Multiple testing with heterogeneous multinomial distributions

Standard false discovery rate (FDR) procedures can 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 moderately or strongly associated with productivity. However, standard procedures discover the most abundant species even when their association is weak and fail to discover many moderate and strong associations when the species are not abundant. This article provides a new FDR-controlling method based on a finite mixture of multinomial distributions and shows that it tends to discover more moderate and strong associations and fewer weak associations when the data are heterogeneous across tests. The new method is applied to the rhizobacteria data and performs favorably over competing methods.