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A tale of two methods: putting biology before statistics in the study of phenotypic evolution

The matrix diagonalization methods advocated by Blows have a number of statistical advantages, especially for the study of natural selection, but these advantages are usually outweighed by the disadvantage that the results are not very biologically interpretable. This lack of interpretability has been the major impediment to the adoption of these methods over the past 15 years. Blows (2007) makes a number of valuable points that deserve wider attention from evolutionary biologists. The paper advocates a more wholly multivariate approach to understanding adaptive evolution of phenotypic traits as an alternative to the now-standard selection gradient and G-matrix analysis. In the standard analysis, multiple regression techniques are used to measure selection on individual traits, correcting for correlations among all traits included in the analysis. With this correction, selection gradients are an extremely useful way of studying adaptive evolution, as they aim to measure direct selection after removing indirect selection caused by phenotypic correlations. More ambitious studies using the standard analysis include both linear terms to estimate the strength of directional selection, and quadratic (squared) terms to estimate the degree of curvature in the fitness function, which might indicate stabilizing and disruptive selection if the maximum or minimum fitness value occurs at intermediate phenotypic values. Still more ambitious studies include the cross-product terms in the regression to measure correlational selection, that is, whether pairs of traits interact to determine fitness in ways not described by the linear and quadratic terms alone. This accumulation of terms in the regression model is perhaps the biggest problem with standard selection gradient analyses, and thus the biggest advantage of the multivariate methods like canonical analysis (one kind of diagonalization) advocated by Blows. To estimate all the quadratic and correlational selection gradients requires a large number of predictor variables in the analysis with only a modest number of traits. With n traits, the number of predictors in the full model (all linear, quadratic and cross-product terms) is 2n+[n(n)1)]/2, so with only five traits there are 20 variables in the model! With canonical analysis, this number is reduced to 2n, because the number of new variables created is the same as the number of original variables in the analysis, and there is a linear and quadratic term for each. The second term in the formula above represents the correlational selection estimates, which drop out when new axes are created that are orthogonal (i.e. uncorrelated). The main reason that correlational terms are often not included in selection gradient analyses (Kingsolver et al., 2001) is not their difficulty in interpretation (more on this below), but rather that people rarely have the large sample sizes needed to fit so many variables in their regression models. Thus, a method that allows people to explore the multivariate nature of selection with the more modest sample sizes of most studies is a real advantage. Therefore, the multivariate methods that Blows champions will prove to be useful additions to the evolutionary biologist’s toolbox in some cases, providing new insight not available from standard selection gradient analysis. However, it is not yet clear how often this will be the case, and these methods will not replace the nowfamiliar selection gradient and G-matrix analysis. This is not just due to tradition; there are good reasons why the selection gradients and G-matrix framework has become one of the most widely adopted in all of evolutionary biology (allowing meta-analyses like that of Kingsolver et al., 2001): the approach can be undertaken by empiricists with modest quantitative skills, and more importantly it produces results that are eminently interpretable, leading to robust conclusions about the nature of the adaptive process. Selection gradients are often presented in standardized form, that is, in standard deviation units, making them readily comparable across different traits and studies. This makes it possible to make comparisons between, for example, the strength of direct selection on tarsus length in different bird species, or to compare the strength of selection on morphological vs. life-history traits (Kingsolver et al., 2001). Blows performs a good service by reminding us of the potential pitfalls of selection gradient analysis; however, as long as practitioners are aware of these and exercise due caution in analysis and interpretation, they are almost never a problem. The diagonalization techniques advocated by Blows will not replace selection gradient and G-matrix analyses because they have a fundamental problem, which is pointed out at the very end of Blows’ article: the difficulty of biological interpretation. This is the Achilles’ heel of these methods, because the central goal of selection and G matrix analyses has to be biological understanding. Once new axes through multivariate space are created for statistical, rather than biological reasons (for example, to maximize variance explained and to make axes orthogonal), then it is often very difficult to understand what these axes mean in terms of adaptation or organismal function (Endler, 1986, p. 192; Correspondence:Jeffrey K. Conner, Kellogg Biological Station, 3700 East Gull Lake Drive, Hickory Corners, MI 49060, USA. Tel.: 269-671-2269; fax: 269-671-2104; e-mail: connerj@msu.edu