When the assumptions underlying analytical distributions are not met, permutation tests are a popular method for computing significance levels. In a genome-wide association study, the problem is that genotypes are correlated due to LD; indeed, the correlations are necessary for the design to be successful. The standard procedure is to reassign trait values among study subjects, while keeping their genotypes fixed, thereby preserving the correlation structure across the multiple genes and realising the exchangeability conditions for a valid test [42]. When performing thousands of tests on thousands of subjects, however, a permutation procedure using thousands of replicates becomes extremely time-consuming, with possible running times of days or weeks. Therefore, more efficient approaches to permutation testing have recently been proposed.
