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Comparing Evolutionary Rates Using An Exact Test for 2x2 Tables with Continuous Cell Entries

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.