If there really is an effective number of independent tests nE, then the minimum P–value should follow a Beta distribution with parameters (1,nE), as [Šidák, 1967] (7) We fitted the Beta distribution to the minimum P–value of the permutation replicates, with the first parameter set to 1 and also with both parameters free. This would allow us to test whether the minimum P–value is consistent with an effective number of independent tests, by testing whether the first parameter is 1 [Dudbridge and Koeleman, 2004], and whether Patterson's estimator is accurate, by testing whether np = nE. The moment estimators for the parameters of the Beta (a,b) distribution are (8) (9) where and s2 are the sample mean and variance of observations, respectively. When a is set to 1, the moment estimate of b is (10) We used the moment estimates as starting points for numerical maximum likelihood estimation, using the optim function in R.
