Calculating independent contrasts for the comparative study of substitution rates

Phylogenetic comparative methods have been used to study patterns of correlated evolution between biological traits of all kinds, and are increasingly used to identify predictors of the rate of DNA substitution. Substitution rate differs from most traits studied in that it cannot be observed directly, but must be inferred from substitutions accrued over a long period of time. Studying a mathematical model of trait and rate evolution, we show that the special nature of substitution rates can lead to a severe loss of power for standard comparative methods. We further show how strategies designed to maximise power, by increasing the number of data points, can have the opposite effect when substitution rate is involved. We then propose two modifications of the standard methods that can mitigate these problems. First, we show how pre-existing tests for homogeneity of variance can be used to identify and exclude those data from which changes in substitution rate cannot be reliably inferred. Second, we show that power can be increased by comparing substitution rate with the time average of the predictor trait along the history of the lineage, and introduce a maximum likelihood estimator of this quantity.