The corrected χ2 test based on ϕest successfully reduced spurious associations between markers and disease. Q – Q plots of the p-values are shown in Panel E and F of Figure 6. In the Guam data analysis, p-values of the uncorrected χ2 tests are not uniformly distributed, but p-values of the corrected χ2 tests based on ϕest are close to uniformly distributed. These results show that the false positive rate was reduced by correcting for relatedness. If the distribution of the p-values were exactly uniform, the expected 1% and 5% quantiles would be 0.01 and 0.05, but the 1% and 5% quantiles of the χ2 test are 0.0007 and 0.007, which are considerably smaller than the nominal quantiles. For the corrected χ2 test, the 1% and 5 % quantiles are less extreme at 0.02 and 0.098, and are actually conservative. When the uncorrected χ2 test was used, 95 markers were significantly associated with the disease at significance level 0.05, but only 10 markers remained at this significance level with the corrected χ2 test.
