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**The sequential rejection principle of familywise error control.**
*(English)*
Zbl 1204.62140

Summary: Closed testing and partitioning are recognized as fundamental principles of familywise error control. 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 S. Holm [Scand. J. Stat., Theory Appl. 6, 65–70 (1979; Zbl 0402.62058)], J. P. Shaffer [J. Am. Stat. Assoc. 81, 826–831 (1986; Zbl 0603.62087)] and Y. Hochberg [Biometrika 75, No. 4, 800–802 (1988; Zbl 0661.62067)], 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.

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.

### MSC:

62L10 | Sequential statistical analysis |

62H15 | Hypothesis testing in multivariate analysis |

62J15 | Paired and multiple comparisons; multiple testing |

05C90 | Applications of graph theory |

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\textit{J. J. Goeman} and \textit{A. Solari}, Ann. Stat. 38, No. 6, 3782--3810 (2010; Zbl 1204.62140)

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