The concerns raised in the simulated example are further evidenced in the mathematical proof where we calculate the exact distribution of the TATES statistic for a two variable example (Appendix 2). In this example we first defined some number d between 0 and 1, then created two variables with uniform distributions. Then we used the first one directly as the p-value for the first phenotype. For the second phenotype’s p-value, we used a random combination of two initially created uniform variables (note the first uniform variable directly defined the p-value for phenotype 1). We defined the p-value for phenotype 2 by assigning the first uniform variable with probability d, and the second one with probability 1−d. In Appendix 2 we prove that the second p-value also has a uniform distribution and, interestingly, the correlation between the two p-value variables is d. The exact pdf of the TATES variable based on p-values of two phenotypes is calculated as fXT(t)=2−d22−d−2d(1−d)2−dt.
