To minimize the number of adjusted PCs while allowing an effective correction of the confounding effect by PS, our proposed selection procedure identifies PCs that are effective in the reduction of the over-dispersion factor. A PC is chosen for the correction of PS whenever the over-dispersion factor in the model with the PC included is significantly lower than the expected value of the factor from including an additional randomly generated covariate. When applied to the two original studies, this new procedure identified a single PC to achieve the optimal PS correction. Interestingly, the procedure selected the first PC (corresponding to the largest genetic variation) for the PLCO prostate cancer study, but the second PC for the NHS breast cancer study suggesting that the control selection procedure used in the NHS study effectively removed the confounding that might have been caused by the axis with the largest genetic variation. When the controls were exchanged between the two studies, the procedure required the 1st, 2nd, and 4th PCs in PLCOca-NHSco, and the 1st, 2nd, and 3rd PCs in NHSca-PLCOco. Thus, the increased
