A formula for the correlation between random-set enrichment scores

Given constants s1, s2, . . . , sG, we consider variables X = ∑ sg1[g ∈ A] and Y = ∑ sg1[g ∈ B] as random variables with a joint distribution determined by taking (A,B) as uniformly random among set pairs in which A has fixed cardinality m, B has fixed cardinality n, and the intersection A ∩ B has fixed size q. Arguments about without-replacement sampling give the marginal mean and variance of X and Y separately. Here we extend those calculations and compute the correlation between X and Y . The joint distribution so determined is helpful in gene set enrichment analysis.