In 1992, David Houle showed that measures of additive genetic variation standardized by the trait mean, CVA (the coefficient of additive genetic variation) and its square (IA), are suitable measures of evolvability. CVA has been used widely to compare patterns of genetic variation. However, the use of CVAs for comparative purposes relies critically on the correct calculation of this parameter. We reviewed a sample of quantitative genetic studies, focusing on sire models, and found that 45% of studies use incorrect methods for calculating CVA and that practices that render these coefficients meaningless are frequent. This may have important consequences for conclusions drawn from comparative studies. Our results are suggestive of a broader problem because miscalculation of the additive genetic variance from a sire model is prevalent among the studies sampled, implying that other important quantitative genetic parameters might also often be estimated incorrectly. We discuss the most prominent issues affecting the use of CVA and IA, including scale effects, data transformation, and the comparison of traits with different dimensions. Our aim is to increase awareness of the potential mistakes surrounding the calculation and use of evolvabilities, and to compile general guidelines for calculating, reporting, and interpreting these useful measures in future studies.
The ability of populations to respond to natural or sexual selection, termed “evolvability” in quantitative genetics (Houle 1992; Lynch and Walsh 1998; Hansen 2006; Sniegowski and Murphy 2006; Hansen and Houle 2008; Pigliucci 2008), is contingent on the level of additive genetic variation underlying trait expression. Consequently, a common practice in quantitative genetics studies is to derive standardized measures of evolvability that allow comparisons among traits and taxa. In a landmark paper, Houle (1992) proposed that a dimensionless statistic (see also Charlesworth 1987), termed the coefficient of additive genetic variation (CVA), is appropriate for such purposes. CVA is simply
that is, the square root of the additive genetic variance (VA) divided by the phenotypic mean of the trait (note that Houle (1992) expressed this quantity using a 100 multiplier). Unlike heritability, CVA is a measure of additive genetic variation that is standardized by the trait mean and therefore independent of other sources of variance. It is precisely these properties that make CVA suitable for comparative purposes.
Houle (1992) addressed a long-standing difficulty with the interpretation of patterns of genetic variation in fitness traits. Traits closely related to fitness, such as survival or fecundity, typically exhibit lower narrow-sense heritabilities (i.e., the ratio of additive genetic variance to total phenotypic variance) than traits under weak or stabilizing selection, such as morphological traits (Gustafsson 1986; Charlesworth 1987; Mousseau and Roff 1987; Roff and Mousseau 1987; Houle 1992; Falconer and Mackay 1996; Kruuk et al. 2000; Merila and Sheldon 2000). This pattern was traditionally interpreted as resulting from the depletion of genetic variation due to strong directional selection (Fisher 1930). By contrast, Houle (1992) showed that traits closely associated with fitness generally exhibit higher CVAs, and thus higher, not lower additive genetic variability than those under weaker selection. Houle's (1992) data supported the view that traits closely associated with fitness have higher levels of residual variation (e.g., nonadditive genetic, maternal, and environmental variation, including error variation), thereby explaining their low heritabilities (Barton and Turelli 1989; Price and Schluter 1991).
Since the publication of Houle's (1992) paper, there has been a proliferation of studies reporting either mean-scaled additive genetic variances (predominantly CVA but see below) or using these evolvability measures for comparisons, and this work has improved our understanding of the factors that contribute toward the maintenance of genetic variation. This work has led to the general consensus that traits tightly linked to fitness exhibit high levels of both genetic and residual variation (e.g., Merila and Sheldon 1999, 2000; Kruuk et al. 2000; McCleery et al. 2004; Coltman et al. 2005; Hansen et al. 2011). Furthermore, the realization that some traits harbor considerable additive genetic variance despite strong directional selection has fuelled the development of new theory, such as the genic-capture model used to address the lek paradox (Kotiaho et al. 2008), and more generally the maintenance of genetic variation in fitness traits (Rowe and Houle 1996; Tomkins et al. 2004).
Given the utility of CVA for comparative studies of genetic variation, it is crucial that primary studies employ correct and consistent methods to estimate this parameter (or suitable alternatives; see below and Discussion). If mistakes are frequent, incorrectly calculated CVAs are likely to have been reported in reviews or studies that compile or compare these values, thus potentially biasing and/or confounding the conclusions drawn from such studies. In an attempt to determine the extent to which mistakes in the calculation of CVA occur in the literature, and their potential consequences, we have reviewed recent quantitative genetic studies that have reported this statistic. We also review important issues in relation to the use and limitations of CVA.
Although CVA and IA are related, they are distinct quantities (Houle 1992), and a key advantage of IA is that its numerical value can be interpreted as the expected proportional change under a unit strength of selection (see Hansen et al. 2003; Hereford et al. 2004; Hansen et al. 2011). For this reason, Hansen et al. (2011) recommend the use of IA as a measure of evolvability. It is therefore likely that future research will shift focus from CVA to IA. Our review, however, focuses on CVA because until now this coefficient has been used predominantly to report and compare evolvabilities. Nevertheless, given the relationship between CVA and IA and the fact that both involve mean scaling, both measures suffer from similar limitations and are prone to similar calculation errors. Therefore, our results and guidelines can be extended to both measures of evolvability.
We used the Web of Science to identify all articles published between 2000 and 2010 (11 years) that cited Houle (1992). Specifically, we selected papers appearing in the top 40 journals ranked according to impact factors (Journal Citation Reports 2009) within each of the following four areas: Evolutionary Biology, Genetics and Heredity, Multidisciplinary Sciences, and Biology. These filters limited the results to 346 papers published in 29 journals. During the process, we noticed that a high number of papers citing Houle (1992) were published in the journal Genetica (n= 18 between 2000 and 2010); we therefore also included this journal in our Web of Science search. Including papers from this journal does not change results. Thus, the total number of papers covered in our initial screening was 364.
We classified studies by quantitative genetic design. We decided a priori to focus exclusively on studies employing nested full-sib half-sib designs (Lynch and Walsh 1998; termed half-sib designs after Roff 1997) because this is the single most common quantitative genetic design and thus provides a simple limit to the breadth of the literature review. Our study is therefore based on a sample of quantitative genetic studies, and assumes that this sampling yields a nonbiased picture of the use of CVA in general.
Of the 364 papers screened, 49 were empirical studies using nested full-sib half-sib designs and 38 of these reported coefficients of additive genetic variation. From these 38 papers, we recorded whether the study provided sufficient information to calculate CVA. This information is, as recommended by Houle (1992), the phenotypic mean of the trait , and either the sire variance component (Vsire) or the additive genetic variance, VA, which is four times Vsire for a full-sib half-sib design on a diploid organism (Becker 1984; Falconer and Mackay 1996; Roff 1997; Lynch and Walsh 1998). In the absence of these statistics, a study can report other parameters (which we refer to as “cryptic” information) that can be used to calculate CVAs:
1The output of the analysis of variance. In these models, the mean squares (MS) and degrees of freedom are informative, because Vsire and therefore VA can be calculated from such data, provided that the output refers to analyses using untransformed data (CVAs have little relevance if they are calculated using transformed data; see Discussion).
2CVAs can also be calculated when narrow sense heritability (h2) and the mean of the trait are provided together with a measure of dispersion such as the standard deviation (SD) or the variance. This is because, in general, VA can be calculated as , where VP is the total phenotypic variance. Caution is needed, however, because h2 estimates can be dependent on the structure of the quantitative genetic model used to infer them; the inclusion of fixed effects in the model reduces the phenotypic variance that is partitioned thereby increasing the heritability value (Wilson 2008).
3Likewise, if the standard error (SE) and the sample size for the phenotypic measurements are given, VP can be calculated and from here VA can be inferred from the formula above. In some cases, h2 might not exactly correspond to , for instance when dealing with threshold traits, and so caution needs to be taken when inferring CVAs from VA values obtained from VP and heritabilities.
4VA can be calculated if the coefficient of residual variation is provided
on the assumption that .
5Finally, if a study uses log-transformation of data, it is still possible to calculate CVA on the untransformed scale. This is because the additive genetic variance calculated on the log-transformed scale is an estimate of IA for the trait on the original scale as long as IA << 1 (Hansen et al. 2011). CVA (as in eq. 1) can be then calculated as the square root of IA.
When possible, we calculated CVA and contrasted this value with the reported CVA. If the study reported genetic parameters for several traits, we focused on the first trait reported in the paper, unless the calculation of CVA for this trait was not straightforward (e.g., if VA was zero, or if the trait had zero mean or had been otherwise transformed).
Of the 38 studies reporting CVA that we scrutinized, nearly half (44.7% or 17 studies) miscalculated CVA, and 36.8% of studies lacked information on either Vsire or VA. All results can be extracted from Table S1. Error rates were not higher in the studies for which “cryptic” information (see Methods) was used (43.75% of studies that provided the mean and Vsire or VA were incorrect; 44.75% of studies were wrong among those that failed to provide these parameters).
The number of studies miscalculating CVA is probably an underestimation for several reasons. First, for some studies categorized as calculating CVA correctly, we had to assume that the reported Vsire or VA was correct. Second, we avoided looking at traits for which it was obvious that residuals were used (see Discussion). Third, some mistakes were only obvious after reanalyzing the original data of those studies (we were able to do this in papers authored by us; see Table S1). Thus, it is possible that similar mistakes would be uncovered in other studies assumed to be correct, if reanalysis of the raw data in such studies was possible.
Our review reveals a general lack of consistency surrounding the calculation and reporting of evolvabilities, the most common being the incorrect use of Vsire rather than VA (which is four times Vsire for a full-sib half-sib design on a diploid organism) in the calculation of CVA (18.4% of studies reviewed reporting CVAs, or 41.2% of the studies with incorrect CVA), which underestimates the actual CVA by half (and will also result in a fourfold reduction in the heritability; Fig. 1). A further source of error revealed in 10.5% of the studies (23.5% of studies with incorrect CVA) was the use of the square root of the ratio of additive genetic variance to the trait mean, instead of the square root of the additive genetic variance. This mistake may lead to significant overestimation of CVA (see Fig. 1), sometimes by orders of magnitude. Problems with transformation of variables, which undermine the utility of CVA for comparative purposes, were detected in 13.2% of studies (29.4% of studies with incorrect CVA), but this is an underestimation of this problem. Undetermined errors were found in 13.2% of studies (29.4% of studies with incorrect CVA).
Finally, although we have here focused on CVAs, it is important to note that we encountered similar errors (e.g., using Vsire instead of VA) in the reporting of evolvabilities as IA.
THE PROBLEM AND ITS CONSEQUENCES
Comparing evolvabilities within or among populations or species requires standardized measures of additive genetic variation. CVA and IA, which standardize VA by the mean of the trait, are suitable measures of evolvability that do not suffer from the problems arising when using variance-standardized alternatives such as the heritability (Houle 1992; Hansen et al. 2011). Heritability is not a suitable measure of evolvability for comparative purposes because, among other reasons, it standardizes VA by VP, which not only contains the former but also comprises other components of variance that can correlate with VA (Hansen et al. 2011).
Coefficients of additive genetic variation have been reported, or interpreted, widely but our review highlights two areas in need of attention. The first surrounds the calculation of the parameter itself. We found that almost half of studies that reported CVAs had calculated these parameters incorrectly. The most common error involved taking the sire variance component Vsire as VA, thereby underestimating the actual CVA by half in a full-sib half-sib analysis of diploid organisms. We suspect that this mistake arises because of a misunderstanding of the VA notation in Houle (1992), or because of misunderstandings regarding quantitative genetic models. Thus, the high frequency of this mistake may reflect a more serious and broader problem affecting the accuracy of other quantitative genetic parameters (including h2, which depend on VA) in the literature. We are also aware of the existence of typographic errors in the formula for the calculation of coefficients of genetic variation in some papers (e.g., mutational coefficient of variation in Houle 1998), which may lead to further accumulation of errors. We reiterate that these coefficients should be calculated as in equation (1) where VA stands for additive genetic variation (but may be replaced by any other desired variance component; e.g., VR provides CVR).
The second issue surrounds the quality or quantity of information reported in papers, which in turn are needed to determine whether CVA and IA can be calculated, or assessed for accuracy (i.e., whether authors report phenotypic means, and additive genetic variance or phenotypic variances, along with clearly defined methods for how data were handled—e.g., transformations, scales, etc.). We found that a high proportion of studies that report CVAs did not provide sufficient information to assess whether these parameters were calculated accurately; approximately, 37% of the studies reviewed here did not provide VA or Vsire parameters.
What are the consequences of calculating evolvabilities incorrectly? The magnitude of the errors vary to the extent that the actual CVA value can be grossly over- or underestimated depending on the nature of the error (see Table S1 and Fig. 1). Our analysis suggests that nearly 50% of papers based on half-sib designs calculate CVAs incorrectly. Clearly, this is likely to severely compromise studies that use such estimates for comparative purposes.
MEANINGFUL MEASURES OF EVOLVABILITY: A DIMENSIONLESS MEASURE INFLUENCED BY DIMENSIONALITY, AND ISSUES OF SCALE AND TRANSFORMATION
The property making CVA and IA suitable measures of evolvability is a consequence of scaling to the mean. The influence of the mean on these two measures is obvious, but often not fully appreciated. Figure 2 illustrates the effects of mean scaling upon coefficients of variance with an example of two equal-variance different-skew distributions.
Houle (1992) pointed out the necessity to correct for scaling effects in comparative analysis of variability. Scaling effects deal with the relationship between means and variances, and although they may have biological relevance they can also be statistical artifacts. Thus, the interpretation surrounding mean-scaled additive genetic variances (CVA and IA) can be complicated unless the scaling effects are properly accounted for or eliminated. Where higher measurement errors are associated with small means (as one might expect), traits with smaller means will generally have comparatively higher coefficients of variance (or IA; Houle 1992 and references therein). The same problem of a negative relationship between means and variances applies, regardless of measurement errors, when analyzing meristic traits (Lande 1977; Houle 1992; Lynch and Walsh 1998 pp. 302–305).
Houle (1992) also emphasized Lande's (1977) point that the relationship between means and variances is influenced by the covariance between the different components of a unit, and the extent to which these parts or components combine multiplicatively (e.g., length, area, and volume), or additively, to make up the whole. A well-known consequence of these purely mathematical influences on CVs is that, for instance, CVs of body volume would be larger than those of linear body measures (e.g., length, width, and height) as long as these linear traits are positively correlated (Lande 1977). To compare CVAs among traits with different dimensionalities, some authors scale downward the CVAs of areas and volumes by a factor of 2 and 3, respectively, based on the assumption that the relative magnitude of the CVs of linear, area, and volume measurements approximates 1, 2, 3, respectively (see Lande 1977). However, in most cases dividing CVs by their dimensionalities is not an adequate correction (but see Houle 1992), as it would only apply to more or less perfectly geometrically proportioned objects (i.e., where the correlation between the linear measurements is close to 1), and even in such cases it is only an approximation when dealing with objects that have low coefficients of variation for the linear dimensions (<10%; Lande 1977). In most cases of biological variation, where variation in shape is likely, the ratio 1:2:3 for linear, area, and volume dimensions needs to be considered an upper limit for the comparison of CVs of these different dimensions (Lande 1977), and not a rule. Thus, correcting for the effects of dimensionality, beyond acknowledging that due to mathematical constraints some CVs are expected to be higher than others, is not straightforward (see Fig. 3 for an illustration of this point; and see also Milner et al. 2000).
These considerations raise interesting questions in regards to the causes of variation in evolvabilities. For example, if fitness depends strongly on body size, fitness can be expected to have higher evolvability than linear morphological traits (e.g., leg length) for several reasons. First, fitness would be affected by a higher number of genetic events, which would capture higher levels of genetic variance than less polygenic traits (Houle 1992; Houle et al. 1996; Rowe and Houle 1996). Second, fitness would be determined by a higher number of dimensions, and of relationships among traits. The extent to which the evolvability of fitness is affected by these factors would depend on the sign and strength of the relationships between the components that make up fitness (Lande 1977; Price and Schluter 1991; and see Kirkpatrick 2009), something that in most cases is unknown.
A crucial issue deserving attention relates to measurement scale and scale transformation. Only data on ratio and log-interval scales produce meaningful CVA and IA (Hansen et al. 2011; Houle et al. 2011). The key point here is to confirm that mean-standardized measures of additive genetic variation are based on data that have scales with meaningful zero values and that ratios of the data values are relevant. For example, length, mass, duration, or age measured in any unit of time meet these two requirements (a mass gain of 0 g means that there is no mass gain; a 20-cm long wing is twice as long as a 10-cm wing, etc.). Temperature in degrees Celsius or dates are counter examples: they do not have an absolute, nonarbitrary, zero value (a temperature of 0°C does not mean “no temperature”; Houle et al. 2011). Moreover, the ratio between their measurements has no meaning (5°C is not five times hotter than 1°C). Critically, transformations of variables that are in ratio or log-interval scales lead to different scales that no longer meet the criteria above (Hansen et al. 2011; Houle et al. 2011). Thus, CVA and IA have no meaning if they are calculated on transformed scales.
It is easy to see then that the problem of scale also applies to standardization of variables, to the use of scales yielding negative means, and to the use of residuals. CVA and IA cannot be calculated on variables with zero means (e.g., z-scores, principal components, and relative warps from geomorphic morphometric analyses) or residuals (see Kotiaho 1999; Birkhead et al. 2006 for discussion on residuals).
In summary, the utility of mean-scaled additive genetic variances as measures of evolvability in a broad range of situations is unquestionable. However, these parameters, as many others, are subject to errors of calculation, assumptions, and limitations, and all of these issues need to be accounted for when reporting and comparing evolvabilities. We encourage researchers to examine in detail the existing literature discussing the advantages, use, and limitations of evolvability measures, and in particular Houle's (1992), Lande's (1977), Roff's (1997), Lynch and Walsh's (1998), Hansen et al.'s (2011), and Houle et al.'s (2011) collective advice before undertaking a comparative study of evolvabilities.
While highlighting several common errors surrounding the calculation and use of CVAs, we also advocate some very simple guidelines for avoiding the miscalculation of evolvability estimates. The broad solution is that empiricists should strive for clarity and accuracy when reporting evolvabilities while supplementing these estimates with fundamental summary statistics on the raw scale, and by explicitly noting the scales of measurement. Our findings also highlight the utility of depositing data files in public archives, such as the Dryad data repository (http://datadryad.org/). As for researchers using CVA or IA for review studies, our results suggest that the probability that the evolvability value in any given paper is incorrect may be considerable, and so they underscore the need to confirm the accuracy and validity of these statistics before they are included in any formal analysis.
We advocate that researchers adopt the following practices when reporting quantitative genetic data:
1Consistency in the calculation of CVA, as in equation (1), and of IA as in equation (2). Note that using a 100 multiplier when reporting CVA is optional, although we recommend reporting CVA as in equation (1), as there is no justified reason to use the 100 multiplier. Moreover, it is easier to relate CVA without the 100 multiplier to IA, which in turn translates directly into the standardized Lande (1979) equation (Hansen et al. 2003; Hereford et al. 2004; Hansen et al. 2011). In any case, we emphasize that studies need to state clearly whether CVA is expressed with or without the 100 multiplier.
2CVA and IA must be calculated using the raw (untransformed) scale, and data need to be on ratio or log-interval scale (see above).
3Transparency in the reporting of methods, which should cover the inclusion of the formula used for the calculation of evolvabilities (important given some of the mistakes in the literature). Also critical, complete clarity on the use of transformations and clear descriptions of scale details and measurement units (including whether traits are measured in absolute units, proportions, or percentages), or otherwise, of any methods that may affect the estimation of variability.
4Reporting of all summary statistics necessary for the calculation of CVA and IA (Houle 1992). As a minimum, we suggest that the phenotypic mean and SD (or variance, or SE together with the sample size) is always provided (see also Wilson 2008) together with observational components of variation, causal components of additive genetic variance (VA) and residual variance (VR), narrow-sense heritability estimates (h2), coefficients of additive genetic, residual, and phenotypic variation (CVA, CVR, CVP) and IA.
5In addition, we recommend, where possible, reporting the SEs of CVA and IA. SEs of these derived statistics would allow researchers to carry out unbiased meta-analyses of data on evolvabilities. An approximation to the sampling variance of a CV is given in Appendix 1 of Lynch and Walsh (1998; pp. 819–821); the SE of CV is simply the square root of such (large-sample) sampling variance (Lynch and Walsh 1998; p. 812; note that the SE of an estimate, the square root of the error variance of such estimate, is different from the SE of a sample, the sample's SD divided by the square root of the sample size). Based on the same procedure used by Lynch and Walsh (1998) to approximate the sampling variance of CV (The Delta Method, consisting of a Taylor series expansion), a normal approximation to the SE of CVA (measured as in eq. 1), assuming multivariate normality of the sampling error of VA and the mean (μ) is
where denotes the SE of VA, denotes the SE of the mean, and r[VA, μ] is the error correlation of VA and μ.
If r[VA, μ] is lacking or not easily obtained, an approximation would be
Also, the sampling error of the mean will typically be modest. In fact, lacking , an approximation is
For IA, the equations equivalent to equations (4)–(6) are
The SE of VA, (the square root of the large sample variance of VA), can be obtained in most REML software for animal models. Also, Lynch and Walsh (1998) provide equations to calculate large sample variances of variance components for sib designs (see eq. 18.20b in page 561 and see also page 577). For nested full-sib half-sib mode, would be four times (twice in the case of full-sib models) the SE or the large sample variance of Vsire. The Delta Method assumes that the sampling errors of VA and μ are multivariate normal, and additionally, that errors of CVA and IA are normal.
Markov chain Monte Carlo (MCMC)-based Bayesian analysis of quantitative genetic breeding experiments will also allow the calculation of the SEs of evolvabilities. Bootstrapping and jackknifing can also be considered as alternatives for some breeding designs. All these methods lead to appropriate characterization of uncertainty of CVA and IA estimates, and they would be of particular value in cases where the sampling error of the additive genetic variance does not distribute normally or symmetrically.
Adopting the five measures above will enable researchers to have a complete understanding of the meaning and utility of the genetic variance estimates reported in individual studies. They will allow the independent calculation of evolvabilities measured as CVA or IA, and will enable performing unbiased meta-analyses of these data. We anticipate that the adoption of these practices will broaden the scope and value of future investigations on variability in evolvabilities.
Associate Editor: L. Kruuk
We are very grateful to D. Houle, M. Morrissey, and an anonymous reviewer for useful suggestions that greatly improved the final version of the manuscript. We are most grateful to M. Morrissey for pointing out the approaches relating to the calculation of sampling errors and for discussion on their applications. We thank the Australian Research Council for financial support to FG-G, LWS, JLT, and JPE, and to the Academy of Finland's Centre of Excellence in Evolutionary Research for support to JSK.