Lineage-dependent rates of evolutionary diversification: analysis of bivariate ellipses

1. A method based on the calculation of major and minor axes of bivariate ellipses to estimate relative rates of evolutionary diversification of two traits is presented. The advantage of ellipse analysis over the more common allometric or regression approaches to variation in comparative data is that diversification of each trait is estimated relative to that of the other. This can reveal differences in the relative rates of diversification of two traits among sister groups or, through hierarchical analysis, over the history of a lineage. 2. Equations are provided for calculating the size, shape and orientation of bivariate ellipses based on the variance-covariance matrix of the original data. Standard errors and biases of these parameters are also quantified. 3. Applications of ellipse analysis to the relationship between egg size and incubation period in birds illustrate differences in the diversification of two traits between independent lineages. Nested analysis of covariance based on taxonomic levels further illustrates how relative rates of evolutionary diversification may vary through the history of a monophyletic group. 4. A simple model incorporating both independent (special) and correlated (general) change in two traits shows how different shapes and orientations of bivariate ellipses can be produced by different rates of evolutionary diversification in special and general genetic factors. 5. Ellipse analysis is a descriptive tool that can clarify patterns of diversification. It cannot distinguish differences in evolutionary responsiveness (phylogenetic constraint) from differences in the selective environment affecting the shape and orientation of the bivariate ellipse. It can, however, provide a more detailed characterization of trait evolution than other comparative methods, taking advantage of the additional information provided by the shape and, in some cases, the dimensions of the bivariate ellipse.