  Multiple testing literature contains ample research on controlling false
discoveries for hypotheses classified according to one criterion, which we
refer to as one-way classified hypotheses. Although simultaneous classification
of hypotheses according to two different criteria, resulting in two-way
classified hypotheses, do often occur in scientific studies, no such research
has taken place yet, as far as we know, under this structure. This article
produces procedures, both in their oracle and data-adaptive forms, for
controlling the overall false discovery rate (FDR) across all hypotheses
effectively capturing the underlying one- or two-way classification structure.
They have been obtained by using results associated with weighted
Benjamini-Hochberg (BH) procedure in their more general forms providing
guidance on how to adapt the original BH procedure to the underlying one- or
two-way classification structure through an appropriate choice of the weights.
The FDR is maintained non-asymptotically by our proposed procedures in their
oracle forms under positive regression dependence on subset of null $p$-values
(PRDS) and in their data-adaptive forms under independence of the $p$-values.
Possible control of FDR for our data-adaptive procedures in certain scenarios
involving dependent $p$-values have been investigated through simulations. The
fact that our suggested procedures can be superior to contemporary practices
has been demonstrated through their applications in simulated scenarios and to
real-life data sets. While the procedures proposed here for two-way classified
hypotheses are new, the data-adaptive procedure obtained for one-way classified
hypotheses is alternative to and often more powerful than those proposed in Hu
et al. (2010).
