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 significance level (i.e. without lower bound). In a final stage, a false discovery rate (FDR) procedure as implemented in the R/qvalue (Storey and Tibshirani, 2003) package is used on the set of adjusted P-values obtained either from (1), (2) or (3) to extract all significant phenotype-variant pairs at a given FDR, usually chosen to be 5% or 10% (see Section 2.6). All this, plus other optional functionalities, have been implemented in the FastQTL software package (see Section 2.7).
