Multiple Testing with Heterogeneous Multinomial Distributions

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.