In the analysis of gene expression data, Tian et al. [27] formulated two statistical hypotheses for testing coordinated association between a group of genes with a phenotype of interest. In the context of GWAS analysis, they are Competitive null hypothesis (Q1) - The genes in a gene-set show the same magnitude of associations with the disease phenotype compared with genes in the rest of the genome;Self-contained null hypothesis (Q2) - The genes in a gene-set are not associated with the disease phenotype. A third null hypothesis (Q3) - none of the gene sets considered is associated with the phenotype - has also been proposed recently [28,29]. In contrast to Q1 and Q2, which test for individual gene sets, Q3 tests the entire dataset. For tests of individual gene sets, Goeman and Buhlmann [30] classified tests corresponding to Q1 and Q2 as competitive and self-contained tests, respectively. While a competitive test compares disease association test statistics for genes in the gene set versus that for genes in the rest of the genome, a self-contained test directly tests gene set association with
