Detecting (non)parallel evolution in multidimensional spaces: angles, correlations and eigenanalysis

Parallelism between evolutionary trajectories in a trait space is often seen as evidence for repeatability of phenotypic evolution, and angles between trajectories play a pivotal role in the analysis of parallelism. However, properties of angles in multidimensional spaces have not been widely appreciated by biologists. To remedy this situation, this study provides a brief overview on geometric and statistical aspects of angles in multidimensional spaces. Under the null hypothesis that trajectory vectors have no preferred directions (i.e. uniform distribution on hypersphere), the angle between two independent vectors is concentrated around the right angle, with a more pronounced peak in a higher-dimensional space. This probability distribution is closely related to t- and beta distributions, which can be used for testing the null hypothesis concerning a pair of trajectories. A recently proposed method with eigenanalysis of a vector correlation matrix can be connected to the test of no correlation or concentration of multiple vectors, for which simple test procedures are available in the statistical literature. Concentration of vectors can also be examined by tools of directional statistics such as the Rayleigh test. These frameworks provide biologists with baselines to make statistically justified inferences for (non)parallel evolution.