  We consider the problem of multiple hypothesis testing when there is a
logical nested structure to the hypotheses. When one hypothesis is nested
inside another, the outer hypothesis must be false if the inner hypothesis is
false. We model the nested structure as a directed acyclic graph, including
chain and tree graphs as special cases. Each node in the graph is a hypothesis
and rejecting a node requires also rejecting all of its ancestors. We propose a
general framework for adjusting node-level test statistics using the known
logical constraints. Within this framework, we study a smoothing procedure that
combines each node with all of its descendants to form a more powerful
statistic. We prove a broad class of smoothing strategies can be used with
existing selection procedures to control the familywise error rate, false
discovery exceedance rate, or false discovery rate, so long as the original
test statistics are independent under the null. When the null statistics are
not independent but are derived from positively-correlated normal observations,
we prove control for all three error rates when the smoothing method is
arithmetic averaging of the observations. Simulations and an application to a
real biology dataset demonstrate that smoothing leads to substantial power
gains.
