### Abstract

- Top of page
- Abstract
- Two simple equations to predict adaptive phenotypic microevolution
- Decomposition of selection
- Causation
- Expectations for genetic covariances with fitness
- Testing for genetic consequences of natural selection
- Distinguishing between studying selection and predicting evolution
- Summary
- Acknowledgments
- References

The breeder's equation, which predicts evolutionary change when a phenotypic covariance exists between a heritable trait and fitness, has provided a key conceptual framework for studies of adaptive microevolution in nature. However, its application requires strong assumptions to be made about the causation of fitness variation. In its univariate form, the breeder's equation assumes that the trait of interest is not correlated with other traits having causal effects on fitness. In its multivariate form, the validity of predicted change rests on the assumption that all such correlated traits have been measured and incorporated into the analysis. Here, we (i) highlight why these assumptions are likely to be seriously violated in studies of natural, rather than artificial, selection and (ii) advocate wider use of the Robertson–Price identity as a more robust, and less assumption-laden, alternative to the breeder's equation for applications in evolutionary ecology.

### Two simple equations to predict adaptive phenotypic microevolution

- Top of page
- Abstract
- Two simple equations to predict adaptive phenotypic microevolution
- Decomposition of selection
- Causation
- Expectations for genetic covariances with fitness
- Testing for genetic consequences of natural selection
- Distinguishing between studying selection and predicting evolution
- Summary
- Acknowledgments
- References

Evolutionary change in a phenotypic trait under selection can be predicted using the univariate breeder's equation:

- (1)

where *R* is the per generation response of a quantitative, or continuous, trait, to selection, *h*^{2}, the heritability, is the proportion of phenotypic variance in the trait that is attributable to additive genetic differences among individuals, and *S* is the selection differential (Lush, 1937, chapter 12). The selection differential is the difference between the population mean before and after selection, and more generally, the selection differential is the covariance between the trait and the relative fitness. This equation plays a fundamental role in how we think about and study natural selection and its microevolutionary consequences; it features prominently in many of the texts we read and cite (Falconer, 1981; Futuyma, 1998; Lynch & Walsh, 1998; Roff, 2002), and papers and lectures we write and give. The breeder's equation, as its name implies, traces its roots to animal breeding and artificial selection (Lerner, 1958; Lush, 1937), where it has had a long and successful history, at least in univariate studies (Clayton *et al.*, 1957; Falconer, 1953; Roff, 2007). However, the breeder's equation has been less successful in predicting evolutionary change in natural, rather than agricultural or laboratory, systems (Merilä*et al.*, 2001a). Evolutionary biologists have made many estimates of both the form of selection, especially quantitative estimates of directional selection (Endler, 1986; Kingsolver *et al.*, 2001), and also many estimates of heritabilities, both in the laboratory and in the field, for ecologically important traits (Mousseau & Roff, 1987). In general, hypotheses that any given traits are under directional selection are very often true, and furthermore variation in these traits often, if not generally, has a demonstrable genetic basis. Thus, the breeder's equation suggests that we should expect adaptive phenotypic microevolution of quantitative traits to be common. However, well-supported examples of microevolution in natural populations are very difficult to produce, and on careful inspection, stasis, or even counterintuitive phenotypic trends (e.g. Gienapp *et al.*, 2008; Kruuk *et al.*, 2003; Larson *et al.*, 1998; Wilson et al., 2007; Merilä*et al.*, 2001b) are at least as common as changes consistent with breeder's equation-based predictions (e.g. Grant & Grant, 1995). Demonstrating that phenotypic change – whether consistent with predictions or not – has an underlying genetic component is a technically challenging task (see Hadfield *et al.*, 2010 for a critique of the statistical procedures most commonly used to date). Although this difficulty undoubtedly contributes to the scarcity of well-documented cases of adaptive phenotypic selection responses, we also propose that there is a more general failure to appreciate the limitations of the breeder's equation when applied outside of the context of artificial selection.

An alternative simple equation to predict evolutionary change is the Robertson–Price identity:

- (2)

where Δ*z* is the expected change in mean phenotype *z* between generations and *w* is relative fitness. *σ*_{a}(*z*,*w*) represents the additive genetic covariance between *z* and *w*, or the extent to which heritable genetic differences among individuals co-determine the trait and fitness (Price, 1970; Robertson, 1966). The Robertson–Price identity is also sometimes called the ‘secondary theorem of natural selection’ (Crow & Nagylaki, 1976; Robertson, 1968). The left-hand sides of the breeder's equation and the Robertson–Price identity are the same: they are both the per generation change in mean phenotype. However, the equivalence of the right-hand sides is not so apparent.

Translating these equations into words, the breeder's equation states that the response to selection is equal to the proportion of variance in a trait that is caused by genetic effects, multiplied by the strength of (directional) selection on the trait. The Robertson–Price identity states that the response to selection is equal to the genetic covariance between a trait and (relative) fitness. The key difference between these two statements is that the breeder's equation implies we should, or at least can, measure the relationship between a trait and fitness (i.e. selection) and the relationship between a trait and genes (i.e. heritability) separately. In contrast, the Robertson–Price identity contains only a single term, requiring that we consider fitness variation and the genetics of a trait jointly to predict evolution. Expanding the *h*^{2} term in eqn 1 and setting the right-hand sides of eqns 1 and 2 to be equal, as their left-hand sides are identical, we obtain

- (3)

The quotient of a covariance and a variance is a regression, and thus in words eqn 3 states that the regression, or more loosely the relationship, between fitness and genes must be the same as the regression/relationship between fitness and phenotypes in order for the predictions of the breeder's equation and the Robertson–Price identity to be the same. The result in eqn 3 is derived (differently) in the appendix to Hadfield (2008). Each of these parameters can in fact vary independently, and so eqn 3 does not represent an exact theorem, but rather a narrow condition. This narrow condition is the situation in which the two primary tools at our disposal for predicting evolution will give us consistent results. Below we develop a graphical model of the genetics of selection on a heritable trait to explore the implications of the distinction between genetic and phenotypic patterns of trait–fitness covariance.

### Decomposition of selection

- Top of page
- Abstract
- Two simple equations to predict adaptive phenotypic microevolution
- Decomposition of selection
- Causation
- Expectations for genetic covariances with fitness
- Testing for genetic consequences of natural selection
- Distinguishing between studying selection and predicting evolution
- Summary
- Acknowledgments
- References

An individual's phenotype (*P*) is determined by genes and environmental effects such that

- (4)

where *G* is the genetic merit (i.e. what the phenotype would be in the absence of environmental influences) and *E* is the effect of the environment.^{1} This simple decomposition holds at the individual level for phenotypic traits, and at the population level for variances in (and covariances between) traits. We can therefore apply this decomposition to selection which is simply the covariance between two traits in the particular case that one of those traits is fitness. Although we donot often conceptualize fitness as a trait, it is a measurable feature of individuals and so can be treated in this way.

Using simulated data, we can explore different scenarios that lead to a phenotypic covariance between a trait and fitness (Fig. 1). The first column of plots in Fig. 1, scenarios 1–3, shows a simple and commonly observed occurrence in studies in evolutionary biology: the trait appears to be under directional selection, i.e. it covaries with fitness. In the case of the simulated data sets in scenarios 1–3, the phenotypic covariances are all positive and we would conclude that selection favours individuals with greater phenotypic values. The next two columns in the figure depict alternative ways in which this phenotypic covariance might arise from genetic and environmental sources of covariance. In all these scenarios, the trait under apparent selection has genetic variance and is heritable (i.e. there is variance along the *x*-axis in all plots in the middle column). However, the scenarios differ with respect to the presence and sign of genetic covariance between the trait and the fitness. Scenario 4 in Fig. 1 applies the same principle of decomposing a phenotype–fitness relationship into plausible genetic and environmental components. However, in this final example, the net phenotypic covariance between a trait and fitness is zero. Consequently, we might conclude an absence of selection (and expected change) in this case, despite the fact that trait–fitness relationships do exist at the genetic and environmental levels.

The consequence of this exercise of plotting hypothetical decompositions of a covariance representing selection may not be immediately apparent from the scatter plots in Fig. 1 alone. Genes do not interact with environments in ways that influence fitness, only whole organisms do. However, whereas selection only ‘sees’ phenotypes, the colours in Fig. 1 illustrate how it is that the relationship between trait and fitness at the genetic level is what matters for adaptive evolution. Red points denote those 10% of simulated individuals with the largest genetic merits for the trait. Blue points correspond to those 10% of individuals with the smallest genetic merit. Given the first column (and remembering that this is the only one where selection ‘sees’ traits), we can see that the pattern of trait–fitness covariation at the genetic level provides the only correct description of what genes will be represented in the next generation. For instance, in scenario 1, individuals with genes that promote large phenotypic values (red upward-pointing triangles) will be increasingly represented in future generations, given that the scenario illustrated in the top panels exists. In contrast, there will be no adaptive evolutionary change in scenario 2, despite apparent positive directional selection on the heritable trait. In scenario 3, evolution would actually occur in the opposite direction to that expected based on the positive covariance between trait and fitness. That is to say that in this scenario, individuals with genes promoting smaller than average phenotypic values (blue downward-pointing triangles) will have increased genetic representation in the future. Clearly, the representation of different genotypes in the next generation is a consequence of the genetically based relationship between the trait and the fitness.

The patterns in Fig. 1 are intended to show the qualitative range of genetic and environmental patterns of covariance that might potentially underlie an observation of apparent directional selection on a heritable trait. However, an additional axis of quantitative variation likely exists in the genetic variance of fitness. In general, variation in fitness is expected to be low or absent in populations at equilibrium (Fisher, 1930), and empirical results support this, at least in a relative sense, i.e. the heritability of fitness is generally low (Coltman *et al.*, 2005; Merilä and Sheldon, 1999, 2000; Mousseau & Roff, 1987; Price & Schluter, 1991; Teplitsky *et al.*, 2009). In the extreme equilibrium scenario, where the additive genetic variance of fitness is zero, it is not possible for a genetic covariance of a trait with fitness to exist. This has two important implications for how we relate observations of phenotypic covariances of traits with fitness. First, if populations generally harbour little variance for fitness, genetic covariances of traits with fitness may often necessarily be smaller than naïve application of the breeder's equation might implicitly require. Specifically, the minimum genetic variance in relative fitness required to support a given response to selection, *h*^{2}*S*, or in other words the genetic variance caused by selection on heritable phenotypic variation, is

- (5)

and thus this quantity must be equal to or less than the genetic variance in relative fitness (Walsh & Blows, 2009).^{2} This provides a test of the plausibility of any given prediction of evolutionary change. As the right-hand side of eqn 5 can take any nonnegative value, one can calculate its value and determine whether or not it is consistent with an estimated or plausible value of the genetic variance of relative fitness. Second, this highlights the important role played not only by variation in fitness (Orr, 2009), but in particular by the genetic variance of fitness in adaptive evolution, and so by extension to potential benefits to bringing its consideration more thoroughly into our approaches for studying microevolution.

So how does this graphical model relate to the two equations for predicting change? The breeder's equation predicts an increase in a trait's mean if the trait has a positive phenotypic covariance with fitness and is genetically variable. As these criteria are both met in scenarios 1–3 of Fig. 1, the prediction would be erroneous in two cases. Furthermore, in scenario 4, the breeder's equation has a zero term (i.e. *S* = 0) and therefore predicts no evolutionary change, despite the fact that a genetic covariance between the trait and the fitness exists. The breeder's equation would fail in scenarios 2–4 because in these situations the environmentally based association between trait and fitness differs from the relationship at the genetic level. This is a fundamental violation of the model's assumptions (as we discuss further below). In contrast, the Robertson–Price identity makes predictions that agree with our graphical model in all cases because it explicitly addresses the genetic basis of the trait–fitness relationship. We hope this graphical presentation will promote the appreciation of the range of true compositions of the relationship between an apparently selected heritable trait and fitness. We do not intend to suggest, however, that we are the first to present this phenomenon, rather we aim at presenting a means by which to make it more accessible. More general mathematical expressions of the principle are illustrated by Rausher (1992)’s eqn 5 and its derivation, by Queller (1992)’s eqn 6, and by the third equation in the appendix to Hadfield (2008).

Several studies have obtained estimates of the form of selection on estimated individual-level effects of genes on phenotype for ecologically important traits. The basic premise has been to regress fitness on predicted breeding values from linear mixed models. Predicted breeding values (estimates of the effect of an individual's genes; obtained by a technique called BLUP for Best Linear Unbiased Prediction) are predictions of the contribution of genes to an individual's deviation from a phenotypic mean, based on phenotypic data of the individual and its relatives. In plant systems, this kind of approach has provided evidence that environmentally induced relationships between phenotypic traits and fitness differ from genetically induced covariances in 25–30% of studies investigated (Rausher, 1992, Stinchcombe *et al.*, 2002). This type of approach has also been used several times in animal systems to test whether the selection is actually acting on the genotype in situations where selection on heritable traits has not resulted in demonstrable evolutionary changes as predicted (Kruuk *et al.*, 2000; 2002). These studies have repeatedly demonstrated a lack of correlation between predicted breeding values and fitness for otherwise apparently selected traits. Recently, this application of breeding values as predicted from mixed linear models has been criticized on statistical grounds. This is because, despite the terms ‘best’ and ‘unbiased’ used to describe them, (i) predicted breeding values have smaller variance than true breeding values, limiting comparison of selection differentials assessed at phenotypic and genetic levels, (ii) predicted breeding values are inevitably predicted with some error, and this error represents environmental effects on phenotype (Postma, 2006) and (iii) the very relatedness that makes the estimation of quantitative genetic parameters possible results in nonindependence of predicted breeding values and can make the main methods applied to date for statistical hypothesis testing anti-conservative (Hadfield *et al.*, 2010). Thus, although predicted breeding values are more closely correlated with true breeding values than are phenotypic trait measurements, they nonetheless remain confounded with environmental differences among individuals. The statistical problems that have become apparent mean that this use of predicted breeding values is no longer considered an appropriate way to estimate selection on the genotype (we advocate an alternative below). Nevertheless, on balance, the inherent problems with the methods probably render it conservative with respect to detecting a difference in selection at the levels of genotype and phenotype. Therefore, it should be safe to conclude that a range of existing empirical studies have at least supported the strong possibility that environmentally induced covariation between traits and fitness often drives patterns of apparent natural selection on the phenotype.

### Causation

- Top of page
- Abstract
- Two simple equations to predict adaptive phenotypic microevolution
- Decomposition of selection
- Causation
- Expectations for genetic covariances with fitness
- Testing for genetic consequences of natural selection
- Distinguishing between studying selection and predicting evolution
- Summary
- Acknowledgments
- References

No model can be expected to provide accurate predictions if its assumptions are seriously violated. The simple approaches we discuss here for predicting evolutionary change actually make many simplifying assumptions (e.g. constant population demography, discrete generations, constant environmental conditions; Merilä*et al.*, 2001a), and we are certainly not the first to highlight the potential for predictions to fail when when these models are applied to natural systems (Hadfield, 2008; Price *et al.*, 1988; van Tienderen & de Jong, 1994) or to call for caution in this regard. However, we would argue that the most fundamental assumption of the breeder's equation – that of causation (Hadfield, 2008; Kruuk *et al.*, 2003; Pigliucci, 2006) – is very often both poorly recognised and seriously violated. In its univariate form, the assumption required to generate valid predictions is one of a causal relationship between trait and fitness. In other words, whereas we may estimate natural selection as the covariance between fitness and a trait of interest, our prediction depends on the trait differences being the cause of the fitness variation.

We will see that causation with respect to the breeder's equation needs to be considered much more closely if we are to be able to make and interpret predictions of evolutionary change in natural populations. Relationships between traits and fitness at the genetic level, as in the first and third scenarios of Fig. 1, could result from causation. Alternatively, either could arise as a result of selection acting directly on a genetically correlated trait. Note that whereas the pattern depicted by even the first scenario is consistent with the trait being the cause of fitness variation, it is also consistent with the existence of noncausal genetic and environmental factors that happen to be similarly affecting the trait and fitness. Similarly, the trait in the third scenario could have a causal negative effect on fitness, but may be phenotypically correlated with an unmeasured trait that has a positive effect on fitness. The critical feature of these scenarios is that the environmental contributions to the phenotype–fitness covariance, whatever they may be, are stronger and have overwhelmed the genetic agents. Thus, the critical assumption of the breeder's equation is in fact not only causation but is *sole* causation.

A useful alternative, or rather a reformulation, to the condition in eqn 3 is to state that the breeder's equation and the Robertson–Price identity are the same when

- (6)

i.e. beyond the effect of the phenotype *P* on fitness, the correlation of breeding value *G* and fitness *W* is zero (Queller, 1992; van Tienderen & de Jong, 1994; see also appendix 1 to Hadfield, 2008 for complementary formulations in terms of the conditions under which the breeder's equation is quantitatively predictive in the presence of unmeasured variables). In other words, breeder's equation-based predictions of evolutionary change in a heritable trait with a causative effect on fitness can be wrong if other unidentified sources of covariance between the trait and fitness exist. This is particularly problematic in natural systems where unmeasured variables commonly generate the majority of variation (Moller & Jennions, 2002; Peek *et al.*, 2003). Failure to explain more than a small proportion of the variance is not a problem in and of itself. However, if most of the variance in a trait (or in fitness) is caused by unknown factors, then it would seem prudent to recognise that these unknown factors may also be contributing to the observed trait–fitness covariance.

One approach to improving the predictions of the breeder's equation is to extend it to multiple traits with potential causal links to fitness. The multivariate breeder's equation is

- (7)

where **R** is a vector of responses to selection, **S** is a vector of selection differentials and **G** and **P** are the genetic and phenotypic variance–covariance matrices of the phenotypic traits. Descriptions and graphical representations of how the multivariate breeder's equation works are provided elsewhere (Futuyma, 1998; Walsh & Lynch, 2011) but for current purposes, a simple two-trait example is useful to illustrate how its application can alleviate the problem of causation. Consider two traits, only one of which (trait 1) has a causative effect on fitness. If these two traits are phenotypically correlated, both will be correlated with fitness and thus the selection differential associated with trait 2 will be nonzero. However, if the traits do not covary genetically, then we do not expect phenotypic evolution of trait 2 because it has no causal effect on fitness. Assuming trait 2 is heritable, then the univariate breeder's equation would erroneously predict an evolutionary response, but this failure is corrected by application of the bivariate model. Alternatively, the multivariate breeder's equation can be formulated as

- (8)

the Lande equation (Lande, 1976, 1979), where *β* is a vector of selection gradients (Lande & Arnold, 1983). Selection gradients are classically obtained as partial regression coefficients of fitness on a trait, and as such account for selection on all other measured and phenotypically covarying traits. Thus, with the multivariate breeder's equation, we can relax the assumption that any one trait is the sole cause of its covariance with fitness. Conceptually, selection gradients are very important, because in principle they identify causal effects of traits on fitness. However, our assumption must now be that we have identified (and meaningfully measured and appropriately modelled) all the traits and environmental factors that cause traits of interest to covary with fitness, or in practical terms, all the important traits and factors. At the phenotypic level, selection gradients will correctly reflect causative effects of phenotypic variation of fitness if all sources of covariance are adequately measured and modelled. At the genetic level, the post-multiplication of **G** by *β* will correctly predict evolution if all genetically correlated traits have known selection gradients and genetic parameters. Let us call this new assumption ‘‘joint-sole’’ causation.

We will never achieve, nor likely approach, sole or joint-sole causation in descriptions of fitness covariation in studies of natural populations. Causative agents of fitness variation and covariation need not even be factors we would traditionally consider phenotypic traits. For example, in discussing this problem, Stinchcombe *et al.* (2002) used a hypothetical example of variation in the nitrogen content of soil. If soil nitrogen content independently has positive effects on fitness and the biochemical composition of plant leaves, then they will appear to be under selection. However, this nitrogen-induced component of the relationship between leaf composition and fitness will have no evolutionary consequences. In principle, inclusion of soil nitrogen measurements in a multiple regression analysis (Lande & Arnold, 1983) or path analysis (Scheiner *et al.*, 2002) could alleviate this problem. For analytical purposes, there is really no necessary distinction between an environmental variable and an additional phenotypic trait, provided that the environmental variable can be assigned to the individual (Wilson & Nussey, 2010). However, other factors of huge potential to jointly influence phenotypic traits and fitness include, but are by no means limited to, soil content of other nutrients, soil moisture content, light availability and disease. The possibility that all such factors can be identified, much less measured meaningfully (as their ecological relevance probably depends on variation at multiple spatial and temporal scales), is remote in studies conducted in natural or semi-natural settings (Kruuk *et al.*, 2003). In animals, the situation may generally be even more difficult. The environments experienced by more mobile organisms are particularly difficult to measure meaningfully. Robertson (1966) described this from the perspective of genetic improvement by example:

[S]uppose a cow eats a piece of wire. Then, assuming that wire-eating has no genetic basis, the milk yield will be reduced, though the cow may well be culled because of her general loss of condition rather than because of the milk yield itself. Here we have a positive environmental contribution to the [phenotype-fitness] covariance which will make the phenotypic regression larger than the genetic and the genetic gain will be less than that predicted [by the breeder's equation].

The wire-eating tendencies of cattle may seem trivial, but it is this triviality multiplied by the preponderance of such environmental influences on phenotype and fitness that makes the breeder's equation a poor predictor of genetic change. In fact, a myriad of such influences must act and interact to create the phenotypes and fitness of most organisms, especially outside of the environmental controls of agriculture and breeding programs. Most importantly, though, Robertson (1966) provided the solution: the expected response to selection is the genetic covariance between phenotype and relative fitness. George Price discovered this relation independently (Price, 1970) and presented the equation in a more general form (see Gardner, 2008 for an introduction to the Price equation).

### Expectations for genetic covariances with fitness

- Top of page
- Abstract
- Two simple equations to predict adaptive phenotypic microevolution
- Decomposition of selection
- Causation
- Expectations for genetic covariances with fitness
- Testing for genetic consequences of natural selection
- Distinguishing between studying selection and predicting evolution
- Summary
- Acknowledgments
- References

Little theory exists to inform our expectations of the genetic covariances between traits and fitness or to suggest whether or not genetic and phenotypic covariances should differ systematically. However, by adapting existing theory on trade-offs among traits, we can make some predictions about the relationships between traits and fitness. van Noordwijk & de Jong (1986) provided a general model for visualizing covariances between two traits that are involved in a life history trade-off. This model is easily adapted to the quantitative genetic decomposition of trait values, in a similar way to that which proved useful in Fig. 1.

Both scenarios in Fig. 2 show hypothetical decompositions of (co)variance for a trade-off between two traits. In both scenarios presented, the two traits are under positive selection (i.e. fitness is increased by higher values of both trait 1 and trait 2). Additionally, traits 1 and 2 covary positively with each other (column 1 of Figure 2). Note that for energetically costly traits, such as most ecologically–important characters, it is reasonable to assume that an individual allocating more to one trait will have less to allocate to others, but this does not necessarily lead to a clear expectation that the traits should show negative covariance at the population level. In fact, positive phenotypic correlations are common between (positively selected) life-history traits (Kruuk *et al.*, 2008) and are expected if individuals vary in the amount of resource they can acquire (de Jong & van Noordwijk, 1992; van Noordwijk & de Jong, 1986).

In scenario 1, the genetic and environmental influences on variation in both allocation and acquisition of resources are similar, and so the relationships between the trait and fitness at the phenotypic and genetic levels are similar (i.e. both traits show positive genetic covariance with fitness, and with each other). However, such a pattern is unlikely to be maintained for very long in any population, as selection should erode the genetic variation in fitness. Such a post-selection pattern is depicted in scenario 2. Here, the positive covariance between the traits arises from environmental influences (i.e. there is environmnetally induced variance in resource acquisition). However, the genetic variance along the axis of fitness that we depicted in scenario 1 has been largely eroded. As a consequence, there is little genetic variance in the direction of phenotypic change favoured by selection, and in fact, there is negative genetic covariance between the two traits. When we plot the traits against fitness, it becomes apparent that there is no genetic covariance between traits and fitness in the second scenario. We should note here, too, that the graphical model in Fig. 2, scenario 2 describes a situation where the genetic variance in fitness is low, and so the maximum potential for genetic covariances of the traits with fitness (if any existed at all) is necessarily small.

The extent to which the kind of thinking behind Fig. 2 is applicable to expectations for genetic covariances of traits with fitness depends on (i) whether traits that we suspect are under directional selection and are also involved in trade-offs and (ii) whether the genetic and environmental sources of covariance between traits involved in trade-offs vary (in magnitude and sign). Many traits that interest us are likely to be resource-limited, whether the resource in question is energy, time, space or substrates, and thus the potential for widespread trade-offs exists. Similarly, genetic architectures in nature are likely to have been affected by past selection. Thus, consequent to the past fixation of any alleles that positively affect multiple selected traits, trade-offs are most likely to be manifested at the genetic level. In other words, past selection is expected to have generated genetic constraints that will determine the current and future evolutionary potential of traits. In fact, our conjecture that there will be a lack of genetic covariance between traits and fitness is a corollary of the general theory that adaptive phenotypic evolution is constrained by multivariate genetic correlations, as discussed by Walsh & Blows (2009).

### Testing for genetic consequences of natural selection

- Top of page
- Abstract
- Two simple equations to predict adaptive phenotypic microevolution
- Decomposition of selection
- Causation
- Expectations for genetic covariances with fitness
- Testing for genetic consequences of natural selection
- Distinguishing between studying selection and predicting evolution
- Summary
- Acknowledgments
- References

Selection is a phenotypic phenomenon, but it only has evolutionary implications when fitness differences among individuals relate directly to genetic differences. Clearly, we must investigate the genetic basis of variation in fitness and the covariation of fitness with traits to predict the course of adaptation. However, this is more easily said than done. Quantitative genetic parameters are generally difficult to measure, and genetic covariances (or correlations) have particularly large sampling errors. Genetic covariances are especially difficult to estimate precisely when one or more of the traits involved has low heritability, which is likely for fitness (Lynch & Walsh, 1998). Nonetheless, the core assumption of causality in the breeder's equation is likely to be seriously violated in studies of natural systems. Consequently, it would be highly desirable to better inform our expectations for evolutionary change by parameterising the Robertson–Price identity. We therefore propose the following framework as an extension of existing common practice for predicting the response of heritable traits to selection.

Given its elegant simplicity and the widespread empirical applicability of techniques used to estimate natural selection on phenoptype (Lande & Arnold, 1983), we would argue that the breeder's equation does represent a reasonable starting point for predicting trait evolution. However, all applications of the breeder's equation should be complemented by discussion of the ability of the study to infer causality. In a typical selection analysis involving no experimental manipulation of phenotype, this ability is, strictly speaking, nil. Sometimes, however, mechanistic arguments may be useful. For example, if the primary cause of variation in viability is known to be predation by a gape-limited predator, arguments that size or aspects of size are likely to have causal effects on fitness may be convincing. Obviously, if correlative analyses can be bolstered by *in situ* experimental manipulations, then more powerful insights into causal agents of selection are possible (e.g. Parachnowitsch & Caruso, 2008).

Following the application of the breeder's equation and providing that heritable variation for the trait(s) in question exists, the next step is to estimate the genetic parameters of fitness (co)variation. At first, it might seem odd to suggest proceeding based only on evidence for heritability of the trait in question. What if no selection on the phenotype has been detected? Whereas we have primarily focussed our discussion on the conditions that lead the breeder's equation to predict evolutionary change when none will happen, it is useful to recognise that the same reasoning can cause a prediction of no change to be erroneous. That is to say an absence of a phenotypic trait–fitness covariance (i.e. *S* = 0) does not have to mean that *σ*_{a}(*z*,*w*) = 0 (as depicted in Fig. 1, scenario 4).

Although imprecise, the genetic variance of fitness and the genetic covariance of traits with fitness should be estimated, reported and checked for qualitative agreement with the predictions of the breeder's equation. Note that these estimates are best made using quantitative genetic techniques to directly infer the genetic basis of the relationship between trait and fitness, i.e. estimation of the genetic covariance, rather than regression of fitness on predicted breeding values, because the latter will be substantially biased towards environmental patterns (Hadfield, 2008; 2010; Postma, 2006; and see above). In studies reporting multiple estimates of selection on multiple traits through multiple components of fitness, it may be useful to break this step of the framework into two component steps. If point estimates of genetic variance of fitness (or in practical terms of different proxies for fitness) are made first, estimation of genetic covariances of traits with relative fitness can be limited to those cases where some indication exists that the genetic variance in fitness may be nonzero. This two-step procedure reduces the number of imprecise parameter estimates that have to be generated and interpreted, while still allowing the application of analyses that are relatively assumption-free with respect to the prediction of evolutionary change.

Detailed discussion of analytical considerations for producing quantitative genetic estimates of fitness-related parameters is beyond the scope of this paper. In general, some manifestation of a mixed model (Bolker *et al.*, 2009; Kruuk, 2004) will be required. In the simplest but general cases, such models might take the form

- (9)

where **y** is a matrix of observations of fitness and trait values, **u** is a vector of means, **Z** is a design matrix relating individuals to **a**, a matrix additive genetic effects, and **e** is a matrix of environmental effects. In a bivariate analysis of a trait and fitness, the variance–covariane matrix associated with **a** will provide an estimate of the genetic covariance between the trait and the fitness. In this bivariate analytical approach, one cay bypass the use of predicted breeding values and obtain direct estimates of the components of the covariance between trait and fitness. Specically, one can directly test whether *σ*_{a}(*z*,*w*)≠0. The interested reader should also consult other resources for information on estimating variance components in natural populations, including (Bolker *et al.*, 2009; Garant & Kruuk, 2005; Kruuk, 2004; Lynch & Walsh, 1998; Wilson *et al.*, 2010).

Additionally, one can move beyond simply obtaining an estimate of the genetic covariance between the trait and relative fitness and can examine the genetically induced relationship relative to the environmentally induced relationship. We can test a corollary of Hadfield (2008; and eqn 3) and Queller (1992; and eqn 6) results using mixed model analysis. We can decompose the numerator and denominator of eqn 3 into genetic and environmental components using eqn 4, i.e. *σ*_{p}(*z*,*w*) = *σ*_{e}(*z*,*w*) and , to obtain

which with rearrangment yields

- (10)

Each of the terms in this formulation of the conditions for the breeder's equation to be predictive can be estimated directly from the solution of the mixed model descibed in eqn 9. Thus, in addition to allowing estimation of the Robertson–Price identity, eqn 10 provides a tool with which to test the null hypothesis that the assumptions of the breeder's equation hold. This can be accomplished by comparing constrained models in a likelihood ratio test framework. For example, the software asreml (Gilmour *et al.*, 2002) can implement a flexible range of constraits on covariance matrices. These analyses could also be conducted in a Bayesian framework, where, for example, one could fit the model specified by eqn 9 using the R package MCMCglmm (Hadfield *et al.*, 2010) and test equality of the genetic and residual regressions of fitness on traits by examination of the posterior distributions of the covariance matrices associated with **a** and **e**.

Within the context of the framework we propose here for providing more robust predictions of responses to selection, one technical point is worth discussing. Transformation of fitness to the relative scale is desirable at the phenotypic level because then the breeder's equation works quantitatively (assuming sole causation of course!). Analysis of relative fitness may not always be desirable in genetic analyses. For example, no known distribution exists for relative fitness, but some standard statistical distributions will be useful for approximating distributions of fitness or fitness components such as counts of offspring, i.e. the (overdispersed) Poisson distribution. Calculation of parameters on different scales (i.e. covariances with absolute rather than relative fitness) will make only qualitative comparison of trait–fitness relationships at the phenotypic and genetic levels possible.

The evolutionary parameters associated with the Robertson–Price identity, i.e. and *σ*_{a}(*z*,*w*), are difficult to estimate and will have substantial uncertainty associated with them. The components of the breeder's equation can be estimated with higher precision, and so seem statistically more desirable as a means to predict evolution (although it is notable that while standard errors are typically estimated and presented for *h*^{2} and *S*, the uncertainty around the prediction is more rarely quantified). However, it is necessary to consider both the uncertainty that can be quantified, for example by estimating standard errors, and the uncertainties that can only be qualified, which are broken assumptions. The difficulty of the very strong assumption of sole causation in the application of the breeder's equation in natural systems means that the total amount of uncertainty in breeder's equation-based estimates of responses to selection is very large. Thus, especially because both and *σ*_{a}(*z*,*w*) can be estimated in any study where estimates of selection and heritability come from the same data set, little additional effort will often be necessary to evaluate these fundamental evolutionary parameters.

We hope it is clear at this point that the Robertson–Price covariance should provide robust predictions of evolutionary change. However, in arguing these benefits, it must also be made clear that we should not expect this approach to predicting evolutionary trajectories to solve all of the potential problems that we can encounter. For example, the predictions of both the breeder's equation and the Robertson–Price covariance are for the change in mean breeding value over one generation. This does not equate to the change in mean phenotype if the environment changes (Dickerson, 1955; Merilä*et al.*, 2001a; Pemberton, 2010). Similarly, temporal and spatial variation in selection and the expression of genetic variation (Wilson *et al.*, 2006), nonlinear parent–offspring phenotypic relationships (Heywood, 2005), unmeasured changes in the age-, stage- or state-structure of populations (Ozgul *et al.*, 2009) and nonrandom immigration or emigration (Garant *et al.*, 2005) can all contribute to phenotypic change (or stasis). Thus, for predicting the evolutionary trajectories of natural populations, continued attention to the ecological theatre in which change is occurring will often be at least as important as estimating the quantitative genetic parameters (Pemberton, 2010).

### Distinguishing between studying selection and predicting evolution

- Top of page
- Abstract
- Two simple equations to predict adaptive phenotypic microevolution
- Decomposition of selection
- Causation
- Expectations for genetic covariances with fitness
- Testing for genetic consequences of natural selection
- Distinguishing between studying selection and predicting evolution
- Summary
- Acknowledgments
- References

The breeder's equation provides a useful framework for conceptualizing the process of adaptive evolution by natural selection: selection causes phenotypic changes in a population, and genetic variation transmits these changes to future generations. This is not wrong, but given the assumptions we have discussed, it may generally be very inappropriate to apply this framework as a predictive tool in nature. There is consequently a problem with the enthusiastic way in which evolutionary biologists, ourselves included, have transferred this model from being a tool to conceptualise trait evolution and analyse artificial selection experiments (a context in which the inherent assumptions can be more readily approximated) to a predictive model in the field. Estimates of selection based on selection gradients are undoubtedly a conceptual step forward, but they do not escape reliance on the assumption that all traits and environmental factors that jointly influence studied traits and fitness have been identified, meaningfully measured and adequately modelled. We simply should not expect that estimates of phenotypic selection differentials, multiplied by values of estimates of heritability or **G** (whether estimated *in situ*, extrapolated from other studies, or simply assumed), will yield generally accurate predictions of phenotypic change in natural populations. Multivariate extensions acknowledging the importance of phenotypic and genetic correlations among traits are entirely sufficient when we can assume that all relevant factors have been properly handled; however, this assumption is naïve in the analysis of most data from largely unmanipulated populations.

The two equations for predicting evolutionary change are surprisingly different, despite their common left-hand sides. The use of these two equations to help us conceptualise the process of microevolution should occur at different levels. When our interest is in natural selection and the ecological factors that cause individuals with some traits or sets of traits to vary in their fitness, the breeder's equation and associated methodologies for quantifying selection, especially in their multivariate forms, provides a coherent framework. With the breeder's equation, we can begin to piece together how finite sets of traits, potentially correlated with one another and potentially varying in the degrees to which they cause fitness variation, might evolve. Clearly, understanding what traits influence fitness and how their evolution affects other aspects of phenotype is complementary and very closely related to the business of predicting evolutionary change. However, the direct cause of microevolutionary change is a relationship between fitness and genetic variation, regardless of how that relationship arises. The most robust prediction of evolutionary change will therefore be obtained by applying theory and analysis, i.e. the Robertson–Price equation, that matches this causal level.