The dangers of diagonalization

Dr Mark W. Blows (A tale of two matrices: multivariate approaches in evolutionary biology; Blows, 2007) rightly describes the importance of multivariate methods in evolutionary research. While matrix diagonalization is an important method in statistical research, one needs to exercise caution to avoid over-interpreting or reifying the mathematical results of the procedure. Dr Blows has written an excellent summary of the importance and utility of considering the structure of two square matrices, the genetic variance–covariance matrix (G) describing the patterns of heritable variation and the stabilizing selection matrix (W) describing the pattern of nonlinear selection gradients for a set of traits. He clearly points out that interpretation of individual traits in isolation from the morphological or physiological processes in which they participate is not productive and can lead to misinterpretations of evolutionary processes. He then argues that the procedures of matrix diagonalization can be an important aid in understanding and interpreting the evolution of multivariate trait sets and notes the prevalence of diagonalization procedures in multivariate statistical analyses. Even though diagonalization has some desirable properties, such as statistical independence of the eigenvectors, there are also some dangers involved in overinterpretation of the eigenstructure of these matrices. One point made by the author is that even when none, or only a few, of the traits are significantly heritable, it is quite possible that certain eigenvectors, the first is singled out for special attention, have higher heritabilities and hence more chance of being statistically significant with a smaller sample, than other aspects of variation. However, Lin & Allaire (1977) showed that the heritability of a linear combination is the linear combination of the heritabilities of the traits, so heritability is not necessarily enhanced for first eigenvectors unless highly heritable traits happen to have higher loadings than lowly heritable traits on that eigenvector. Nothing in the spectral decomposition of a square symmetrical matrix guarantees this. As mentioned briefly in this review paper, Cheverud (1981) showed that one can identify highly and lowly heritable linear combinations by performing a quantitative genetic analysis on the phenotypically independent principal components. The genetic and environmental variance–covariance matrices of these phenotypic principal components are mirror images of one another (as they add to a diagonal matrix) and will have the same eigenvectors but arranged in opposite eigenvalue order. This property allows one to concentrate heritable or nonheritable effects into a few independent factors each. Over interpretation of the principal components of G can also be misleading. There is nothing in the mathematical operation of spectral decomposition of square symmetric matrices that has any necessary relationship with the biology underlying the traits. It is a purely mathematical operation, like taking a logarithm, not a biological interpretive tool. It is tempting to reify these mathematical constructs in biological terms. For example, Cheverud et al. (1983) found that the spectral decomposition of G for age-specific weights identified a first component with all age-specific weights having positive loadings and second component contrasting early and later age-specific weights. It is tempting to interpret this result as identifying separate gene effects, some genes affecting all weights the same while other genes have opposite effects on early and later growth. However, Quantitative Trait Locus (QTL) analysis of growth indicates that age-specific effects are modular, affecting either early or late growth, and that there are no genes with opposite effects at different ages, corresponding to the pattern of the second eigenvector of the variance–covariance matrix (Cheverud et al., 1996; Cheverud, 2005). The contrast between early and late weights indicated by the second eigenvector is statistical and does not correspond to the effects of any single gene. Likewise, in a study of QTLs affecting cranial morphology (Leamy et al., 1999), we again found that individual gene effects were modular, rather than contrasting. QTLs mapped for the second eigenvector (contrasting the face with the neurocranium) really only affected one of the units, the face or the neurocranium. Genes with opposite effects on the face and neurocranium were not found. Again, over-interpretation of the eigenvectors can be misleading. Therefore, while matrix diagonalization can be a useful procedure and plays a central role in the multivariate general linear model, it should not be expected to provide special biological insight.