  Often in multiple testing, the hypotheses appear in non-overlapping blocks
with the associated $p$-values exhibiting dependence within but not between
blocks. We consider adapting the Benjamini-Hochberg method for controlling the
false discovery rate (FDR) and the Bonferroni method for controlling the
familywise error rate (FWER) to such dependence structure without losing their
ultimate controls over the FDR and FWER, respectively, in a non-asymptotic
setting. We present variants of conventional adaptive Benjamini-Hochberg and
Bonferroni methods with proofs of their respective controls over the FDR and
FWER. Numerical evidence is presented to show that these new adaptive methods
can capture the present dependence structure more effectively than the
corresponding conventional adaptive methods. This paper offers a solution to
the open problem of constructing adaptive FDR and FWER controlling methods
under dependence in a non-asymptotic setting and providing real improvements
over the corresponding non-adaptive ones.
