  False discovery rate (FDR) procedures provide misleading inference when
testing multiple null hypotheses with heterogeneous multinomial data. For
example, in the motivating study the goal is to identify species of bacteria
near the roots of wheat plants (rhizobacteria) that are associated with
productivity, but standard procedures discover the most abundant species even
when the association is weak or negligible, and fail to discover strong
associations when species are not abundant. Consequently, a list of abundant
species is produced by the multiple testing procedure even though the goal was
to provide a list of producitivity-associated species. This paper provides an
FDR method based on a mixture of multinomial distributions and shows that it
tends to discover more non-negligible effects and fewer negligible effects when
the data are heterogeneous across tests. The proposed method and competing
methods are applied to the motivating data. The new method identifies more
species that are strongly associated with productivity and identifies fewer
species that are weakly associated with productivity.
