This article discusses problems with and solutions to performing valid permutation tests for quantitative trait loci in the presence of polygenic effects. Although permutation testing is a popular approach for determining statistical significance of a test statistic with an unknown distribution--for instance, the maximum of multiple correlated statistics or some omnibus test statistic for a gene, gene-set, or pathway--naive application of permutations may result in an invalid test. The risk of performing an invalid permutation test is particularly acute in complex trait mapping where polygenicity may combine with a structured population resulting from the presence of families, cryptic relatedness, admixture, or population stratification. I give both analytical derivations and a conceptual understanding of why typical permutation procedures fail and suggest an alternative permutation-based algorithm, MVNpermute, that succeeds. In particular, I examine the case where a linear mixed model is used to analyze a quantitative trait and show that both phenotype and genotype permutations may result in an invalid permutation test. I provide a formula that predicts the amount of inflation of the type 1 error rate depending on the degree of misspecification of the covariance structure of the polygenic effect and the heritability of the trait. I validate this formula by doing simulations, showing that the permutation distribution matches the theoretical expectation, and that my suggested permutation-based test obtains the correct null distribution. Finally, I discuss situations where naive permutations of the phenotype or genotype are valid and the applicability of the results to other test statistics.
Keywords: QTL; family studies; permutation test; polygenic effect; population structure; type I error rate.
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