Nonparametric multiple test procedures with data-driven order of hypotheses and with weighted hypotheses

Abstract

Recently, Kropf (Shaker Verlag, Aachen, 2000) and Kropf and Läuter (Biometrial J. 44(7) 2002 789) proposed a procedure for testing separate hypotheses concerning the single variables of a multivariate normal population with strong control of the familywise type I error. The procedure is particularly focussed on situations where all variables have approximately equal variances. It establishes a suitable data-driven order of hypotheses and carries out unadjusted tests in that order until the first nonsignificant result. This proposal has been generalised by Westfall et al. (In: Benjamini, Y., Bretz, F., Sarkar, S.K. (Eds.), Recent developments in multiple comparison procedures, IMS Lecture Notes and Monograph Series, IMS, Haywood CA, 2004) in a weighted Bonferroni–Holm procedure that can be tuned with a free parameter η and involves the above procedure as limiting case for η→∞. Here, a nonparametric counterpart is given. Whereas the original parametric procedures utilise properties of spherically distributed matrices to maintain the exact type I error despite the data-dependent sorting or weighting of the variables, the nonparametric procedure exploits the independence of rank and order statistics under the null hypothesis to arrange the variables in a suitable order for unadjusted sequential testing or to weight the variables. Again, the procedures are sensible if the variables have a similar variance. The procedures are demonstrated with data from gene-expression studies in hot or cold thyroid nodules using Affymetrix GeneChips. Furthermore, simulation results are shown to consider the power under various conditions.