  In many large scale multiple testing applications, the hypotheses often have
a known graphical structure, such as gene ontology in gene expression data.
Exploiting this graphical structure in multiple testing procedures can improve
power as well as aid in interpretation. However, incorporating the structure
into large scale testing procedures and proving that an error rate, such as the
false discovery rate (FDR), is controlled can be challenging. In this paper, we
introduce a new general approach for large scale multiple testing, which can
aid in developing new procedures under various settings with proven control of
desired error rates. This approach is particularly useful for developing FDR
controlling procedures, which is simplified as the problem of developing
per-family error rate (PFER) controlling procedures. Specifically, for testing
hypotheses with a directed acyclic graph (DAG) structure, by using the general
approach, under the assumption of independence, we first develop a specific
PFER controlling procedure and based on this procedure, then develop a new FDR
controlling procedure, which can preserve the desired DAG structure among the
rejected hypotheses. Through a small simulation study and a real data analysis,
we illustrate nice performance of the proposed FDR controlling procedure for
DAG-structured hypotheses.
