Rank tests are known to be distribution-free for simple linear models, where the observations are i.i.d.
For general linear models with nuisance parameters, however, the alignment principle can be applied to
obtain asymptotically distribution-free rank tests. This is especially so when the centered design
matrices have full rank and the alternatives are local.

We apply the Chernoff - Savage approach to derive the asymptotic distribution of a test statistic for a
repeated observations model with an orthonormal design, given that the centered design matrices are not
of full rank. The distribution turns out to be chi-square under the null hypothesis, regardless of the
choice of the aligner. Simulation studies regarding the Type I error rate and power in testing for
linearity of a nonparametric regression model with standard Cauchy random errors corroborate this
theoretical result. Additional insight into the independence on the aligner is gained in scale models.

This approach can be extended to the multivariate case using a projection method.