Rotation-based multiple testing in the multivariate linear model

Permutation methods are very useful in several scientic elds. They have the advantage of making fewer assumptions about the data and of providing more reliable inferential results. They are also particularly useful in case of high-dimensional problems since they easily account for dependence between tests, thereby allowing for more powerful multiplicity control procedures. Indeed, Westfall and Young's min-p procedure often improves on the Holm procedure by providing more rejections. The advantage of being able to make fewer assumptions about the process generating the data unfortunately involves an inherent limitation in the way a process can be modeled (e.g. through multiple linear models). In this work, we propose a permutation (and rotation) method which allows the inference in the multivariate linear model even in the presence of covariates (i.e. nuisance parameters, i.e. confounders). Also, the method allows for the immediate application of the min-p procedure. We make clear how permutations are a particular case of rotations of the data. Permutation tests are exact, while rotation tests retain exactness under multiple-multivariate linear model with normal errors. When errors are not normal, the rotation tests are weakly exchangeable (i.e. approximated and asymptotically exact). A real application to genetic data is presented and discussed.