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Closure properties of classes of multiple testing procedures

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  • Georg Hahn

    (Imperial College London)

Abstract

Statistical discoveries are often obtained through multiple hypothesis testing. A variety of procedures exists to evaluate multiple hypotheses, for instance the ones of Benjamini–Hochberg, Bonferroni, Holm or Sidak. We are particularly interested in multiple testing procedures with two desired properties: (solely) monotonic and well-behaved procedures. This article investigates to which extent the classes of (monotonic or well-behaved) multiple testing procedures, in particular the subclasses of so-called step-up and step-down procedures, are closed under basic set operations, specifically the union, intersection, difference and the complement of sets of rejected or non-rejected hypotheses. The present article proves two main results: First, taking the union or intersection of arbitrary (monotonic or well-behaved) multiple testing procedures results in new procedures which are monotonic but not well-behaved, whereas the complement or difference generally preserves neither property. Second, the two classes of (solely monotonic or well-behaved) step-up and step-down procedures are closed under taking the union or intersection, but not the complement or difference.

Suggested Citation

  • Georg Hahn, 2018. "Closure properties of classes of multiple testing procedures," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 102(2), pages 167-178, April.
    • Handle: RePEc:spr:alstar:v:102:y:2018:i:2:d:10.1007_s10182-017-0297-0
    DOI: 10.1007/s10182-017-0297-0
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    References listed on IDEAS

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    1. Axel Gandy & Georg Hahn, 2016. "A Framework for Monte Carlo based Multiple Testing," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 1046-1063, December.
    2. Guo Wenge & Peddada Shyamal, 2008. "Adaptive Choice of the Number of Bootstrap Samples in Large Scale Multiple Testing," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 7(1), pages 1-21, March.
    3. Axel Gandy & Georg Hahn, 2014. "MMCTest—A Safe Algorithm for Implementing Multiple Monte Carlo Tests," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 1083-1101, December.
    4. Gordon, Alexander Y., 2007. "Unimprovability of the Bonferroni procedure in the class of general step-up multiple testing procedures," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 117-122, January.
    5. Joseph P. Romano & Michael Wolf, 2005. "Exact and Approximate Stepdown Methods for Multiple Hypothesis Testing," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 94-108, March.
    6. Gordon, Alexander Y. & Salzman, Peter, 2008. "Optimality of the Holm procedure among general step-down multiple testing procedures," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1878-1884, September.
    7. van Wieringen, Wessel N & van de Wiel, Mark A & van der Vaart, Aad W, 2008. "A Test for Partial Differential Expression," Journal of the American Statistical Association, American Statistical Association, vol. 103(483), pages 1039-1049.
    8. Ajit C. Tamhane & Lingyun Liu, 2008. "On weighted Hochberg procedures," Biometrika, Biometrika Trust, vol. 95(2), pages 279-294.
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