  We introduce a multiple testing procedure (TreeBH) which addresses the
challenge of controlling error rates at multiple levels of resolution.
Conceptually, we frame this problem as the selection of hypotheses which are
organized hierarchically in a tree structure. We describe a fast algorithm for
the proposed sequential procedure, and prove that it controls relevant error
rates given certain assumptions on the dependence among the p-values. Through
simulations, we demonstrate that TreeBH offers the desired guarantees under a
range of dependency structures (including one similar to that encountered in
genome-wide association studies) and that it has the potential of gaining power
over alternative methods. We also introduce a modified version of TreeBH which
we prove to control the relevant error rates under any dependency structure.
  We conclude with two case studies: we first analyze data collected as part of
the Genotype-Tissue Expression (GTEx) project, which aims to characterize the
genetic regulation of gene expression across multiple tissues in the human
body, and secondly, data examining the relationship between the gut microbiome
and colorectal cancer.
