A simple procedure for the comparison of covariance matrices

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.

Figure 1

Contributions (S1₁, S2₁, S3₁) of the first eigenvectors of two sample matrices to the three sums used to measure the differentiation between these matrices in six hypothetical two-variable situations differing in matrices’ shape and orientation. The ellipse axes’ lengths in the graphics represent the magnitude of the eigenvalues and the orientation of the eigenvectors in the two samples. The straight lines mark the first eigenvectors. The tables in the middle column contain the variances explained by the first eigenvectors obtained in each sample when calculated in the two data sets. Details about the generation of the used matrices are given in Appendix 2.

Figure 2

S1, S2 and S3 statistics values (y axis; black, grey and white points respectively; they are slightly displaced for clarity) in five matrix-shape differences and six relative orientations of the matrices’ first eigenvectors, from zero to 90º (x axis). A) two matrices with very different shape, one with eigenvalues equal to 95 and 5% of total variance and the other with eigenvalues equal to 55 and 45% of total variance; B) two same-shape “elongate” matrices, both with eigenvalues explaining 95 and 5% of total variance; C) two same-shape “rounded” matrices both with eigenvalues explaining 55 and 45% of total variance; D) two “elongate” matrices with slightly different shapes, one with eigenvalues explaining 95 and 5% of total variance and the other, 90 and 10% of total variance; E) two “rounded” matrices with slightly different shapes, one with eigenvalues explaining 60 and 40% of total variance and the other, 55 and 45% of total variance. Matrices are schematically represented at left in a zero degrees relative orientation, with ellipses’ axes equal to the matrices’ eigenvalues. Note that the scale varies between plots.

Figure 3

Examples of population samples used in the simulations (two variables case, size = 100): (a) from the reference population, (b) from the population resulting in a covariance matrix with changed orientation, (c) from the population resulting in a covariance matrix with changed shape, and (d) from the population resulting in a covariance matrix with both orientation and shape changed. Each axis in the graphs corresponds to one of the two variables.

Figure 4

Proportions (from 0 to 100%; the lower and upper dotted lines mark the 5 and 95% respectively) of simulation replicates in which a difference between covariance matrices was found by the S1, S2, S3 (black, grey and white circles), RS (rhombs) and T method (squares) in comparisons involving matrices of samples taken from the same reference population (reference) or one from the reference population and another from a population resulting in matrices with altered orientation(orientation) or with altered shape (shape) or both (orientation + shape) in situations involving 2, 4 or 7 variables and sample sizes of 25, 50 or 100 individuals. The sign positions in each sample size were slightly displaced to improve clarity. The top of each graph shows the results of CPC-based comparisons of two samples taken at random from each of the two populations considered in that graph (E: equal, P: proportional, C: CPC result, meaning that all eigenvectors were common but the matrices were not proportional –i.e., same orientation but differences in shape- and U: unrelated matrices). Note: the CPC program considers the possibility that only a subset of eigenvectors are in common, but that result was never found in these simulations.

Figure 5

CPC results and bootstrap distributions for five statistics to compare Littorina data covariance matrices in comparisons within sample (grey-lined boxplots; 1 to 3, Rbs; 4 to 6, Sus), between locations within morph (black-lined boxplots; 7 to 9, between Rbs; 10 to 12, between Sus) and between morphs (grey-filled boxplots; 13 to 18, between morphs of different locations; 19 to 21, between morphs of the same location). The T% values were divided by 100 to make them comparable with the other statistics. The CPC box shows the number of common principal components; U: unrelated. No CPC analysis was done for the comparisons within samples. Plots do not include outliers. Circles mark the observed values for the statistics. No observed values are printed in the case of within sample comparisons (i.e., of matrices with themselves) because they were always equal to one for the RS and equal to zero for the other statistics. Comparison codes: 1, Rb1-Rb1; 2, Rb2-Rb2; 3, Rb3-Rb3; 4, Su1-Su1; 5, Su2-Su2; 6, Su3-Su3; 7, Rb1-Rb2; 8 Rb1-Rb3; 9, Rb2-Rb3; 10, Su1-Su2; 11, Su1-Su3; 12, Su2-Su3; 13, Rb1-Su2; 14, Rb2-Su1; 15, Rb1-Su3; 16, Rb3-Su1; 17, Rb2-Su3; 18, Rb3-Su2; 19, Rb1-Su1; 20, Rb2-Su2; 21, Rb3-Su3.

Figure 6

Representation of the contribution (vertical axes) of each of the seven eigenvector pairs (1 to 7 from left to right in the horizontal axis) to the S1 (black points), S2 (solid lines) and S3 (dashed lines) statistics in each comparison between samples. All graphs are drawn to the same scale (minimum 0, maximum 0.0058) to ease comparison.

Figure 7

Proportions of the total variance (vertical axis; log-transformed for clarity of representation) of each sample explained by the eigenvectors (1 to 7 from left to right in the horizontal axis) obtained in the analysis of the reciprocal sample in each between-samples comparison. The gray circles mark the increases in variance explained by higher order reciprocal eigenvectors. The asterisks correspond to bootstrap tests of the change in proportion of variance explained (average of the two reciprocal comparisons) by successive eigenvectors. They mark changes in which the 97.5 percentile of the bootstrapped distribution was negative (i.e., the third reciprocal eigenvector explained more variance than the second).

Figure 8

Back and opercular view of shells of the lower-shore Su (left) and upper shore Rb (right) morphs of Littorina saxatilis form the Galician coasts. The seven measures used are shown on the Su shell. Note: Rb snails are on average larger than Sus; shells of similar sizes were chosen to ease comparison on the image.