The probability of drawing intersections: extending the hypergeometric distribution

The wide availability of biological data at the genome-scale and across multiple variables has resulted in statistical questions regarding the enrichment or depletion of the number of discrete objects (e.g. genes) identified in individual experiments. Here, I consider the problem of inferring enrichment or depletion when drawing independently, and without replacement, from two or more separate urns in which the same $n$ distinct categories of objects exist. The statistic of interest is the size of the intersection of object categories. I derive a probability mass function describing the distribution of intersection sizes when sampling from N urns and show that this distribution follows the classic hypergeometric distribution when N = 2. I apply the theory to the intersection of genes belonging to a set of traits in three different vertebrate species illustrating that the use of P-values from one-tailed enrichment tests enables accurate clustering of related traits, yet this is not possible when relying on intersection sizes alone. In addition, intersection distributions provide a means to test for co-localization of objects in images when using discretized data, allowing co-localization tests in more than two channels. Finally, I show how to extend the problem to variable numbers of objects belonging to each category, and discuss how to make further progress in this direction. The distribution functions are implemented and freely available in the R package 'hint'.