A survey on multivariate chi-square distributions and their applications in testing multiple hypotheses

Abstract

We are concerned with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and consider the computation of multiplicity-adjusted p-values under the respective global hypothesis. By means of numerical examples, we demonstrate how much gain in level exhaustion or, equivalently, power can be achieved with corresponding multivariate multiple tests compared with approaches which are only based on univariate marginal distributions and do not take the dependence structure among the test statistics into account. As a further contribution of independent value, we provide an overview of essentially all analytic formulas for computing multivariate chi-square probabilities of the considered types which are available up to present. These formulas were scattered in the previous literature and are presented here in a unified manner.

Keywords:

• contingency tables
• Kruskal–Wallis test
• multiple Wald tests
• multivariate analysis
• multivariate gamma distributions
• statistical genetics

AMS Subject Classification:

• 62J15
• 62E15

Acknowledgements

We are grateful to two anonymous referees and the Associate Editor for their constructive comments, and we thank Yuriy Kopanskyy and Jens Stange for help with LATEX and computer simulations.

Additional information

Funding

This research was partly supported by the Deutsche Forschungsgemeinschaft [grant number DI 1723/3-1].