Sharper Confidence Intervals for Hochberg- and Hommel-Related Multiple Tests Based On an Extended Simes Inequality

References

• Alosh, M., Bretz, F., and Huque, M. (2013), “Advanced Multiplicity Adjustment Methods in Clinical Trials,” Statistics in Medicine, DOI: 10.1002/sim.5974.
   Google Scholar
• Bernhard, G., Klein, M., and Hommel, G. (2004), “Global and Multiple Test Procedures Using Ordered p-Values—A Review,” Statistical Papers, 45, 1–14.
   Google Scholar
• Bernhard, G., Klein, M., and Hommel, G. (2013), “Erratum: Global and Multiple Test Procedures Using Ordered p-Values—A Review,” Statistical Papers, DOI: 10.1007/bf02778266.
   Google Scholar
• Daniels, H.E. (1945), “The Statistical Theory of the Strength of Bundles of Threads,” Proceedings of the Royal Statistical Society, Series A, 183, 405–435, DOI: 10.1098/rspa.1945.0011.
   Google Scholar
• Dmitrienko, A., Bretz, F., Westfall, P.H., Troendle, J., Wiens, B.L., Tamhane, A.C., and Hsu, J.C. (2009), “Multiple Testing Methodology,” in Multiple Testing Problems in Pharmaceutical Statistics, eds. A. Dmitrienko, A.C. Tamhane, and F. Bretz, Boca Raton, FL: Chapman & Hall/CRC, pp. 35–98.
   Google Scholar
• Dmitrienko, A., and D’Agostino, R.B. (2013), “Traditional Multiplicity Adjustment Methods in Clinical Trials,” Statistics in Medicine, DOI: 10.1002/sim.5990.
   Google Scholar
• Dmitrienko, A., D’Agostino, R.B., and Huque, M.F. (2013), “Key Multiplicity Issues in Clinical Drug Developments,” Statistics in Medicine, 32, 1079–1111.
   PubMed Web of Science ®Google Scholar
• Dmitrienko, A., Tamhane, A.C., and Wiens, B.L. (2008), “General Multistage Gatekeeping Procedures,” Biometrical Journal, 50, 667–677.
   PubMed Web of Science ®Google Scholar
• Dunnett, C.W., and Tamhane, A.C. (1991), “Step-Down Multiple Tests for Comparing Treatments With a Control in Unbalanced One-Way Layouts,” Statistics in Medicine, 10, 939–947.
   PubMed Web of Science ®Google Scholar
• Finner, H. (1999), “Stepwise Multiple Test Procedures and Control of Directional Errors,” The Annals of Statistics, 27, 274–289.
   Web of Science ®Google Scholar
• Finner, H., and Strassburger, K. (2007), “Step-Up Related Simultaneous Confidence Intervals for MCC and MCB,” Biometrical Journal, 49, 40–51.
   Google Scholar
• Guilbaud, O. (2008), “Simultaneous Confidence Regions Corresponding to Holm's Step-Down Procedure and Other Closed-Testing Procedures,” Biometrical Journal, 50, 678–692.
   PubMed Web of Science ®Google Scholar
• Guilbaud, O. (2009), “Alternative Confidence Regions for Bonferroni-Based Closed-Testing Procedures that are not Alpha-Exhaustive,” Biometrical Journal, 51, 721–735.
   PubMed Web of Science ®Google Scholar
• Guilbaud, O. (2012), “Simultaneous Confidence Regions for Closed Tests, Including Holm-, Hochberg-, and Hommel-Related Procedures,” Biometrical Journal, 54, 317–342.
   PubMed Web of Science ®Google Scholar
• Guilbaud, O. (2013), “Corrections: Simultaneous Confidence Regions for Closed Tests, Including Holm-, Hochberg-, and Hommel-Related Procedures,” Biometrical Journal, 55, 634–635.
   Google Scholar
• Hayter, A.J., and Hsu, J.C. (1994), “On the Relationship Between Stepwise Decision Procedures and Confidence Sets,” Journal of the American Statistical Association, 89, 128–136.
   Web of Science ®Google Scholar
• Hochberg, Y. (1988), “A Sharper Bonferroni Procedure for Multiple Significance Testing,” Biometrika, 75, 800–802.
   Web of Science ®Google Scholar
• Hochberg, Y., Liu, W., and Parmat, Y. (2000), “,” in Perspectives in Statistical Sciences, eds. A. Basu, J. Gosh, P.K. Sen, and B. Sinha, India: Oxford University Press, pp. 179–190.
   Google Scholar
• Holm, S. (1979), “A Simple Sequentially Rejective Multiple Test Procedure,” Scandinavian Journal of Statistics, 6, 65–70.
   Web of Science ®Google Scholar
• Hommel, G. (1988), “A Stagewise Rejective Multiple Test Procedure Based on a Modified Bonferroni Test,” Biometrika, 75, 383–386.
   Web of Science ®Google Scholar
• Hsu, J.C. (1996). Multiple Comparisons—Theory and Methods, London: Chapman & Hall.
   Google Scholar
• Hsu, J.C., and Berger, R.L. (1999), “Stepwise Confidence Intervals Without Multiplicity Adjustment for Dose-Response and Toxicity Studies,” Journal of the American Statistical Association, 94, 468–482.
   Web of Science ®Google Scholar
• Klein, M. (1998), “Multiple Testprozeduren: Step-Down und Step-Up,” unpublished diploma thesis, University Mainz, Mainz, Germany. . [In German, unpublished, but available from Markus Zervos: [email protected]]
   Google Scholar
• Liu, W. (1996), “Multiple Tests of a Non-Hierarchical Finite Family of Hypotheses,” Journal of the Royal Statistical Society, Series B, 26, 455–461.
   Google Scholar
• Liu, W. (1997), “Control of Directional Errors With Step-Up Multiple tests,” Statistics and Probability Letters, 31, 239–242.
   Google Scholar
• Sarkar, S.K. (1998), “Some Probability Inequalities for Ordered MTP2 Random Variables: A Proof of the Simes Conjecture,” The Annals of Statistics, 26, 494–504.
   Web of Science ®Google Scholar
• Sarkar, S.K. (2008), “On the Simes Inequality and its Generalization,” in Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen (Vol. 1), eds. N. Balakrishnan, E. Pena, and M.J. Silvapulle, Beachwood, Ohio: IMS Collections, 231–242, DOI: 10.1214/193940307000000167.
   Google Scholar
• Sarkar, S.K., and Chang, C.K. (1997), “Simes’ Method for Multiple Hypothesis Testing With Positively Dependent Test Statistics,” Journal of the American Statistical Association, 92, 1601–1608.
   Web of Science ®Google Scholar
• Sarkar, S.K., Sen, P.K., and Finner, H. (2004), “On Two Results in Multiple Testing,” in Recent Developments in Multiple Comparison Procedures (Vol. 47), eds. Y. Benjamini, F. Bretz, and S.K. Sarkar, Beachwood, Ohio: IMS Lecture Notes—Monograph Series, pp. 89–99, DOI: 10.1214/lnms/1196285628.
   Google Scholar
• Simes, R.J. (1986), “An Improved Bonferroni Procedure for Multiple Tests of Significance,” Biometrika, 73, 751–754.
   Web of Science ®Google Scholar
• Shaffer, J.P. (1980), “Control of Directional Errors With Stagewise Multiple Test Procedures,” The Annals of Statistics, 8, 1342–1348.
   Web of Science ®Google Scholar
• Stefansson, G., Kim, W.C., and Hsu, J.C. (1988). “On Confidence Sets in Multiple Comparisons,” in Statistical Decision Theory and Related Topics IV (Vol. 2), eds. S.S. Gupta and J.O. Berger, New York: Springer-Verlag, pp. 89–104.
   Google Scholar
• Strassburger, K., and Bretz, F. (2008), “Compatible Simultaneous Lower Confidence Bounds for the Holm Procedure and Other Bonferroni-Based Closed Tests,” Statistics in Medicine, 27, 4914–4927.
   PubMed Web of Science ®Google Scholar
• Westfall, P.H., Bretz, F., and Tobias, R.D. (2013), “Directional Error Rates of Closed Testing Procedures,” Statistics in Biopharmaceutical Research, 5, 345–355.
   Web of Science ®Google Scholar