  Permutation testing in linear models, where the number of nuisance
coefficients is smaller than the sample size, is a well-studied topic. The
common approach of such tests is to permute residuals after regressing on the
nuisance covariates. Permutation-based tests are valuable in particular because
they can be highly robust to violations of the standard linear model, such as
non-normality and heteroscedasticity. Moreover, in some cases they can be
combined with existing, powerful permutation-based multiple testing methods.
Here, we propose permutation tests for models where the number of nuisance
coefficients exceeds the sample size. The performance of the novel tests is
investigated with simulations. In a wide range of simulation scenarios our
proposed permutation methods provided appropriate type I error rate control,
unlike some competing tests, while having good power.
