  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'.
