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Genetics and Molecular Biology

Print version ISSN 1415-4757On-line version ISSN 1678-4685

Genet. Mol. Biol. vol. 21 n. 4 São Paulo Dec. 1998 

Orthogonal projections and bootstrap resampling procedures in the study of infraspecific variation


Luiza Carla Duarte1, Fernando José Von Zuben2 and Sérgio Furtado dos Reis3
1Programa de Pós-Graduação em Zoologia, Universidade Estadual Paulista, Caixa Postal 179, 13506-900 Rio Claro, São Paulo, Brasil.
2Departamento de Computação e Automação Industrial, Universidade Estadual de Campinas, 13083-970 Campinas, São Paulo, Brasil.
3Departamento de Parasitologia, IB, Universidade Estadual de Campinas, Caixa Postal 6109, 13083-970 Campinas, SP, Brasil. Send Correspondence to S.F.R.




The effect of an increase in quantitative continuous characters resulting from indeterminate growth upon the analysis of population differentiation was investigated using, as an example, a set of continuous characters measured as distance variables in 10 populations of a rodent species. The data before and after correction for allometric size effects using orthogonal projections were analyzed with a parametric bootstrap resampling procedure applied to canonical variate analysis. The variance component of the distance measures attributable to indeterminate growth within the populations was found to be substantial, although the ordination of the populations was not affected, as evidenced by the relative and absolute positions of the centroids. The covariance pattern of the distance variables used to infer the nature of the morphological differences was strongly influenced by indeterminate growth. The uncorrected data produced a misleading picture of morphological differentiation by indicating that groups of populations differed in size. However, the data corrected for allometric effects clearly demonstrated that populations differed morphologically both in size and shape. These results are discussed in terms of the analysis of morphological differentiation among populations and the definition of infraspecific geographic units.




A description of the morphological variation among populations of a species is a fundamental step in the search for the geographical structure that underlies the recognition of infraspecific evolutionary units (Barrowclough, 1982; Thorpe, 1987; Smith et al., 1997). From conceptual and empirical standpoints it has been argued that such evolutionary units should include populations that share character complexes which are diagnostic in the sense that they define the boundaries of variation at this level (Barrowclough, 1982; Smith and Patton, 1988; Smith et al., 1997).

From a morphological perspective, variation among populations has been traditionally assessed using quantitative continuous traits as a means of uncovering differences in shape that may be useful in defining geographic units (Rohlf and Bookstein, 1987; Patton and Smith, 1990; DeQueiroz and Good, 1997). Such units can be distinguished on the basis of character and geographic discontinuity, uncovered from the ordination pattern of populations obtained from multivariate analyses of quantitative continuous traits (Smith and Patton, 1988; Sullivan and Best, 1997; DeQueiroz and Good, 1997). Nevertheless, ordination, based on the relative positions of population centroids, can be perturbed by variability in size as a result of different growth stages among individuals within a sample (Thorpe, 1983). This, in turn, may jeopardize the analysis of geographic variation and, consequently, the taxonomic and systematic conclusions drawn from the data (Thorpe, 1976, 1983; Bookstein, 1991; Klingenberg, 1996). This effect is well established for heterogeneous samples of juvenile and adult individuals (Thorpe, 1983; Rohlf and Bookstein, 1987). One solution to this problem is to use techniques that remove within-group allometric size effects (Rohlf and Bookstein, 1987; Klingenberg, 1996).

Another scenario that has not been fully appreciated occurs with organisms that continue to grow after they become adults, that is, organisms with indeterminate growth (Gaillard et al., 1997). Such a situation is common in morphometric analyses of systematic and evolutionary problems, and raises the question as to whether the analysis of differentiation patterns among populations is perturbed in organisms with indeterminate growth.

Here, we investigate the effect of size variation in quantitative continuous traits of adult individuals with indeterminate growth on the assessment of population differentiation. This question is relevant to analyses of population differentiation in general, especially when dealing with character complexes that can be expressed as shape differences that diagnose infraspecific geographic units. In the present study, variation in quantitative continuous traits due to indeterminate growth was quantified by partitioning the variance into within- and among-group components using random effect analysis of variance. The effect of within-group size attributable to growth on the analysis of population differentiation patterns was then investigated by comparing a set of quantitative continuous traits not corrected for size effects with the same data corrected for growth effects using Burnaby's (1966) procedure of orthogonal projections. Burnaby's procedure removes the confounding effects of allometric size variation attributable to growth in quantitative continuous traits by constructing a size vector and then projecting the original data points onto a space orthogonal to the size vector. The corrected and uncorrected data are then assessed for patterns of variation among populations using canonical variate analysis.

The usefulness of canonical variate analysis arises from the fact that it was mathematically designed to describe and summarize variation among populations defined a priori while taking into account variation within populations (Mardia et al., 1979; Krzanowski, 1988). Although canonical variate analysis has become an established procedure in systematic and evolutionary biology, it has been employed mostly in an exploratory manner in which individual scores or group means (= centroids) are simply plotted on the first few canonical axes (Mardia et al., 1979; Neff and Marcus, 1980). In the present study we have added inferential information to the graphical displays of population ordination based on recent developments in parametric bootstrap theory as applied to canonical variate analysis (Ringrose, 1996).

The combination of orthogonal projections, to remove the within-group size effect, with parametric boostrap canonical variate analysis is illustrated here in a study of infraspecific differentiation in a rodent, Thrichomys apereoides, a species in which variation in quantitative continuous traits is markedly influenced by indeterminate growth (Moojen et al., 1988). Although the present contribution focuses on infraspecific differentiation, the approach can in fact be applied to any level of morphological differentiation among organisms with indeterminate growth.



Thrichomys apereoides is an echimyid rodent that inhabits the savannas of eastern and central Brazil and Paraguay. Currently five subspecies are recognized (see Moojen, 1952; Petter, 1973; Mares et al., 1981; Alho, 1982; Mares and Ojeda, 1982; Bandouk and Reis, 1995; Bandouk et al., 1996). The present study assesses regional patterns of variation in T. apereoides in northeastern Brazil. Three hundred and fifty-tree specimens of T. apereoides from 10 sites were analyzed. All specimens were adults as defined by Moojen et al. (1988) based on the patterns of dental eruption and wear. The specimens are housed in the mammal collection of the Museu Nacional, Rio de Janeiro. The sites and sample sizes (n) were as follows: State of Ceará - Itapagé (3o 41'S, 39o34'W: n = 23) and Campos Sales (7o04'S, 40o23'W: n = 65); State of Paraíba - Princesa Isabel (7o44'S, 38o00'W: n = 14); State of Pernambuco - Bodocó (7o47'S, 39o55'W: n = 33), Triunfo (7o50'S, 38o07'W: n = 72), Caruaru (8o17'S, 35o38'W: n = 27), and Floresta (8o36'S; 38o34'W: n = 12); State of Alagoas - Santana do Ipanema (9o22'S, 37o14'W: n = 38); State of Bahia - Feira de Santana (12o15'S, 38o57'W: n = 37) and Palmeiras (12o31'S, 41o34'W: n = 32).

Quantitative continuous traits were defined in this study as the cranial distance measurements that form the basis for studies of the systematic relationships and evolutionary biology of mammals (Carleton, 1989; Voss et al., 1990). Fourteen cranial distance measurements defined by Bandouk and Reis (1995) were obtained for each specimen using digital calipers interfaced to a microcomputer. The measurements were occipito-nasal length (ON), basilar length (BL), zygomatic breadth (ZB), mastoid breadth (MB), rostral length (RL), nasal length (NL), rostral width (RW), diastema (DI), maxillary tooth row length (TR), bulla length (BU), rostral depth (RD), cranial depth (CD), mandible length (ML) and mandible depth (MD).

The variation in cranial distance measurements within populations generated by indeterminate growth was quantified by variance component analysis (Montgomery, 1984; Winer et al., 1991). Locality was treated as a random variable in the random effects ANOVA (Montgomery, 1984; Winer et al., 1991), and the variation in cranial measurements was partitioned into within- and among-locality components of variance. The within-locality component of variance thus quantified the effect of indeterminate growth on the cranial measurements. Because of unequal sample sizes, the following correction was used in the computation of variance components (Montgomery, 1984; pp. 74)

21n4a13fo1.GIF (1936 bytes)

where g is the number of random factor levels and ni is the size of the i-th sample.

The effect of indeterminate growth in T. apereoides on the variation in cranial measurements was removed using the approach described by Burnaby (1966). In this procedure, within-population growth of cranial distances was represented as a general size vector constructed using the quantitative continuous traits without any reference to the measurements themselves or to any external standard for age (Rohlf and Bookstein, 1987; Bookstein, 1991). The size vector was the eigenvector associated with the largest eigenvalue extracted from the variance-covariance matrix of the log-transformed cranial distance measurements (Bookstein et al., 1985; Rohlf and Bookstein, 1987). The pooled within-group covariance matrix, W, was computed as (Krzanowski and Radley, 1989)

21n4a13fo2.GIF (1859 bytes)

where n = S n1, g is the number of groups defined a priori, xij is the vector of cranial distances for the j-th individual in the i-th group, 2036f1.JPG (2086 bytes), and the symbol T denotes the operation of transposition.

The intragroup (growth) size vector, F1, was computed from the W matrix. The effect of size variation that resulted from growth within samples was partitioned out by projecting the data points onto an orthogonal subspace, independent of the size vector, F1. A projection matrix, L, that corrected for the effect of within-group size was computed as (Burnaby, 1966)

21n4a13fo3.GIF (1601 bytes)

where I is the identity matrix, and p is the number of variables. The original data matrix, X, was then multiplied by the projection matrix L to produce the ajusted, growth-invariant data matrix X' = XL. Both matrices, X and X', were used as the input for canonical variate analysis to describe the ordination of populations and to assess the pattern of covariation among quantitative continuous traits in order to permit description of the nature of the morphological differences among populations.

Inferential information was added to the canonical variate analysis by superimposing confidence regions on the plotted means in the canonical diagram (Ringrose, 1996), thereby producing the expected bounds of craniometric variability for the populations sampled. According to Ringrose (1996), the confidence regions are based on deviations from the sample group means projected onto the sample axes 2036f2.JPG (2375 bytes) around the population group means projected onto the sample axes 2036f3.JPG (2418 bytes) . In these expressions, a is an eigenvector, 2036f4.JPG (2303 bytes) and mi is the vector of population means. Since the expression for 2036f5.JPG (1232 bytes) allows for variability in the sample axes, the correct variance for the sample axes is (Ringrose, 1996) 2036f6.JPG (2472 bytes).

The variance of 2036f7.JPG (1198 bytes) is obtained from parametric bootstrap replicates of the data matrices X and X' that are obtained from the 2036f8.JPG (2160 bytes) distribution, where p is the number of variables, 2036f9.JPG (1210 bytes) is the vector of the i-th group means and 2036f10.JPG (1305 bytes) is the pooled within-group covariance matrix scaled by n - g. Assuming that the 2036f11.JPG (1198 bytes)are normal, the 2036f12.JPG (1188 bytes)distribution can be used to construct the confidence regions using equations given in Krzanowski and Radley (1989; see also Ringrose, 1996). In this study, 1,000 replicates of matrices X and X' were generated by sampling with replacement and 95% confidence regions were constructed around the centroids for each sample.

The relationship between the original cranial distances and the canonical axes was expressed as the Pearson product moment correlation coefficient. The values obtained were displayed in a bivariate plot that could be inspected for the contribution of the original variables to the canonical axes (Jolicoeur, 1959; Neff and Marcus, 1980; Patton and Smith, 1991).

All computations were carried out with MatLab (Moler et al., 1987). A program that implements Burnaby's (1966) orthogonal projection and Ringrose's (1996) parametric bootstrap of canonical variate analysis written for MatLab can be obtained from the authors.



The variation in cranial distances attributable to indeterminate growth within populations of T. apereoides relative to the effect of locality was partitioned by variance component analysis. Variation attributable to locality was high only for occipito-nasal length, intermediate for basilar length and mandibular length and low for all remaining cranial distance measurements (Table I). Variance component analysis shows that most variation in the cranial distances was residual, that is, attributable to the indeterminate growth of adult individuals within localities (mean = 79.13%; range, 19.89-99.19%; Table I).

21n4a13t1.GIF (17501 bytes)


The effect of variation within localities in cranial distances attributable to indeterminate growth on the ordination of populations and the pattern of covariation in the continuous traits that differentiate the populations of T. apereoides was evaluated by performing parametric bootstrap canonical variate analysis on both the uncorrected data and on the data corrected for growth effects (Burnaby, 1966). The expected bounds of craniometric variability included in the 95% confidence regions, as derived from the parametric bootstrap, are represented by ellipses for each population; the population centroids are indicated by dots.

For the uncorrected data set, the pattern of ordination revealed two groups of population separated along canonical variate 1, which explained 53.28% of the variation between groups (Figure 1A). One group consisted of the populations from Feira de Santana and Palmeiras in Bahia. These two populations were sharply separated from a major group that included all of the remaining populations (Figure 1A). Pearson product moment correlations between the original cranial variables and the first two canonical axes were positive for all cranial variables and showed a significant correlation with canonical variate 1, except for rostral width (RW; Table II). This result is depicted in Figure 1B where the correlations are represented as vectors that point in the same direction, indicating that all cranial distances increase jointly but at different rates. As a result, the populations are arrayed in a gradient of increasing size along this axis. Thus, populations of T. apereoides from Bahia located to the left of the canonical plot are characterized by individuals with a smaller overall size, whereas those in the major group of populations on the right are of larger size (Figure 1A,B).

21n4a13f1a.GIF (9019 bytes)

21n4a13f1b.GIF (5226 bytes)

Figure 1 - A, Bivariate plot of centroids (dots) and 95% parametric bootstrap confidence ellipses of the first two canonical variates, CV1 and CV2, for population samples of the rodent species Thricomys apereoides. The data have not been corrected for within-group size effects. The localities are indicated in the plot. B, Vector plots portraying the principal direction of variation in cranial distance variables in the plane of the first two canonical variates. See text for character abbreviations.


21n4a13t2.GIF (20778 bytes)


The procedure of Burnaby (1966) to correct for within-group size variation in cranial distances caused by growth was applied to the T. apereoides data, and the resulting F1 vector extracted from the pooled within-group covariance matrix explained 78.52% of the total variation. This vector had a highly significant positive correlation with all cranial distances, and therefore represents a general size vector (Table III; Rohlf and Bookstein, 1987). As expected from the dynamics of variation in quantitative continuous traits (Bookstein, 1991; Lynch and Walsh, 1998), each of the cranial distances in the size vector varied at different rates (Table III). Such a pattern of allometric variation is to be expected whenever sufficient size variation is present in a sample (Bookstein et al., 1985), as in the present study. The F1 vector was used to construct the projection matrix that produced the cranial distances adjusted for within-group growth effects. Parametric bootstrap canonical variate analysis applied to the adjusted data set yielded an ordination of T. apereoides populations identical to that obtained for the uncorrected data set, as evidenced by the absolute and relative positions of the population centroids (Figures 1A, 2A). This was verified quantitatively by calculating Mantel's matrix correlation between the matrices of Mahalanobis D2 statistic (Manly, 1985; Mardia et al., 1979) for the uncorrected and corrected data (r = 0.9929, P < 0.001). Cranial distances showed positive and negative correlations with canonical variate 1 so that this axis can be interpreted as a shape vector (Table IV; Rohlf and Bookstein, 1987). Again, the correlations between cranial distances and the canonical variate axes can be pictorially displayed as vector correlations (Figure 2B). That the vectors point in the direction of positive and negative correlations indicates that the two groups of T. apereoides populations differ in cranial shape. Discrimination between these two groups along canonical variate 1 results primarily by a contrast between rostral length (RL), rostral width (RW) and nasal length (NL) which influence the projection of the samples from Feira de Santana and Palmeiras, versus tooth row length (TR), mandibula length (ML) and mastoid breadth (MB) which project the remaining populations in the opposite direction (Figure 2A,B). The populations from Feira de Santana and Palmeiras have a proportionally broader and longer rostrum but shorter tooth row and mandible combined with a narrower mastoid. Individuals from the major group of populations typically have a longer mandible and tooth rows combined with a wider mastoid but a slender and shorter rostrum (Figure 2A,B).

21n4a13t3.GIF (17924 bytes)


21n4a13f2a.GIF (8537 bytes)

21n4a13f2b.GIF (6056 bytes)

Figure 2 - A, Bivariate plot of centroids (dots) and 95% parametric bootstrap confidence ellipses of the first two canonical variates, CV1 and CV2, for population samples of the rodent species Thricomys apereoides. The data have been corrected for within-group size effects. The localities are indicated in the plot. B, Vector plots portraying the principal direction of variation in cranial distance variables in the plane of the first two canonical variates. See text for character abbreviations.



21n4a13t4.GIF (20323 bytes)




Variance component analysis showed that although only adult T. apereoides were sampled in this study there was considerable variability (mean of 79.13%) in cranial measurements within populations. This variability, which resulted from indeterminate growth, increased the cranial distances over time. Indeed, distance measurements are highly correlated because of their joint dependence on growth (Rohlf and Bookstein, 1987; Bookstein, 1991). However, this substantial component of variability in and covariation between cranial distances did not influence the pattern of population ordination derived from canonical variate analysis. Thus, the same two groups of T. apereoides populations were obtained by canonical variate analysis of the uncorrected and the growth-corrected data. The absolute and relative positions of the centroids were identical in both cases (Figures 1A and 2A) as confirmed quantitatively by the very high value for Mantel's matrix correlation between matrices of the Mahalanobis D2 statistic. Thus, the variance in size associated with indeterminate growth did not influence population ordination, and no bias was introduced by the presence of adult individuals that differed substantially in size. Such a bias can occur when the sample includes a mixture of juveniles and adults, and may alter the ordination pattern of population centroids in the reduced space of canonical variates (Thorpe, 1983). The uncorrected and growth-corrected data used here predicted the same ordination for populations of T. apereoides, indicating that indeterminate growth did not compromise the taxonomic and systematic conclusions drawn from morphometric data.

The component of variability in and covariation between cranial distances attributable to the indeterminate growth of individuals can nevertheless obscure the nature of the morphological differences between populations uncovered by canonical variate analysis. The presence of such covariances among cranial distance variables that reflects their joint dependence on growth within populations is mirrored by the vector plot in Figure 1B, which, taken at face value, suggests that the two groups of populations differ in cranial size alone. As emphasized by Bookstein (1991), the joint dependence of distance variables on an increase in overall size attributable to growth does prevent the demonstration of shape differences among groups. This effect was clearly seen in the present study. When the effect of growth on cranial distances within populations of T. apereoides was removed by Burnaby's procedure, the pattern of covariance that emerged among the cranial measurements could be interpreted in terms of shape. Thus, while individuals from the populations from the State of Bahia were indeed smaller in cranial size, they also differed from the other major group of populations in their cranial shape.

The basis for infraspecific systematics is currently regarded as comprised of evolutionary units with a structure that involves groups of populations having both character and geographic discontinuity (Barrowclough, 1982; Thorpe, 1987; Smith and Patton, 1988; Patton and Smith, 1990; Smith et al., 1997). For quantitative continuous characters, the ordination of populations by multivariate statistical procedures, such as canonical variate analysis, can be taken as evidence for the existence of geographic units.

Canonical variate analysis has been employed with great success in studies of systematic and evolutionary divergence among groups sampled for quantitative continuous traits (e.g., Thorpe, 1987; Brown and Perez-Mellado, 1996; Ferreira and Rezende, 1997). However, such analysis has been used typically in an exploratory manner in which individual scores or group means (centroids) are simply plotted on the first few canonical variates and the low-dimensional display of the data is used to investigate possible relationships such as similarities or differences among groups (Mardia et al., 1979). The recent application of resampling procedures to problems in systematics and evolutionary biology (Felsenstein, 1988; Crowley, 1992; Li, 1997) has made it possible to move from an exploratory to an inferential approach. The parametric boostrap confidence regions for canonical variates developed by Ringrose (1996) and introduced here are particularly suited to the study of infraspecific differentiation for two reasons. First, population variation in quantitative continuous traits is expressed in terms of the expected bounds of variability included in the elliptic confidence regions. Second, since the parametric bootstrap involves resampling from the distribution of the data, one is effectively assessing the statistical stability of the ordination of populations in canonical space (Ringrose, 1992). This should lead to patterns that are robust to sampling, and therefore provide greater reliability to the evolutionary units uncovered by canonical variate analysis.

The results obtained here for adult organisms with indeterminate growth show that the assessment of population differentiation can be jeopardized if the effect of growth upon quantitative continuous variables is not removed. The effect on the patterns of covariance among quantitative continuous traits is marked, although the discrimination and ordination of populations in canonical space does not appear to be influenced. If the data are not adjusted for growth effects, only covariance among traits attributable to their joint dependence on growth is expressed, thereby confounding interpretation of the nature of the morphological differentiation. In this case, size alone becomes the dominant theme of morphological variation and the interesting aspects of shape that may be important in diagnosing infraspecific evolutionary units are masked by allometric effects. The application of orthogonal size correction and parametric bootstrap canonical variate analysis to the data for T. apereoides avoided these problems and provided evidence for the existence of two evolutionary units that share character cohesiveness in terms of shape and geographic continuity. Given the importance attached to shape, it is of paramount importance that the effect of size attributable to growth be taken into consideration in any search for patterns of systematic and evolutionary differentiation. As demonstrated here, these confounding effects can be detected and removed, and the corrected data then used in conjunction with inferential procedures and standard multivariate statistical analyses.



This research was supported by a grant from the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP, No. 1996/3429-0). We are grateful to L.R. Monteiro (Departamento de Zoologia, UNESP, Rio Claro) and L.B. Klaczko (Departamento de Genética, UNICAMP) for their suggestions and careful reviewing of the manuscript. We are indebted to an anonymous reviewer for the revision of the manuscript. The authors also thank Stephen Hyslop (Departamento de Farmacologia, UNICAMP) for reviewing the English of the manuscript. The specimens examined in this study are deposited in the Museu Nacional, Rio de Janeiro. We thank L.B.F. Oliveira, J.A. de Oliveira and L.O. Salles (Museu Nacional, Rio de Janeiro) for allowing access to the specimens. We also thank S.M. Franco (Museu Nacional, Rio de Janeiro) for help in handling the specimens. L.C.D. and S.F.R. are recipients of by fellowships from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil. Publication supported by FAPESP.




A influência do aumento em caracteres quantitativos contínuos devido ao crescimento indeterminado sobre a análise de diferenciação entre populações foi investigado utilizando como exemplo um conjunto de dados de variáveis craniométricas em 10 populações de uma espécie de roedor. Dois conjuntos de dados, um não corrigido para o efeito alométrico do tamanho e um outro corrigido para o efeito alométrico do tamanho utilizando um método de projeção ortogonal, foram analisados por um procedimento "bootstrap" de reamostragem aplicado à análise de variáveis canônicas. O componente de variância devido ao crescimento indeterminado dentro das populações foi significativo para a maioria das medidas de distâncias, o que não influenciou a ordenação das populações, conforme evidenciado pela posição relativa dos centróides. O padrão de covariância entre as variáveis de distância que foi utilizado para inferir a natureza das diferenças morfológicas foi, no entanto, fortemente influenciado pela variação nas medidas de distâncias dentro das populações. O conjunto de dados não corrigido resultou em uma interpretação errônea sobre a natureza da diferenciação morfológica, sugerindo que as populações diferiram somente em tamanho. O conjunto de dados corrigido para o efeito alométrico, por sua vez, demonstrou claramente que as populações diferiram, não somente no tamanho, mas também na forma. Os resultados são discutidos em termos da diferenciação das populações em forma e tamanho no contexto da definição das unidades geográficas infraspecíficas.




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(Received March 5, 1998)

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