Measuring Morphological Integration Using Eigenvalue Variance

Abstract

The concept of morphological integration describes the pattern and the amount of correlation between morphological traits. Integration is relevant in evolutionary biology as it imposes constraint on the variation that is exposed to selection, and is at the same time often based on heritable genetic correlations. Several measures have been proposed to assess the amount of integration, many using the distribution of eigenvalues of the correlation matrix. In this paper, we analyze the properties of eigenvalue variance as a much applied measure. We show that eigenvalue variance scales linearly with the square of the mean correlation and propose the standard deviation of the eigenvalues as a suitable alternative that scales linearly with the correlation. We furthermore develop a relative measure that is independent of the number of traits and can thus be readily compared across datasets. We apply this measure to examples of phenotypic correlation matrices and compare our measure to several other methods. The relative standard deviation of the eigenvalues gives similar results as the mean absolute correlation (W.P. Cane, Evol Int J Org Evol 47:844–854, 1993) but is only identical to this measure if the correlation matrix is homogenous. For heterogeneous correlation matrices the mean absolute correlation is consistently smaller than the relative standard deviation of eigenvalues and may thus underestimate integration. Unequal allocation of variance due to variation among correlation coefficients is captured by the relative standard deviation of eigenvalues. We thus suggest that this measure is a better reflection of the overall morphological integration than the average correlation.

Appendices

Appendix 1

Let R be a homogenous correlation matrix of size N with ${\mathbf{R}}_{ij} = r$ for all i ≠ j. If 1 = (1,…,1) is an eigenvector of R and λ ₁ = 1 + (N − 1)r the corresponding eigenvalue, then ${\mathbf{R1}} = \lambda_{1} {\mathbf{1}}$. Considering this equation component wise:

$$ \begin{gathered} ({\mathbf{R1}})_{i} = (\lambda_{1} {\mathbf{1}})_{i} \hfill \\ ({\mathbf{R1}})_{i} = \sum\limits_{j = 1}^{N} {{\mathbf{R}}_{ij} {\mathbf{1}}_{j} } \hfill \\ \end{gathered} $$

we see that the right hand side of the latter equation is simply the row sum of the i-th row. Because of the symmetry of R this sum is the same for all rows in R, and which has (N − 1) terms of size r and one equal to 1:

$$ \sum\limits_{j = 1}^{N} {{\mathbf{R}}_{ij} 1_{j} } = 1 + (N - 1)r = \lambda_{1} 1_{i} $$

The other eigenvalues of R can be obtained by considering the constraint $\sum\nolimits_{i = 1}^{N} {\lambda_{i} } = \text{tr} {\mathbf{R}} = N$, from which we see that

$$ \begin{aligned} \sum\limits_{i = 1}^{N} {\lambda_{i} } & = \lambda_{1} + \sum\limits_{i = 2}^{N} {\lambda_{i} } = N \hfill \\ \sum\limits_{i = 2}^{N} {\lambda_{i} } & = N - \lambda_{1} \hfill \\ \sum\limits_{i = 2}^{N} {\lambda_{i} } & = N - 1 - (N - 1)r \hfill \\ \sum\limits_{i = 2}^{N} {\lambda_{i} } & = (N - 1)(1 - r) \hfill \\ \end{aligned} $$

Due to the symmetry of the R matrix the N − 1 non-leading eigenvalues are all the same and λ _i > 1 = 1 − r.

Appendix 2

To derive the maximum variance corresponding to a particular $\bar{r}$, we consider maximal symmetrical dispersal of r. Maximum dispersal is reached when half of the sample size is located at outermost values, in equal distances to both sides from the $\bar{r}$. This can be expressed as

$$ {\text{Var}}_{\max } (r) = \frac{{\frac{{N^{2} - N}}{4}\left( {r_{i\max } - \bar{r}} \right)^{2} + \frac{{N^{2} - N}}{4}\left( {r_{i\min } - \bar{r}} \right)^{2} }}{{\frac{{N^{2} - N}}{2}}} = \frac{1}{2}\left[ {\left( {r_{i\max } - \bar{r}} \right)^{2} + \left( {r_{i\min } - \bar{r}} \right)^{2} } \right] $$

where r _imax, r _imin are the maximal and minimal values of r _i , given the symmetrical distribution. For example, if $\bar{r} = 0.2$, the r _max = 1 and r _min = 0.2 − (1 − 0.2) = −0.6. In general:

$$ r_{\max } = \bar{r} + \left( {1 - \left| {\bar{r}} \right|} \right) $$

$$ r_{\min } = \bar{r} - \left( {1 - \left| {\bar{r}} \right|} \right) $$

And therefore, the correlation variance of the maximal symmetrical dispersion for a given average r is:

$$ {\text{Var}}_{\max } (r) = (1 - \left| {\bar{r}} \right|)^{2} = \bar{r}^{2} - 2\left| {\bar{r}} \right| + 1 $$

Appendix 3

Here we investigate the eigenvalue variance in two special cases of non-hierarchical modular matrices. General equation for eigenvalue variance in matrices with k homogeneous submodules, each of size N _i , and of no correlation between modules is:

$$ {\text{Var}}(\lambda ) = \frac{{\sum\nolimits_{i = 1}^{k} {\left( {\left( {\left( {1 + \left( {N_{i} - 1} \right) r_{wi} } \right) - 1} \right)^{2} + \left( {N_{i} - 1} \right) \left( {\left( {1 - r_{wi} } \right) - 1} \right)^{2} } \right)} }}{N} = \frac{{\sum\nolimits_{i = 1}^{k} {N_{i} r_{wi}^{2} \left( {N_{i} - 1} \right)} }}{N} $$

It is helpful to consider a case where all modules are of the same size, $N_{i} = \frac{N}{k}\hbox{; }{\text{and}}\;\sum\limits_{i = 1}^{k} {N_{i} } = N$, and equal within-module correlation r _w . In this case the equation simplifies to

$$ {\text{Var}}(\lambda ) = r_{w}^{2} \left( {\frac{N}{k} - 1} \right) $$

N > k, thus eigenvalue variance (i.e., integration) is lower than in a homogeneous (unimodular) matrix by a factor

$$ \frac{N - k}{k(N - 1)} $$

To compare it with integration estimated from the overall mean correlation, we calculate overall mean correlation:

$$ \bar{r} = \frac{{r_{w} k \left( {N_{i} (N_{i} - 1)} \right)}}{N(N - 1)} = r_{w} \frac{{N_{i} - 1}}{N - 1} $$

And therefore

$$ {\text{Var}}(\lambda ) \ge \bar{r}^{2} = \;r_{w}^{2} \;\left( {\frac{{N_{i} - 1}}{N - 1}} \right)^{2} $$

Here, the two integration levels differ by a factor

$$ \frac{N - k}{{k(N - 1)^{2} }} $$

the factor is smaller than 1, indicating that the estimate by the mean squared correlation is lower than the true integration of modular matrix. In fact we can see that the smaller the module size (and the greater thus the k), the smaller this factor will be.

The second case considered here is a case with different-sized modules of equal within-module correlation.

$$ {\text{Var}}(\lambda ) = r_{w}^{2} \frac{{\sum\nolimits_{i = 1}^{k} {N_{i}^{2} - N_{i} } }}{N} = r_{w}^{2} \left( {\frac{{\sum\nolimits_{i = 1}^{k} {N_{i}^{2} } }}{N} - 1} \right) $$

In which case only the term of the eigenvalue variance describing the effect of submodule size is averaged.

Appendix 4

Eigenvalue Variance for the Three-Trait Case with Unsymmetrical Distribution of Correlation

The correlation coefficients in matrices underlying this consideration are normally distributed on a Fisher’s z-transformed scale (Fisher 1928), but not on the Pearson’s correlation scale −1 < r < 1. Note that in the following we refer to the result of the normalizing Fisher’s z-transformation as to the z-scale. This is not to be confused with z-standardization of normal distribution. The Pearson’s product–moment correlations (r) are referred to as being on the r-scale.

Consider a given mean correlation on r-scale $(\bar{r})$ and given maximum deviation from the mean (d) on z-scale. The $\bar{r}$ has a corresponding value when transformed to z-scale, and the deviation d is added and subtracted to gain the lowest and the highest values of the correlation coefficients on the z-scale. The two values were back-transformed onto the r-scale and the deviations ε₁, ε₂ from $\bar{r}$ were calculated as differences of the extreme values from the mean. The third correlation coefficient is e ₃ = 0 − ε₁ − ε₂ due to the constraint of $\bar{r}$, i.e., $\sum {\varepsilon_{i} } = 0$. The three off-diagonal elements of the correlation matrix are now $(\bar{r} + \varepsilon_{1} ),(\bar{r} - \varepsilon_{2} ),(\bar{r} + \varepsilon_{3} )$. The corresponding eigenvalue polynomial (as in previous symmetrical case, see text), substituting x for (1 − λ) and multiplying out reduces to

$$ \begin{gathered} \det \,\left( {{\mathbf{R}}_{N = 3} - \lambda {\mathbf{I}}} \right) = x^{3} - x\,\left( {3r^{2} + 2r\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} } \right) + \varepsilon_{1}^{2} + \varepsilon_{2}^{2} + \varepsilon_{3}^{2} } \right) \\ + 2r\,\left( {r^{2} + r\,\left( {\varepsilon_{1} - \varepsilon_{2} + \varepsilon_{3} } \right) + \varepsilon_{1} \varepsilon_{3} - \varepsilon_{1} \varepsilon_{2} - \varepsilon_{2} \varepsilon_{3} } \right) - 2\varepsilon_{1} \varepsilon_{2} \varepsilon_{3} \\ \end{gathered} $$

Figure 6 shows plot of a family of polynomials with different combinations of parameters $\bar{r}$ and d; the three solutions representing the three x _i (=1 − λ _i ). Bundles of polynomials share the same mean correlation of matrix $(\bar{r})$, but differ in dispersion of correlations (d).

Fig. 6

The plot of the polynomial with solutions equal to the three (1 − λ) of the heterogeneous correlation matrix with N = 3, but with relaxed constraint on symmetry of the correlation distribution. The three solutions of the polynomial are the intersections of the curve with x-axis. The two groups of curves show the mean correlations of r = 0.3 and 0.7. The within-group variability is due to variable deviations from the mean correlation (ε = 0.05, 0.10, 0.15, 0.20, 0.25, and 0.30). Note that, in contrast to the analogous plot in Fig. 3 where the correlation distribution in underlying matrices was constrained to be symmetrical, in this plot the solution to the left of 0 is not invariant with the varying ε. Rather, similar to the two solutions >0, it varies dependent on ε

Analogous to the matrices with symmetrical distribution of correlation coefficients (main part of the text), we do not present the exact solutions, but rather approximate the three solutions here. Note that the three solutions follow the same pattern as above in symmetrical matrices. The following can be concluded from the plot: (i) when allowing the asymmetrical distribution of correlation, the heterogeneity affects all three eigenvalues; (ii) the effect of heterogeneity of matrix on first eigenvalue is smaller at higher $\overline{\left| r \right|}$; (iii) the first eigenvalue increases with increased heterogeneity (i.e., x ₁ decreases) while the remaining two diverge.

Approximating the three solutions in a way similar to symmetrical situation, but noting that all three eigenvalues vary depending on the heterogeneity, rather than only the second and the third, we consider the following solutions:

$$ x_{1} \approx - 2\bar{r} - \delta_{\varepsilon 1} $$

$$ x_{2} \approx \bar{r} - \delta_{\varepsilon 2} $$

$$ x_{3} \approx \bar{r} + \delta_{\varepsilon 3} $$

and the resulting eigenvalues are

$$ \lambda_{1} \approx 1 + 2\bar{r} + \delta_{\varepsilon 1} $$

$$ \lambda_{2} \approx 1 - \left( {\bar{r} - \delta_{\varepsilon 2} } \right) $$

$$ \lambda_{3} \approx 1 - \left( {\bar{r} + \delta_{\varepsilon 3} } \right) $$

If we substitute these into the equation for eigenvalue variance, it yields for three traits:

$$ {\text{Var}}(\lambda ) = \frac{1}{N}\left( {(N - 1)^{2} \bar{r}^{2} + 2\bar{r}^{2} + 2\bar{r}\left( {(N - 1)\delta_{1} - \delta_{2} - \delta_{3} } \right) + \sum\limits_{i = 1}^{N} {\delta_{i}^{2} } } \right) $$

And after some rearrangement for a general case:

$$ \begin{gathered} {\text{Var}}(\lambda ) = (N - 1)\,\bar{r}^{2} + \frac{1}{N}\left[ {\sum\limits_{i = 1}^{N} {\delta_{i}^{2} } + 2\bar{r}\left( {(N - 1)\delta_{1} - \sum\limits_{i = 2}^{N} {\delta_{i} } } \right)} \right] \\ = (N - 1)\,\bar{r}^{2} + \frac{1}{N}\left[ {\sum\limits_{i = 1}^{N} {\delta_{i}^{2} } + 2\bar{r}\sum\limits_{i = 2}^{N} {(\delta_{1} - \delta_{i} )} } \right] \\ \end{gathered} $$

This version corresponds to the eigenvalue variance equation for the heterogeneous matrices with symmetrical distribution of correlations, but involves an additional term (that yields zero if the distribution is symmetrical). This equation also shows that the eigenvalue variance in heterogeneous matrices will be greater than the corresponding eigenvalue variance of the homogeneous matrix of the same average correlation.

Appendix 5

Phenotypic Datasets

Tamarin Cranial Dataset

This dataset is comprised of 16 linear lateral cranial measurements on the skull of 275 individuals of the pure bred cotton-top tamarin (Saguinus oedipus). Detailed description of the population, measurement method and the list of measurements can be found in Hutchison and Cheverud (1995). The linear measurements were derived from 3D coordinate data for 16 landmarks per each side of the skull. All landmarks were taken twice, to estimate the measurement error and improve repeatability of short measurements by using their average. The average repeatability per trait of these data was 0.788 on the right and 0.776 on the left side.

Bird Wing Dataset

The lengths of seven wing feathers have been measured on 64 museum specimens of Eurasian Goshawk (Accipiter gentilis gentilis, data are available in online supplementary material, Table S1). This dataset comprises the first five primary—(‘hand’) as well as the third and the fifth secondary (“forearm”) flight feathers, counted from distal end of the wing towards the body. The feathers were measured as a distance from bend of the wing (i.e., wrist joint) to the tip of the feather along the shaft, measured over the folded wing. The sample size consists of individuals covering most of the subspecies range. All measurements were taken three-fold. The average repeatability of single measurements in these data is 0.978.

Mouse Skeletal Dataset

Phenotypic traits comprise 70 skeletal measurements representing the cranium, axial and appendicular skeleton, as well as body weight at 10 weeks and at necropsy. The experimental population results from an intercross of inbred mouse strains LG/J and SM/J, selected for large and small body weight at 60 days of age, respectively. The data stems from two consecutive analogous intercrosses, comprising a total of 1040 F₂ mice. For the list of measurements and details on measurement techniques see Kenney-Hunt et al. (2008). Extreme outliers were identified by being more than 2.7 standard deviations from the sample mean (SYSTAT 10.2) and were eliminated to avoid biasing the data. The data were corrected for the effects of dam, litter size, experimental block, sex, age at necropsy, and intercross. The correction was done by regressing each of the phenotypic trait measurements on all of the respective predictor variables (dam, litter size, sex, etc.) and taking the residuals as corrected scores for the trait. The repeatability was assessed for each trait by repeated measurements on 30–50 individuals and amounts on average 0.93, with the range from 0.82 to 0.992 for single traits.

Appendix 6

Several Other Indices of Integration

Some of the listed measures require the matrices to be non-negative, or even positive definite. Errorless phenotypic matrices are always at least non-negative definite, however due to error, either sampling or measurement error, empirical matrices are often negative. To apply the methods 2 and 5 to the empirical datasets in the paper, we used only dimensions with eigenvalues greater than zero, in case the matrices were not positive definite.

• 1. Van Valen (1965) suggested calculating the average Fisher’s z-transformed correlation, and transforming it back to the scale 0–1.

• 2. Cheverud et al. (1983) defined the index of integration as one minus the geometric mean of eigenvalues of a correlation matrix:

  $$ I = 1 - \sqrt[N]{{\prod\limits_{i = 1}^{N} {\lambda_{i} } }} $$

• 3. Wagner (1984) suggested the variance of eigenvalues of a correlation matrix. We provide also the relative eigenvalue variance, as described in the main body of this paper:

  $$ \text{Var}_{\text{rel}} (\lambda ) = \frac{{\text{Var} (\lambda )}}{{\max \text{Var} (\lambda )}} = \frac{{\text{Var} (\lambda )}}{N - 1} $$

• 4. Cane (1993) used an average absolute pair wise correlation

• 5. Hansen and Houle (2008) suggested a measure that is based on covariance matrix and applies the harmonic mean of eigenvalues (H(λ)):

  $$ i = 1 - \left[ {\frac{H\left( \lambda \right)}{{\overline{\lambda } }}\left[ {1 + \frac{{1 + I\left( {\frac{1}{\lambda }} \right) + I(\lambda ) - \frac{H(\lambda )}{{\overline{\lambda } }}}}{k + 1}} \right]} \right] $$

  where

  $$ I\left( {\frac{1}{\lambda }} \right) = \frac{{{\text{Var}}\left( {\frac{1}{{\lambda_{i} }}} \right)}}{{\left( {\overline{{\frac{1}{{\lambda_{i} }}}} } \right)^{2} }}\quad I(\lambda ) = \frac{{{\text{Var}}(\lambda )}}{{(\overline{\lambda } )^{2} }} $$

The data for this procedure need to be on similar scale, which can be accomplished by variance or mean standardization. We used variance standardization in the examples, which corresponds to correlation.