Measuring Morphological Integration Using Eigenvalue Variance

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.