The following simulation further supports this observation. For each of the four considered studies, we rearranged each subject's case/control status so that new designated case and control groups were completely separately on a chosen PC direction (e.g., the first PC direction), and evaluated the over-dispersion factor and empirical type I error rate in the new generated dataset. Results are summarized in Table S4). When the mismatch (to the most extreme extend) between cases and controls occurred on a PC with large genetic variation, the resultant inflation in type I error was much more severe than the one observed when the disparity occurred on a PC with relatively low variation (high Tracy-Widom test P-value). For all 4 considered studies, the over-dispersion factor was around or less than 1.05 when the mismatch (in the worst scenario) of cases and controls occurred on a PC with the associated Tracy-Widom test P-value larger than 0.05.
