Röhmel, Joachim On familywise type I error control for multiplicity in equivalence trials with three or more treatments. (English) Zbl 1404.62127 Biom. J. 53, No. 6, 914-926 (2011). Summary: For the all pairwise comparisons for equivalence of \(k\) \((k\geq 2)\) treatments C. Lauzon and B. Caffo [Am. Stat. 63, No. 2, 147–154 (2009; Zbl 1404.62124)] proposed simply to divide the type I error level \(\alpha \) by \(k-1\) to achieve a Bonferroni-based familywise error control when declaring pairs of two treatments equivalent. This rule is shown to be too liberal for \(k\geq 4\). It works for \(k=3\) yet for reasons not considered by Lauzon and Caffo. Based on the two one-sided testing procedures and using the closure test principle we develop valid alternatives based on Bonferroni’s inequality. The set \(H\) of intersection hypotheses reveals a rich structure, leading to the possibility to present \(H\) as a directed acyclic graph (DAG). This in turn allows using some graph theoretical theorems and eases proving properties of the resulting multiple testing problems. Cited in 2 Documents MSC: 62P10 Applications of statistics to biology and medical sciences; meta analysis Keywords:all pair comparisons; Bonferroni adjustment; directed acyclic graph (DAG); equivalence; familywise error rate (FWER) Citations:Zbl 1404.62124 PDFBibTeX XMLCite \textit{J. Röhmel}, Biom. J. 53, No. 6, 914--926 (2011; Zbl 1404.62127) Full Text: DOI References: [1] Aho, The transitive reduction of a directed graph, SIAM Journal on Computing 1 pp 131– (1972) · Zbl 0247.05128 [2] Bang-Jensen, Digraphs Theory, Algorithms and Applications pp 178– (2007) [3] Barnett, Magnetic resonance-guided, real-time targeted delivery and imaging of magnetocapsules immunoprotecting pancreatic islet cells, Nature Medicine 13 pp 986– (2007) [4] Bergmann, Multiple Hypothesis Testing Symposium pp 100– (1987) [5] Bofinger, Equivalence with respect to a control: stepwise tests, Journal of the Royal Statistical Society Series B 57 pp 721– (1995) · Zbl 0827.62099 [6] Chen, On testing the equivalence of treatments using the measure of range, Computational Statistics and Data Analysis 55 pp 603– (2011) · Zbl 1247.62272 [7] Dunnett, A multiple comparison procedure for comparing several treatments with a control, Journal of the American Statistical Association 50 pp 1096– (1955) · Zbl 0066.12603 [8] Dunnett, Step-down multiple tests for comparing treatments with a control in unbalanced one-way layouts, Statistics in Medicine 10 pp 939– (1991) [9] Giani, Some general results on least favourable parameter constellations with special reference to equivalence testing and the range statistic, Journal for Statistical Planning and Inference 28 pp 33– (1991) · Zbl 0745.62018 [10] Hochberg, A sharper Bonferroni procedure for multiple tests of significance, Biometrika 75 pp 800– (1988) · Zbl 0661.62067 [11] Hommel, A stagewise rejective multiple test procedure based on modified Bonferroni test, Biometrika 5 pp 383– (1988) · Zbl 0639.62025 [12] Hothorn, Proof of hazard and proof of safety in toxicological studies using simultaneous confidence intervals for differences and ratios to control, Journal of Biopharmaceutical Statistics 18 pp 915– (2008) [13] Lauzon, Easy multiplicity control in equivalence testing using two one sided tests, The American Statistician 63 pp 147– (2009) · Zbl 1404.62124 [14] Marcus, On closed testing procedures with special reference to ordered analysis of variance, Biometrika 63 pp 655– (1976) · Zbl 0353.62037 [15] Schuirman, The two sided test procedure versus the power approach, Journal of Pharmacokinetics and Biopharmaceutics 15 pp 657– (1987) [16] Sonnemann, Allgemeine Lösungen multipler Testprobleme, EDV in Medizin und Biologie 13 pp 120– (1983) [17] The ALLHAT Officers and Coordinators for the ALLHAT Collaborative Research Group, Major Outcomes in high-risk hypertensive patients randomized to angiotensin-converting enzyme inhibitor or calcium channel blocker vs diuretic; the antihypertensive and lipid-lowering treatment to prevent heart attack trial (ALLHAT), Journal of the American Medical Association 288 pp 2981– (2002) [18] Wellek, Testing Statistical Hypotheses of Equivalence pp 162– (2003) · Zbl 1019.62001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.