  Closed testing and partitioning are recognized as fundamental principles of
familywise error control. In this paper, we argue that sequential rejection can
be considered equally fundamental as a general principle of multiple testing.
We present a general sequentially rejective multiple testing procedure and show
that many well-known familywise error controlling methods can be constructed as
special cases of this procedure, among which are the procedures of Holm,
Shaffer and Hochberg, parallel and serial gatekeeping procedures, modern
procedures for multiple testing in graphs, resampling-based multiple testing
procedures and even the closed testing and partitioning procedures themselves.
We also give a general proof that sequentially rejective multiple testing
procedures strongly control the familywise error if they fulfill simple
criteria of monotonicity of the critical values and a limited form of weak
familywise error control in each single step. The sequential rejection
principle gives a novel theoretical perspective on many well-known multiple
testing procedures, emphasizing the sequential aspect. Its main practical
usefulness is for the development of multiple testing procedures for null
hypotheses, possibly logically related, that are structured in a graph. We
illustrate this by presenting a uniform improvement of a recently published
procedure.
