The recently proposed simple but powerful sign depth tests depend on the order of the residuals. While one-dimensional explanatory variables provide a natural order, there exists no canonical order for multidimensional explanatory variables. For this scenario, we present different approaches for ordering multidimensional explanatory variables and compare them regarding their performance with respect to the stability of the ordering, the usability for non-metric explanatory variables, the computational time complexity, and in the context of testing in linear models including high-dimensional multiple regression. It is shown that the sign depth tests based on orderings given by pairwise distances perform best. They are much more powerful than the classical sign test and also than the F-test when not having normally distributed errors. They are competitive to the much more complicated robust Wald test based on efficient MM-estimation. Additionally, the sign depth tests are more appropriate for outlier robust model checks.
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