Fortunato Pesarin (University of Padova, Italy)

Nonparametric combination of dependent partial tests

We deal with permutation approach of a variety of multidimensional problems of testing of hypotheses in a nonparametric framework. There are many multidimensional complex problems, frequently encountered in most applicational fields (agriculture, biology, clinical trials, engineering, the environment, experimental design, genetics, pharmacology, psychology, quality control, zoology, etc.), which are rather difficult to solve outside the permutation context, and in particular outside the method of nonparametric combination of dependent partial tests (Pesarin, 1992, 1999). Moreover, within parametric solutions based on normality of errors, it is sometimes impossible to obtain proper solutions. We mention, for instance, three such testing problems. One is related to the paired observations problem when scale coefficients are dependent on units, another is related to the two-way ANOVA, and the third to some multidimensional tests when the number of observed variables is higher than the sample size. In the first, within a parametric framework it is impossible to obtain estimates of standard deviations for each unit, whereas an exact effective permutation solution does exist. In the second it is impossible to obtain independent or even uncorrelated separate inferences for main factors and interactions, because all related statistics are compared with the same estimate of the variance of error components. Within the permutation approach, it is possible to obtain uncorrelated exact inferences in the general case and independent inferences under normality of errors. In the third, it is impossible to find estimates of the covariance matrix with more than zero degrees of freedom, whereas the nonparametric combination method allows for a proper solution, which is often asymptotically efficient. In a great variety of statistical analyses of complex hypotheses testing, when many response variables are involved or many different aspects are of interest, to some extent it is natural, and often convenient, first to process data by a finite set of *k* > 1 different *partial tests* (note that *k* is not necessarily equal to dimensionality *q* of responses). Therefore, they may be useful in a marginal or disjoint sense. But, when they are jointly considered, they provide information on a general overall (or global) hypothesis, which typically represents the true objective of the majority of multidimensional testing problems. combination in one (unidimensional) *combined* or *second-order* test, naturally arises. Multiple comparisons extensions of the above methodology are also discussed (Westfall and Young, 1993).

References:

- Edgington E.S. (1995)
*Randomization tests*, 3rd ed., Marcel Dekker, New York. - Good P. (1994)
*Permutation Tests*. Springer-Verlag, New York. - Manly B.F.J. (1997)
*Randomization, bootstrap and Monte Carlo methods in biology*. 2nd edition, Chapman and Hall, London. - Pesarin, F. (1992) A resampling procedure for nonparametric combination of several dependent tests.
*Journal of the Italian Statistical Society*, 1, 87-101. - Pesarin, F. (1994) Goodness of fit testing for ordered discrete distributions by resampling techniques.
*Metron*, LII, 57-71. - Pesarin, F. (1997a) An almost exact solution for the multivariate Behrens-Fisher problem,
*Metron*, LV, 85-100. - Pesarin, F. (1997a) A nonparametric combination method for dependent permutation tests with application to some problems with repeated measures.
*Proceedings of the ISI Satellite Meeting on Industrial Statistics*, C.P. Kitsos and L. Edler Eds., Pysica-Verlag, Heidelberg, 259-268. - Pesarin F. (1999).
*Permutation testing of multidimensional hypotheses by nonparametric combination of dependent tests*. CLEUP, Padua. - Sprent P. (1998)
*Data driven statistical methods*. Chapman and Hall, London. - Westfall P.H., Young S.S. (1993)
*Resampling-based Multiple Testing*. Wiley, New York.