  The introduction of the false discovery rate (FDR) by Benjamini and Hochberg
has spurred a great interest in developing methodologies to control the FDR in
various settings. The majority of existing approaches, however, address the FDR
control for the case where an appropriate univariate test statistic is
available. Modern hypothesis testing and data integration applications, on the
other hand, routinely involve multivariate test statistics. The goal, in such
settings, is to combine the evidence for each hypothesis and achieve greater
power, while controlling the number of false discoveries. This paper considers
data-adaptive methods for constructing nested rejection regions based on
multivariate test statistics (z-values). It is proved that the FDR can be
controlled for appropriately constructed rejection regions, even when the
regions depend on data and are hence random. This flexibility is then exploited
to develop optimal multiple comparison procedures in higher dimensions, where
the distribution of non-null z-values is unknown. Results are illustrated using
simulated and real data.
