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Open Access Methodology article

A simple procedure for the comparison of covariance matrices

Carlos Garcia

Author affiliations

Department Xenética, CIBUS Campus Sur, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia 15782, Spain

Citation and License

BMC Evolutionary Biology 2012, 12:222  doi:10.1186/1471-2148-12-222

Published: 21 November 2012

Abstract

Background

Comparing the covariation patterns of populations or species is a basic step in the evolutionary analysis of quantitative traits. Here I propose a new, simple method to make this comparison in two population samples that is based on comparing the variance explained in each sample by the eigenvectors of its own covariance matrix with that explained by the covariance matrix eigenvectors of the other sample. The rationale of this procedure is that the matrix eigenvectors of two similar samples would explain similar amounts of variance in the two samples. I use computer simulation and morphological covariance matrices from the two morphs in a marine snail hybrid zone to show how the proposed procedure can be used to measure the contribution of the matrices orientation and shape to the overall differentiation.

Results

I show how this procedure can detect even modest differences between matrices calculated with moderately sized samples, and how it can be used as the basis for more detailed analyses of the nature of these differences.

Conclusions

The new procedure constitutes a useful resource for the comparison of covariance matrices. It could fill the gap between procedures resulting in a single, overall measure of differentiation, and analytical methods based on multiple model comparison not providing such a measure.

Keywords:
Eigenvectors; Principal component analysis; Littorina Saxatilis; Matrix orientation; Matrix shape; Hybrid zone