Simultaneous confidence regions for closed tests, including H olm‐, H ochberg‐, and H ommel‐related procedures

The derivation of simultaneous confidence regions for some multiple‐testing procedures ( MTP s) of practical interest has remained an unsolved problem. This is the case, for example, for H ochberg's step‐up MTP and H ommel's more powerful MTP that is neither a step‐up nor a step‐down procedure. It is shown in this article how the direct approach used previously by the author to construct confidence regions for certain closed‐testing procedures ( CTP s) can be extended to a rather general setup. The general results are then applied to a situation with one‐sided inferences and CTP s belonging to a class studied by W ei L iu. This class consists of CTP s based on ordered marginal p ‐values. It includes H olm's, H ochberg's, and H ommel's MTP s. A property of the confidence regions derived for these three MTP s is that no confidence assertions sharper than rejection assertions can be made unless all null hypotheses are rejected. Briefly, this is related to the fact that these MTP s are quite powerful. The class of CTP s considered includes, however, also MTP s related to H olm's, H ochberg's, and H ommel's MTP s that are less powerful but are such that confidence assertions sharper than rejection assertions are possible even if not all null hypotheses are rejected. One may thus choose and prespecify such an MTP , though this is at the cost of less rejection power.