Bernt-Erik Sæther, Department of Biology, Centre for Conservation Biology, NTNU, NO-7491 Trondheim, Norway. Tel.: +47 9057 8544; fax: +47 7359 6100; e-mail: firstname.lastname@example.org
In age-structured populations, viability and fecundity selection of varying strength may occur in different age classes. On the basis of an original idea by Fisher of weighting individuals by their reproductive value, we show that the combined effect of selection on traits at different ages acts through the individual reproductive value defined as the stochastic contribution of an individual to the total reproductive value of the population the following year. The selection differential is a weighted sum of age-specific differentials that are the covariances between the phenotype and the age-specific relative fitness defined by the individual reproductive value. This enables estimation of weak selection on a multivariate quantitative character in populations with no density regulation by combinations of age-specific linear regressions of individual reproductive values on the traits. Demographic stochasticity produces random variation in fitness components in finite samples of individuals and affects the statistical inference of the temporal average directional selection as well as the magnitude of fluctuating selection. Uncertainties in parameter estimates and test power depend strongly on the demographic stochasticity. Large demographic variance results in large uncertainties in yearly estimates of selection that complicates detection of significant fluctuating selection. The method is illustrated by an analysis of age-specific selection in house sparrows on a fitness-related two-dimensional morphological trait, tarsus length and body mass of fledglings.
Although Lande & Arnold's (1983) method greatly improved our understanding of selection as a process, this approach is still based on several simplifying assumptions that may influence the interpretation of the results. One of these is that the effects of age-structure are ignored. In most vertebrate species, significant age-specific variation has been found in several fitness-related traits (Sæther, 1990; Forslund & Pärt, 1995; Gaillard et al., 2000). Thus, the response to selection will be a complex result of temporal variation in selection on fitness-related traits acting at different stages of the life history. Accordingly, several studies of different species have shown large age-specific differences in the pattern and strength of selection (McCleery et al., 2004; Charmantier et al., 2006a,b). This seriously complicates the interpretation of the evolutionary consequences of these selective processes because there is no single selection differential in age-structured populations.
Fisher (1930) preceded the derivation of his fundamental theorem of natural selection by a discussion of deterministic age-structured dynamics in continuous time, defining the Malthusian parameter r as the asymptotic growth rate on the log scale. He showed that the population will approach a stable age distribution and then grow asymptotically linear on the log scale with rate r. The value of r for a hypothetical population of identical individuals then serves as a measure of fitness for these individuals. To deal with populations that have not yet reached the stable age distribution, Fisher introduced the reproductive value. Each age has a reproductive value, and the population has a total reproductive value V that is the sum of the reproductive values of all individuals. Fisher then showed that V grows exactly exponential with rate r even if the population deviates from its stable age distribution. We have previously extended this approach to model selection acting on a single allele in an age-structured diploid population (Engen et al., 2009a) as well as to describe fluctuating stabilizing and directional age-specific selection on a single quantitative trait constant with age in a variable environment (Engen et al., 2011). In the latter case, assuming weak selection and fitness components with Gaussian shape, the response turned out to be a first-order autoregressive model with temporally correlated noise, characterized by simple weighted means of age-specific selection parameters defined separately for each vital rate.
Previously, analyses of evolutionary responses to selection in age-structured populations have been based on the net reproductive rate (Lande, 1982; Charlesworth, 1994) or the specific population growth rate at a single point of time (Coulson & Tuljapurkar, 2008) as a measure of fitness. The approach of Engen et al. (2011) based on reproductive value represents an important advance because it allows partitioning selection acting on a quantitative trait into age-specific components that can be estimated from a sample of individuals in the population. Another advantage of using the reproductive value can be illustrated by the effects of fluctuating environments on selection in natural populations with overlapping generations. For instance, a large number of studies have recently examined how fluctuations and trends in climate are likely to induce changes in the distribution of the phenotypes of fitness-related characters in natural populations (e.g. Gienapp et al., 2006; Ozgul et al., 2009, 2010). Following Engen et al. (2011), one effect of changes in the environment may be that the phenotype with the largest contribution to future generations may differ among years. As a consequence, the fitness contributions of two individuals with the same phenotype and life history, but born in different years will then differ. In addition, the strength of selection may also differ among years and age classes, which makes it difficult to compare contributions from individuals based on measurement of fitness components estimated at a single point of time (Wilson et al., 2006).
Another stochastic effect affecting the dynamics of populations is random differences among individuals in vital rates within a year, known as demographic stochasticity (Lande et al., 2003). Such random individual variation in fitness contributions has stronger effects on the dynamics of small populations (Lande et al., 2003). However, it will affect estimates of directional and fluctuating selection even in an infinite population because estimates of selection inevitably must be based on finite samples of individual survival and reproduction. Thus, estimates will be strongly affected by demographic stochasticity, which in turn influences estimates of fluctuating selection caused by variation in the environment. Such sampling variation caused by demographic stochasticity must also be accounted for when estimating uncertainties in the strength of directional selection.
Here, we will develop methods for estimating weak directional as well as fluctuating selection based on individual data on age, fecundity, survival and fitness-related quantitative characters fixed at birth. Our statistical approach applies Fisher's (1930) concept of the total reproductive value as well as the concept of individual reproductive value introduced by Engen et al. (2009b). Selection in an age-structured population then acts through individual reproductive values. This leads to the correct combination of all age-specific components of selection within and among years that determines the evolutionary response to selection (Engen et al., 2011). Having first estimated the mean projection matrix through time, individual reproductive values can be computed for all individuals of known age for which the survival and number of recruits produced are known. These estimates can then be used as dependent variables in age-specific regression models with measured phenotypes of the individuals as independent covariates, leading to estimates of directional and fluctuating selection.
Age-structured populations in a stochastic environment without density regulation can be described by stochastic projection matrices with expected elements that are independent of the present population size (Caswell, 2001; Engen et al., 2005). If the population vector a given year is n = (n1, n2, …, nc)′, where the superscript′ denotes matrix transposition, the expected population vector in the next year is ln, where l is a square matrix with non-negative elements describing transitions between stages. In an age-structured model, the nonzero elements are those in the first line representing mean fecundities of the different age classes 1, 2, …, c, defined as the mean number of offspring surviving to the next census, and the subdiagonal elements being survival probabilities. More generally, for stage-structured populations, the matrix may have other nonzero elements (Caswell, 2001).
Following Engen et al. (2011), we consider selection on a vector of phenotypes z = (z1, z2,…, zk) that determines the expected elements lij = lij(z), assuming that the phenotype z does not change with age. We assume weak selection, that is, variation in z among individuals only induces small variation in the elements lij(z) so that its dominant eigenvalue can be approximated by a linear function. If there is a temporal additive effect on the phenotype generated by fluctuations in the environment, we assume that this is the same for all individuals regardless the value of z. Although this term will affect the stochastic growth rate of the population, it will not influence either selection or genetic drift, so this temporal component is ignored in the following. For simplicity of notation, we assume that z is centred by subtraction of its mean value across years so that the population mean is the zero vector.
Let λ(0) be the real dominant eigenvalue of the mean matrix l(0) with right and left eigenvectors u and v defined by l(0)u = λ(0)u and vl(0) = λ(0)v. Provided that the eigenvectors are scaled so that ∑ui = 1 and ∑viui = 1, u is the stable age distribution and v the vector of reproductive values associated with the projection matrix l(0). The eigenvalue λ(0) represents the deterministic multiplicative growth rate of a pure population of individuals with z = 0. Assuming that z≠0 causes small changes in the expected elements lij(z), we may apply the first-order approximation to the growth rate of a pure population with phenotype z, giving
where the derivatives are evaluated at the population mean z = 0. Using the fact that ∂λ/∂lij = viuj (Charlesworth, 1994; Caswell, 2001), and by the definition of the eigenvectors ∑ijviujlij(0) = λ(0), we find to the first order of approximation that
The stochastic projection matrix operating a given year is composed by individual contributions to the population the next year (Lande et al., 2003). These contributions are dependent on survival of the individual itself as well as the production of offspring surviving to the next year. In the simple age-structured model, an individual of age j contributes with its number of offspring Bj to the first age class and adds one to age class j+1 if it survives. Engen et al. (2009b) defined the individual reproductive value as the contribution from the individual to the total reproductive value of the population the next year (Fisher, 1930), that is
Here vj+1 = vj and Ij = 1 if the individual survives and otherwise zero, and the z in Wj(z) indicates that its distribution depends on the phenotype. The individual reproductive value Wj(z) has expectation v1l1j(z) +vj+1lj+1,j(z) = ∑ivilij(z). The relation EWj(z) = ∑vilij(z) is easily seen to be valid for any stage-structured model. From eqn (1), it follows that, to the first order of approximation, the deterministic growth rate of a hypothetical pure population of individuals with phenotype z can be expressed by the expected individual reproductive values for the different age classes,
Now, because the Wj(z) are stochastic quantities that can be recorded when samples of individuals with known age, survival and reproduction are available (Engen et al., 2009b, 2010), eqn (3) is a fundamental equation for studying weak selection of the phenotype z in a stage-structured model using linear regression models with individual reproductive values Wj(z) as dependent variables and individual phenotype as covariates. The eigenvectors, u and v, must first be estimated by estimating the mean projection matrix from temporal mean values of observed vital rates. An advantage of this approach is that it is based on reproductive values Wj(z) from samples of individuals over a period of time and does not require observations of individuals throughout their whole life to record their lifetime reproductive rate. Another important advantage is that possible correlations between individual survival and reproduction are accounted for by introducing the single independent variable Wj(z). Such correlations may be positive due to large stochastic fluctuation in the environment, or negative due to a trade-off in resource allocation between survival and reproduction (Engen et al., 2011). These correlations will confound analyses based on separate use of individual fecundity and survival as measure of fitness (Wilson & Nussey, 2010).
To develop methods for estimation and testing as well as allowing correct interpretation of variances and uncertainties, it is necessary to include the stochastic properties of the individual reproductive values, as introduced by Engen et al. (2009b). Writing ɛt for the vector of environmental variables at time t affecting the vital rates, the age-specific demographic variance components for a constant z are defined as , where the conditional variance is the variance among individuals within a year, and the expectation is the temporal expectation representing the mean value of through time. Similarly, the environmental covariance components are defined as τeij =cov[E(Wi | ɛt),E(Wj | ɛt)]. The total demographic and environmental variance for the population is then and . These variances may in general depend weakly on the phenotype, but under our assumption of weak selection, they can be approximated by their values evaluated at z = 0. The demographic and environmental variance defines the between year variance in the total reproductive value V of the population by
Furthermore, the process V will have approximately white noise (Engen et al., 2007a). The total population size N will fluctuate around its total reproductive value V with a return time at the order of a few generations. Hence, N will show transient fluctuation, whereas V serves as a filter removing these fluctuations. Furthermore, it is V that contains the information about future population sizes (Fisher, 1930), and thus, the process V rather than N should be used for predictions.
Fitness and selection differentials
Let z be some component of the phenotype vector z. In populations with no age-structure, the selection differential is given by the covariance between phenotype and individual relative fitness (Lande, 1982; Coulson & Tuljapurkar, 2008; Morrissey et al., 2010). In a constant environment, this also holds for the present model with weak selection, giving the selection differential . Here is the mean fitness in the population, that is, the growth rate defined by the mean projection matrix. Because we measure growth and fitness using reproductive value weighting, all age classes have the same absolute fitness in this model because the total reproductive value of any subpopulation always grows exactly exponentially with the same growth rate as the whole population (Fisher, 1930), which is the dominant eigenvalue. From eqn (3), it now follows that . In a fluctuating environment, the expected individual reproductive values and the growth rate may be time dependent, giving the selection differential at time t on the form
Here the subscript j in covj is added to emphasize that this is the covariance for individuals of age j, whereas subscript t indicates that mean survivals and fecundities may fluctuate through time.
To express results in terms of age-specific fitnesses and selection differentials, we consider the subpopulation of individuals of age j at time t. Weighted by their reproductive value, individuals in this age class with phenotype z have multiplicative growth rate λjt(z) = [pjt(z)vj+1 + fjt(z)v1]/vj = EWjt(z)/vj and relative age-specific fitness . Here pjt(z) is the probability of survival lj+1,j(z) at time t, whereas fjt(z) is the mean fecundity l1j(z). From this, the selection differential produced by age class j is the covariance between phenotype and relative fitness, that is,
and it follows that
Hence, the total selection differential is the weighted mean of the age-specific differentials with weight equal to the Fisherian stable age distribution ujvj, as defined by Engen et al. (2011).
Although this expression can be used to estimate selection differentials, the statistical inference is complicated by the presence of demographic stochasticity in the observed individual reproductive values Wj(z) combined with small temporal fluctuations in EWjt(z) and weak relationship between fitness and phenotypes.
If the relative fitness of individuals could be observed, the covariances could be estimated by random sampling. Notice then that estimation of age-specific covariances only would require random sampling of individuals within age classes, whereas estimation of the overall unconditional covariance must be based on random samples from the whole population, which is usually quite difficult to achieve in age-structured populations. This emphasizes the importance of the above decomposition of the overall covariance into age-specific components.
Response to selection in the linear model
Below we present the statistical analysis for linear models with fluctuating selection given by
The mean fitness of all individuals with phenotype z (including all age classes) at time t is then the expectation of the individual reproductive values
where . From this, we see that the vector of selection differentials is
where Pt is the phenotypic (k × k) covariance matrix at time t and βt is the vector with components defined for m = 1, 2, …, k. According to standard theory of evolution of quantitative characters (Lande, 1979, 1982), the response vector is then
where Gt is the additive genetic covariance matrix.
We consider individual reproductive values Wjt(z) of individuals of age j with vital rates observed at time t at environmental conditions ɛt, which are independently identically distributed given time and environments. We assume that the expected individual reproductive values are linear functions of the phenotypes z1, z2, …, zk leading to eqn (4) with p = k + 1 unknown regression coefficients that in general depend on the environment ɛt. The expectation refers to demographic stochastic variation among individuals in survival and reproduction at time t and is conditioned on ɛt (Engen et al., 1998). Write njt for the number of observations of individual reproductive value and phenotype for age j at time t. By fixing age and time, the model then becomes a standard linear regression EW = zα, where W is the vector of observed individual reproductive values and z is the n × p matrix with the individual phenotype vectors z = (z0, z1, z2, …, zk) as rows, where z has now been redefined by including the component z0 which is one by definition. Under weak selection, the expectations EW(z) change little with z, and therefore, small changes in the variance are a reasonable assumption and therefore can be approximated by a constant. The temporal distribution of this conditional variance generated by temporal fluctuations in the environment ɛt then has a mean which is the age-specific components of the demographic variance , and the total demographic variance (Engen et al., 2009b). The least squares estimate of the regression coefficients in this model are , whereas the variance estimate is the residual sum of square divided by n − p. The covariance matrix for with (kl)-elements is estimated by . Performing this estimation for age class j at times t = 1, 2, …, τ, the age-specific demographic variance is finally estimated by the relevant weighted mean over years as , where Nj = ∑tnjt is the total number of observations of individuals of age j. Although there may be temporal fluctuations in the , these are likely to be small compared with the standard deviations of their sampling distributions. Hence, assuming that the variances are the same each year, we obtain improved estimators for the yearly covariance matrices for given above as , where z is the matrix of independent variables at time t.
The parameters determining the response to selection at time t are accordingly the weighted means αmt estimated as divided by . The sampling variance of is only demographic because the αjmt are defined conditional on the environment. Hence, for two different age classes i≠j, and have independent sampling distributions and the (lm)-element of the autocorrelation matrix At for is accordingly
where (lm) denotes the (lm)-element of the matrices.
Under fluctuating selection, we assume that the vectors αt fluctuate among years with temporal covariance matrix M and no temporal autocorrelation. Including this temporal variation, the covariance matrix for the yearly estimates are At + M. In Appendix A, we show how to estimate M and the temporal mean coefficients α = Eαt, assuming initially that the yearly estimates αt are multinormally distributed. However, this approximation is not crucial because it is only used to construct the estimators and the properties of all estimates are finally checked by resampling methods.
On the other hand, to find yearly estimates of αt corrected for sampling errors, we will have to use the normal approximation. The estimator can then be based on the conditional mean , known as the best linear predictor, which takes the form
Finally, an estimator for αt is obtained replacing α by the estimate in this expression.
Using the general expression for the environmental variance and the expression for λt(z), we find that the environmental variance for a hypothetic population with phenotype z is
If the phenotypes are centred to fluctuate around zero with moderate variances, the major contribution to the environmental variance comes from the intercepts (the term proportional to ), and the total environmental variance in the populations is therefore approximately var(α0t).
Uncertainties and testing
To find approximations for uncertainties or for testing hypotheses resampling is required because the sampling distributions of the estimators are non-normal due to the fact that the dependent variables in the regressions (individual reproductive values) have discrete distributions very different from normal. First, we need to explore the uncertainties introduced by the demographic variances used to define the elements of the covariance matrices Ajt which in turn determine the uncertainty in At. The estimates of the demographic variance are all sum of squares of residuals in the regression divided by njt − p so that is a sum of njt squared residuals. Consequently, is a sum of Nj squared residuals. Although the residuals are not independent due to the linear relations defining the estimated coefficients, the squared residuals are very weakly correlated and thus can be considered as approximately independent. Consequently, writing for the total sum of squared residuals, the variance of this sum can be estimated by
where the bar defines mean values. This expression divided by (Nj − pτ)2 provides an estimate of the variance of . Accordingly, in each resampling of estimates, we may include the uncertainty in the estimation of At by choosing bootstrap replicates of the defining this matrix as independent normal variables with means and the above variance. A complete bootstrap replication including the temporal variation in the regression coefficients is then obtained by parametric bootstrapping of the αt using the overall estimated mean and covariance matrices At + M for the yearly estimates. Standard bootstrapping by resampling among the αt with replacement can alternatively be performed provided that τ is large enough to avoid bias. From these simulations, we can compute confidence intervals and sampling variance and covariances for any of the parameters we estimate.
Statistical inference under the assumption of no fluctuating selection
It follows from the derivation in Appendix A that the estimate of α under the assumption of no fluctuating selection (M = 0) is
with covariance matrix . The (co)variances will usually be rather accurate because they are based on a large number of regressions. Accordongly, testing the hypothesis of no selection under the assumption of no fluctuating selection can be performed by simply using the normal approximation for considering the covariance matrix as known equal to the estimated one.
An interesting null hypothesis is that there is directional but no fluctuating selection, that is M = 0. We then first perform estimation of α by eqn (5). Under the null hypothesis, the covariance matrix for is At which we have estimated as . Hence, we can simulate replicates of for t = 1, 2, …, τ using the overall estimated mean and assuming that the vectors of estimated regression coefficients are multinormally distributed. From each resampling of τ regression vectors, we then estimate M obtaining a multivariate bootstrap distribution of the temporal covariance matrix for the regression vector under the null hypothesis which can be compared to the estimates found from the real data.
An example: selection on the morphology of house sparrows
As a methodological example, we analyse selection on two fitness-related morphological traits of house sparrows Passer domesticus living at the small island of Aldra off the coast of northern Norway (66○24’N, 13○05’E). This population, located within a larger metapopulation (Jensen et al., 2007; Pärn et al., 2009, 2012), was founded by four individuals (one female and three males) in 1998. Afterwards the population increased rapidly to reach a maximum breeding population size of 26 pairs in 2005 (Billing et al., 2012). During the period 1998–2008, all juvenile and adult individuals on the island have been banded with a numbered aluminium ring and three coloured plastic rings for individual identification and measured for morphological traits. The birds live in close association with human settlements and during the breeding season, nests were localized and visited repeatedly until hatching. Number of eggs and fledglings were recorded for each nest. Hatching date was determined either directly or based on a subjective estimate of nestling age at the first visit after hatching. Several morphological traits of fledgelings were measured and standardized to a 10-day-old measure by regression techniques (see Ringsby et al., 1998), including tarsus length to the nearest 0.1 mm by a sliding caliper and body mass to the nearest 0.1 g by a Pesola spring balance (see Ringsby et al., 1998 and Jensen et al., 2008 for further details). A fledgling was considered to have recruited to the breeding population if it was recorded during the breeding season the following year. House sparrows in this area reach reproductive age the year after hatching. The number of female recruits produced was determined by genetic parenthood analyses as described in Billing et al. (2012). Emigrants to surrounding islands are rare (Tufto et al., 2005) and were considered as dead individuals.
Previous studies have shown that both the body mass and tarsus length at fledging are related to the probability of first-year survival of house sparrows in this study area (Ringsby et al., 1998, 2002) and therefore represent two quantitative characters fixed at an early stage of life which are related to individual differences in fitness. To illustrate our approach, we here analyse how differences in these two morphological traits affect variation among 65 female fledglings from the cohorts 1999–2008 in their contribution to the total reproductive value of the population.
We use two age classes, birds in their first year of life and birds older than 1 year. Surviving individuals in age class 2 remain in this age class. Morphological measurements of each bird were standardized as deviations from the overall mean across years. Over the whole study period, there was no significant directional selection on either body mass (Fig. 1a, , two-tailed: P = 0.36, n = 104) or tarsus length (Fig. 1b, , two-tailed: P = 0.28, n = 104) of fledgling house sparrows based on resampling under the null hypothesis of no fluctuating selection. Although there was large annual variation in the (Fig. 1a, b), there was still no significant fluctuating selection (P > 0.3). As expected from the large demographic variance in this population () as well as in other house sparrow populations in this area (Engen et al., 2007b), the uncertainty in the estimates of the directional selection αj (Figs 1c–f and 2a,b) and the fluctuation selection σj (Fig. 2c,d) are large. This is illustrated by the large reduction in the selection coefficients after accounting for the uncertainties in the estimates of directional selection (Fig. 1c,d) Accordingly, several of the realizations obtained by parametric bootstrapping of the model show no temporal variation in the αj corresponding to (Fig. 1e,f). The power of detecting significant selection coefficients was strongly influenced by the demographic variance (Fig. 3).
In the simple case of purely directional selection and characters not varying with age, we here provide methods using the concept of individual reproductive value for estimation and testing fluctuating and directional selection on multiple quantitative characters in age-structured populations. Components of selection are estimated by simple regression models for each age class within years. These are combined using results from the theoretical analysis by Engen et al. (2011) to provide estimates of how selection in all age classes jointly within a year affects the total selection on the trait, which in turn determine the evolutionary response to selection. Our analyses are based on the concept of individual reproductive value, that is the contribution of an individual to the total reproductive value of the population the next year, which varies in a stochastic way among individuals within as well as among years, thus determining the demographic and environmental variance of the population (Engen et al., 2009b). In agreement with Fisher (1930), we show that selection acts through this quantity rather than the unweighted contribution to the next generation measured in number of individuals as generally used in classical theory.
The present method provides an extension of the approach by Lande & Arnold (1983) in four important ways. First, we can estimate age-specific components of selection resulting from selection on viability or fecundity, or both (Fig. 1). Lifetime production of offspring has generally been considered an appropriate measure of fitness in age-structured populations, treating the population as one with new discrete generations at time steps T (Lande, 1982; Charlesworth, 1994). Several studies of vertebrate age-structured populations have used this measure of fitness (e.g. Gustafsson, 1986; Merilä & Sheldon, 2000; Brommer et al., 2004; McCleery et al., 2004; Jensen et al., 2008). However, this approach makes it difficult to handle correctly the variability in survival and fecundity among age classes and deviations from a stable age distribution (Grafen, 1988). Here we use the result obtained by Fisher (1930), who showed that the total reproductive value always grows exactly exponentially with growth rate equal to the Malthusian parameter, regardless of the actual age distribution. Thus, the problem of age-structure in relation to Fihers's fundamental theorem of natural selection could be overcome simply by weighting individuals by their reproductive value rather than just counting them in calculations of allele frequencies (Engen et al., 2009a). In this way, all age classes could be treated jointly and selection considered at each time step as in the case of no age-structure. We extend this approach to estimate parameters describing selection from samples of individuals of different ages that can be included in stochastic models of evolutionary processes in age-structured populations (Engen et al., 2010, 2011).
Second, we estimate the temporal covariance matrix for the vectors of selection coefficients αt, which can be used for statistical inference on fluctuating selection based on bootstrap methods developed for this purpose. In contrast to Lande & Arnold (1983), selection episodes do not need to be independent.
Third, available evidence suggests that estimates of selection coefficients in natural populations often are uncertain (Morrissey & Hadfield, 2012). The uncertainty in the estimates of temporal variation in selection may be large (Fig. 2c,d), making it difficult to detect significant variation among years in selection (Fig. 1c,d). Thus, our approach provides estimates of uncertainties as well as bias corrections based on bootstrapping. Our analyses illustrate the importance of considering uncertainties when deriving conclusions from analyses of selection based on samples of individuals (Mitchell-Olds & Shaw, 1987). Our results indicate that large sample sizes in terms of number of individuals and long time series are required to obtain sufficient power in tests for directional and fluctuating selection. Accordingly, Morrissey & Hadfield (2012) argued that much of the evidence for fluctuating selection in natural populations (e.g. Siepielski et al., 2009) in fact could be explained by uncertainties in the estimates of the selection coefficients.
Fourth, our method takes into account demographic stochasticity which induces random variation in realized fitness components among individuals in a sample. Such individual differences in demography produce uncertainty that can erroneously be interpreted as directional and fluctuating selection. In particular, actual temporal fluctuations in the coefficients may become invisible due to the stochastic sampling noise in the estimates. The possibility of detecting statistically significant fluctuating selection is therefore small when demographic variance is large, unless extremely large data sets are available or temporal variation in selection is large. Figure 3 illustrates how the power of tests for selection, under the assumption of no fluctuating selection, strongly depends on the demographic variance.
We have proposed using resampling to find uncertainties in estimates, confidence limits and p-values in statistical tests. This implies resampling from different empirical distributions of individual reproductive values defined by eqn (2). The demographic noise in these quantities generated by correlated noise in survival and reproduction is an essential component of the stochasticity leading to uncertainties in yearly estimates as well as estimates of parameters describing fluctuating selection. Alternatively, the statistical analysis of the model can be carried out using MCMC methods, but this may be rather difficult to implement because the distribution of individual reproductive values rarely follows any well-known class of distributions that can be parameterized. One possibility may be that all probabilities describing these distributions are considered as unknown parameters with some parameterized temporal fluctuations and that relevant prior distributions are defined for all these parameters.
Because our approach is based on standard linear regressions, it can also be used to study models where the effect of phenotypes is nonlinear, such as for example second degree polynomials with a maximum, representing stabilizing selection (Mitchell-Olds & Shaw, 1987). However, when the function is linear in the phenotypes, as in our example, plasticity (Lande, 2009) will not have any effect on the estimated selection coefficients, whereas for a second degree polynomial, plasticity will affect the coefficients (Engen et al., 2011). Thus, if plasticity occurs, a more complex approach including more parameters is necessary, which will further increase the uncertainty in the parameter estimates.
The present theory is based on the important simplification that the characters are constant through life although fitness may fluctuate through time. Fisher's (1930) weighting of individuals by their reproductive values ensures that the mean fitness does not change with age because the total reproductive value of any subpopulation has the same expected exponential growth as the whole population. The fitness of a given type z, however, will in general differ among ages. Hence, selection also differs at different ages. We have expressed this by defining age-specific selection differentials with temporal fluctuations, Sjt, as covariances between the phenotype and relative age-specific fitness (Engen et al., 2011), analogous to models with no age-structure (e.g. Lande, 1976, 1979). The selection differential for the total population is then the sum of these differentials weighted by the Fisherian stable age distribution, that is St = ∑ujvjSjt. This decomposition allows us to perform estimation for each age class separately based on age-specific vital rates and then to combine these estimates to provide a total selection differential determining the overall response to selection through time (Engen et al., 2011).
Several approaches have recently appeared estimating selection and evolution of quantitative traits in natural populations using modifications of Price's (1970, 1972) equation. Basically, this involves separating the total change in a character into two components (Gardner, 2008). One component is the change that can be ascribed to selection, described by covariance between individual phenotypic values and relative fitness. The remaining term describes to what extent offspring differ from their parents, either due to genetic causes or changes in the environment. This avoids the problem of unaccounted effects of selection on unmeasured traits (Morrissey et al., 2010) and allows analyses of selection on characters that change throughout the life of an individual taking the deterministic components of temporal phenotypic changes into account (Coulson & Tuljapurkar, 2008; Ellner et al., 2011). In practice, these components are estimated relying heavily on retrospective analyses on the covariance between variation in the character and relative fitness (Coulson & Tuljapurkar, 2008; Ozgul et al., 2009, 2010; Coulson et al., 2010; Ellner et al., 2011). Because many mechanisms affect the degree of parent–offspring similarity, prediction of future evolutionary changes may become difficult. In contrast, our approach is extended to age-structured populations within a similar general theoretical framework as previously developed for evolution of quantitative characters in unstructured populations (Lande, 1976, 1979, 1982).
We are grateful to financial support from the Research Council of Norway (FRIBIO), European Research Council (Advanced Grant) and the Norwegian University of Science and Technology. R. Lande provided valuable discussion. The R package lmf for the estimation procedures is described in Appendix C. The package can be obtained from T. Kvalnes (thomas.kval email@example.com) and is also available from CRAN (http://CRAN.R-project.org/).
Appendix A: Estimation of fluctuating selection
We now model fluctuating selection by assuming that the vectors αt at τ different times are identically independently distributed among years with mean α and covariance matrix M. The vectors are then also independent with the same mean α but covariance matrices At + M differing among years due to different number of individuals sampled. One will often have a large number of individual observations for each age class so that At may be considered as known equal to in the construction of an estimation method for (α, M). Finally, the properties of the estimation method derived by this assumption, including the effects of the sampling distributions of the At, can be investigated by stochastic simulations.
Because the regression coefficients are linear combinations of observations of the dependent variable with a large number of terms, rather efficient estimates are obtained by the maximum likelihood method based on the assumption that the αt have a multivariate normal distribution. Because the estimators are not exactly normal, this does not lead to the maximum likelihood estimators, but is still likely to give estimators with high precision relative to what is possible with demographic noise in survival and fecundity. Ignoring the trivial constant, the log likelihood multiplied by 2 based on the yearly estimates is then
where |At + M| is the determinant of At + M. Here we may reduce the number of dimensions in the numerical maximization of the likelihood by first substituting α by the vector α(M) maximizing log likelihood for a given value of M, or equivalently minimizing . The solution to this problem is
Inserting this for α in the expression for log likelihood gives an expression 2lnL(M) to be maximized numerically with respect to M.
It is preferable to write the symmetric covariance matrix using the Cholesky decomposition (Ripley, 1987) M = DD′, where D is a lower triangular matrix with positive diagonal elements. Then M is positive definite for any choice of elements of D, and there is an equivalence between D and M. For a given D the elements of M are given by M = DD′, whereas the elements of D for a given M can be computed recursively as shown in Appendix B.
Now, we use the p(p + 1)/2 elements in D as variables determining 2lnL, which for any values of dij corresponds to a positive definite matrix M. Hence, maximization can be carried out numerically by some procedure maximizing functions of a given number of variables with no constraints on the elements. If the maximization procedure chooses a negative diagonal element dii, we simply replace it by the corresponding positive number |dii|.
Using the same Cholesky decomposition writing At + M = CC′ where C is lower triangular, we also obtain a very simple expression for the log of the determinant occurring in the likelihood function,
where cii are the diagonal elements of C at time t.
Appendix B: The Cholesky decomposition
To find the elements of the lower triangular matrix D by the elements mij of the symmetric covariance matrix M, we first observe that , d21 = m21/d11 and . If p > 2 we go on recursively for i = 3, 4 …, p first computing di1 = mi1/d11 and then for j = 2, 3, …, i − 1,
Appendix C: A worked example with the R package lmf
In this appendix, we first go through the estimation procedures in the paper step-by-step, then we work through the methodological example with selection on the morphology of house sparrows and provide R codes using the R package lmf.
Here are the procedures, step-by-step, to estimate selection with the approach described in the paper:
(a) We begin by calculating the mean projection matrix (l(0)) and accompanying stable age distribution (u), reproductive values (v) and the deterministic multiplicative growth rate (λ).
l (0): The projection matrix with mean age-specific fecundities (fj) across years in the first row and mean age-specific survival probabilities (pj) across years on the subdiagonal. The survival probability for the final age class (pc) enters as the lcc element of the matrix.
u: The stable age distribution is calculated as the right eigenvector of l(0) scaled so that ∑ui = 1.
v: The reproductive values are calculated as the left eigenvector of l(0) scaled so that ∑viui = 1.
λ: The deterministic multiplicative growth rate is calculated as the dominant eigenvalue of l(0).
(b) The next step is to calculate the individual reproductive values (Wjt(z)) for each individual in our data set.
Wjt(z): The individual reproductive values are given by Wj(z) = v1Bj + vj+1Ij, that is the sum of the number of offspring contributed by an individual of age j to the first age class (Bj) weighted by the reproductive value of the first age class (v1) and the survival of the individual to the next reproductive event (Ij) weighted by the reproductive value for the following age class (vj+1).
(c) Then we are in position to estimate the yearly age-specific selection components (αjt), covariance matrices (Ajt), demographic variances () and individual residual values (ei) by standard least square regression of Wj(z) (fitness) on the individual pheno-types (z).
αjt: The yearly age-specific selection components are the parameters from the linear regressions.
A jt: The yearly age-specific covariance matrices contain the variance for each estimated selection component on the diagonal and the covariance between the selection components on the off-diagonal elements.
: The yearly age-specific demographic variances are estimated as the residual standard errors from the linear regressions.
(d) We now have what we need to calculate the yearly selection components (αt). They are obtained by the sum of αjt within years weighted by u, the stable age distribution.
(e) Furthermore, the age-specific demographic variances () can be calculated as the mean within each age class weighted by the degrees of freedom for each linear regression.
(f) With and αt at hand, the total demographic variance () can be calculated as the sum of weighted by u, and the environmental variance () can be estimated as the variance of α0t, the first element of the αt for all years (This corresponds to the intercepts of the yearly linear regressions).
(g) To account for uncertainty in the estimation of which affect our estimate of Ajt, we assume that the variances are the same each year and improve the estimated Ajt by scaling with (and not which give the standard covariance matrix for any regression).
(h) The variances of , which are needed when resampling estimates of selection, can be estimated using the residuals (ei) from the least square regressions and the sample size for each age class (Nj) when applying the equation given under Bootstrapping in the Uncertainties and testing section.
(i) At this point, we can obtain the yearly covariance matrices (At). These are calculated as the sum of Ajt within years weighted by u, the stable age distribution.
(j) Finally, we are in position to estimate the temporal selection components. Under fluctuating selection, the temporal covariance matrix (M) and the temporal mean selection components (α) given αt can be estimated through a numerical maximization of twice the log likelihood function (2ln L(α, M)). Using the analytical solution for α given M, the log likelihood function can be maximized with respect to M, after replacing M by the lower triangular matrix of its Cholesky decomposition. Thus, assuring that the solution for M remains positive definite.
M : The temporal covariance matrix provides the temporal variance (σj2) across all years for each estimated selection component on the diagonal and the temporal covariance across all years between the selection components on the off-diagonal elements.
α: The estimated mean selection components across all years.
(k) If we assume that there is no fluctuating selection M = 0 the mean selection components α(M) can be estimated by inserting M = 0 into the analytical solution for α (see eqn 5), and the corresponding covariance matrix is found by .
(l) Confidence intervals and statistical inference on the estimates of selection can be performed by parametric bootstrapping accounting for demographic variance as described in the Uncertainties and testing section of the main text.
We have made all the procedures above available through the R package lmf. Now we will use this package and provide the R codes to work through the methodological example in the main text to show how the procedures above are implemented in the statistical software R. The data set (sparrowdata) is available with the distribution of the R package.
After loading the data set into R, we first fit the desired model to estimate selection acting on the fledgling mass and tarsus length of house sparrows and view the output from the model (outputs are not printed in the appendix). All the steps from (a) through (k) are performed as we fit the model.
> model <- lmf(formula = cbind(recruits, survival) ∼ weight + tars, age = age, year = year, data = sparrowdata)
Next we look at the summary of the fitted model to see the estimated projection matrix, variance components, temporal mean selection components and temporal covariance matrix.
To extract the yearly or the yearly and age-specific estimates, we can specify an additional argument to summary() as shown below.
> summary(model, what.level = ‘year’)
> summary(model, what.level = ‘age’)
Now, as mentioned in step (l), confidence intervals for the estimated parameters can be estimated through parametric bootstrapping using the function boot.lmf(). Using this function, we specify the number of bootstraps (nboot), whether we want to include uncertainty in the parameters due to demographic variance (sig.dj), what parameters to bootstrap (what) and whether we want to perform a parametric or ordinary bootstrap (asim).
Additional insight into the distribution of the bootstrapped parameters can be gained by density plots that are available using the function plot(). The optional what argument can be used to plot density plots for subsets of the parameters.
> plot(mod.boot, what = ‘all’)
> plot(mod.boot, what = ‘projection’)
> plot(mod.boot, what = ‘alpha’)
Now the confidence intervals can be generated using the function ci.boot.lmf() as shown below.
The final step that remains is testing of hypotheses. The p-values provided in the summary of the model are only to be considered as suggestive, and tests of hypotheses should be performed by bootstrapping. The boot.lmf() function has additional arguments to this end. By specifying the expected parameter values under the null hypothesis (H0exp) and the conditions, we want to test hypotheses under (H0con) the bootstrap function resamples parameter estimates under the null hypothesis and compares with the estimates from the data. Again, results of the bootstrap are available through the summary() function.