Simultaneous Confidence Intervals Compatible with Sequentially Rejective Graphical Procedures

ABSTRACT

The Bonferroni-based sequentially rejective graphical procedures of Bretz et al. (2009) and Burman et al. (2009) include well-known multiple testing procedures (MTPs) such as the fixed-sequence MTP, the Fallback MTP, the Holm MTP, various gatekeeping MTPs, and quite elaborated variants/combinations of these MTPs. One-sided simultaneous confidence intervals (SCIs) compatible with such graphical MTPs have previously been based on results for more general MTPs involving closed-testing or partitioning arguments. If the MTP is alpha-exhaustive, then these SCIs are quite simple, but no confidence assertions sharper than rejection assertions are possible unless all hypotheses are rejected. If the MTP is not alpha-exhaustive (as the Fallback MTP), then some confidence assertions sharper than rejection assertions are possible even when not all hypotheses are rejected. These SCIs are, however, far from simple and transparent, so their description is typically omitted in methodological accounts in this context. It is shown in this article how SCIs can be formulated in a simple and transparent way for any MTP in the Burman et al. (2009) class. Various illustrations are given. A direct proof of the validity of these SCIs is provided that is based on simple monotonicity relations and does not involve closed-testing or partitioning arguments. Supplementary materials for this article are available online.

KEYWORDS:

• Alpha exhaustive
• Clinical trial
• Default graph
• Entangled graphs
• Multiple tests

Supplementary Materials

The supplementary materials for this article consist of three appendices (A–C). Appendix A provides the detailed numerical illustration of confidence bounds for a Fallback MTP with different a priori weights mentioned in Section 4.3.3. Appendix B provides the comparisons between alternative confidence bounds for a Fallback MTP with equal a priori weights also mentioned in Section 4.3.3. Appendix C provides the direct proof of the main result (21(21) $$\begin{array}{c}\hfill Pr\left[{\tilde{L}}_i<{\theta }_i\mathrm{for}\mathrm{all}i=1,\dots ,m\right]\ge 1-\alpha .\end{array}$$(21) ).