  Complex large-scale studies, such as those related to microarray data and
fMRI studies, often involve testing multiple hierarchically ordered hypotheses.
However, most existing false discovery rate (FDR) controlling procedures do not
exploit the inherent hierarchical structure among the tested hypotheses. In
this paper, we first present a generalized stepwise procedure which generalizes
the usual stepwise procedure to the case where each hypothesis is tested with a
different set of critical constants. This procedure is helpful in creating a
general framework under which our hierarchical testing procedures are
developed. Then, we present several hierarchical testing procedures which
control the FDR under various forms of dependence such as positive dependence
and block dependence. Our simulation studies show that these proposed methods
can be more powerful in some situations than alternative methods such as
Yekutieli's hierarchical testing procedure (Yekutieli, \emph{JASA} \textbf{103}
(2008) 309-316). Finally, we apply our proposed procedures to a real data set
involving abundances of microbes in different ecological environments.
