Informative Simultaneous Confidence Intervals in Hierarchical Testing^[*]

Summary

Background and Objectives: In clinical trials involving multiple tests it is often difficult to obtain informative simultaneous confidence intervals (SCIs). In particular in hierarchical testing, no quantification of effects is possible for the first tested (and most important) hypothesis after its rejection. Our goal is a construction of SCIs that are always informative.

Methods: We present an approach where the level is split after rejection of each hypothesis to obtain an informative confidence bound. The splitting weights are continuous functions of the parameters. Our method is realizable by a simple algorithm and is illustrated by an intuitive graphical representation.

Results: We show theoretically and by an example that the new SCIs always provide information when a hypothesis is rejected. The power to reject the first hypothesis is not smaller than for the classical fixed-sequence procedure. The price for the extra information is a small power loss in the hypotheses proceeding the most important one.

Conclusions: Given the substantial gain in information, a small loss of power for the non-primary hypotheses seems often acceptable. Especially in the context of non-inferiority trials, this method is a useful alternative. The flexibility in the choice of the weight functions makes the procedure attractive for applications.

Keywords

Confidence interval - fixed-sequence test - family wise error rate - simultaneous coverage probability

^* Supplementary material published on our website www.methods-online.com

• References

• 1 Hochberg E, Tamhane AC. Multiple Comparison Procedures. New York: Wiley; 1987
• 2 Bretz F, Hothorn T, Westfall P. Multiple Comparisons Using. RCRC Press; 2011
• 3 Marcus R. et al On closed testing procedures with special reference to ordered analysis of variance. Biometrika 1976; 63: 655-660.
• 4 Maurer W. et al Multiple comparisons in drug clinical trials and pre-clinical assays: a-priori ordered hypotheses. In. Vollmar J. editor. Biometrie in der chemisch-pharmazeutischen Industrie. Stuttgart: Fischer Verlag; 1995: 3-18.
• 5 Bretz F. et al A graphical approach to sequentially rejective multiple test procedures. Stat Med 2011; 28: 586-604.
• 6 Stefansson G. et al On confidence sets in multiple comparisons. In. Gupta SS, Berger JO. editors Statistical Decision Theory and Related Topics IV. Vol. 2. New York: Springer; 1988: 89-104.
• 7 Hsu JC, Berger RL. Stepwise confidence intervals without multiplicity adjustment for dose-response and toxicity studies. J Am Stat Ass 1999; 94: 468-482.
• 8 Strassburger K, Bretz F. Compatible simultaneous lower confidence bounds for the Holm procedure and other Bonferroni-based closed tests. Stat Med 2008; 27: 4914-4927.
• 9 Guilbaud O. Simultaneous confidence regions corresponding to Holm’s step-down procedure and other closed-testing procedures. Biometrical Journal 2008; 50: 678-692.
• 10 Guilbaud O. Alternative confidence regions for Bonferroni-based closed-testing procedures that are not alpha-exhaustive. Biometrical Journal 2009; 51: 721-735.
• 11 Wiens BL, Dmitrienko A. The fallback procedure for evaluating a single family of hypotheses. J Biopharm Stat 2005; 15: 929-942.
• 12 Brannath W, Schmidt S. A new class of powerful and informative simultaneous confidence intervals. Stat Med 2014, early view.
• 13 Holm S. A simple sequentially rejective multiple test procedure. Scand J Stat 1979; 6: 65-70.
• 14 Tu YH, Cheng B, Cheung YK. A note on confidence bounds after fixed sequence multiple tests. J Stat Plann Inference 2012; 142: 2993-2998.
• 15 Rauch G, Kieser M. Multiplicity adjustment for composite binary endpoints. Methods Inf Med 2012; 51: 309-317.