1. A method based on the calculation of major and minor axes of bivariate ellipses to estimate relative rates of evolutionary diversification of two traits is presented. The advantage of ellipse analysis over the more common allometric or regression approaches to variation in comparative data is that diversification of each trait is estimated relative to that of the other. This can reveal differences in the relative rates of diversification of two traits among sister groups or, through hierarchical analysis, over the history of a lineage.
2. Equations are provided for calculating the size, shape and orientation of bivariate ellipses based on the variance–covariance matrix of the original data. Standard errors and biases of these parameters are also quantified.
3. Applications of ellipse analysis to the relationship between egg size and incubation period in birds illustrate differences in the diversification of two traits between independent lineages. Nested analysis of covariance based on taxonomic levels further illustrates how relative rates of evolutionary diversification may vary through the history of a monophyletic group.
4. A simple model incorporating both independent (special) and correlated (general) change in two traits shows how different shapes and orientations of bivariate ellipses can be produced by different rates of evolutionary diversification in special and general genetic factors.
5. Ellipse analysis is a descriptive tool that can clarify patterns of diversification. It cannot distinguish differences in evolutionary responsiveness (phylogenetic constraint) from differences in the selective environment affecting the shape and orientation of the bivariate ellipse. It can, however, provide a more detailed characterization of trait evolution than other comparative methods, taking advantage of the additional information provided by the shape and, in some cases, the dimensions of the bivariate ellipse.
Phenotypic values of the traits of organisms are influenced by selective factors in the contemporary environment and by evolutionary history. The influence of ancestry is variously called evolutionary inertia, phylogenetic effect and phylogenetic constraint (for reviews see McKitrick 1993; Miles & Dunham 1993). Several statistical techniques have been devised to quantify phylogenetic effects. These techniques include phylogenetic autocorrelation (Cheverud, Dow & Leutenegger 1985), phylogenetic regression (Grafen 1989) and hierarchical analysis of variance (Stearns 1983, 1984; Dunham & Miles 1985; Bell 1989; Herrera 1992; Miles & Dunham 1992; Smith 1994). Such analyses partition variation among taxa into components associated with phylogenetic relationship and components that are uniquely derived in each taxon. Phylogenetic components of phenotypic variation have been interpreted as representing the past history of evolution, in contrast to adaptation to contemporary conditions (Cheverud et al. 1985), although stasis in the selective environment confounds ancestry and adaptation (Harvey & Pagel 1991; Pagel 1993). Phylogenetic effects have also been thought of as representing constraints on evolutionary adaptation to contemporary conditions (Gould & Lewontin 1979).
Phylogeny influences phenotypic values of traits in part because close relatives have had less time than more distant relatives to diverge from a shared ancestral condition. (In this paper we use diversification to mean increase in the variance of trait values within a clade over time.) However, phylogenetic effects by themselves convey no information about rates of evolutionary diversification. Indeed, it has been argued that ‘comparative’ studies of variation in traits of organisms can provide little insight into processes of evolution (Leroi, Rose & Lauder 1994). The course of evolution is determined by a tension between selective factors in the environment and the evolutionary responsiveness of the phenotype, in addition to random changes arising from mutation, drift and gene flow. Changes in rate or direction of diversification can be brought about by variation in either selective pressures or responsiveness, but the history of diversification alone cannot distinguish between these components (Lande 1979).
Although inferences from phylogenetic analysis of trait data are limited, such analyses can provide an important empirical basis for further inquiry into the processes of evolutionary diversification. In one approach to this problem, Derrickson & Ricklefs (1988) used taxonomically organized data to detect variation in relative rates of evolutionary diversification of several traits within a lineage; Bell (1989) has advocated a similar approach. Because taxonomic arrangements and most phylogenetic hypotheses do not specify the ages of ancestral nodes, it is generally not possible to estimate the rate of divergence of a single character. Derrickson & Ricklefs (1988) suggested that one may nonetheless estimate relative rates of diversification by using each one of a pair of traits as a standard for the other. Their approach was to estimate the size, shape and orientation of bivariate ellipses relating two traits (X and Y) within a lineage, and then to compare the shape and orientation of these ellipses between lineages (e.g. sister taxa) or at different levels of taxonomic distinction within lineages (i.e. monophyletic groups).
Derrickson & Ricklefs (1988) based their analysis on the idea that taxa diverge by small steps in random directions with respect to a common ancestral point. Accordingly, distributions of trait values of taxa within lineages assume normal distributions whose variances increase in direct relation to time since divergence (Felsenstein 1985). Depending on their relative rates of change, simultaneous evolution of two traits should produce two orthogonal axes of variation. When the evolution of each trait is correlated with that of the other, these axes will be oriented at some angle to the measured traits. Thus, diversification results in a bivariate normal distribution, with a characteristic shape and orientation. As long as rates of evolution are invariant, the variances of the traits and their covariance increase isometrically with time. In reality, evolutionary diversification may involve strong directional selection away from the centroid of the clade, perhaps the result of competitive interactions among sister taxa, and the model of small, random steps – often referred to as Brownian motion – may not be wholly appropriate. However, the idea of isometrically increasing variance and covariance is equally applicable to diversifying selection, except that increase in variance is proportional to the square of time when the evolutionary response is constant. Regardless of what propels evolutionary change, however, the bivariate ellipse nonetheless provides a useful description of variation within a clade. Random change and normal distribution of variation are not critical, although statistical interpretation is based on normality. In general, however, multivariate statistical inference is robust to departures from normality (Harris 1975).
The size, shape and orientation of an ellipse may be described by the variances of traits X and Y and the covariance between them. The covariance indicates the degree of correlated variation in X and Y, which may be quantified by the correlation coefficient (r) and further characterized by the slope of the major axis of the bivariate ellipse (Rayner 1985). A difference in the orientation of two bivariate ellipses indicates differences in relative rates of diversification of X and Y, their interdependence or both. The potential causes of such differences include shifts in selection pressures, the relative responsiveness of X and Y to selection, and genetic pleiotropy or functional relationships (perceived as genetic covariances) that connect the evolution of X and Y. In contrast to analyses of covariance and hierarchical (nested) analyses of covariance, Derrickson & Ricklefs’ approach explicitly included the dimensions of variation in two measurements as well as their orientation, and used principal axes rather than regression slopes to characterize the orientation of the covariation (Rayner 1985). This paper extends Derrickson & Ricklefs’ approach to comparative analysis by (1) providing simple methods of calculating bivariate ellipse parameters, (2) illustrating the approach with an empirical example, that of egg size and incubation period in birds, and (3) exploring further biological implications of changes in the size, shape and orientation of ellipses. In the case of any pair of traits, such as egg size and incubation period, we may ask whether their evolution relative to each other has been homogeneous among sister taxa or over the history of a monophyletic group. Rejecting the hypothesis of homogeneity provides an empirical basis for further study of the nature of ‘phylogenetic constraint’.
Evolutionary inferences from statistical analyses of trait values
Comparisons that can be made with taxonomically organized data sets are summarized in Table 1. In using such data, one assumes that taxonomic distributions reflect phylogenetic history. Three levels of statistics are available: mean values of traits, variances of traits and covariances among traits. These may be compared either between independent lineages of similar rank (e.g. between sister taxa), or at different levels of taxonomic distinction within a monophyletic group (e.g. genera within families compared with families within orders). Different inferences are provided by each of the six combinations of statistic and comparison, as indicated in Table 1. Mean values provide the least information about evolutionary diversification within clades. In comparisons among sister taxa, which by definition are of identical age, variances of traits among contemporary taxa are assumed to be related to average rates of diversification. When these differ significantly among sister groups, one may infer that diversifying selection pressures or evolutionary responsiveness also have differed between the two groups. Compared over different levels of taxonomic distinction, differences in variances are informative only when taxonomic distinction can be related to time. When taxonomic levels are defined by genomic divergence, as in the case of Sibley & Ahlquist’s (1990) taxonomy of birds, and when one assumes that divergence is directly related to time, taxonomic levels provide a metric against which variances in mensural traits can be scaled, as shown below. In general, however, use of taxonomic classifications in comparative analysis is weakened by taxonomic groups with polyphyletic origins and by lack of comparability of taxonomic levels between groups.
Table 1. . Comparisons that may be obtained from taxonomically or phylogenetically organized data sets
The most powerful inferences from comparative analysis come from variances and covariances of two or more variables. In this case, each variable serves as a standard of comparison for the other, and changes in the relative diversification of two or more variables may be readily identified. Differences among sister groups may reflect occupation of different adaptive landscapes, shifts in the genetic correlation between two traits, or changes in the functional relationship between two traits. Differences with respect to level of taxonomic distinction may reflect changes in the diversifying forces of selection at different times in the evolution of a lineage, or changes in the responses of traits to selective forces. Hierarchical analysis may be performed in two ways. The first approach is to compare traits across all taxa at each taxonomic level within a lineage. This defines the generalizable pattern of diversification at particular levels within the lineage. The second approach is to examine one taxon of low rank and all the larger taxa within which it is included. For example, Derrickson & Ricklefs (1988) characterized bivariate ellipses relating various life-history traits to body size in the genus Peromyscus, family Cricetidae, order Rodentia, class Mammalia.
Characterizing bivariate ellipses
The relationship between measurements X and Y can be characterized by the size, shape and orientation of a bivariate ellipse describing the pattern of variance and covariance (Table 2, Fig. 1). These parameters are obtained from a principal components analysis – more generally, singular value decomposition (Green 1978) – of the variance–covariance matrix of X and Y, which produces eigenvalues (λi) and eigenvectors (directional cosines aij) of each axis i with respect to each original measurement j (Harris 1975). Size (L1, L2) is estimated by the square roots of the eigenvalues of the major and minor axes; these are the standard deviations of the sample data projected onto the two axes, which are commensurate with the units of measurement of variables X and Y. Shape (S) of an ellipse is the square root of the ratio of its eigenvalues, which corresponds to the ratio of the length to the width of the ellipse (L1/L2). Orientation (A) is the slope of the major axis with respect to X, which may be calculated from the directional cosines (eigenvectors) of each axis with respect to the original measurements. These cosines measure the angles of the major and minor axes to the X and Y axes (θ1,X, θ1,Y, θ2,X, θ2,Y, respectively). The slope of the major axis is the tangent of θ1,X; the angle θ1,X is equal to sin–1 (a12). Statistical comparisons among ellipses are based on standard errors of the eigenvalues, shape parameter and directional cosines, whose estimation is described in the Appendix.
Because ellipse analysis is based on the dispersion of measurements about a centroid, the approach presented here cannot make use of phylogenetically ‘corrected’ data. However, use of original measurements – the so-called TIP values referring to the tips of the phylogenetic tree – in comparative analyses rarely biases interpretations of covariation between traits (Ricklefs & Starck 1996; Price 1997). This is not surprising inasmuch as both TIP and PIC approaches use the same measurements (Pagel 1993). In addition, the calculation of PICs introduces variance at each step by having to estimate ancestral states (Schluter, Price & Mooers 1997). Furthermore, results of PIC analyses seem to be little affected by gross variations in the topology of a phylogeny (Garland & Adolph 1994), which suggests that patterns of covariation recovered in the absence of a phylogeny are useful.
Conservatively interpreted TIP analyses do not produce biased or contrary results compared with PIC analyses (Pagel 1993), and the risk of accepting differences where none exists is small. Moreover, the ability to apply comparative analyses in the absence of a phylogenetic hypothesis and the ability to use taxonomic data to conduct hierarchical analyses would seem to outweigh the potential problems produced by lack of independence of data. Even when a phylogenetic hypothesis is available, we feel that there are advantages to defining hierarchical levels by the structure of the tree but continuing to use TIP values in nested analyses of variance and covariance based on these levels. Indeed, the methods described here, which include hierarchical analyses of variance, are phylogenetic methods to the extent that taxonomy reflects phylogenetic relationship. Variance components at each level of the taxonomic hierarchy are equivalent to variances calculated for estimated nodal values within specified ranges of ages assigned according to genetic divergence or character differences.
Two examples show how the calculation of bivariate ellipses can be applied to comparative data. The data set from which these examples are drawn includes egg mass (grams) and length of incubation period (days) of a large sample of birds (unpublished compilation). (Additional analyses of this kind for body mass and growth rate may be found in Starck & Ricklefs 1998b.) In the first example, the relationship between incubation period and egg size in two families of birds, the plovers and their allies (Charadriidae) and the pheasants and their allies (Phasianidae), is compared. The Charadriidae and Phasianidae are not sister taxa; they were chosen simply for heuristic purposes. All measurements were converted to logarithms (base 10) to reduce skewness and to provide a scale of factorial rather than absolute differences, which portrays the allometric relationship between two variables. Accordingly, the relationships between incubation period and egg size differ between the families (Fig. 2). Indeed, incubation period is independent of egg size in the Charadriidae.
Variances and covariances of the data sets are presented in Table 3. None of the distributions of log-transformed egg mass or incubation period exhibited significant skewness (g1) or kurtosis (g2) (Sokal & Rohlf 1981, p. 139). Ellipse parameters calculated from the variances and covariances are presented in Table 4. As a quick rule of thumb, differences between two values that do not exceed the sum of their standard errors are not significant. Accordingly, the first and second eigenvalues and the shape parameter do not differ significantly between the two families. The orientation of the ellipses can be compared using the directional cosines and their associated standard errors. These clearly differ (t = 3·0, P < 0·01) between Charadriidae (a12 = 0·034 ± 0·031; A = 0·031) and Phasianidae (a12 = 0·173 ± 0·035; A = 0·176); the value for the Charadriidae does not differ significantly from 0 (t = 1·1, P > 0·2).
Table 3. . Variance–covariance matrix for egg size and incubation period in the Charadriidae and Phasianidae
Table 4. . Ellipse parameters for the relationship between incubation period (Y) and egg size (X) in the Charadriidae and Phasianidae
A second example using a larger compilation (n = 796 species) of incubation periods and egg sizes employs a nested analysis of variance and covariance to characterize the relationships between these traits at taxonomic levels of order, family, genus and species. The taxonomy is that of Sibley & Monroe (1990), who defined each rank of taxonomic distinction by a particular range of genomic divergence. The nested analysis estimates the variance and covariance across taxa at one level nested within the next higher level; for example, genera within families (Bell 1989; Herrera 1992). This is equivalent to calculating the dispersion of generic means about the mean value of the family to which each of the genera belongs (Stearns 1983; Harvey & Pagel 1991). The variance and covariance components for these relationships are presented in Table 5. The distribution of incubation periods of species within genera exhibited significant negative skewness (g1 = – 0·44). Incubation period exhibited significant leptokurtosis at the level of family (g2 = 1·22), genus (1·13) and species (7·19), and egg mass exhibited significant leptokurtosis at the level of genus (0·89) and species (4·41). Calculated parameters for bivariate ellipses are presented in Table 6.
Table 5. . Nested analysis of variance and covariance for egg size and incubation period in birds
Table 6. . Ellipse parameters for the relationship between incubation period (Y) and egg size (X) at different levels of taxonomic distinction in birds
Variance components of egg size and incubation period show that most of the variance in both parameters resides at the taxonomic level of order (77 and 69%, respectively). The next largest component is the 21% of variance in incubation period that resides at the level of family within orders. All other components of variance are less than 10% of the total. We may ask whether the generation of variance in either egg size or incubation period corresponds to genomic divergence attributable to each step in the taxonomic hierarchy. Sibley & Ahlquist (1990) assigned ordinal-level distinction to lineages that diverged from any node having a contemporary genetic distance corresponding to 20–22 °C ΔTH50 melting point of hybridized (heteroduplex) DNA. Values for families and genera were 9–11 and 0–2·2 °C, respectively. Sibley & Ahlquist placed the base of the avian phylogeny at ΔTH50 = 28 °C. Therefore, the amount of genomic divergence within each level (species within genera, genera within families, families within orders, and orders) can be apportioned as 4, 32, 39 and 25% of the total. The distribution of variance in both egg size and incubation period matches poorly the expectation of uniform increase in variance with time (Fig. 3). Compared with genomic diversification estimated by DNA hybridization, egg size and incubation period appear to have evolved rapidly during the early diversification of avian orders, and much more slowly thereafter. The match is even worse under an assumption of uniform rate of diversification, according to which variance should increase as the square of time.
The major axis calculated from the variance and covariance portrays correlated variation in both egg size and incubation period. Because the allometric slope of the relationship between egg size and incubation period is relatively low at all taxonomic levels (A = 0·12–0·30), the size of this axis (L1) is determined primarily by variation in egg size, which in turn is closely related to body size (Rahn, Paganelli & Ar 1975). The minor axis (L2) represents variation primarily in incubation period in a direction perpendicular to the major axis, that is, it mostly represents increase or decrease in incubation period for a given egg size.
The dimension of the major axis (L1) is large at the ordinal level compared with lower levels of taxonomic distinction, which is consistent with a rapid diversification of egg size (and body size) among birds in conjunction with the establishment of taxonomic orders. In contrast, the minor axis (L2) is greatest at the level of families within orders, suggesting diversification in incubation period, independently of egg size, associated with the establishment of avian families (Table 6). Orientations of ellipses are higher at the ordinal (a12 = 0·23 ± 0·02) and familial (0·28 ± 0·04) levels, than at the generic (0·12 ± 0·01) and species levels (0·13 ± 0·01) (e.g. ordinal vs specific levels, t = 4·8, P < 0·001). These trends may be visualized in a plot of the values for taxa at each level centred upon the means of the values for that taxonomic level (Fig. 4).
The examples presented above revealed significant differences between lineages, and between taxonomic levels within lineages, in the size, shape and orientation of bivariate ellipses describing covarying traits. In the case of the comparison between the Charadriidae and Phasianidae, sizes (and consequently shapes) of ellipses did not differ, but their orientation did. A nested analysis of variance over all birds revealed differences in dimensions of ellipses, differences in one dimension relative to the other, hence the shapes of ellipses, and differences in orientation of ellipses. The most striking phenomena were (1) the greatly elongated ellipse associated with the diversification of orders and (2) the relatively compact ellipse with a similarly high angle of orientation associated with the diversification of families within orders. These variations indicate that the relative rates of diversification of egg size and incubation period, and the covariation between the diversification of the two traits, have varied between monophyletic lineages and within a monophyletic lineage over time.
To explore the relationship between character evolution and changes in shape and orientation of ellipses, some simple simulations with bivariate ellipses were performed. It was supposed that simultaneous diversification of two variables can be characterized as having two components: one general, represented by concerted or strictly correlated change in two variables, the other special, representing independent change in each of the two variables (Fig. 5). General variation may be viewed as constrained diversification of a pair of traits, but it should not be equated with genetic correlation, which may include the effects of general and special genetic factors. Special variation in each variable is unconstrained by variation in the other. These two types of variation would arise if genetic factor B influenced both traits X and Y, and perhaps other attributes, and if genetic factors A and C independently influenced traits X and Y, respectively (Fig. 6). The relationship between two variables may also be subject to more than one general factor, and various genetic factors that comprise B may produce different slopes for the general relationship in isolation.
The general component of diversification (XG and YG) is represented by the distribution of taxa along a line having slope b. Thus, when the variance in XG is V(XG), the variance in YG is V(YG) = b2V(XG). V(G) is equal to 1 + b2V(XG). The covariance between YG and XG is V(XG,YG) = bV(XG); the correlation between YG and XG is 1. Special components of diversification were scaled as XS = aXG and YS = cXG; therefore their variances were V(XS) = a2V(XG) and V(YS) = c2V(XG). Finally, V(X) = (1 + a2)V(XG), V(Y) = (b2 + c2)V(XG) and V(X,Y) = bV(XG). When the variances and covariances are scaled to the general variance in X[V(XG)], these are related as V(X) = 1 + a2, V(Y) = b2 + c2 and V(X,Y) = b. The value of b in this case has an upper bound of V(X,Y)/V(X), the regression slope of Y on X.
Values for the orientation (a12) and shape (S) of bivariate ellipses were then generated from various combinations of values for a and c. In these calculations, b was set at a value of 0·25 which is approximately the greatest orientation observed in ellipses relating incubation period to egg size. The value of a was varied between 0·2 and 4, by increments of 0·2, and c between 0·02 and 0·40, by increments of 0·02. One can see from Fig. 7 that a wide range of ellipse shapes and orientations can be produced without changing the general (constrained) relationship between Y and X. Indeed, all the ellipse parameters calculated in the examples given above are included within the boundary of values plotted in Fig. 7.
To determine the values of a and c which yield each ellipse calculated from the sample data sets under the assumption b = 0·25, various combinations of values of a11 and S were simply boxed, and then the combinations of parameters a and c that produced these values (Fig. 8) were plotted. The low orientation of the ellipse for the Charadriidae compared with that for the Phasianidae can be produced under b = 0·25 by a high value of a, the special component of variation in egg size (X). These values of a are ≈ 2·4–2·6 and 0·6–0·8, respectively, and the value for the Charadriidae is large compared with the genera-within-families and species-within-genera ellipses produced by nested analysis of variance of birds as a whole (a≈ 1·0). Ellipses resembling the one that characterizes orders of birds in the nested analysis can be produced only by very low values of both a (0·2–0·4) and c (0·02–0·08). Under the assumption of b = 0·25, diversification of orders was dominated by the general component, which produced an ellipse with a high shape parameter having nearly the same orientation as the slope of the general component.
According to this simulation, the family level ellipse represents a large increase in the special component of variation in incubation period (c), which is mirrored by the large value for the minor axis (L2) at the family level. At the genus and species levels, the special component of variation in egg size increases relatively to a = 1. Variation among genera within families appears to have a low special component of variation in incubation period (c), but this component is much higher at the species level.
Clearly, differences in orientation of ellipses also can result from differences the slope of their general components of variation. The only constraint is that the slope of the general component (b) of variation cannot exceed the regression slope of variable Y with respect to X. The model presented above also shows that dramatic differences in ellipse parameters may be obtained without changing the general relationship of Y to X, but by altering special components of variance in X and Y.
The biological nature of general and special components of variation is outside the scope of comparative work. General components of variation may represent the linking of traits by physical functions or processes that are beyond the reach of evolutionary modification. For example, increasing egg size has implications for embryo growth and incubation period because of systematic changes in relationships of surfaces to volumes (Kooijman 1993; Ricklefs & Starck 1998). Thus, the early diversification of avian orders may have simply involved diversification of body and egg size, with incubation period following according to physical laws governing embryo growth without strong selection for variation about this relationship. Special components of variation may arise in one variable when the other is maintained at a fixed value by stabilizing selection, even though such evolution might have to work against genetic covariances linking two traits. One can imagine that egg size may be stabilized by selection on body size, while length of time in the egg may vary as a consequence of differences in the degree of maturity of the hatchling, or the thickness of the egg shell, which determines gas diffusion. Alternatively, body and egg size may diversify while strong selection from time-dependent egg mortality pushes embryonic development rate to a physiologically determined maximum.
Differences in the orientation and shape of bivariate ellipses may also result from differences in the slope of the general component of variation (b). For example, higher-order diversification (ordinal, familial) of birds is associated with the evolutionary diversification of development mode (altricial vs precocial) (Starck & Ricklefs 1998a). Smaller taxa (having smaller eggs) tend to have altricial (rapid) development, which results in shorter incubation periods and a steeper slope of the regression of incubation period on egg size. Because development type is generally fixed below the family level, the slope of the incubation period–egg size regression is dominated by other factors and becomes less steep. Various special components of variance, for example associated with selection on incubation period irrespective of egg size by sibling competition (Ricklefs 1993), may also be diversified at higher taxonomic levels but relatively uniform at lower levels.
Another source of special variation is measurement and sampling error (Rayner 1985; Riska 1991). Incubation period is measured independently of egg size. Errors may occur through inaccuracies in measurement, inadequate sampling of a particular population, or failure to sample an entire evolutionary unit adequately. Such errors are likely to be most important at the level of species within genera because individual values are used rather than means of included taxa. Furthermore, in birds, genetic distances between species within genera are less than genetic distances between genera, families or orders, and so measurement errors are likely to be large compared with evolved components of general and special diversification. Measurement errors may explain relatively high estimates of a and c at the species level in the nested analysis; variances in the general component of variation associated with differences among species within genera are small relative to variances at higher taxonomic levels, hence measurement variance may be relatively large.
One advantage of ellipse analysis is the standardization of evolution in one trait by that in another. This standardization allows one to visualize differences in the pattern of diversification between lineages or within a monophyletic group over its evolutionary history. One attribute of a bivariate ellipse is the allometric relationship between the two variables. By means of ellipse analysis, one may relate the allometric slope to evolutionary change in each character, taking into account both the orientation and the shape of the bivariate ellipse. It is important to understand how this perspective compares with other approaches to character evolution. The slopes of allometric relationships often differ among taxa of similar rank, or among ranks in a hierarchical analysis (Gould 1966; Gould 1977). Lande (1979), and later Riska & Atchley (1985), outlined a general explanation for allometric relationships based on the outcome of selection on genetically correlated traits. Lande (1979) suggested that a low allometric slope relating brain to body weight within genera of mammals (0·2–0·4) could be accounted for by selection only upon weight, with a correlated response in brain size. The high slope observed among orders (0·67) could only have occurred following strong selection on brain size, as well. From values of realized heritabilities of brain and body mass calculated from selection experiments, Lande (1979) estimated the genetic correlation between brain and body mass to be 0·68. The genetic correlation between two traits could also be interpreted as the slope of the general relationship discussed in this treatment, in which case it would not be surprising that the ordinal-level allometric slope of brain mass to body mass is close to this level.
As an alternative to Lande’s (1979) conclusion that a high ordinal-level allometric slope implies strong selection on brain mass, one could interpret the steep slope as representing a high degree of diversification of general factors, which influence both traits, compared with special factors that select upon each trait independently. For example, Riska & Atchley (1985) suggested that a general increase in, or prolongation of, mitotic activity during development could produce both larger body size and larger brain size. Strong selection on brain mass (large value of c in this case) does not preclude a high allometric slope but neither is it required. All that is necessary is that the special component of body mass diversification is small.
Genetic-covariance and general–special factor approaches represent somewhat different concepts of evolutionary diversification. Ellipse analysis emphasizes diversification of a lineage within a phenotypic space rather than development of allometric relationships. In ellipse analysis, the shape of an occupied space is as informative about the history of diversification as the orientation of the occupied space, and diversification can result from general (correlated) and special (independent) components for each measured trait. This interpretation of ellipse parameters emphasizes long-term evolutionary responses of traits, which include both selective pressures and responsiveness of phenotypes to these pressures. In the genetic-covariance approach of Lande (1979), response is the outcome of selection acting through the contemporary genetic architecture of the phenotype. Although this approach is appropriate for predicting short-term trajectories of evolutionary change, especially under strong selection, behaviour of a lineage under weak selection for long periods may be essentially independent of the genetic architecture of the phenotype at a particular moment (Zeng 1988; Schluter 1996).
Ellipse analysis, which may be extended to multivariate comparisons (Rayner 1985), provides a more complete empirical basis for understanding evolutionary diversification than the more traditional allometric analysis because it considers the size and shape, as well as the orientation, of a bivariate distribution. Its application may also help to place quantitative genetic interpretations of diversification in a more limited context of short-term response to evolution and to reverse a prevalent trend of artificially coupling microevolutionary processes and macroevolutionary patterns.
* program ellipse.sas;
* this program calculates estimates of eigenvalues and directional cosines for a bivariate normal distribution;
sample = ’sample name’; * name of sample, to be supplied;
x = ’variable 1’; * name of variable 1, to be supplied;
y = ’variable 2’; * name of variable 2, to be supplied;
n = 000; * number of cases, to be supplied;
varx = 0.000000; * variance of variable x, to be supplied;
vary = 0.000000; * variance of variable y, to be supplied;
covxy = 0.000000; * covariance between x and y, to be supplied;
d = sqrt((varx+vary)**2 - 4*(varx*vary-covxy**2));
g1 = (varx + vary + d)/2; * first eigenvalue;
L1 = sqrt(g1); * standard deviation of major axis;
seg1 = g1*sqrt(2/n); * standard error of first eigenvalue;
g2 = (varx + vary - d)/2; * second eigenvalue;
L2 = sqrt(g2); * standard deviation of minor axis;
seg2 = g2*sqrt(2/n); * standard error of second eigenvalue;
a11 = sqrt(varx/g1); * directional cosine of x on first eigenvector;
a12 = sqrt(1 - a11**2); * directional cosine of y on first eigenvector;
a21 = a12; * directional cosine of x on second eigenvector;
a22 = -a11; * directional cosine of y on second eigenvector;
f = n*(g1-g2)**2;
vara11 = g1*g2*a21**2/f;
sea11 = sqrt(vara11); * standard error of a11;
vara12 = g1*g2*a22**2/f;
sea12 = sqrt(vara12); * standard error of a12;
shape = sqrt(g1/g2); * ratio of the ellipse dimensions;
seshape = shape/sqrt(n); * standard error of the shape parameter;
bias = 2.5*shape/n**1.25; * deviation of shape from true ratio of L1/L2);
proc print; var sample n x y;
proc print; var varx vary covxy;
proc print; var g1 seg1 g2 seg2 L1 L2;
proc print; var a11 sea11 a12 sea12;
proc print; var shape seshape bias;
We are grateful for helpful comments from J. Cheverud, W. Ewens, S. B. Heard, R. Lande, J. Losos, T. Price, D. Schluter, J. M. Starck and an anonymous reviewer. RER is supported by NSF OPP-9423522.
(Jackson 1991, p. 81; Morrison 1976, p. 293). The length (Li) of each axis i is the square root of λi.
The directional cosines aij estimate the cosine of the angle between axis i and variable j. Thus, a11 and a12 pertain to the major axis and a21 and a22 pertain to the minor axis. a11 and a12 are the elements of the first eigenvector of the covariance matrix. a11 estimates the cosine of the angle (θ1,X) between the major axis and variable X. a12 estimates the cosine of the angle (θ1,Y) between the major axis and the variable Y, which is equal to the sine of θ1,X. The directional cosines are calculated by
The sign of a11 is always positive. The sign of a12 (= sin[θ1,X]) is positive if the covariance is positive, and negative if the covariance is negative. The directional cosines of the second eigenvector are a21 = a12 and a22 = –a11. The angle between the major axis and variable X is θ1,X = sin–1(a12). The slope (A) of the major axis relating variation in Y to variation in X is calculated by
The standard errors of the directional cosines of the first eigenvector are
ERRORS AND BIASES IN ESTIMATES OF BIVARIATE ELLIPSE PARAMETERS
Taxa sampled within a clade represent an underlying evolutionary process of diversification that can be idealized by a multivariate normal distribution. The parameters of this distribution are estimated, as shown above, from the variances and covariances of the variables with uncertainty (error) and, in some cases, bias. Because small samples are not representative, the calculated major and minor axes may differ markedly from those of the underlying distribution. The estimated orientation of the axes is not biased. The estimated size of the major axis (L1) and shape of the ellipse (S), which is the ratio of the major to the minor axis, have a positive bias because the major axis produced by a principal components analysis maximizes the variance of data projected onto it.
Analytically derived standard errors for eigenvalues and eigenvectors (see above) apply to large samples. For small samples, these standard errors underestimate variation in the values of both the eigenvalues and eigenvectors. Furthermore, the standard error of the shape of a bivariate ellipse does not have an analytical solution. To examine the magnitude of the standard errors of ellipse parameters, and the magnitude of bias in estimating the shape of an ellipse, randomized data sets were generated from which ellipse parameters were calculated. Distributions of the calculated parameters were then compared with underlying statistical distributions used to generate the randomized data sets.
Pairs of X,Y-values were generated using random normal deviates with mean 0 and standard deviations equal to the square roots of λ1 and λ2, that is, L1 and L2. For each of 100 sets of randomly generated data, a covariance matrix and ellipse parameters were calculated. Means and variances of these parameters were then calculated across the 100 data sets. In addition, bias in S was calculated as the difference between the value of S for the randomized data set (S′) and L1/L2. Data were generated for sample sizes (number of taxa) of 4, 9, 16, 25, 49 and 100, and for values of S of 1, 1·5, 2, 2·5, 3, 4, 5, 6, 7, 8, 9 and 10. These values cover the range of sample sizes and shape parameters commonly found in life-history data (Derrickson & Ricklefs 1988).
Because the first eigenvalue maximizes the variance of data projected onto the major axis, λ1 tends to be overestimated, especially when the shape of the ellipse is close to 1. As a result, λ2 tends to be underestimated (Jackson 1991). This bias in the eigenvalues is relatively small, however, and can be ignored for ellipses with shape parameters exceeding 2 (Fig. A1, left). Although estimated standard errors for the eigenvalues apply strictly only to large samples, they nonetheless predict standard errors calculated from eigenvalues of the sample data sets very closely (Fig. 1, right). Thus, standard errors calculated with equation A4 provide a reliable basis for statistical comparison of eigenvalues.
Directional cosines appear to be estimated without bias. In the randomized data sets, the value of the covariance was set to 0 and the expected value of the directional cosine a12 (= sin[θ1,X]) was therefore also 0. Calculated values were for the most part reasonably close to 0 (Fig. A2, left), although large deviations occurred when S was close to 1, that is, when an ellipse was nearly circular. Estimated standard errors of a12 (equation A6) predict calculated standard errors closely for sample sizes of 9 or more observations (Fig. A2, right).
The shape of a bivariate ellipse is always overestimated, and this was a serious problem for sample sizes of 4 (Fig. A3, left). As sample size increased, the calculated shape more closely approached the underlying shape of the sample distribution. Because there is no analytical solution for the bias and standard error of shape, simulation data were used to devise estimates of these parameters. In the case of the standard error of the shape parameter, the logarithm of the standard error of shape (SE[S]) was compared with the logarithms of sample size (n) and the shape itself (S) for sample sizes of 16 and above; the resulting regressions estimated shapes for smaller sample sizes poorly. Variation in SE(S) was independent of any interaction between log10(S) and log10(n) (F1,36 = 0, P = 0·96). With no interaction term included in the regression model, the intercept was 0·0187 (0·051 SE), and coefficients were 1·076 (0·030) for log10(S) and – 0·540 (0·030) for log10(n). This suggests that a reasonable estimate of SE(S) is S/√n, as shown in Fig. A311, right.
Bias in shape proved more difficult to predict accurately. This poses relatively little problem in comparative studies because the magnitude of the bias is generally less than the standard error of the estimate of shape and, as long as sample sizes are comparable, groups being compared should be subject to similar bias. From regressions of bias on shape and sample size, we determined that the bias (B) could be reasonably estimated by B = 2·5 S/n1·25 for S≥ 1·5 (Fig. A4). Notice that bias decreases rapidly with increasing sample size, so that for most samples this will not be a problem.
Having obtained estimates for the ellipse parameters, differences among taxa can be compared statistically by calculating t-values with unequal sample sizes and unequal variances (see, for example, Sokal & Rohlf 1981, p. 226, eqn 9·2).
The following is the program ELLIPSE.SAS, which contains statements for use with the SAS® statistical package to calculate the parameters of the bivariate ellipse. These calculations could also easily be built into a spreadsheet.