A note on multiple testing for composite null hypotheses

Multiple hypothesis testing literature has recently experienced a growing development with particular attention to the control of the false discovery rate (FDR) based on p-values. While these are not the only methods to deal with multiplicity, inference with small samples and large sets of hypotheses depends on the specific choice of the p-value used to control the FDR in the presence of nuisance parameters. In this paper we propose to use the partial posterior predictive p-value [Bayarri, M.J., Berger, J.O., 2000. p-values for composite null models. J. Amer. Statist. Assoc. 95, 1127–1142] that overcomes this difficulty. This choice is motivated by theoretical considerations and examples. Finally, an application to a controlled microarray experiment is presented.