| Abstract: | Statistical methods are used to test the hypothesis that rate of gain is equal to rate of loss for a single character on a cladogram. Ancestral character states are used as input for maximum likelihood (ML) rate estimation. Two markovian models of character evolution are considered: one has equality of rate across branches; the other permits variation in rate according to predetermined weights for branches. ML estimates are derived for both models, and their properties in large and small trees are investigated. Bias and error are significant in small trees. Error is greatest for characters in which rate of gain is low, and is greater for the loss estimate than for the gain estimate. Likelihood ratio (LR) tests of the null hypothesis of equality of gain/loss rate are derived, and their properties investigated. The distribution of -2 log LR is close to χ2 with 1 df with as few as 32 taxa. However, the power of the test is low unless the character is evolving rapidly. Methods for increasing power are examined, including selection of rapidly evolving subsets of characters, and pooling across characters. A goodness of fit test is presented to determine if pooling is justified. An example using published restriction site data on the Asteraceae demonstrates significant deviation from the null model in the direction predicted on the basis of the molecular biology of restriction enzyme site recognition, but only for one large subset of the data in which pooling is warranted. |
