A simple procedure for the comparison of covariance matrices
- Carlos Garcia^{1}Email author
https://doi.org/10.1186/1471-2148-12-222
© Garcia; licensee BioMed Central Ltd. 2012
Received: 18 June 2012
Accepted: 2 November 2012
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
Background
Covariance matrices are key tools in the study of the genetics and evolution of quantitative traits. The G matrix, containing the additive genetic variances and covariances for a set of characters, summarizes the genetic architecture of traits and determines their short-term response to multivariate selection along with the constraints this response will face. The more easily estimated matrix of phenotypic variances and covariances P can be used as a surrogate for G, especially in the case of high heritability morphological characters [1–4]. Comparisons between covariance matrices are carried out in the study of a wide array of evolutionary problems, such as the stability of G in the presence of selection and drift [5–7], the role of genetic constraints on determining evolutionary trajectories in adaptive radiations [8], the response of genetic architecture to environmental heterogeneity [9], the evolution of phenotypic integration [4, 10], multi-character phenotypic plasticity [11] and sexual dimorphism [12, 13].
Several methods for the comparison of covariance matrices are available (reviewed in [14]). They range from the most mathematically sophisticated, such as those using maximum likelihood [15] or Bayesian frameworks [16], to simple methods that are useful for exploratory analyses and are not dependent on distributional assumptions. The simplest methods [17–20] are based on pair wise comparisons of the matrices’ elements, so that they typically ignore the lack of independence between these values, cannot detect proportionality between matrices, and consider two matrices only. More recent procedures, also using the matrix elements [21], take into account these elements’ lack of independence and permit the joint consideration of several matrices, making it possible to study the contribution of identified external factors to the magnitude of the differentiation.
Other simple procedures [22, 23] are based on comparisons between vectors resulting from the product of the studied matrices and sets of test vectors, their rationale being that similar matrices would produce similar results when multiplied by the same sets of vectors. However, most of these procedures result in a single measure of the divergence between matrices that does not provide information about the nature of this divergence. Such information is provided by CPCA (Common Principal Components Analysis [24]), which uses the Flury [25] hierarchy, a maximum likelihood-based procedure to compare the structure of two or more matrices and sequentially test if the matrices are “unrelated” (have no eigenvectors in common); if they have some number of eigenvectors in common, if they are proportional (have all their eigenvectors in common and their eigenvalues keep the same proportions), and finally if they are equal (have all eigenvectors and eigenvalues in common). Then it determines which of these descriptions best fits to the relationship between the matrices’ structures.
Among the limitations of CPCA are, first, that it is based on the assumption of multivariate normality, and second, that it results in categorical, not continuously varying measures of matrix similarity [26]. The CPCA consists in a series of yes/no comparisons between ordered eigenvectors, which allow testing a full series of hypotheses about the relationship between matrices in a hierarchical way, but idoes not have an associated parameter measuring the degree of similarity, relying only on the results of the significance tests. This limitation can be serious in some situations. Two matrices are declared as “unrelated” when that is the best fit of all null hypotheses tested, but this result does not preclude the existence of any similarities between them [14]. In fact, the procedure may declare two matrices as “unrelated” when many data are available and there is great power to detect differences, even if these differences are trivial from a biological point of view [26, 27]. Thus, there is no simple relationship between matrix similarities measured by CPCA and other matrix comparison procedures [27].
In the present work I propose a new, simple and distribution-free procedure for the exploration of differences between covariance matrices that, in addition to providing a single and continuously varying measure of matrix differentiation, makes it possible to analyse this measure in terms of the contributions of differences in matrix orientation and shape. I use both computer simulation and P matrices corresponding to snail morphological measures to compare this procedure with some widely used alternatives. I show that the new procedure has power similar or better than that of the simpler methods, and how it can be used as the basis for more detailed analyses of the nature of the found differences.
Pairwise matrix comparison
The S statistics are easier to compare between studies if they are made to vary between zero and one by making them relative to their maximum possible value. This maximum would occur in the extreme situation in which single eigenvectors explain all variation in each of the compared samples, and the eigenvectors of the two samples are orthogonal. In that case, S1 is equal to 2 times the sum of twice the square of the total variance of the first sample and twice the square of the total variance of the second sample. When the variances explained by each eigenvector are expressed as proportions of the total variance, so that the sum of all the proportions is equal to one, the maximum possible value for S1 is equal to eight. In the computer simulation and real data example shown in this article, the explained variances are expressed as proportions, and the S1, S2 and S3 statistics divided by eight so that they could vary between zero and one.
Results
I contrasted the results obtained with the proposed procedure with those from other widely used ones, namely CPCA and two simpler procedures providing single measures of matrix differentiation: one, the Random Skewers, based on products with test vectors, and the other, the T method, based on the comparison of matrix elements (see Methods). I followed two approaches. First, I studied the procedures’ power and Type I error through computer simulations that considered covariance matrices differing in shape, orientation or both, in different number of variables and sample size situations. Second, I compared their ability to detect differences between covariance matrices of shell measures from different morphs and populations of the seashore snail Littorina saxatilis.
Computer simulation
Figure 4 shows also the results obtained with CPCA. These results cannot be directly compared with those of the remaining methods because they were based on a single replicate instead of an average of replicates for each case. However, the results suggest that CPCA could produce an excess of false positives as the number of variables increases (it declared as not equal the matrices from the same population in the seven variables cases). In these simulations, CPCA should declare the matrices as “unrelated” when they differed in orientation, because in that case they would not share any eigenvector in common and as “CPC”, i.e., sharing all eigenvectors but being not proportional when the matrices differed in shape. The procedure tended to correctly detect changes in shape even for small sample sizes, except for the two variables case. Its performance tended to be lower in the detection of changes in orientation in the smaller sized samples. Overall, CPCA tended to agree with the S and RS procedures for the largest (100) sample size.
Littorinadata
Eigenvector analysis
Sample | EG1 | EG2 | EG3 | EG4 | EG5 | EG6 | EG7 |
---|---|---|---|---|---|---|---|
Rb loc 1 | |||||||
0.349 | 0.178 | 0.046 | 0.428 | −0.049 | −0.596 | 0.551 | |
0.297 | 0.464 | 0.231 | 0.492 | 0.139 | 0.614 | −0.064 | |
0.323 | 0.350 | 0.209 | −0.711 | 0.396 | 0.008 | 0.260 | |
0.365 | 0.010 | −0.033 | −0.260 | −0.870 | 0.170 | 0.047 | |
0.427 | −0.068 | −0.867 | 0.011 | 0.219 | 0.107 | −0.049 | |
0.499 | −0.765 | 0.354 | 0.041 | 0.127 | 0.146 | 0.039 | |
0.347 | 0.176 | 0.152 | 0.011 | 0.026 | −0.454 | −0.790 | |
21e-2 | 28e-3 | 62e-4 | 11e-4 | 76e-5 | 28e-5 | 18e-5 | |
85.48 | 11.14 | 2.48 | 0.42 | 0.30 | 0.11 | 0.07 | |
Rb loc 2 | |||||||
0.349 | 0.125 | 0.133 | 0.432 | −0.335 | −0.568 | 0.473 | |
0.306 | 0.305 | 0.323 | 0.464 | 0.235 | 0.648 | 0.122 | |
0.314 | 0.226 | 0.339 | −0.567 | 0.551 | −0.259 | 0.208 | |
0.377 | 0.124 | 0.137 | −0.490 | −0.700 | 0.304 | −0.032 | |
0.447 | 0.250 | −0.843 | −0.020 | 0.154 | 0.050 | 0.027 | |
0.471 | −0.865 | 0.023 | 0.027 | 0.125 | 0.098 | 0.058 | |
0.357 | 0.185 | 0.127 | 0.005 | −0.001 | −0.748 | −0.512 | |
21e-2 | 27e-3 | 91e-4 | 12e-4 | 71e-5 | 49e-5 | 11e-5 | |
84.24 | 11.00 | 3.66 | 0.50 | 0.29 | 0.20 | 0.11 | |
Rb loc 3 | |||||||
0.362 | 0.143 | 0.124 | 0.481 | −0.094 | −0.599 | 0.483 | |
0.306 | 0.411 | 0.334 | 0.406 | 0.020 | 0.678 | 0.004 | |
0.317 | 0.284 | 0.209 | −0.616 | 0.567 | −0.068 | 0.264 | |
0.363 | 0.099 | 0.060 | −0.464 | −0.797 | 0.048 | 0.049 | |
0.417 | 0.131 | −0.881 | 0.062 | 0.111 | 0.126 | −0.008 | |
0.489 | −0.828 | 0.152 | 0.037 | 0.124 | 0.180 | 0.050 | |
0.360 | 0.129 | 0.161 | 0.060 | 0.085 | −0.354 | −0.832 | |
18e-2 | 17e-3 | 87e-4 | 79e-5 | 49e-5 | 21e-5 | 18e-5 | |
83.44 | 12.85 | 2.20 | 0.93 | 0.40 | 0.13 | 0.05 | |
Su loc 1 | |||||||
0.325 | 0.176 | 0.147 | 0.272 | 0.196 | −0.375 | 0.767 | |
0.245 | 0.291 | 0.395 | 0.414 | 0.379 | 0.585 | −0.204 | |
0.288 | 0.309 | 0.346 | −0.198 | −0.788 | 0.178 | 0.100 | |
0.360 | 0.120 | 0.063 | −0.817 | 0.426 | 0.055 | 0.016 | |
0.493 | −0.838 | 0.200 | 0.076 | −0.071 | 0.056 | −0.037 | |
0.523 | 0.154 | −0.796 | 0.152 | −0.099 | 0.184 | −0.043 | |
0.324 | 0.221 | 0.158 | 0.138 | 0.014 | −0.668 | −0.597 | |
20e-2 | 25e-3 | 14e-3 | 15e-4 | 99e-5 | 27e-5 | 15e-5 | |
82.97 | 10.07 | 5.76 | 0.62 | 0.40 | 0.11 | 0.06 | |
Su loc 2 | |||||||
0.343 | 0.101 | 0.186 | 0.305 | 0.2812 | −0.529 | 0.620 | |
0.277 | 0.128 | 0.474 | 0.356 | 0.224 | 0.711 | 0.014 | |
0.281 | 0.181 | 0.365 | −0.119 | −0.853 | −0.059 | 0.100 | |
0.365 | 0.108 | 0.145 | −0.848 | 0.327 | 0.062 | 0.060 | |
0.469 | −0.866 | −0.126 | 0.039 | −0.094 | 0.066 | 0.000 | |
0.506 | 0.408 | −0.723 | 0.125 | −0.086 | 0.176 | −0.001 | |
0.343 | 0.114 | 0.217 | 0.170 | 0.144 | −0.414 | −0.775 | |
16e-2 | 33e-3 | 13e-3 | 17e-4 | 11e-4 | 35e-5 | 16e-5 | |
76.48 | 15.88 | 6.06 | 0.80 | 0.54 | 0.17 | 0.08 | |
Su loc 3 | |||||||
0.338 | 0.124 | 0.204 | 0.308 | −0.222 | −0.378 | −0.736 | |
0.255 | 0.192 | 0.451 | 0.459 | −0.202 | 0.634 | 0.202 | |
0.282 | 0.200 | 0.356 | −0.344 | 0.767 | 0.124 | −0.176 | |
0.322 | 0.126 | 0.152 | −0.737 | −0.555 | 0.070 | 0.035 | |
0.425 | −0.899 | 0.073 | 0.015 | 0.051 | 0.044 | 0.031 | |
0.594 | 0.237 | −0.739 | 0.097 | 0.089 | 0.161 | 0.038 | |
0.322 | 0.166 | 0.231 | 0.153 | 0.044 | −0.638 | 0.618 | |
24e-2 | 17e-3 | 14e-3 | 16e-4 | 13e-4 | 24e-5 | 96e-6 | |
87.32 | 6.74 | 5.15 | 0.38 | 0.27 | 0.08 | 0.06 |
The overall agreement between CPCA and the other procedures found in the simulations was lost in the analysis of Littorina data. (upper side of Figure 5). According to the RS, T% and S statistics, the three Rb samples had very similar covariance matrices, but the CPC procedure determined that Rb2 and Rb3 were “unrelated”, and that Rb1 and Rb2 had only one eigenvector in common. The few comparisons finding some eigenvectors in common did not correspond to particularly low measures of differentiation by the other methods. This shows that the CPC analysis did not consider the matrices’ properties in the same way.
Discussion
The S statistics constitute sensitive tools for the detection of differences between covariance matrices. In the Littorina example used here, it was found that the local differentiation was clearly higher for the Su than for the Rb morph. This could be due to lower genetic connectivity for populations of the Su morph, and also to environmental differences between localities. However, the absence of such differences between the Rbs in the same localities would imply that any environmental differences would exist not between whole localities but only between the micro-habitats the Su snails use in the mid-shore. The differences between morphs, and specially those due to changes of orientation in the covariance matrices, were the largest in the analysis, and if not of the same size, they were of the same nature in all locations. Since shell morphology variation is adaptively important in Littorina[29], this suggests that these differences in covariance could be relevant for the evolutionary dynamics of the hybrid zone. They could simply result from the environmental differences between the two morphs’ microhabitats within the midshore, but even these environmental differences could have a genetic origin because individuals of the two morphs, even when living within extremely short distances of each other, tend to actively choose microhabitats with different physical characteristics [30].
The computer simulations shown in Figure 4 are limited to measure the power of the considered procedures to detect the consequences of changes in matrix orientation and shape. They show that such differences can be detected even when moderate in magnitude and when sample sizes are not too large, but cannot be taken as complete or definitive comparisons between the procedures. Matrices may differ in many relevant aspects, and different comparison procedures may have different aims and take different aspects into consideration. For example, while the S measures consider the differences in eigenstructure between two matrices, the RS procedure focuses on the related, but not equivalent problem of the differences between the evolutionary responses generated by these matrices. Comparisons would be even more difficult with procedures such as the set of evolvability measures proposed by Hansen and Houle [31], which consider the magnitude of the populations’ responses to different natural selection regimes.
Since the S statistics introduced here simply measure what proportion of variation exists in a given sample along the axis of variation defined by the eigenvectors in the compared sample, they are similar to the RS and T% ones in that they do not compare and are not dependent on the matrices’ sizes. They focus instead on the more interesting differences in matrix shape and orientation. In any case, S statistics-based comparisons could use raw covariance components instead of proportions as in the example shown, so that the results would depend on between-matrix size differences. However, in that case the S statistics would not be able to separate the effect of size from those of other sources of differentiation between matrices. Similarly, the basic version of the T method proposed in [19] reflects the differences in matrix size, as it is based on raw variance components instead of the proportions used by the T% statistic.
Calculating the amount of variance explained by a set of eigenvectors in a given dataset is straightforward in the case of datasets containing the phenotypic measures used to obtain P matrices. In the case of G matrices, the comparison would have to be based on additive genetic value estimates for individuals or families.
Since the proposed procedure is limited to two-sample comparisons, it cannot be used to make higher order analyses of the divergence among several populations (see [32]). However, and as shown in the Litttorina example, it can be useful for the study of evolutionary relevant situations such as hybrid zones. The S measures appear to be similar to the measure of distance between covariance matrices used by Mitteroecker and Bookstein [33], based on the calculation of relative eigenvalues, i.e., the eigenvalues of the product of one matrix premultiplied by the inverse of the second matrix, and therefore on expressing the variances and covariances of one sample relative to those of the other sample. I calculated the correlation of this measure with S1, S2 and S3 in a computer simulation considering the same cases as in Figure 4 and found that, while the measures were clearly related, they were not equivalent. For example, in the simulation of four variables and sample size 50, the correlation across replicates of the Mitteroecker and Bookstein measure with S1, S2 and S3 were: in the comparison of one population with itself, 0.206, 0.399 and 0.140 respectively; in the case of divergence in orientation, 0.396, 0.533 and 0.040^{NS}; in the case of divergence in matrix shape, 0.568, 0.281 and 0.554; and in the case of divergence in both orientation and shape, 0.454, 0.572 and 0.341 (all correlations, P < 0.001 except when indicated).
The Littorina example supports the view that CPCA might lead to misleading conclusions about the overall similarity between matrices [26], as pairs of matrices found very similar by other procedures were declared as “unrelated” by CPCA, and there was no clear correspondence between the two sets of results. The observed between-morphs reversal in importance of the second and third eigenvectors could play a role in this discrepancy. CPCA is based on a series of paired comparisons between eigenvectors of the same rank. Two matrices may share their axes of variation, but not the amount of variance in each axis. For example, the ith eigenvector of one matrix might be the same as the i+1 eigenvector of the other, and the i+1 of the first, the same as the ith of the second. In that case, the two matrices would have the same eigenvectors, but in a reverse order. A comparison between their ith eigenvectors would find them orthogonal, and this would also be the case for their i+1 eigenvectors. Thus there may be considerable similarity between the two matrices, but this similarity is overlooked by the comparison procedure which finds the paired eigenvectors very different. The CPC software [34] enables users to compare the eigenvectors in any order, but this does not fix this particular limitation, as the order chosen for the two samples must be the same. The T and RS methods, based on matrices’ elements and product vectors, would provide a more balanced measure of similarity in this situation because the differences between these elements and these vectors would not depend on the existence of reversals in eigenvector order per se, but on the magnitude of the differences involved. However they don’t allow further analysis of the pattern of differentiation. The three S statistics are affected by different patterns of divergence, so that their joint use provided a deeper view of the differences between the two morphs’ P matrices. The S1 statistic is not dependent on the eigenvectors’ ordering per se because it is based on comparisons within eigenvector, i.e., on the difference between the amount of variance explained by one eigenvector from one sample in the original and reciprocal samples. These differences do not change with eigenvector order. But S2 changes when the order of eigenvectors in one of the samples is reversed (see formulas) because this would be considered as a change in matrix orientation. In case the reversal in eigenvectors’ importance was complete, so that there were no changes in overall shape, S3 would remain unaffected (see the second row in Figure 1). However, the reversal of the second and third eigenvectors between morphs cannot fully explain the disagreement between CPCA and the remaining methods because the results for S2 were not particularly similar to those of CPCA (Figure 5). This suggests that other aspects of covariance matrix structure might control the degree of agreement between different comparison procedures.
Conclusions
The S-statistics procedure provides a simple and continuously-varying overall measure of differentiation that is distribution free and interpretable in terms of changes in matrix orientation and shape. In addition, it makes it easy to study the contribution of the different eigenvectors to the statistics values, which could provide further details on the nature of the differentiation, as was the case of the Littorina example used. This procedure could thus fill the gap between simpler statistics such as T% and RS, and more analytical methods like CPCA or Bayesian MCMC. The S-statistics procedure is not based on a formal analysis of matrices’ properties. Instead, it could serve for a simple and fast exploration of the magnitude and nature of the differentiation.
Methods
Compared procedures
and the measure of similarity between matrices as the average correlation for all vectors.
where c is the number of nonredundant elements in the matrices, M_{i1} and M_{i2} are such elements in the two matrices, and ${M}_{1}^{\u2015}$ and ${M}_{2}^{\u2015}$ their averages. The proportional nature of T% makes the comparison between matrices of different sizes easier. It is unreliable when there are both positive and negative elements in any of the two matrices [35], but this was not the case in the examples used in this work to compare the different methods’ performances.
Finally, I used the CPC (Common Principal Components) software of Phillips and Arnold [34] carrying out CPCA. I chose the “step-up” option, in which the likelihood of any model is tested against the likelihood of the next lowest model in the hierarchy, and used the Akaike Information Criterion to select the best description.
Simulations
In each sample, matrix orientation was controlled by the relative contribution (fixed within sample) k_{i} of the common component to each variable’s value, and matrix shape, by the s_{i} variances. Four kinds of sample matrix comparisons were made: between samples taken at random from the same population, between samples from populations whose covariance matrices differed in orientation, whose matrices differed in shape, and whose matrices differed in both orientation and shape. One sample was taken at random from each of the two populations compared in each simulation case, and their covariance matrices and comparison statistics calculated. The observed value of each statistic was compared with the distribution obtained by comparing 50 pairs of resamples of the same size taken from the first sample, and with that obtained by comparing 50 pairs of resamples of the same size taken from the second sample. If the observed value was greater than these 100 resampled values, I concluded that the statistic found differences between the two samples’ matrices. This process was repeated 1000 times for each simulation case. I considered three sample sizes, 25, 50 and 100, and three numbers of variables, 2, 4 and 7 (the number of variables in the Littorina data). The particular changes in shape and orientation considered in each simulation case were chosen so that at least one of the three S statistics had nearly (but not exactly) 100% power to detect differences. The list of sample parameters used in the simulations is shown in Appendix 3.
Littorinadata
Appendix
Appendix 1
Amounts of variance explained by th eith eingeinvectors of each compared sample
A ith eigenvector | B ith eigenvector | |
---|---|---|
Sample A | v_{i11} | v_{i12} |
Sample B | v_{i21} | v_{i22} |
Appendix 2
where e_{ ij } and e_{ im } are eigenvectors i from samples j and m, V_{im} is eigenvalue i from sample m and the apostrophe indicates transposition.
Appendix 3
Summary of cases
Detailed list of cases
List of parameter sets used in every simulated case and resulting covariance matrices, eigenvectors and eigenvalues. The expected compositions of eigenvectors were obtained via eigenvector analyses applying R function eigen to random samples of size 10^{6}. Note that for four and seven variables cases it was not possible to obtain a constant set of eigenvector coefficients (beyond the first eigenvector) even for such large samples. In any case, The S statistics recognized their equivalence despite differences in eigenvectors’ coefficients (see the S3 and S2 values in the second row and third rows respectively of Figure 3).
In the two variables case we had:
and the expected eigenvalues were: 1.8 and 0.2.
and the expected eigenvalues were: 1.8 and 0.2.
and the expected eigenvalues were: 1.5 and 0.5.
and the expected eigenvalues were: 1.5 and 0.5.
The expected total variance in all two variable samples was = 2.
In the four variables case, we had:
and the expected eigenvalues were: 3.4, 0.2, 0.2, 0.2.
and the expected eigenvalues were: 3.4, 0.2, 0.2, 0.2.
and the expected eigenvalues were: 2.8, 0.4, 0.4, 0.4.
and the expected eigenvalues were: 2.8, 0.4, 0.4, 0.4.
The expected total variance in all four variable samples was = 4.
In the seven variables case, we had:
and the expected eigenvalues were: 5.8, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2.
and the expected eigenvalues were: 5.8, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2.
and the expected eigenvalues were: 4.6, 0.4, 0.4, 0.4, 0.4, 0.4, 0.4.
and the expected eigenvalues were: 4.6, 0.4, 0.4, 0.4, 0.4, 0.4, 0.4.
The expected total variance in all seven variable samples was = 7.
Declarations
Acknowledgements
I thank Raquel Cruz, Javier Mosquera and Carlos Vilas for allowing me to use the data from our previous experiments, which had been funded by Spain’s DGICYT grant PB94-0649, and Paul Hohenlohe and David Houle for helpful criticism of the manuscript. This work was funded by Ministerio de Ciencia y Tecnología (CGL2009-13278-C02),) and Fondos Feder.
Authors’ Affiliations
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