Rank test statistics for unbalanced nested designs

We formulate rank statistics for testing hypotheses in unbalanced, and possibly heteroscedastic, two-factor nested designs with independent observations. These include Wald-type statistics based on the theory introduced by Akritas, Arnold and Brunner, as well as a Box-type approximation which is intended to improve the accuracy of approximation to asymptotic distributions. We also present statistics based on a recent theory of weighted FF-statistics for ranks. The actual sizes of the statistics at various nominal levels are compared in a simulation study. Our main conclusion is that the Box-adjusted Wald-type statistic is the only statistic that is accurate across all the situations considered and therefore we recommend it for general use.