Introduction

Additive genetic variances and covariances play an important role in evolutionary change, but we are still surprisingly ignorant of how these genetic components themselves vary in natural populations. Whereas, we have abundant evidence that genotypic and phenotypic means vary in response to selection, we have little empirical information on the extent to which additive genetic variances and covariances are moulded by natural selection and/or genetic drift. Evolutionary change in quantitative traits can be described by the multivariate extension of the breeder's equation Δz̄=GP−1S where Δz̄ is the vector of mean responses, G is the matrix of additive genetic variances and covariances, P is the matrix of phenotypic variances and covariances and S is the vector of selection differentials (Lande, 1979). The matrix combination GP−1 can be viewed as the multivariate ‘equivalent‘ of h2, with G replacing the additive genetic variance in the numerator and P−1 replacing the phenotypic variance in the denominator. Note that although the nonadditive genetic variance components (dominance, epistasis) do not enter into the G matrix, they do impact on evolutionary change by their inclusion in the phenotypic variance.

Without further qualification, as stated, the multivariate evolutionary response equation assumes that the genetic and phenotypic matrices remain constant. This assumption has been justified by two further assumptions, namely that population size is sufficiently large that genetic drift can be ignored and that selection is sufficiently weak that variation eroded by selection is replaced by mutation (Lande, 1980). Both of these assumptions are highly controversial. Effective population sizes are frequently small enough that significant drift can be expected to occur, particularly over long periods of time (Lande, 1976; Lynch, 1990; Chapter 8 in Roff, 1997). Estimates of selection coefficients in wild populations show that the strength of selection varies widely from weak to very strong (Endler, 1986; Kingsolver et al, 2001). Thus, these data indicate that the assumption that the G or P matrix will be invariant cannot be justified on theoretical grounds alone but must be verified empirically (Turelli, 1988; Arnold, 1992).

A number of studies have estimated and compared the G matrices from different populations of the same organism. In the cases of Thamnophis ordinoides (Brodie, 1993) and Holcus lanatus (Shaw and Billington, 1991), Allonemobius fasciatus and A. socius (Roff and Mousseau, 1999; Roff et al, 1999) no differences were observed. Significant differences were found in populations of Gammarus minus (Fong, 1989; Jernigan et al, 1994; Roff, 2002), Thamnophis elegans (Arnold and Phillips, 1999) and Drosophila melanogaster (van ‘T Land et al, 1999). With one exception, in all of the preceding cases and those involving higher taxa the offspring generation and typically the parental generation were raised in the laboratory. Arnold and Phillips (1999) brought in gravid females from the wild and estimated genetic (co)variances using the regression of mean offspring value of dam value. In this case it might reasonably be argued that the estimated G matrix is equivalent to that obtained for a wild population.

So far as we are aware, the Arnold and Phillips analysis is the only study of G matrix variation in free-ranging populations of animals, although several studies have estimated genetic components of individual wild populations. Estimates of morphological heritabilities for traits in free-ranging individuals are very similar to and not significantly different from those obtained under laboratory conditions (lab=0.50, field=0.56, see Table 1 in Weigensberg and Roff, 1996). Thus, we might expect that laboratory estimates of G also to be similar to those estimated from natural populations. Nevertheless, it is still preferable to estimate the G matrix within the environment in which the organism has evolved (Orengo and Prevosti, 1999; Blanckenhorn, 2002). On the other hand, this approach can potentially confound G × E interactions with genetic variation per se. Thus, in principle, an analysis would include a common garden experiment in which the G matrices were estimated under the same conditions. Such an experiment is not possible with the organism studied herein and hence conclusions must be tempered by considerations of environment on genetic expression. In the present paper, we present an analysis of variation in the phenotypic, environmental, and genetic (P, E, and G) matrices of morphological traits in three wild populations of the barn swallow, Hirundo rustica.

Table 1 Results of the Flury hierarchical test on the P, E, and G matrices of three barn swallow populations
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Estimates of the matrix elements were made using offspring on female parent regression and the matrices then compared using two statistical approaches, the Flury approach (Phillips and Arnold, 1999) and the jackknife followed by MANOVA method (Roff, 2002). The first method examines the structure of the matrices (equality, proportionality, common principal components, etc), whereas the second tests for equality and correlations between the matrices and other variables such as sex, latitude of origin, etc.

While the G matrix is the focus of this paper, we present an analysis of the P matrix for two reasons: First, there is considerably less error in the estimation of the P matrix compared to the G matrix and hence the P matrix will be a much more sensitive indicator of variation, although it cannot itself tell us the source of the variation. Significant variation in the P matrix would imply variation in either or both the G and E matrices. The second reason for examining the P matrix is that, because the G matrix is part of the elements of the P matrix (P=G+E), it may be possible to use phenotypic (co)variances as surrogate measures of the genetic (co)variances (Cheverud, 1988; Roff, 1995, 1996; Koots and Gibson, 1996). Thus, we can ask two questions: ‘Are the two matrices highly correlated?’ and ‘Do we get the same statistical answer using the P matrix as the G matrix?’.

Materials and methods

Study sites

We studied barn swallows at Badajoz (38°50′ N, 659′ W), Southern Spain (since 1976), Milano (45°28′ N, 9°10′ E), Northern Italy (since 1993), and Kraghede (57°12′ N, 10°00′ E), Denmark (since 1970), as part of a long-term project. The study site at Badajoz consists of open farmland with pastures, cereals, and fruit plantations, and most barn swallows breed in barns. The study site at Milano consists of open farmland with pastures, cereals, and hedges. Barn swallows are mainly restricted to dairy farms, where they breed inside buildings. The study site at Kraghede consists of open farmland with pastures, cereals, potatoes, and rape with mixed plantations, hedges, and ponds. As at the other two sites, barn swallows breed within barns and other buildings.

General field procedures

Barn swallows were captured at least weekly from the arrival in early spring until the end of the breeding season using mist nets at all entrances to the barns with breeding birds. Mark-recapture analyses of the data have revealed that the capture probability of birds exceeds 98% in Spain and Denmark (F de Lope, AP Møller and T Szép, unpublished manuscripts), with a presumably equally high capture probability in Milano (based on the presence of birds without individually numbered aluminium rings) (N Saino, unpublished data).

Measurements

Seven linear traits were measured: (1) beak length, BL, (2) beak width, BW, (3) beak height, BH, (4) mean wing length, W, (5) tarsus length, T, (6) mean outermost tail length, TL, (7) length of the shortest central tail feathers, TS. Not all traits were measured on all birds: in particular, beak width was often omitted and for this reason, we deleted this measurement from the analysis. Sample sizes for measurements on all six remaining traits for both female parent and offspring were 91 for Spain, 51 for Denmark, and 69 for Italy. In addition, there were five Spanish and 25 Danish families for which partial measurements were available.

Statistical methods

The genetic variance–covariance matrices were estimated using offspring on parent regression. Extra-pair paternity is common in barn swallows (Møller and Tegelström, 1997) but brood parasitism is not (none was found in the Italian sample, N Saino, unpublished data). The possibility that the male was not the father of all offspring precludes in principle the use of either offspring on midparent or offspring on father regressions. Estimates from mean offspring on mother are also confounded by the uncertain relationship between offspring (full or half siblings). Therefore, we took the most conservative approach and used the regression of a single male offspring on the mother. Cross-fostering experiments with song sparrows, blue tits, willow tits, collared flycatchers, pied flycatchers, starlings, and tree swallows found no evidence of maternal effects on offspring morphological traits (reviewed in Weigensberg and Roff, 1996). Møller (1994, pp 152–156), using a series of indirect measures of maternal effects, found in barn swallows no effect of the mother on tail length of her offspring. While we have no direct estimates from cross-fostering in the present experiments, the unanimous findings cited above indicate that the offspring on female parent approach can be expected to provide valid estimates.

A number of techniques for comparison of G matrices have been proposed (Roff, 2000; Steppan et al, 2002). The most widely used technique in the last few years is that known as the Flury model (Phillips and Arnold, 1999). This method uses a stepwise approach to examine the matrices for equality, proportionality, common principal components, or variation in one or more of the principal components. To increase the robustness of the analysis, Phillips incorporated a randomization test rather than using the likelihood ratio directly. We, therefore, report the probability as ‘Prand’. Several problems arose when we implemented the Flury method: first, the method cannot readily accommodate missing data and second, because of nonpositive-definite matrices, we had to employ the ‘bending’ option in the Phillips program CPCrand (available at <http://www.uta.edu/biology/philipps/software.html>). ‘Bending’ makes the matrix positive definite by adjusting the eigenvalues of the matrix just enough to eliminate the negative eigenvalues (Hayes and Hill, 1981). Unfortunately, bending changes the error structure of the comparison matrix and hence may invalidate the statistical test (Phillips and Arnold, 1999).

An alternate approach is the ‘jackknife followed by MANOVA’ method of Roff (2002), hereafter referred to simply as the MANOVA method. The Flury and MANOVA methods bear strong statistical kinship and in the one case in which they have been compared reach the same statistical conclusion (Roff, 2002). An advantage of the MANOVA method for the present analysis is that it permits the incorporation of those offspring–parent combinations for which not all measurements were taken (see below) and does not suffer the problem arising from nonpositive-definite matrices. On the other hand, it does not provide the analysis of principal component structure that the Flury method does. We used both methods, the MANOVA method providing a check on the reliability of the initial comparison of equality given by the Flury method.

The ‘jackknife followed by MANOVA’ method is described in detail in Roff (2002). Briefly, the method is as follows. Taking each country separately, first the G matrix was estimated using the single offspring on female parent, Gobs. Next, the first family was deleted and the G matrix estimated using the reduced data set, G−1. The first pseudovalue G matrix, φ−1, was calculated as φ−1=n Gobs−(n−1)G−1. The deleted family was added back to the data set, the next family deleted and the procedure repeated to obtain, φ−2. The jackknife mean estimate of G is given by φ̄=∑i=1nφ−i/n. More importantly for the present analysis the set of pseudovalues have the same distribution as G and hence these may be used in a MANOVA analysis with population as an independent variable (Roff, 2002). Multivariate techniques such as MANOVA may be sensitive to outliers (Tabachnick and Fidell, 2001), and so we additionally estimated the probability for this test using the following randomization procedure. To understand the process it is necessary to understand the data layout: the data set consists of n1+n2+n3 rows, where ni is the number of families in the ith population and each row is the set of (co)variance pseudovalues for a single deletion. The last column in the data set contains the country designator. After initial calculation of Wilks' λ, which we denote as λobs, we randomized the country designations and recalculated Wilks' λ, which we denote as λrand. This procedure was repeated 999 times and the probability of obtaining a Wilks' λ as small or smaller than observed (note that in contrast to most statistical measures variance accounted for increases with a decrease in Wilks' λ) estimated as (1+N)/1000, where N is the number of cases in which λrand < λobs (Manly, 1997). We designate this estimate of the probability as Pλ: in no case did the two probability values differ significantly, confirming that our data satisfied the assumptions of MANOVA.

Because of the varying number of families per trait in some offspring–parent combinations the total number of pseudovalues differed among the variances and covariances. We retained all families in the initial estimation of the pseudovalues as this minimizes the standard error, but we reduced this set of pseudovalues such that the set used in the MANOVA had equal numbers per (co)variance (MANOVA requires that there be no missing variables). We achieved this by random deletion of pseudovalue sets (ie random φ−i). We also analysed the data using only that set used in the Flury analysis. In general, both sets of MANOVA analyses gave the same result and for ease of comparison between the MANOVA and Flury method, unless otherwise indicated, we present the results using the same data set (ie complete measures per family).

Because the P matrix is guaranteed to be positive definite, the analysis of the phenotypic matrix does not present the same problem as the G matrix. For assurance, we used both methods of analysis. Similarly, we also analysed the environmental covariance matrix, which, like the G matrix also required bending for the Flury method.

Results

Variation in phenotypic means

There is highly significant multivariate variation among countries in phenotypic means (Wilks' λ=0.421, F14,442=17.088, P<0.0001). Previous work has established that morphological trait means increase with latitude (Møller, 1994). The present data also show this pattern with a general size increase northwards (Figure 1), but there are too few data points to statistically test for clinal variation. Variation in trait means does not imply variation in trait variances or covariances, but it does suggest the action of natural selection, genetic drift, and/or environmental effects, all of which could affect the G matrix (Roff, 1997, 2000).

Figure 1
figure 1

Variation (mean±SE) in morphological traits in three populations of barn swallows as a function of latitude. Traits are displayed sequentially in the order as presented in the methods. For visual purposes, each trait was standardized to a mean of zero and a SD 1 using the combined sample (ie all three countries). Statistical analysis was done using the original data.

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Variation in the P, E, and G matrices

There is highly significant variation among countries in the phenotypic (co)variances (Prand=0.012). but the matrices do not differ with respect to proportionality (Prand=0.06, Table 1). This result suggests that the variation among countries might be a scaling phenomenon. The typical relationship between means and variances is allometric, which can be removed by a logarithmic transformation. Such a transformation makes no qualitative difference to the statistical conclusion (for equality Prand=0.014, for proportionality Prand=0.061), indicating that the variation is not due to allometric scaling. The MANOVA analysis also indicates highly significant variation among populations (Wilks' λ=0.633, F42,376=2.299, P<0.001, Pλ<0.001)

The environmental matrices differ with respect to equality (Prand <0.0001), proportionality (Prand <0.0001), common principal components (Prand=0.0167), but not with respect to partial principal components (Table 1). Using the log-transformed data made little difference except that the hierarchy now becomes significant at CPC(4), the partial common principal component model with four components shared in common. However, the MANOVA method did not show a significant difference among countries (Wilks' λ=0.758, F42,376=1.329, P=0.090, Pλ=0.073).

There is considerable variation in the estimated components of the G matrices (Table 2). This variation is reflected in significant statistical differences in the G matrices (Table 1). Both the Flury and MANOVA methods agree in finding significant variation with respect to equality (Flury, Prand=<0.001, MANOVA, Wilks' =0.732, F42,376=1.511 P=0.025, Pλ=0.022 for complete data set Wilks' =0.718, F42,376=1.613, P=0.012, Pλ=0.009). The Flury method suggests that there are no shared principal components (Table 1). Analysis of the log-transformed data show statistically significant differences with respect to equality (Prand <0.001) and proportionality (Prand <0.001) but not with respect to common principal components (Prand=0.072).

Table 2 Genetic variances and covariances of six morphological traits in three wild populations of barn swallow
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Table 3 summarizes pairwise analyses of the three matrices and Figure 2 shows the standardized (co)variances as a function of latitude. The two latitudinally closest pair of countries, Spain and Italy, do not statistically differ in their P matrices, but the other two comparisons show a statistically significant proportional difference, as in the full analysis. Both the E and G matrices differ in equality and proportionality in all comparisons. The MANOVA results for the pairwise matrix comparisons are not the same as the Flury results (Table 3). The MANOVA analysis of the P matrices give all pairwise comparisons to be significant, although the Spain vs Italy comparison is marginal (nonsignificant for the restricted data set). The MANOVA analysis of the G matrices gives results most similar to those of the Flury analysis of the P matrices (Table 3): pairwise comparison of countries reveals a significant difference between the geographically most separate swallow populations of Denmark and Spain, but there is no significant difference between Denmark and Italy or Spain and Italy. Similar discrepancies exist for the E matrix, with the Flury method giving highly significant pairwise differences (P<0.001 in all cases), whereas the MANOVA analysis of the same data set gives none to be significant (P=0.077, Pλ=0.073 for Denmark vs Spain, P=0.368, Pλ=0.374 for Denmark vs Italy, and P=0.375, Pλ=0.355 for Spain vs Italy).

Table 3 Summary of Flury hierarchical analysis using pairwise contrasts
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Figure 2
figure 2

Standardized genetic (co)variances of six morphological traits in three populations of barn swallow. For visual purposes each trait (co)variance was standardized to a mean of zero and a SD of 1 using the combined sample (ie all three countries). Statistical analysis was done using the original data.

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The phenotypic (co)variances are made up of the genetic and environmental (co)variances. Morphological traits typically have high heritabilities and genetic correlations (Mousseau and Roff, 1987; Roff, 1996). Consequently, we would expect that the phenotypic and genetic (co)variances will also be highly correlated. Comparison both within and between countries indicates that the P and G matrices are indeed highly correlated (Table 4). Because of the problem of part-whole correlation the statistical tests associated with these correlations are unreliable, but they are sufficiently high to indicate statistical significance. More importantly, the high correlation between the P and G matrix elements suggests that the former might be used as a surrogate for the latter, at least as a first approximation.

Table 4 Pearson product moment correlations (below diagonal, probabilitya above the diagonal) between the G and P matrices of morphological traits in the barn swallow
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Discussion

Selection on morphological components has been demonstrated to occur in a wide range of organisms (Endler, 1986; Fairbairn and Reeve, 2001; Kingsolver et al, 2001) and in various species of birds (Holland and Yalden, 1995; Brown and Brown, 1998; Larsson et al, 1998; Merila et al, 1999; Balmford et al, 2000; Barbraud, 2000; Nowakowski, 2000; Przybylo et al, 2000), including barn swallows (Møller, 1993; Møller and Tegelstrom, 1997; Møller et al, 1998; Brown and Brown, 1999). In Nebraska, a period of severe weather resulted in size-related mortality of barn swallows (Brown and Brown, 1999) indicating that, as with other bird species natural selection acts upon at least some morphological components. Of particular interest would be an analysis of the pattern of genetic variances and covariances in relation to estimates of selection on the traits. The present data are not suitable for such an analysis. However, a quantitative genetic analysis of morphometric traits in the water strider Aquarius remigis suggests that patterns of selection may be reflected in patterns of variation in the G matrix (Preziosi and Roff, 1998). Thus, a fruitful avenue of empirical investigation is the relationship between the structure of the G matrix and the structure of selection (Arnold et al, 2001).

Using data on three geographically distinct populations of barn swallow, we have shown using the Flury method that the populations differ in mean phenotype and in their phenotypic, environmental, and genetic variance–covariances matrices. These results differ from those obtained by Arnold and Phillips (1999) for two populations of garter snakes: in their analysis, Arnold and Phillips observed equality of the environmental matrices and differences in the P and G matrices. Further, Arnold and Phillips found that, in general, the results from the phenotypic analysis were the same as those from the genetic analysis. In contrast, in the present study using the Flury method, we observed significant variation in the environmental matrices and obtained different conclusions from the phenotypic and genetic analyses (Table 1). There is generally far more statistical power in the analysis of phenotypic than genetic variation and hence the observation of only proportional differences in the P matrices whereas the G matrices differed in their principal components suggests that in this case, the two matrices are quite different. However, the correlations between the elements of the two matrices are high (Table 4) as typically found for morphological traits (Roff, 1996). It was necessary to apply a bending coefficient to the G and E matrices to carry out the Flury hierarchical test: it is possible that this may produce spurious results. However, both the Flury method and the MANOVA method indicate significant differences among the G matrices and the latter method does not require the adjustment used in the Flury method. Further, the randomization procedure applied to the MANOVA method provided confirmation of the estimated probabilities. We recommend use of this randomization procedure as a general approach to the MANOVA analysis. Given the correspondence between the Flury and MANOVA methods, the inequality among the G matrices is a robust result, and agrees with the P matrix analysis. Because of the possible problem associated with ‘bending’ differences found by the Flury method at a deeper level of structure should be regarded with caution. Similarly, the significant differences found by the Flury method for the E matrix but not for the MANOVA method may be a result of bending the E matrices for the former analysis. The general findings of the MANOVA method in the present analysis do agree with the qualitative findings of Arnold and Phillips (1999), that is, differences in the P and G matrices but not the E matrices. More work on the statistical properties of the several methods of matrix analysis are required.

While the present analysis does indicate overall variation in the G matrices, much of this variation appears to come from differences between the population of Denmark vs the other two (Figure 2, Table 3). Because the birds were free ranging, we cannot say if this variation is a consequence of genotype by environment interaction or differences in the G matrices that would be found under a common garden rearing regime. Regardless of the source of variation, the differences do indicate that evolutionary trajectories in the field populations will be different under the same selection regime. We do not suggest that the present results demonstrate clinal variation – there are too few populations and the geographic distribution insufficiently great – but note that it is consistent with the observed clinal variation in mean morphological trait values and hence the hypothesis of clinal variation is worth investigating further.