  Assessing the statistical significance of an observed 2x2 contingency table
can easily be accomplished using Fisher's exact test (FET). However, if the
cell entries are continuous or represent values inferred from a continuous
parametric model, then FET cannot be applied. Such tables arise frequently in
areas of biostatistical research including population genetics and evolutionary
genomics, where cell entries are estimated by computational methods and result
in cell entries drawn from the non-negative real line R+. Simply rounding cell
entries to conform to the assumptions of FET is an ill-suited approach that we
show creates problems related to both type-I and type-II errors. Pearson's
chi^2 test for independence, while technically applicable, is not often
effective for these tables, as the test has several limiting assumptions that
make application of this method inadvisable in many common instances
(particularly with small cell entries). Here we develop a novel method for
tables with continuous entries, which we term continuous Fisher's Exact Test
(cFET). Through simulations, we show that cFET has a close-to-uniform
distribution of p-values under the null hypothesis of independence, and more
power when applied to tables where the null hypothesis is false (compared to
FET applied to rounded cell entries). We apply cFET to an example from
comparative genomics to confirm an overall increased evolutionary rate among
primates compared to rodents, and identify several genes that show particularly
elevated evolutionary rates in primates. Some of these genes exhibit signatures
of continued positive selection along the human lineage since our divergence
with chimpanzee 5-7 million years ago, as well as ongoing selection in modern
humans.
