Confidence regions for Bonferroni-based closed tests extended to more general closed tests

Simultaneous confidence regions were recently derived for a large class of Bonferroni-based closed testing procedures (CTPs). This class and a very useful subclass of graphical procedures are described, and it is then shown how the direct approach of Guilbaud (2008) for such CTPs can be extended to cover also CTPs that are not Bonferroni based and that may utilize dependencies not utilized by Bonferroni-based procedures. The simultaneous confidence regions that can be derived through this extension include the known confidence bounds for the step-down version of one-sided Dunnett-type multiple comparisons versus a control in the balanced case. As a simple illustration of how the general results can be applied, similar bounds are derived for the unbalanced case. As a further, more elaborated, illustration, confidence bounds are derived for the Dunnett gatekeeping procedure of Xu et al. (2009) based on a primary and a secondary variable. In this procedure, primary rejections are made through an ordinary one-sided Dunnett procedure, and secondary rejections are restricted to treatment comparisons with primary rejections. It turns out that ordinary Dunnett-type confidence assertions can be made for all primary comparisons without rejections. Interestingly, informative confidence assertions can be made also for secondary comparisons without rejections, but only in case of matching primary rejections. The gatekeeping is thus naturally extended to confidence assertions.