Testing for Constant Diversification Rates Using Molecular Phylogenies: A General Approach Based on Statistical Tests for Goodness of Fit

Abstract

Several recent studies have shown the usefulness of phylogenies based only on living species to study evolutionary past events such as mass extinctions or adaptive radiations (Hey 1992; Nee, Mooers, and Har-vey 1992; Harvey, May, and Nee 1994; Kubo and Iwasa 1995; Cooper and Penny 1997). In this paper, I introduce two statistical tests which can be used to compare the observed distribution of branching times inferred from a molecular phylogeny against the one predicted from a null model of diversification. Wollenberg, Arnold, and Avise (1996) propose a statistical method to test whether the diversification rate of a lineage was constant through time. They used a branching-and-extinction process with constant and equal branching and extinction probabilities to derive the expected distribution of branching times under this null model. It is valid to use this particular stochastic process as a null model of diversification, but in a general perspective, one may wish to test the validity of any other model of diversification. I use the following notations: , speciation (or branching) probability; , extinction probability; , diversification rate (). Wollenberg, Arnold, and Avise's (1996) null model assumes , and hence 0. A natural alternative for a null model of diversification is the case with. In this case, the probability of extinction of a single lineage after an amount of time t is (Kendall 1948): (e t 1) P(n 0) , (1) t e t where n t is the number of species in the lineage at time t. It follows from this that the probability of observing a divergence (or branching event) of age t in a phylog-eny of living species can be formulated as a (compli-cated) function of , , and the number of species living at time t (which is also a random variable depending on and). It is then possible to derive the expected distribution of branching times and test whether this distribution fits significantly to an observed distribution from an actual phylogeny. Note that it is necessary to integrate the probability of branching times to obtain the corresponding theoretical cumulative distribution function (CDF) in order to perform a goodness-of-fit (GOF) test, but the formulation of this probability is quite complicated , and its integration is not really straightforward. An approximation for this CDF is given by an expo