Post hoc false positive control for structured hypotheses

Guillermo Durand (1) , Gilles Blanchard (2, 3, 4, 5) , Pierre Neuvial (6) , Etienne Roquain (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 Institut für Mathematik [Potsdam]
3 LMO - Laboratoire de Mathématiques d'Orsay
4 CNRS - Centre National de la Recherche Scientifique
5 DATASHAPE - Understanding the Shape of Data
6 IMT - Institut de Mathématiques de Toulouse UMR5219

In a high‐dimensional multiple testing framework, we present new confidence bounds on the false positives contained in subsets S of selected null hypotheses. These bounds are post hoc in the sense that the coverage probability holds simultaneously over all S, possibly chosen depending on the data. This article focuses on the common case of structured null hypotheses, for example, along a tree, a hierarchy, or geometrically (spatially or temporally). Following recent advances in post hoc inference, we build confidence bounds for some pre-specified forest‐structured subsets and deduce a bound for any subset S by interpolation. The proposed bounds are shown to improve substantially previous ones when the signal is locally structured. Our findings are supported both by theoretical results and numerical experiments. Moreover, our bounds can be obtained by an algorithm (with complexity bilinear in the sizes of the reference hierarchy and of the selected subset) that is implemented in the open‐source R package sansSouci available from https://github.com/pneuvial/sanssouci, making our approach operational.

Mots clés: Dvoretzky Kiefer Wolfowitz inequality; multiple testing; post hoc bounds; forest structure; selective inference

Domaines: Statistiques [math.ST]

Guillermo Durand, Gilles Blanchard, Pierre Neuvial, Etienne Roquain. Post hoc false positive control for structured hypotheses. Scandinavian Journal of Statistics, 2020, 47 (4), pp.1114-1148. ⟨10.1111/sjos.12453⟩. ⟨hal-01829037v3⟩