Evolutionary biologists have been interested in genetic covariances because they can alter the response to selection from the simple univariate prediction. A genetic correlation provides a pathway that allows the evolution of one trait to be influenced by selection acting on another trait. Such correlations can either facilitate or restrain the rate of adaptation. Our primary goal was to provide a metric for quantifying the net effect of all the correlations among a set of traits, holding the trait variances constant.

For any pair of traits,

*z*_{i} and

*z*_{j}, it is simple to determine whether the correlation between them will have a positive or negative effect on the rate of adaptation. If the correlation is of the same (opposite) sign as the product of the directional selection gradients,

*β*_{i}×

*β*_{j}, the effect of the correlation will be positive (negative). Using this approach, one can consider the qualitative effect of each correlation individually (

Etterson & Shaw 2001). However, this approach does not provide a quantitative measure of the importance of the correlation. Nor does it allow the joint effects of all correlations among a set of traits to be assessed simultaneously.

When *n* traits are under consideration, there will be *n*(*n*−1)/2 genetic correlations. Some of these may facilitate the rate of adaptation and while others may constrain it not all of them will affect the rate of adaptation equally. Correlations that are large in magnitude or involve traits that are under strong selection, or both, will be more important than correlations that are weak in magnitude and/or involve traits that are only weakly selected. The metric *R* provides the net effect of all correlations, appropriately weighting important ones more so than unimportant ones. *R* measures effect of the correlations in terms of fitness, allowing for comparisons to be made between different populations or different species.

Although

*R* quantifies the net effect of all correlations, it is also possible to slightly modify this approach to identify individual traits whose correlations to other traits are particularly important in affecting the rate of adaptation. For example, suppose we are interested in assessing the extent to which trait

*j* alters the rate of adaptation through its genetic correlations with other traits. This can be accomplished by calculating

, where

is the change in the fitness of the mean phenotype assuming that all genetic covariances involving trait

*j* are zero. Specifically,

is calculated from equations

(2.3*a*) and

(2.3*b*) with

but with all elements

*G*_{ij} set to zero for

*i*≠

*j*.

*R*_{j} is the ratio of the rate of adaptation with the observed correlations relative to the predicted rate of adaptation if trait

*j* was genetically independent from all other traits.

For example,

Caruso (2004) measured selection and

*G* on seven floral traits in

*Lobelia siphilitica*. Using the data from the CERA population in 2000, we find that

*R*=0.41, i.e. the predicted rate of adaptation is only 41 per cent of what it would be in the absence of any genetic correlations. Using the method above to calculate

*R*_{j}, we can ask how each trait individually affects the rate of adaptation through its correlations with other traits. For the traits: (i) lobe length, (ii) lobe width, (iii) corolla length, (iv) corolla width, (v) stigma–nectary distance, (vi) stigma exsertion, and (vii) flower number, the corresponding values are

*R*_{1}=1.00,

*R*_{2}=1.01,

*R*_{3}=0.59,

*R*_{4}=0.96,

*R*_{5}=0.34,

*R*_{6}=1.35,

*R*_{7}=0.68. These results reveal that correlations involving lobe length, lobe width or corolla width have little effect on the rate of adaptation under this selection regime. However, the predicted rate of adaptation is reduced relative to what it would be if corolla length, stigma–nectary distance or flower number were genetically independent of other traits (i.e. correlations involving any of these three traits tend to constrain the rate of adaptation). By contrast, correlations involving stigma exsertion tend to facilitate the rate of adaptation.

One could extend this approach to examine the effect of correlations between sets of traits by calculating

after setting all covariances between traits from different trait suites to zero. For example, consider a hypothetical study involving four traits. The first trait suite (e.g. floral traits) comprise traits 1 and 2, whereas traits 3 and 4 belong to a separate trait suite (e.g. vegetative traits). To examine the effect of correlations

*between* floral and vegetative trait suites one would set

*G*_{13}=

*G*_{14}=

*G*_{23}=

*G*_{24}=0 to do the calculation. Analysis of data in this way may serve as a useful approach to study the phenotypic integration and modularity (

Cheverud 1982;

Arnold 1992).

In different contexts, other authors have used other measures to consider how correlations affect evolution.

Schluter (1996) measured the angle between the major axis of genetic variation

*g*_{max} (the leading eigenvector of the

*G*) and major axis of divergence among species within lineages having undergone adaptive radiations. He found that this angle was smaller than expected by chance, leading him to infer that patterns of correlations bias patterns of long-term evolution (

*ca* 4

Myr).

McGuigan *et al*. (2005) applied a related but more advanced approach with a set of fish populations that had repeatedly adapted to lake and stream environments. They found that the divergence of traits that were likely to be under strong selection showed less evidence of being affected by correlations than traits that were less likely to be targets of strong selection.

Blows *et al*. (2004) examined the relationship between

*G* and selection by measuring the angle between

*β* and the projection of the ‘major’ subspace of

*G* onto

*β*, and by comparing the angles between the axes of major subspaces of

*G* and

*γ*. As with our metric, this approach is intended to give a sense of the extent to which patterns of genetic variation constrain or facilitate evolution. This approach can be useful when applied to systems where the biology is well understood and the results are carefully interpreted. However, it has several limitations. First, the resulting metrics are not simple to understand or interpret. Second, the choice of principal component axes used to define the subspaces is somewhat arbitrary, and not all of the principal component axes can be used (

Blows *et al*. 2004). Third, it is possible for this approach to give misleading results when only one or a few of the correlated traits are under selection unless results are interpreted with great caution. For example, if arm and leg length are strongly genetically correlated but only arm length is under selection, there will be a large angle between

*β* and the

*g*_{max} (the one-dimensional subspace of

*G*). A large angle is interpreted as a mismatch between selection and

*G*. However, in this example, the correlation between the traits is meaningless with respect to the rate of adaptation because only arm length is under selection.

Our metric *R* is intuitively meaningful and can be easily calculated and interpreted. This is because *R* measures the effects of correlations in terms of fitness. However, this property also limits the usefulness of *R* in some respects. In the arm/leg length example above, we would find that *R*=1 (or *L*=0), correctly indicating that the correlation between arm and leg length has no effect on the rate of increase in fitness. However, this correlation is clearly of great importance with respect to understanding the evolution of leg length. In this example, leg length, a neutral character evolves because it is genetically correlated to a selected trait (arm length). Because *R* measures effects with respect to fitness, it does not describe how correlations affect the phenotypic evolution of individual traits. For this reason, we advocate that *R* be used in addition to, rather than instead of, other ways of understanding how correlations affect evolution.

Some particularly interesting approaches have been recently described by

Hansen & Houle (2008). Building from earlier work (

Houle 1992;

Schluter 1996;

Hansen *et al*. 2003), these authors have suggested several measures for quantifying constraint and evolutionary potential in multidimensional space. Evolvability is a standardized measure of the amount of genetic variation in the direction of selection,

*β*. Conditional evolvability gives the amount of genetic variation in this direction assuming that movement in any other direction is strongly prohibited by stabilizing selection, i.e. it is the amount of variation along

*β* that is independent of variation in other directions.

These measures quantify the combined effects of trait variances as well as the correlations, and in this sense provide a more complete picture of genetic constraint. By contrast, our goal was to isolate the effect of genetic covariances. The simple logic of our metric could be applied to their elegant measures to determine how correlations affect conditional evolvability. This can be easily accomplished by calculating the ratio of the evolvability (or conditional evolvability) when the genetic correlations are set to zero to the actual evolvability (or conditional evolvability) using the observed *G*. In fact, calculating this ratio for evolvability will be equivalent to the ratio *R* discussed here if there is no nonlinear selection.

One important limitation of our approach is that the value of *R* depends on how traits are defined. This dependency on trait definitions is an inherent part of our question: how do the genetic correlations among a specified set of traits affect the rate of adaptation? If traits are redefined along an alternative set of axes in the same multidimensional space (i.e. an orthogonal transformation to a new coordinate system), then there will be a new set of genetic correlations among these newly defined traits. The value of *R* can be re-calculated to measure how these correlations affect the rate of adaptation, but the new value of *R* will be different from the original since the new *R* refers to a different set of correlations. While we can measure *R* with respect to any defined set of traits, is it meaningful to do so?

A reasonable argument can be made that the way in which traits are defined is somewhat arbitrary and that natural selection may view traits in a very different way than humans do (see the target review by

Blows (2007) and ensuing commentaries for further discussion of trait identification in studies of multivariate evolution). Even though selection may perceive them differently, the traits chosen for study usually are significant to those biologists measuring them. Often, something is known about the function of a trait or a trait is meaningful to how the observer perceives the study organism, e.g. arm and leg length have more meaning than two alternate traits defined by orthogonal linear combinations of these traits. Moreover, some aspects of organisms need to be measured or chosen for measurement by biologists in the first place before any coordinate or orthogonal transformations can take place, suggesting that escaping human perceptual biases is likely to be quite difficult or impossible. Biologists identify measurable traits and are often interested in how the relationships between these measurable traits affect evolution. Our metric helps to understand this relationship better. Our metric is not designed to show selection views on an organism in multivariate space. (It should be noted that the evolutionary hypotheses discussed previously for the average value of

*R* should be reasonably robust to how traits are defined. For example, if genetic variation is exhausted in the direction of selection, genetic correlations should be constraining regardless of the coordinate system used.)

In addition to the metric *R* that requires information on both selection and *G*, we have proposed alternative perspectives from which the potential for correlations to affect the rate of adaptation can be assessed. First, we can ask whether the rate of adaptation under a specified pattern of selection is likely to be sensitive to correlations by measuring the evenness of selection. If the evenness of selection is high across a set of traits, then there will be high potential for a random set of correlations among these traits to affect the rate of adaptation. This metric depends only on selection and thus can be measured even in the absence of any information on *G*.

We can also ask whether the covariance structure is such that covariances are likely to affect the rate of adaptation. This can be assessed by the evenness of the eigenvalues of the *G*. When the evenness of eigenvalues is low, it means that correlations cause genetic variation to be reduced in some directions. Consequently, there is potential for covariances to affect the rate of adaptation if selection occurs in these directions. Measuring potential from only this perspective requires an estimate of *G*. However, estimates of the evenness of eigenvalues should be regarded with caution because of the downward bias in their estimation. Alternatively, we can avoid relying on the estimates of the evenness of the eigenvalues of *G* and instead determine the minimum and maximum possible values of *R* for a given *G* from the range of eigenvalues of *M*=*σ*^{−1}*Cσ*.

Our survey of the literature revealed a distribution of *L*-values centred close to zero. Most datasets were only weakly affected by correlations. There were examples of datasets where correlations were fairly strongly constraining the rate of adaptation but these were balanced by datasets where correlations were facilitating the rate of adaptation. Neither the mean nor variance of *L* was significantly different from that expected by chance.

As stated above, basic quantitative genetics theory makes no prediction about whether correlations should tend to constrain or facilitate the rate of adaptation. However, there are several secondary sets of ideas that do make predictions in this regard (e.g. exhaustion of variation in the direction of selection, long-term shaping of mutational/developmental pathways to facilitate the generation of variation along selected directions, condition-dependence of strongly-selected traits). In light of these hypotheses, there are at least three different ways to view our result that correlations seem to have little or no effect on average. First, it is possible that none of the three hypotheses are important. Second, all three hypotheses may be important in individual cases but none of them are sufficiently common or strong to generate a clear signal across all the datasets. The different hypotheses were suggested in the context of different types of traits (e.g. morphological traits versus life-history traits and/or sexually selected traits) but our dataset is not large enough to meaningfully partition these studies into different trait types for separate analyses. Third, our data may be too noisy to perform a reasonable test. As described above and in the electronic supplementary material, our data are far from perfect. Both estimates of selection and *G* are notorious for their large confidence intervals. Moreover, we had to make a number of ‘approximations’ (e.g. using *P* as an approximation to *G*) to obtain a reasonable number of studies for examination. Even with all these levels of uncertainty, the estimates we used for selection and *G* should be closer to the true values than randomly chosen estimates would be. If there was a strong tendency for *R* to differ from unity, we should have been able to detect it through the noise. However, we have no way to assess the power of our test; it is possible that there is a fairly strong effect of correlations on average but our data are too noisy to detect it. Regardless, we could not know whether a sufficiently strong effect existed without attempting to look for it.

Genetic correlations have intrigued evolutionary biologists since it became well-understood that correlations affect how traits respond to multivariate selection. Genetic correlations became the perspective by which evolutionary quantitative geneticists considered constraint. While some have focused on the idea of absolute constraint, it is more useful to view constraint quantitatively rather than qualitatively. The metric *R* does so with respect to fitness. Our literature survey reveals examples with strong and weak constraint (*R*<1) as well as cases of negative constraint, i.e. patterns of correlations that facilitate the rate of adaptation (*R*>1). Though the average effect of correlations was not significantly different from expected by chance, the available data are noisy and we urge caution in interpreting our results. Nonetheless, we can say that our results provide no support for the emphasis on constraint that seems to permeate the literature on genetic correlations.