Erik Postma, Netherlands Institute of Ecology (NIOO-KNAW), Centre for Terrestrial Ecology, P.O. Box 40, 6666 ZG, Heteren, The Netherlands Tel.: +31 26 479 1252; fax: +31 26 472 3227; e-mail: email@example.com
The ability to predict individual breeding values in natural populations with known pedigrees has provided a powerful tool to separate phenotypic values into their genetic and environmental components in a nonexperimental setting. This has allowed sophisticated analyses of selection, as well as powerful tests of evolutionary change and differentiation. To date, there has, however, been no evaluation of the reliability or potential limitations of the approach. In this article, I address these gaps. In particular, I emphasize the differences between true and predicted breeding values (PBVs), which as yet have largely been ignored. These differences do, however, have important implications for the interpretation of, firstly, the relationship between PBVs and fitness, and secondly, patterns in PBVs over time. I subsequently present guidelines I believe to be essential in the formulation of the questions addressed in studies using PBVs, and I discuss possibilities for future research.
Evolutionary biology tries to understand how variation amongst individuals, amongst groups of individuals, and across time has arisen, and how it is maintained (Endler, 1977; Roff, 1997; Schluter, 2000). This requires the separation of phenotypic variation into its underlying environmental and genetic components. Recently, the application of mixed model methodology in combination with a so-called ‘animal model’ has been used to estimate quantitative genetic parameters in natural populations with known pedigrees (see Kruuk, 2004 for an extensive review). In contrast to the more traditional quantitative genetics methods (see Falconer & Mackay, 1996; Lynch & Walsh, 1998), these methods use all the information that is available in a pedigree, and do not require a specific pedigree structure. Furthermore, they are able to accommodate selection and inbreeding, which are common phenomena in the great majority of natural populations (Kingsolver et al., 2001; Keller & Waller, 2002). Finally, it is possible to include additional environmental effects in the model, which are estimated simultaneously. For these reasons, the animal model is in theory highly suitable for the estimation of quantitative genetic parameters in natural populations (Kruuk, 2004).
The animal model may help to answer two types of questions, both of which are central to the study of evolution. First, on a population level it provides a method to separate the phenotypic variance we observe in the field into the underlying genetic and environmental variance components, and thus for estimating heritabilities (Knott et al., 1995; Lynch & Walsh, 1998; Kruuk, 2004). Secondly, and more excitingly, it makes it possible to separate environmental and genetic effects on both an individual and a population level. More specifically, it allows for the quantification of the sum of the additive effects of an individual's genes for a given trait, and thus the expected effect of the genes that it passes on to its offspring, also referred to as its ‘breeding value’ (Lynch & Walsh, 1998; Kruuk, 2004).
The ability to predict breeding values for free-living animals can be considered as one of the more important recent advances in the field of evolutionary genetics of natural populations. However, whereas it is nowadays the most widely accepted method for the genetic evaluation of livestock (Mrode, 1996), its use in studies of natural populations is still in its infancy. Although a huge amount of literature on the prediction of breeding values exists, this is mainly part of the animal breeding literature, and is often of a rather technical nature (see e.g. Mrode, 1996; Cameron, 1997). Although Lynch & Walsh (1998) and Kruuk (2004) do discuss the theory specific to the animal model, they assume that the basics of the prediction of breeding values are known by the reader. A basic understanding of the underlying theory is, however, essential for a correct application of the methods and interpretation of the results.
The subject of this article is the prediction of breeding values in natural populations using animal model methodology, and especially their interpretation. In the first section of the article, I will provide a brief theoretical introduction into the rationale behind the prediction of breeding values. I will then address the accuracy of predicted breeding values (PBVs), and show how an individual's PBV differs from its true breeding value. I will show that, as an animal model also uses the observations on the individual for whom we want to predict the breeding value, PBVs will partly reflect the environmental component of an individual's phenotype. As a consequence, patterns in PBVs may, in the absence of sufficient pedigree information, simply reflect phenotypic patterns and provide little additional information. Furthermore, the variance of PBVs is lower than that of the true breeding values, which has major implications for the quantification of selection pressures acting upon them, and especially for their comparison with phenotypic measures of selection. In the second part, I then take a more applied approach. I will identify the main questions that PBVs have been used for, and critically evaluate their suitability for answering these questions. I will, however, not only point at the limitations of PBVs, but also present solutions and make suggestions for future research.
Predicting breeding values
Every phenotypic observation on an individual (yi) can be written as the sum of the population mean (or the mean for a specific subset of the population) (μ), its breeding value (ai), and one or more environmental effects (ei). So in its simplest form,
We cannot directly measure an individual's breeding value for a given trait (here referred to as its true breeding value, or a). There are, however, two main sources of information available to obtain a prediction of a (here referred to as ), namely phenotypic observations on the individual itself and on its relatives. The prediction of breeding values from any of these groups of information is based on a similar principle and requires the formulation of the relationship between the observation we want to use and the focal individual's true breeding value.
The slope of a regression of breeding values on individual phenotypes is equal to the heritability (Falconer & Mackay, 1996). In the situation where all we have is a single observation on individual i (referred to as yi) and no information on its relatives, our best prediction of its breeding value is therefore
where h2 is the heritability of the trait (Mrode, 1996). In the absence of any additional information, the best prediction of an individual's breeding value is thus the phenotypic deviation from the mean, multiplied by the heritability of the trait. Although for some traits we may be able to obtain a more informative and biologically more interesting prediction by using additional observations on an individual made at different moments in time, observations on relatives provide better and more informative predictors of an individual's breeding value (Falconer & Mackay, 1996; Lynch & Walsh, 1998). To predict the breeding value of individual i (ai) from the phenotype of relative j (yj), we formulate the slope of the regression of ai on yj, and use this in eqn 2 instead of the heritability. This slope will, in addition to the heritability, depend on the proportion of genes shared between i and j, and thus their relatedness, where the slope will decrease with decreasing relatedness (Mrode, 1996).
Often, we have different sources of information, both on the individual for which we want to predict a breeding value and on a range of its relatives, and both on the trait of interest and on genetically correlated traits. After the appropriate weighing, these sources of information can be combined into one. This combined prediction is referred to as a selection index in the animal breeding literature. The procedure by which the factors weighing the individual sources of information are obtained is similar to that employed in obtaining the individual regression coefficients in multiple linear regression and is based on the phenotypic and additive genetic variances and covariances amongst observations (Falconer & Mackay, 1996; Mrode, 1996).
Using selection index methodology, we can address the question how the different sources of information contribute to an individual's PBV, or in other words, what are the relative sizes of the partial regression coefficients? In Fig. 1a, it is illustrated that the contribution of relatives to an individual's PBV declines rapidly with decreasing relatedness. Although the pedigree in Fig. 1a goes back many generations, the average relatedness is low. As a consequence, the PBV of the focal individual is based mainly on its own phenotype. In Fig. 1b on the other hand, the pedigree contains the same number of individuals but spans only two generations, resulting in a higher average relatedness and in a PBV for the focal individual that is based mainly on the observations on its offspring. Also note that the contribution of observations on relatives to the prediction of an individual's breeding value declines with increasing heritability. Consequently, although every individual has two parents and the number of relatives does thus double with each generation we go back in time, the increase in the number of relatives cannot compensate fully for the decrease in the contribution of an observation to an individual's PBV.
Where on the continuum between the two extremes depicted in Fig. 1, the pedigree of a population will be located will depend on the type of trait under investigation, the life-history of the species and the characteristics of the population. For example, the pedigree for a sex-limited trait that can only be measured in adults (e.g. clutch size) in a mainland population of a passerine bird will be more similar to (a), whereas a pedigree for a juvenile trait like birth weight in an island population of red deer will be more similar to (b). On the whole however, Fig. 1 emphasizes that even if observations on a range of relatives are available, the observations on the focal individual itself make a substantial contribution to the prediction of its breeding value, and that especially for traits with a relatively high heritability the remaining information will come mainly from close relatives.
Selection index methodology as employed above assumes that all individuals come from a population with the same mean, and that this mean (and thus the environment) remains constant across both time and space (Falconer & Mackay, 1996; Mrode, 1996). A way around this assumption was provided for by the development of mixed model methodology, and more specifically a method called Best Linear Unbiased Prediction, or BLUP (Henderson, 1949, 1950). BLUP allows for the prediction of random additive genetic effects (the breeding values) as well as any additional random environmental effects, while simultaneously estimating several fixed effects (different population means). Note, however, that in the absence of additional fixed and random effects a BLUP is equivalent to a selection index (Mrode, 1996).
To obtain (best linear unbiased) PBVs, we rewrite and generalize eqn 1 using mixed model and matrix notation into
where y is a vector containing all phenotypic observations, β is a vector of fixed effects, and a is a vector containing the breeding values. X and Z are incidence matrices that link phenotypic observations to fixed and random effects, respectively. Furthermore, the BLUP of the vector of breeding values a is
Note that if we only had a single record on one individual, and no information on relatives, this reduces to eqn 2. Using instead of μ allows for the incorporation of multiple fixed effects, using GV−1 instead of h2 enables us to use information on all related individuals and account for inbreeding, and using Z allows for repeated or missing records on individuals (Mrode, 1996; Lynch & Walsh, 1998; Kruuk, 2004).
In summary, the combination of an animal model in combination with mixed model methodology like best linear unbiased prediction makes it possible to predict an individual's breeding value from a range of sources of information, and simultaneously takes into account potentially confounding environmental effects. However, because of the way in which the different sources of information are weighed, breeding values are predicted for a large part from observations on the individual itself and on close relatives.
The accuracy of PBVs
Above we have seen how we can use different sources of information to obtain a prediction for an individual's breeding value. The quality of this prediction will depend on the amount and the type of information used. Such a quality measure is provided by the correlation between the true and the PBVs (Falconer & Mackay, 1996; Mrode, 1996; Cameron, 1997). This correlation is referred to as the accuracy r of the PBVs. As the covariance between true breeding values and PBVs is equal to the variance in PBVs [so , the accuracy is given by
where the variance in true breeding values (σ2(a)) is equal to the additive genetic variance. Usually, the accuracy of PBVs is, however, expressed in terms of the reliability, which is the squared correlation between true and PBVs (r2). The reliability is equal to the proportion of the additive genetic variance (or the variance in true breeding values) that is accounted for by the PBVs. If we only have a single observation on an individual, and no observations on relatives, the reliability of its PBV is equal to the heritability, and the reliability increases as the number of observations and the average relatedness increases (Falconer & Mackay, 1996; Mrode, 1996). For example, the reliability of the PBV of individual one in Fig. 1a is only 0.29, given a heritability of 0.25. In contrast, the reliability of its PBV in Fig. 1b is 0.49, while if we reduce the number of offspring to four, the reliability goes down to 0.38. It is thus the difference between the heritability and the reliability that provides insight into how much additional information is provided by the PBVs, on top of the phenotypic values.
The proportion of the variance in true breeding values that is not accounted for by the PBVs is referred to as the prediction error variance (PEV), so
From Appendix 1, it follows that the PEV is also equal to the covariance between the PBV and the residual environmental effect (e), so
This relationship implies that an individual's PBV does not only depend on its true breeding value, but also at least partly reflects the environmental component of its phenotype.
As the covariance between true breeding values and phenotypes is equal to the covariance between PBVs and phenotypes [so ; see Appendix 1], the regression of PBVs on individual phenotypes is always equal to the heritability, irrespective of the reliability of the PBVs. The variance of the PBVs, however, goes down with decreasing reliability. As a consequence, the correlation between PBVs and phenotypes is stronger than the correlation between true breeding values and phenotypes.
In this section, I have shown that PBVs differ from true breeding values in two interrelated ways. Firstly, the variance in PBVs is lower than the variance in true breeding values (or the additive genetic variance). Secondly, PBVs are partly reflecting the environmental component of an individual's phenotype, and patterns in PBVs will thus always resemble the pattern in the phenotypes more than the underlying true breeding values. These two characteristics of PBVs have important implications for their interpretation when applied to evolutionary questions, which I address in the following section.
Application to natural populations and its complications
Animal models have been used to predict an individual's additive genetic or breeding value in a range of studies (Table 1, also see Kruuk, 2004). These studies can be split into two groups. Whereas the first focuses on variation at an individual level, generally in relation to fitness, the interest of the second group lies in variation at the population level and focuses on trends across time or differences amongst groups.
Table 1. Overview of studies that have used PBVs using an animal model in the study of natural selection and micro-evolution in wild populations.
Selection on PBV? Change in PBV over time? Difference in PBV between groups?
Forehead patch size, wing patch size
Selection on PBV? Change in PBV over time?
Wytham woods, United Kingdom
Nestling body weight
Selection on PBV? Change in PBV over time?
Wytham woods, United Kingdom
Nestling body weight
Change in PBV over time?
Vlieland, The Netherlands
Difference in PBV amongst groups?
Corsica/La Rouvière, France
Nestling tarsus length, nestling body weight
Selection on PBV? Change in PBV over time?
Kluane, Yukon, Canada
Change in PBV over time?
Rum, United Kingdom
Selection on PBV? Change in PBV over time?
Ram Mountain, Alberta, Canada
Horn length, body weight
Difference in PBV between groups? Selection on PBV? Change in PBV over time?
Variation at an individual level
Selection on breeding values
Natural selection acts directly upon phenotypic variation, and indirectly on the underlying genetic and environmental components of variation. However, only selection on the genetic component of the phenotype will result in an evolutionary response. The response to selection is thus a direct function of the proportion of the phenotypic variation that has a genetic basis, and thus of the heritability (Falconer & Mackay, 1996). This assumes, however, that genetic and environmental effects have an identical effect on fitness. In other words, it is your phenotype that determines your fitness, irrespective of how this phenotype came about. Price et al. (1988) pointed out that this does not necessarily have to be the case, and provided an example for breeding time in birds. They suggested that variation in both reproductive success and breeding time was related to nonheritable variation in nutritional state. This would then give rise to a relationship between breeding time and reproductive success, and thus to selection on the phenotypic level. However, because the relationship between reproductive success and breeding time is mediated solely by environmental effects, it would not result in evolutionary change. It thereby provides an explanation for the micro-evolutionary stasis of laying date and other traits in the face of selection on phenotypes in several populations (Meriläet al., 2001c and references therein).
Animal model methodology has made it possible to test this hypothesis in wild animal populations, because the ability to predict individual breeding values allows for the direct quantification of the strength of selection on the genetic component of an individual's phenotype, so the relationship between fitness and breeding value (Kruuk et al., 2003). We can then compare the strength of selection on the breeding values to the selection pressure acting upon the phenotypes. In other words, we can compare the relationship between fitness and breeding values to the relationship between fitness and phenotypic values. This comparison makes it then possible to test whether selection on the genetic component of an individual's phenotype is weaker than what we would expect on the basis of the heritability of the trait, and thus for the presence of an environmentally induced bias of phenotypic estimates of selection (Stinchcombe et al., 2002). Such a comparison requires the formulation of the ratio between selection pressures based on breeding values and phenotypic values in the absence of such a bias (c.f. Rausher, 1992; Stinchcombe et al., 2002).
A common measure of the strength of phenotypic selection is the selection differential SP, which is equivalent to the covariance between the trait and relative fitness (Price, 1970, 1972). The expected response to selection is given by the product of phenotypic selection differential and heritability (the breeders’ equation), and is equal to the selection differential based on the breeding values (SBV) (Falconer & Mackay, 1996). In the absence of an environmentally induced bias of the phenotypic selection differential, our null-hypothesis would thus be that the ratio of SBV to SP is equal to the heritability (Table 2), whereas in the case of an environmentally induced bias SBV will be smaller than h2SP. We may also want to quantify selection on several traits simultaneously and separate selection acting directly on the trait, and on traits that are correlated to the trait of interest. This can be done by means of a multiple regression of relative fitness on the trait values, in which case a selection gradient β acting upon the trait of interest is defined as the partial regression coefficient (Arnold & Wade, 1984). It can be shown that selection gradients based on phenotypic and true breeding values, and thus the slopes of the regressions of relative fitness (w) on phenotypic and true breeding values, are equal under the null-hypothesis of no bias (Rausher, 1992) (Fig. 2a,b, Table 2). Although selection gradients are often based on a multiple regression, they can just as well be obtained from a univariate analysis, but note that a univariate selection gradient is not the same as a selection differential (Table 2).
Table 2. Quantifying selection pressures. The relationship between relative fitness (w) on the one hand and phenotypes (y), breeding values (a) and PBVs () on the other, all in the absence of an environmentally induced bias. r and r2 are the accuracy and the reliability of the PBVs, respectively. μ is the population mean phenotype.
As the absolute magnitude of a selection differential or gradient depends on the unit of the trait of interest, they cannot be compared amongst traits or species. Such comparisons are, however, possible if the trait values are standardized by subtracting the mean and dividing by the standard deviation, giving standardized selection differentials i and gradients β′, respectively (Lande & Arnold, 1983; Falconer & Mackay, 1996; but see Hereford et al., 2004). However, Stinchcombe et al. (2002) pointed out that as the variance in breeding values is lower than in the phenotypic values standardizing both separately can lead to very different estimates of selection, also in the absence of a bias. Although they therefore argue for the use of unstandardized selection gradients, the difference between standardized selection gradients based on breeding values and phenotypic values can easily be removed by taking into account the heritability of the trait (Table 2) (Kruuk et al., 2003).
Several studies have calculated selection differentials or gradients based on either standardized or unstandardized phenotypic and breeding values. Whereas Kruuk et al. (2002b) concluded that there was no evidence for selection on the breeding values for antler size in red deer, the majority of studies addressing similar questions concluded that selection was indeed acting on the genetic component [e.g. nestling condition or tarsus length in collared flycatchers (Ficedula albicollis), Kruuk et al., 2001; Meriläet al., 2001a, b; nestling weight of great tits (Parus major), Garant et al., 2004a]. It should be noted however that these studies only tested whether selection pressures based on the breeding values were significantly different from zero, and as the power to detect a significant selection gradient based on breeding values (βBV) is lower than the power to detect a significant selection gradient on phenotypic values (βP), the former may also be nonsignificant in the absence of an environmental bias. As yet no study has compared estimates for breeding values and phenotypes against each other, as suggested by Rausher (1992) and Stinchcombe et al. (2002).
Selection on PBVs
So far we have assumed that we know an individual's true breeding value, and just as the studies above, we have ignored the difference between true and PBVs. Although this does not affect the significance of the selection differentials or gradients, the use of breeding values that are predicted from an animal model, and which are thus partly based on an individual's own phenotype, will have important implications for the relative size of the expected selection pressures based on phenotypes and PBVs.
In the previous section, I have pointed out that the variance of the PBVs is a function of the reliability. In the absence of an environmentally induced bias, the covariance between PBVs and fitness is, however, equal to the covariance between true breeding values and fitness, just as the covariance between PBV and phenotype is equal to the covariance between the true breeding value and phenotype (Appendix 1). As a consequence, the slope of the regression of relative fitness on PBVs (βPBV) will be a function of the reliability, and βPBV will in the absence of a bias be greater than βBV (Fig. 2c,d, Table 2). On the other hand, when considering standardized selection gradients, will also in the absence of an environmentally induced bias be smaller than . Only unstandardized selection differentials based on true and PBVs are identical, as SPBV is not a function of the reliability (Table 2).
In summary, showing that selection is acting predominantly upon the environmental component of a phenotype requires a direct comparison between the selection pressure acting upon phenotypes and PBVs. In this comparison, it is crucial to formulate the correct null-hypothesis, and thus the expected ratio between selection differentials or gradients based phenotypes and PBVs in the absence of an environmentally induced bias, provided in Table 2.
Selection on environmental deviations
As the phenotype of an individual (relative to the population mean) is the sum of its breeding value and a residual, largely environmental component (see eqn 1), Stinchcombe et al. (2002) suggested that the size of this random environmental effect, also referred to as the environmental deviation, can be obtained by subtracting an individual's PBV from its phenotypic value. This assumes, however, that the slope of the regression of phenotype on PBV is equal to one, which is not the case if an individual's own phenotype is used in the prediction. Kruuk et al. (2002b) and Garant et al. (2004a) account for this by instead using the residuals of a regression of PBVs on phenotypic values. However, irrespective of the way in which environmental deviations are obtained, quantifying the strength of selection acting upon them does not provide any information in addition to the strength of selection acting on phenotypic and PBVs. Furthermore, the formulation of the correct null-hypotheses is less straightforward than for the PBVs.
Independence of selection pressures on PBVs
As was illustrated in Fig. 1, an individual's breeding value is predicted using observations on relatives, but also on the individual itself. As a consequence, PBVs are more closely correlated to phenotypic values than are the true breeding values, and estimated selection pressures based on PBVs will partly reflect the selection pressures acting on the environmental component of the phenotype. In the extreme case of only one observation per individual and no observations on its relatives, so when r2 = h2, the relationship between PBVs and fitness will simply reflect the relationship between the phenotypic values and fitness. Although the reliability will generally be greater than the heritability, it will never be one, and thus selection estimated on PBVs will always reflect the selection pressures on the phenotypes more than the selection pressures acting on the true breeding values do.
The studies above asked the question whether phenotypic changes across time or phenotypic differences amongst groups are the result of environmental or genetic variation (Fig. 3a,b). By asking this question, these studies acknowledge that individuals may not be randomly distributed across environments. As BLUP methodology allows for the incorporation of additional environmental effects, we can account for such environmental structure, making it possible to obtain unbiased PBVs from an animal model analysis in the presence of nonrandom environmental variation. However, if environmental structure is not explicitly modelled, PBVs will partly reflect this environmental structure (see eqn 7). Although this bias may be small if many additional observations on close relatives are available, it should be kept in mind that these observations may not be randomized across environments either. A trend over time in PBVs may thus be biased towards the trend in the phenotypes if the trend is not explicitly modelled (Fig. 3c). As in their analyses they did not include year, the significant trend in PBVs in itself in Réale et al. (2003) and Coltman et al. (2003) may therefore overestimate the amount of genetic change.
From the above, it follows that if interest is in the question whether a phenotypic change over time reflects environmental or genetic change, year should be included in the animal model as an additional random or fixed effect to account for the possible presence of systematic environmental variation across years. However, how well we are able to separate year effects into environmental and genetic effects will depend on the accuracy of the PBVs, and how individual pedigrees are distributed across the different effect levels. This is also referred to as the connectedness of the pedigree (e.g. Kennedy & Trus, 1993). When the connectedness of the pedigree across the fixed effects is low, so when we have observations on an individual and its relatives in only a subset of years, the genetic component of the variation amongst years will be accounted for by the year effect instead of by the PBVs (Fig. 3d). The same will be true when the interest is in the difference between groups and there is little gene flow between these two groups. When groups are poorly connected, the phenotypic difference between groups will be mainly attributed to the environmental group effect, even if it in fact has a genetic basis.
Both Kruuk et al. (2001, 2002b) and Sheldon et al. (2003) tested for a trend in PBVs and included year as a variable in their animal model. However, the average lineage length of the collared flycatcher population on Gotland is for example 2.92 generations (Kruuk et al., 2002a), which is a fraction of the length of the study period (approximately 20 years). Connectedness is thus relatively low and, if present, at least part of the genetic trend will be accounted for by the year effect. Therefore, it is difficult to draw the conclusion that there was no genetic change from the absence of a significant trend in the PBVs in these studies. On the whole, we find that the studies in Table 1 found a significant trend in PBVs when there was also a phenotypic trend and year was not included in the model, and no trend when year was included. A few studies, however, did find an effect of year on the PBVs, even though year was included (Meriläet al., 2001a; Garant et al., 2004a, b, 2005). Interestingly, this trend was often in the opposite direction of the phenotypic trend. It is these studies that show opposite patterns with regard to phenotypic and genotypic values that demonstrate micro-evolution most convincingly.
This brings us, however, to a potential problem that arises if there is a relationship between the variable of interest that we have included in the animal model, and the number of observations on an individual and its relatives, which both are important correlates of fitness. In the case of fledgling condition on Gotland, selection differentials are increasing with time, which may result in a decreasing reliability of the PBVs. This is expected to downwardly bias the positive change in PBVs. A similar problem may arise at the beginning of a data set, where little information may be available on an individual's ancestors, or towards the end of the study period where we will have less information on an individual's descendants. This may be especially problematic when comparing different groups of individuals that differ systematically in the reliability of their PBVs, for example trophy-harvested and nontrophy-harvested bighorn (Coltman et al., 2003), or immigrant and resident great tits (Sheldon et al., 2003; Postma & Van Noordwijk, 2005). In these, cases including harvested or nonharvested, or immigrant or resident, in the animal model as fixed effect can lead to misleading results.
The role of the base population
Breeding values are expressed relative to the base population, which consists of those individuals with unknown parents, and their mean breeding value is equal to zero (Mrode, 1996). In an animal breeding setting, the base population typically consists of those individuals with which a selection line was started, and from that moment on the population can be considered to be more or less closed, and animals that come in later are assumed to come from the same population as those in the founding population. This is very different from many natural populations, where there is immigration each year of individuals that were born outside the study area. This group of individuals will by definition have unknown parents and may be many times larger than the number of individuals present at the beginning of the study population. Breeding values of locally born birds are thus predicted relative to the breeding value of a base population that mainly consists of immigrants. A change in mean PBV over time may therefore reflect a change in the genetic composition of the immigrants. A simultaneous shift in the breeding value of immigrants on the other hand may obscure a change in the mean PBV of locally born individuals.
Considerations for future research
Formulating the correct null-hypothesis
Earlier I pointed out the importance of formulating the correct ratio between selection differentials or gradients based on phenotypes and PBVs in the absence of an environmental bias. Also when interest is in trends in PBVs over time, the formulation of the correct null-hypothesis is crucial. The main concern expressed here is that if an individual's breeding value is predicted partly on the basis of its own phenotype, then its PBV will partly reflect the environmental component of its phenotype. As a consequence, PBVs are more closely correlated to phenotypic values than true breeding values are. If the effect of interest (e.g. year) is not included in the animal model, our null-hypothesis should therefore be that the PBVs show the same pattern as the phenotypic values, and thus that the phenotypic trend is indeed the result of genetic change (Fig. 3c). Consequently, we can only prove that the observed phenotypic pattern does not have a genetic basis, namely when a phenotypic pattern is not reflected in the PBVs. On the other hand, if the effect of interest was included in the model, the null-hypothesis should be that the PBVs reflect the phenotypes after correction for this effect, and thus that there is, for example, no genetic change over time (Fig. 3d). In this case, we will have to be very careful with drawing conclusions from the absence of a trend in mean PBVs as this can either be the result of the fact that the pedigree did not contain enough information to separate genetic and environmental year effects, or because really no genetic change took place.
Reporting the amount of information in the pedigree
The extent to which an individual's PBV provides an accurate and informative prediction of its true breeding value depends on the amount of information that is available on this individual and its relatives. Providing such information will aid in the interpretation of our own results and those of others. Although of little value in animal breeding, in an ecological context, the overall reliability can provide us with valuable information and can easily be obtained by dividing the variance in the PBVs by the variance in true breeding values. Future research is, however, required to obtain more insight into the pedigree width and depth required to detect an effect of a specific size.
Using accuracy of PBVs
Using eqn 5, it is relatively straightforward to obtain an overall estimate of the accuracy. Animal breeders, however, are mainly interested in the accuracy of individual PBVs. Although its calculation is computationally less straightforward, it is based on the same principle (Henderson, 1975; Meyer, 1989). As we can thus obtain a measure of the accuracy of each individual's PBV, it may be tempting to use this as a criterion in deciding which predictions to include in our analysis, and to include only those breeding values that were predicted with the greatest accuracy. Unfortunately, the accuracy of the prediction depends on the number of observations both on the individual itself, which is often related to its survival, and on its relatives, generally related to its fecundity. So, on the whole, we might expect a positive relationship between the accuracy with which an individual's breeding value is estimated and its fitness. Those individuals with a relatively accurate PBV will therefore not be a random subset of the population. Another and better approach to this problem would be to weigh individual PBVs by their accuracy, placing more weight on relatively accurate PBVs.
Excluding observations on focal individuals
Most of the problems with the interpretation of PBVs stem from the fact that they are partly based on the phenotype of the focal individual. However, BLUP methodology allows for the prediction of breeding values for individuals on which we do not have a phenotypic record, either because the individual has not been measured, because it does not express the trait, or simply because it was excluded from the data set. These breeding values will on average be based on fewer observations, but they will be less biased. If we are interested in the breeding value of only a few individuals, we may set their phenotypic record to missing. These individuals will, however, be the relatives of others, and by doing this we may loose substantial amounts of information. For sex limited traits, we can take a similar approach and use the PBVs for the sex that does not express the trait (Postma & Van Noordwijk, 2005).
Incorporating PBVs in an experimental context
Although in theory animal model methodology may allow for the separation of environmental effects in a nonexperimental setting, in practice the amount of information that is available may be a limiting factor. A potentially powerful approach would be to incorporate animal model methodology into an experimental framework. For example, we may use PBVs as an internal control and so remove part of the natural variation, and increase the likelihood of detecting a significant treatment effect. It would also be possible to group individuals on the basis of their PBV to create genetically different experimental groups without having to create different selection lines.
Another approach was taken by Postma & Van Noordwijk (2005). They separated the phenotypic difference in clutch size of great tits breeding in two different parts of the island of Vlieland into a genetic and an environmental component by comparing clutch sizes of females that bred in their area of birth to those that dispersed to the other area. This natural transplant experiment provided evidence for a substantial genetic component to the observed phenotypic difference, based on a significant effect of where a female was born on her clutch size. They subsequently performed a similar analysis using the PBVs of both males and females and found that whereas the environmental effect (the effect of the area of breeding) was greatly reduced, the effect of origin was of a similar size, strengthening their conclusion that there indeed was a genetic difference in clutch size between both parts of the island.
Animal model methodology has provided a very powerful tool that has opened up several previously unexplored questions. Its application to natural populations is, however, still in its infancy. Now it has been applied to a range of populations, traits and questions we can take the results from these pioneering studies and use them to identify directions for future research. We have to realize that animal models were originally developed for very different systems and questions, and animal model techniques as they are being applied in animal breeding should be adjusted to fit their application in evolutionary biology. Furthermore, more work is required to obtain a detailed quantification of the required amount of information in the pedigree in relation to the size of the effect of interest, taking into account the distribution of environmental effects across relatives. Although it may sound trivial, the statement that one has to know what one is doing is very much true for the prediction of breeding values using animal model methodology. However, when aware of the potential pitfalls, it provides a powerful tool with an exciting range of possibilities.
Loeske Kruuk provided invaluable help in all stages of the preparation of the manuscript. Earlier versions of the manuscript benefited also from comments by, and discussion with, Dave Coltman, Phillip Gienapp, Kate Lessells, Dan Nussey, Arie van Noordwijk and John Stinchcombe, as well as from the comments by two anonymous referees. Furthermore, I thank all participants of the first wild animal model bi-annual meeting for comments and discussion during and after the meeting. This work has been supported by ALW-NWO and a Marie-Curie fellowship.
Proof that the PEV is equal to the covariance between PBV () and the residual environmental effect (e), and thus that the covariance between phenotype (y) and true breeding value (a) is equal to the covariance between phenotype and PBV.
If we have one observation per individual (so Z = I), no fixed effects and μ = 0, then