TestOfAngle: Perform a test of the angle between two multivariate vectors

In fruciano/GeometricMorphometricsMix: Miscellaneous functions useful for geometric morphometrics

Perform a test of the angle between two multivariate vectors

Description

This function performs a test for the angle between two multivariate vector, optionally allowing for "flipping" one of the axes

Usage

TestOfAngle(x, y, flip = TRUE)

Arguments

x, y	numerical vectors
flip	logical stating whether (TRUE, default) axes should be "flipped" in case the angle is larger than 90 degrees (see Details)

Details

This function is useful to perform a test of the angle between two multivariate vectors (e.g., principal component eigenvectors computed from two covariance matrices, such as covariance matrices of two different species). The function uses internally angleTest from the package Morpho by Stefan Schlager. That function, in turn, uses the formulas in Li 2011 to compute significance, rather than using permutations. This is the same approach also implemented in MorphoJ (Klingenberg 2011). There is no special advantage in using this function relative to the original in the package Morpho, except that this function outputs angles both in radians and degrees and that it optionally allow to "flip" one of the two axes. This is useful in the cases where the direction of one axis is arbitrary, such as in PCA (where positive and negative values are interchangable and recomputing PCA might result in positive scores for the observations which were negative (and vice versa). In this situation, angles larger than 90 degrees are not meaningful and one way of dealing with this is, by choosing the option flip=TRUE (default), to change the sign of one of the two vectors ("flip").

Value

The function outputs a list with

angle	angle between vectors in radians
angle_deg	angle between vectors in degrees
p.value	p value
critical_angle	angle below which two vectors of the same number of dimensions as the ones being tested would be significant

Notice

The p value is for the probability that the angle between two random vectors is smaller or equal to the one computed from the vectors provided (x, y). This means that significant values indicate that the two provided vectors are "significantly similar", whereas non-significant values means that the two vectors are substantially different

References

Klingenberg CP. 2011. MorphoJ: an integrated software package for geometric morphometrics. Mol Ecol Resour 11:353-357.

Li S. 2011. Concise formulas for the area and volume of a hyperspherical cap. Asian Journal of Mathematics and Statistics 4:66-70.

Schlager S. 2016. Morpho: Calculations and Visualisations Related to Geometric Morphometrics.

See Also

critical_angle

Examples

library(MASS)
library(clusterGeneration)
set.seed(123)
Data=mvrnorm(500,mu=rep(1,50),Sigma=genPositiveDefMat(dim=50)$Sigma)
A=Data[1:250,]
B=Data[251:500,]
# Create two groups of observations (e.g., specimens)
# from the same multivariate normal distribution
# with the same starting covariance matrix

PCA_A=prcomp(A)
PCA_B=prcomp(B)
# Perform separate PCAs for each of the two groups of observations

TestOfAngle(PCA_A$rotation[,1],PCA_B$rotation[,1], flip=TRUE)
# Test for the angle between the two first eigenvectors

fruciano/GeometricMorphometricsMix documentation built on Nov. 30, 2024, 3:12 a.m.