In order to have a uniform distribution, the coefficient of t, i.e., −2d(1−d)/(2−d) of fXT(t) must be zero. Furthermore, for non-inflated values, we expect that values of the pdf fXT(t) for t will not exceed 1. However, as t approaches 0, we get fXT(0) = (2−d)2/(2−d) > 1 (d > 0). Again, for the Simes test, which corresponds to the case d = 0 (i.e., no correlation between phenotypes), the distribution is correct. This proof shows that the maximum inflation point is d=2−2 and fXT(0) = 1.1716, which corresponds to an inflation of approximately 17% for this two variable case. Thus, we see 17% more p-values than expected around 0. The results from this proof thus provide an additional demonstration that when the univariate GWAS phenotypes/p-values are correlated, the combined TATES p-value violates the approximate uniform distribution assumption around zero.
