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

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.

Diversification of the ruminant skull along an evolutionary line of least resistance

“…6B ), but, of the six ruminant subfamilies with at least 10 species (the previously mentioned subfamilies and Caprinae), the relationship is only significant in Cephalophinae ( P = 0.012, F = 9.18). Note that, in hyperdimensional spaces like the one here ( k = 75), a distribution of random angles will be normally distributed around 90° with a lower standard deviation than lower-dimensional spaces ( 38 ). The angles of divergence and CREA were significantly different than a distribution of 10,000 random k -dimensional angles [two-sided Kolmogorov-Smirnov test, P < 0.001; ( 67 )] (fig.…”

Distribution theories for genetic line of least resistance and evolvability measures

“…Some consequences of such practices have been seen in the above empirical examples. Watanabe (2022a) has documented a similar problem in the context of characterizing parallelism between evolutionary trajectories.…”

“…This distributional result is fairly well recognized in the literature, and there are many possible ways to prove it, especially when q = 1 (see, e.g., Muirhead, 1982; Rice, 1990; Anderson, 2003; Cai et al ., 2013). Watanabe (2022a) restated one of those proofs in a context where the polarities of the vectors are relevant. The result for the special case of q = 1 is frequently used for testing alignment between allometric vectors in the morphometric literature (e.g., Klingenberg & Marugán-Lobón, 2013), where Li (2011) seems to be a preferred citation.…”

AF‐vapeR: A multivariate genome scan for detecting parallel evolution using allele frequency change vectors

“…As shown across empirical datasets, the genome‐wide distribution of eigenvalues is dependent on dataset features, such as SNP density, sampling/vector design and phylogeny. Consequently, a singular null distribution such as the Wishart distribution (De Lisle & Bolnick, 2020) or the various statistical nulls suggested by Watanabe (2022), is not applicable across all possible genomic datasets. However, the interpretation of empirical p‐values derived from the permuted null should be treated with caution.…”