In neuroimaging and other fields, permutation testing in linear models has become increasingly important. In this talk, we propose a permutation-type test that is also defined for GLMs and even more general models.
GLMs are often misspecified due to overdispersion or heteroscedasticity. We provide a semi-parametric test, based on sign-flipping individual score contributions. (Note that the score, the derivative of the log-likelihood, is the sum of n contributions.) Our test is often robust against the mentioned forms of misspecification and provides better type I error control than its competitors, while having good power. Moreover, when multiple outcome variables are considered, the test can be combined with powerful permutation-based multiple testing methods, which take into account the dependence among the outcomes.
When nuisance parameters are estimated, the score contributions become correlated and our basic test becomes conservative. To solve this, we instead sign-flip contributions of the effective score, which is asymptotically unaffected by the nuisance estimation. Our test is asymptotically equivalent to the classical score test, if the model is correct.