Abstract
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.
Highlights
Multivariate approaches have proven to be powerful means to analyse phenotypes, yielding more holistic and nuanced understanding of organismal evolution and development than achievable from univariate approaches
The term parallel evolution is used in the geometric sense; parallelism between trajectories in a trait space between multiple ancestor– descendant pairs [7,8], which typically results in acquisition of similar derived traits in the descendants
Angles have been commonly used in quantitative analyses of parallel evolution, but their properties in multidimensional spaces have not attained due attention
Summary
Multivariate approaches have proven to be powerful means to analyse phenotypes, yielding more holistic and nuanced understanding of organismal evolution and development than achievable from univariate approaches. This paper gives a brief overview of methods to analyse angles in multidimensional spaces It first derives the probability distribution of the angle between random vectors under the null hypothesis that the vectors have no preferred directions. 11]), so, to be strict, the above derivation was partly circular These results can be used for testing the null hypothesis that two phenotypic change vectors have no preferred directions (population means being (0, ..., 0)T) and are independent from each other, by inserting the dimensionality of the trait space into k. De Lisle & Bolnick [48] proposed to use eigenanalysis of the inter-lineage correlation matrix C to detect concentration of phenotypic change vectors in a trait space. If the detection of parallel signal is of specific interest, it is probably more adequate to use the Rayleigh test from the directional statistics (electronic 5 supplementary material, appendix B)
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