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Closed testing and partitioning are recognized as fundamental principles offamilywise error control. In this paper, we argue that sequential rejection canbe considered equally fundamental as a general principle of multiple testing.We present a general sequentially rejective multiple testing procedure and showthat many well-known familywise error controlling methods can be constructed asspecial cases of this procedure, among which are the procedures of Holm,Shaffer and Hochberg, parallel and serial gatekeeping procedures, modernprocedures for multiple testing in graphs, resampling-based multiple testingprocedures and even the closed testing and partitioning procedures themselves.We also give a general proof that sequentially rejective multiple testingprocedures strongly control the familywise error if they fulfill simplecriteria of monotonicity of the critical values and a limited form of weakfamilywise error control in each single step. The sequential rejectionprinciple gives a novel theoretical perspective on many well-known multipletesting procedures, emphasizing the sequential aspect. Its main practicalusefulness is for the development of multiple testing procedures for nullhypotheses, possibly logically related, that are structured in a graph. Weillustrate this by presenting a uniform improvement of a recently publishedprocedure.

The sequential rejection principle of familywise error control