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Controlling the false discovery rate (FDR) is a powerful approach to multiple testing. In many applications, the tested hypotheses have an inherent hierarchical structure. In this paper, we focus on the fixed sequence structure where the testing order of the hypotheses has been strictly specified in advance. We are motivated to study such a structure, since it is the most basic of hierarchical structures, yet it is often seen in real applications such as statistical process control and streaming data analysis. We first consider a conventional fixed sequence method that stops testing once an acceptance occurs, and develop such a method controlling the FDR under both arbitrary and negative dependencies. The method under arbitrary dependency is shown to be unimprovable without losing control of the FDR and unlike existing FDR methods; it cannot be improved even by restricting to the usual positive regression dependence on subset (PRDS) condition. To account for any potential mistakes in the ordering of the tests, we extend the conventional fixed sequence method to one that allows more but a given number of acceptances. Simulation studies show that the proposed procedures can be powerful alternatives to existing FDR controlling procedures. The proposed procedures are illustrated through a real data set from a microarray experiment.

Testing a sequence of pre-ordered hypotheses to decide which of these can be rejected or accepted while controlling the familywise error rate (FWER) is of importance in many scientific studies such as clinical trials. In this paper, we first introduce a generalized fixed sequence procedure whose critical values are defined by using a function of the numbers of rejections and acceptances, and which allows follow-up hypotheses to be tested even if some earlier hypotheses are not rejected. We then construct the least favorable configuration for this generalized fixed sequence procedure and present a sufficient condition for the FWER control under arbitrary dependence. Based on the condition, we develop three new generalized fixed sequence procedures controlling the FWER under arbitrary dependence. We also prove that each generalized fixed sequence procedure can be described as a specific closed testing procedure. Through simulation studies and a clinical trial example, we compare the power performance of these proposed procedures with those of the existing FWER controlling procedures. Finally, when the pairwise joint distributions of the true null p-values are known, we further improve these procedures by incorporating pairwise correlation information while maintaining the control of the FWER. Copyright © 2015 John Wiley & Sons, Ltd.

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