FastQTL performs linear regressions between genotypes and molecular phenotypes with or without covariates in order to find the best nominal association for each phenotype (see Section 2.2). Then, it can correct for the multiple correlated variants tested via three different permutation schemes: (1) a direct permutation scheme that relies on a fixed number of permutations (see Section 2.3), (2) an adaptive permutation scheme which maintains a reasonable computational load by tailoring the number of permutations to the significance of the association (see Section 2.4) and (3) a beta approximation which models the permutation outcome via a beta distribution (see Section 2.5). For (1) and (2), an adjusted P-value per phenotype is calculated as the proportion of null associations found to be more significant than the nominal one. For (3), we model this null distribution of most significant P-values for a phenotype with a beta distribution, learning the parameters from a few permutations (typically 100–1000) by maximum likelihood estimation. As a result, we obtain a reasonably good approximation of the tail of null distribution to estimate small adjusted P-values at any
