genetic model, a 1-df Wald test can be performed by adjusting any chosen PCs and other covariates. Following Devlin and Roeder [27], the over-dispersion factor can be estimated as the median of Wald test statistics over the set of genomic control SNPs divided by the expected median of the chosen test statistics under the null hypothesis (i.e., in the present case 0.456). The intent in our study is to use the over-dispersion factor for the purpose of evaluating the effectiveness in the correction of PS by adjusting for a given set of PCs, but not for the adjustment for association testing results as originally proposed by others [27], [28]. The over-dispersion factor provides a summary measure for the inflation level in the type I error. Although, as we show below, the over-dispersion factor tends to be positively correlated with empirical type I errors under various considered significant levels, there is no simple analytic relationship between them. For example, in our analysis of NHSca-PLCOco described later, the over-dispersion factor at the level of 1.06 corresponds to 14%, 20% and 30% inflation in type I error under the significant level of 0.05, 0.01 and 0.001, respectively.
