  Scientific hypotheses in a variety of applications have domain-specific
structures, such as the tree structure of the International Classification of
Diseases (ICD), the directed acyclic graph structure of the Gene Ontology (GO),
or the spatial structure in genome-wide association studies. In the context of
multiple testing, the resulting relationships among hypotheses can create
redundancies among rejections that hinder interpretability. This leads to the
practice of filtering rejection sets obtained from multiple testing procedures,
which may in turn invalidate their inferential guarantees. We propose Focused
BH, a simple, flexible, and principled methodology to adjust for the
application of any pre-specified filter. We prove that Focused BH controls the
false discovery rate under various conditions, including when the filter
satisfies an intuitive monotonicity property and the p-values are positively
dependent. We demonstrate in simulations that Focused BH performs well across a
variety of settings, and illustrate this method's practical utility via
analyses of real datasets based on ICD and GO.
