  We consider the problem of testing positively dependent multiple hypotheses
assuming that a prior information about the dependence structure is available.
We propose two-step multiple comparisons procedures that exploit the prior
information of the dependence structure, without relying on strong assumptions.
In the first step, we group the tests into subsets where tests are supposed to
be positively dependent and in each of which we compute the standardized mean
of the test scores. Given the subset mean scores or equivalently the subsets
p-values, we apply a first screening at a predefined threshold, which results
in two types of subsets. Based on this typing, the original single test
p-values are modified such that they can be used in conjunction with any
multiple comparison procedure. We show by means of different simulation that
power is gained with the proposed two-step methods, and compare it with
traditional multiple comparison procedures. As an illustration, our method is
applied on real data comparing topological differences between two groups of
human brain networks.
