Bonferroni parallel gatekeeping - transparent generalizations, adjusted p -values, and short direct proofs.
In the two-step version (Dmitrienko, Tamhane, Wang and Chen, 2006) of the Bonferroni parallel-gatekeeping multiple-testing procedure (MTP): (a) a family F1 of null hypotheses H is used as a gatekeeper for another family F2 in that no H in F2 can be rejected unless at least one H is rejected in F1; (b) a Bonferroni MTP is used for F1 at local multiple-level alpha in the first step; and (c) Holm's (1979) step-down MTP is used in the second step for F2 at a local multiple level that depends on the rejections made in the first step. It is shown in this article that this two-step procedure can be generalized in that any MTP with multiple-level control and available multiplicity-adjusted p -values can be used instead of Holm's MTP in the second step. A further generalization related to what Dmitrienko, Molenberghs, Chuang-Stein and Offen (2005) called modified Bonferroni parallel gatekeeping is also given where in case all H s in F2 are rejected, additional rejections in F1 can be made in a third step at local multiple-level alpha through any MTP that is more powerful than the initial Bonferroni MTP, e.g. Holm's MTP. The proofs that these two generalized Bonferroni parallel-gatekeeping MTPs have multiple-level alpha are short and direct, without closed-testing arguments. Multiplicity-adjusted p -values can easily be calculated for these MTPs. The extensions to several successive gatekeeper families are straightforward. An illustration is given.
Guilbaud
Olivier
O
AstraZeneca R&D, S-15185 Södertälje, Sweden. olivier.guilbaud@astrazeneca.com
D000465:Algorithms; D000959:Antihypertensive Agents / Q000494:pharmacology / Q000627:therapeutic use; D001794:Blood Pressure / Q000187:drug effects; D002986:Clinical Trials as Topic / Q000379:methods*; D003627:Data Interpretation, Statistical; D004305:Dose-Response Relationship, Drug; D006973:Hypertension / Q000188:drug therapy
D000959:Antihypertensive Agents