A multiple testing method for hypotheses structured in a directed acyclic graph

We present a novel multiple testing method for testing null hypotheses that are structured in a directed acyclic graph (DAG). The method is a top-down method that strongly controls the familywise error rate and can be seen as a generalization of Meinshausen's procedure for tree-structured hypotheses. Just as Meinshausen's procedure, our proposed method can be used to test for variable importance, only the corresponding variable clusters can be chosen more freely, because the method allows for multiple parent nodes and partially overlapping hypotheses. An important application of our method is in gene set analysis, in which one often wants to test multiple gene sets as well as individual genes for their association with a clinical outcome. By considering the genes and gene sets as nodes in a DAG, our method enables us to test both for significant gene sets as well as for significant individual genes within the same multiple testing procedure. The method will be illustrated by testing Gene Ontology terms for evidence of differential expression in a survival setting and is implemented in the R package cherry.