BOX-TYPE APPROXIMATIONS IN NONPARAMETRIC FACTORIAL-DESIGNS

Linear rank statistics in nonparametric factorial designs are asymptot ically normal and, in general, heteroscedastic. In a comprehensive sim ulation study, the asymptotic chi-squared law of the corresponding qua dratic forms is shown to be a rather poor approximation of the finite- sample distribution. Motivated by this problem, we propose simple fini te-sample size approximations for the distribution of quadratic forms in factorial designs under a normal heteroscedastic error structure. T hese approximations are based on an F distribution with estimated degr ees of freedom that generalizes ideas of Patnaik and Box. Simulation s tudies show that the nominal level is maintained with high accuracy an d in most cases the power is comparable to the asymptotic maximin Wald test. Data-driven guidelines are given to select the most appropriate test procedure. These ideas are finally transferred to nonparametric factorial designs where the same quadratic forms as in the parametric case are applied to the vector of averaged ranks. A simulation study s hows that the corresponding nonparametric ''F-test'' keeps its level w ith high accuracy and has power comparable to that of the rank version of the likelihood statistic.