In the above permutation procedure, each λ (b) is the over-dispersion factor based on the test adjusting for covariates included in E as well as , which is a the permuted version of u (l). All λ (b), b = 1,…,B, provide the reference distribution for the evaluation of . We find in our numerical experiments that the median of λ (b), b = 1,…,B, is roughly at the same level as λ Current, the over-dispersion factor for the test adjusting for only covariates included in E. Thus, if the additional adjustment of u (l) (in addition to the ones already in E) does not lead to an over-dispersion factor that is lower than the current level (λ Current), we skip the permutation procedure and do not choose u (l).
