Variance-invariant inference for regression models

Generalized linear models usually assume a common dispersion parameter, an assumption that is seldom true in practice. Consequently, standard parametric methods may suffer appreciable loss of type I error control. As an alternative, we present a semiparametric group-invariant method based on sign flipping of score contributions. Our method requires only the correct specification of the mean model, but is robust against any misspecification of the variance. We will further extend the methodology to multivariate responses. Indeed, the weaknesses of the standard approaches can dramatically propagate in multivariate settings. We propose a resampling-based method to handle multivariate generalized linear models which adapts to the unknown correlation structure, resulting in an appreciable gain of power compared to general alternatives when such correlation is present. Finally, we will exploit the benefits of the proposed sign-flipping test to semiparametric models by considering the Cox regression model, where standard approaches rely on asymptotic arguments, while they have a slow convergence to the nominal level of the test.