Powerful multiple testing procedures derived from hyperrectangular confidence regions having a minimal volume
[Procédures de tests multiples puissantes dérivées de régions de confiances hyperrectangulaires de volume minimal]
Journal de la société française de statistique, Tome 162 (2021) no. 1, pp. 2-21.

Nous étudions le contrôle de la probabilité d’obtenir un ou plusieurs faux-positifs dans le cadre du modèle linéaire gaussien lorsque la matrice de planification n×p est de rang p. Les procédures de tests multiples non-séquentielles contrôlant cette probabilité sont dérivées de régions de confiance hyperrectangulaires. Dans cet article, nous construisons une procédure basée sur une région de confiance hyperrectangulaires de volume minimal. Nous montrons que la minimisation du volume est un critère judicieux pour augmenter la puissance d’une procédure de tests multiples. Des expériences numériques montrent que notre démarche fournit une procédure plus performante que les procédures séquentielles et non-séquentielles de l’état de l’art. Enfin, nous appliquons cette procédure à la détection de métabolites en métabolomique.

We study the control of the FamilyWise Error Rate (FWER) in the linear Gaussian model when the n×p design matrix is of rank p. Single step multiple testing procedures controlling the FWER are derived from hyperrectangular confidence regions. In this study, we aim to construct procedure derived from hyperrectangular confidence region having a minimal volume. We show that minimizing the volume seems a fair criterion to improve the power of the multiple testing procedure. Numerical experiments demonstrate the performance of our approach when compared with the state-of-the-art single step and sequential procedures. We also provide an application to the detection of metabolites in metabolomics.

Classification : 62H15
Keywords: family wise error rate, multiple testing procedure, confidence region, linear model
Mot clés : probabilité d’avoir un ou plusieurs faux-positifs, procédure de tests multiples, region de confiance, modèle linéaire
Tardivel, Patrick J.C. 1 ; Servien, Rémi 2 ; Concordet, Didier 2

1 Institute of Mathematics, Wrocław University, Wrocław, Poland
2 INTHERES, Université de Toulouse, INRAE, ENVT, Toulouse, France
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Tardivel, Patrick J.C.; Servien, Rémi; Concordet, Didier. Powerful multiple testing procedures derived from hyperrectangular confidence regions having a minimal volume. Journal de la société française de statistique, Tome 162 (2021) no. 1, pp. 2-21. http://www.numdam.org/item/JSFS_2021__162_1_2_0/

[1] Astle, William; De Iorio, Maria; Richardson, Sylvia; Stephens, David; Ebbels, Timothy A Bayesian model of NMR spectra for the deconvolution and quantification of metabolites in complex biological mixtures, Journal of the American Statistical Association, Volume 107 (2012) no. 500, pp. 1259-1271 | DOI | MR | Zbl

[2] Dunnett, Charles W A multiple comparison procedure for comparing several treatments with a control, Journal of the American Statistical Association, Volume 50 (1955) no. 272, pp. 1096-1121 | DOI | Zbl

[3] Dunn, Olive Jean Confidence intervals for the means of dependent, normally distributed variables, Journal of the American Statistical Association, Volume 54 (1959) no. 287, pp. 613-621 | DOI | MR | Zbl

[4] Dunn, Olive Jean Multiple comparisons among means, Journal of the American Statistical Association, Volume 56 (1961) no. 293, pp. 52-64 | DOI | MR | Zbl

[5] Dudoit, Sandrine; Van Der Laan, Mark J Multiple Testing Procedures with Applications to Genomics, Springer, 2007 | MR

[6] Fromont, Magalie; Lerasle, Matthieu; Reynaud-Bouret, Patricia Family-wise separation rates for multiple testing, The Annals of Statistics, Volume 44 (2016) no. 6, pp. 2533-2563 | MR | Zbl

[7] Gronwall, Thomas Hakon Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics (1919), pp. 292-296 | DOI | JFM | MR

[8] Hao, Jie; Astle, William; De Iorio, Maria; Ebbels, Timothy MD BATMAN - an R package for the automated quantification of metabolites from nuclear magnetic resonance spectra using a Bayesian model, Bioinformatics, Volume 28 (2012) no. 15, pp. 2088-2090 | DOI

[9] Holm, Sture A simple sequentially rejective multiple test procedure, Scandinavian Journal of Statistics, Volume 6 (1979) no. 2, pp. 65-70 | MR | Zbl

[10] Lefort, Gaëlle; Liaubet, Laurence; Canlet, Cécile; Tardivel, Patrick; Père, Marie-Christine; Quesnel, Hélène; Paris, Alain; Iannuccelli, Nathalie; Vialaneix, Nathalie; Servien, Rémi ASICS: an R package for a whole analysis workflow of 1D 1H NMR spectra, Bioinformatics, Volume 35 (2019) no. 21, pp. 4356-4363 | DOI

[11] Lehmann, Erich L.; Romano, Joseph P. Testing Statistical Hypotheses, Springer Texts in Statistics, Springer, New York, 2005 | MR

[12] Lehmann, EL; Romano, Joseph P; Shaffer, Juliet Popper On optimality of stepdown and stepup multiple test procedures, Selected Works of EL Lehmann, Springer, 2012, pp. 693-717 | DOI

[13] Pratt, John W Length of confidence intervals, Journal of the American Statistical Association, Volume 56 (1961) no. 295, pp. 549-567 | DOI | MR | Zbl

[14] Ravanbakhsh, Siamak; Liu, Philip; Bjordahl, Trent C.; Mandal, Rupasri; Grant, Jason R.; Wilson, Michael; Eisner, Roman; Sinelnikov, Igor; Hu, Xiaoyu; Luchinat, Claudio; Greiner, Russell; Wishart, David S Accurate, Fully-Automated NMR Spectral Profiling for Metabolomics, PLoS ONE, Volume 10 (2015) no. 5, e0124219 | DOI

[15] Royen, Thomas A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions, Far East Journal of Theoretical Statistics, Volume 48 (2014) no. 2, pp. 139-145 | MR | Zbl

[16] Romano, Joseph P; Shaikh, Azeem; Wolf, Michael Consonance and the closure method in multiple testing, The International Journal of Biostatistics, Volume 7 (2011) no. 1 | MR

[17] Romano, Joseph P; Wolf, Michael Exact and approximate stepdown methods for multiple hypothesis testing, Journal of the American Statistical Association, Volume 100 (2005) no. 469, pp. 94-108 | DOI | MR | Zbl

[18] Sidak, Zbynek On probabilities of rectangles in multivariate Student distributions: their dependence on correlations, The Annals of Mathematical Statistics, Volume 42 (1971) no. 1, pp. 169-175 | DOI | MR | Zbl

[19] Tardivel, Patrick J. C.; Canlet, Cécile; Lefort, Gaëlle; Tremblay-Franco, Marie; Debrauwer, Laurent; Concordet, Didier; Servien, Rémi ASICS: an automatic method for identification and quantification of metabolites in complex 1D 1H NMR spectra, Metabolomics, Volume 13 (2017) no. 10, p. 109 | DOI

[20] Tulpan, Dan; Léger, Serge; Belliveau, Luc; Culf, Adrian; Čuperlović-Culf, Miroslava MetaboHunter: an automatic approach for identification of metabolites from 1H-NMR spectra of complex mixtures, BMC Bioinformatics, Volume 12 (2011) no. 1, p. 400 | DOI

[21] Weljie, Aalim M.; Newton, Jack; Mercier, Pascal; Carlson, Erin; Slupsky, Carolyn M. Targeted Profiling: Quantitative analysis of 1H-NMR Metabolomics Data, Analytical Chemistry, Volume 78 (2006) no. 13, pp. 4430-4442 | DOI

[22] Šidák, Zbyněk Rectangular confidence regions for the means of multivariate normal distributions, Journal of the American Statistical Association, Volume 62 (1967) no. 318, pp. 626-633 | MR | Zbl