Mixed directional false discovery rate control in multiple pairwise comparisons using weighted p-values

In many applications, researchers are interested in making q pairwise comparisons among k test groups on the basis of m outcome variables. Often, m is very large. For example, such situations arise in gene expression microarray studies involving several experimental groups. Researchers are often not only interested in identifying differentially expressed genes between a given pair of experimental groups, but are also interested in making directional inferences such as whether a gene is up- or downregulated in one treatment group relative to another. In such situations, in addition to the usual errors such as false positive (Type I error) and false negative (Type II error), one may commit directional error (Type III error). For example, in a dose response microarray study, a gene may be declared to be upregulated in the high dose group compared to the low dose group when it is not. In this paper, we introduce a mixed directional false discovery rate (mdFDR) controlling procedure using weighted p-values to select positives in different directions. The weights are defined as the inverse of two times the proportion of either positive or negative discoveries. The proposed procedure has been proved mathematically to control the mdFDR at level α and to have a larger power (which is defined as the expected proportion of nontrue null hypotheses) than the GSP10 procedure proposed by Guo et al. (2010). Simulation studies and real data analysis are also conducted to show the outperformance of the proposed procedure than the GSP10 procedure.