are found to be stronger than the observed one, it means that reaching this correlation level by chance is very unlikely and therefore that the QTL candidate is likely to be true. More formally, if r correlations in the null distribution are found to be stronger than the observed, significance of the QTL candidate is assessed by calculating the following empirical adjusted P-value pd of association (Phipson and Smyth, 2010): (1)pd=r+1R+1 By definition, such an adjusted P-value cannot be smaller than 1/(R + 1); meaning that a large number of permutations are needed to get precise estimates of small adjusted P-values, thereby increasing the computational burden. For instance, reaching P-values of ∼10−3 requires thousands of permutations while billions are needed to get P-values of ∼10−9. In practice, it is very difficult to go beyond a few thousand permutations genome-wide with this approach, which forces us to work with adjusted P-values in the range of 10−3 to 1.0. To alleviate this limitation, we improved the direct permutation scheme with two complementary methods (see Sections 2.4 and 2.5).
