Asymptotic permutation tests in general factorial designs

In general factorial designs where no homoscedasticity or a particular error distribution is assumed, the well-known Wald-type statistic is a simple asymptotically valid procedure. However, it is well known that it suffers from a poor finite sample approximation since the convergence to its (2) limit distribution is quite slow. This becomes even worse with an increasing number of factor levels. The aim of the paper is to improve the small sample behaviour of the Wald-type statistic, maintaining its applicability to general settings as crossed or hierarchically nested designs by applying a modified permutation approach. In particular, it is shown that this approach approximates the null distribution of the Wald-type statistic not only under the null hypothesis but also under the alternative yielding an asymptotically valid permutation test which is even finitely exact under exchangeability. Finally, its small sample behaviour is compared with competing procedures in an extensive simulation study.