Closure properties of classes of multiple testing procedures

Georg Hahn ()
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Georg Hahn: Imperial College London

AStA Advances in Statistical Analysis, 2018, vol. 102, issue 2, No 2, 167-178

Abstract: 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.

Keywords: Multiple hypothesis testing; Statistical significance; Step-up procedure; Set operations; Monotonicity; 62G10 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10182-017-0297-0

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