Predictor versus response permutation for significance testing in weighted regression and redundancy analysis

In testing an overall null hypothesis, it does not matter whether to permute the response variables (Y) while keeping predictors fixed or predictor (X) fixed. However, in weighted (univariate and multivariate) regression partial tests these options yield different results. This paper defines evaluates by simulation ordinary standardized versions of X- Y-permutation for which encompass existing Double Semi-Partialling, here called Collins-Dekker, Freedman-Lane permutation methods. When error variance is inversely proportional weights, methods (which use residuals) are most powerful, as expected, but otherwise they can be extremely liberal. contrast, X-permutation permutes residuals X given any covariates ‘as is’) far robust against variability weights variance-weight relationship thus recommended general use.