)	1.036 (0.053)	1.007 (0.051)	1.011 (0.052)
1–10 PCs	1.025 (0.053)	1.036 (0.053)	1.008 (0.051)	1.010 (0.052)
Selected PCse	1.020 (0.052)	1.032 (0.053)	1.003 (0.050)	1.006 (0.052)

Note: The over-dispersion factor (and the empirical type I error under the significant level of 0.05) are estimated by applying the 1-df Wald test statistic with the adjustment for selected PCs on the set of 241,238 genomic control SNPs.

^aPLCOca-PLCOco is the original PLCO prostate cancer study.

^bPLCOca-NHSco is the reconstructed study with prostate cancer cases from the PLCO, and external controls from NHS.

^cNHSca-NHSco is the original NHS breast cancer study.

^dNHSca-PLCOco is the reconstructed study with breast cancer cases from the NHS, and external controls from PLCO.

^eThose PCs were chosen by the proposed permutation procedure for the correction of PS. The selected sets of PCs for individual analyses (from left to right) are: 1^st PC; 1^st, 2^nd, and 4^th PCs; 2^nd PC; and 1^st, 2^nd, and 3^rd PCs. The PCs were ordered according to their corresponding eigenvalues