Zelditch M.L. (и др.) Geometric Morphometrics for Biologists: a primer

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288 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

for Group 1 are shown in black, those for Group 2 are shown in red. The position of

the landmarks is determined by the coordinates of the reference form. You can edit

the plot using the options on the Image Control pull-down menu (the options are to

alter line width, the size of the symbols for the landmarks, and to fill the symbols). To

remove the axes surrounding the plots, use the Axis Controls pull-down menu (also

located on the toolbar up top). You may need to rotate the plots if the orientation is

not interpretable. If so, use the Reference Rotation Active radio button at the bottom

center of the interface to rotate the reference interactively. Alternatively, if you know

the angle of rotation you need, you can type it into the Default Ref Angle window,

in the yellow field below the red Exit button.

If you would like to plot the difference between the two SA1s using a different plot-

ting style, such as a deformation grid, you can save the SAs (along with the reference

form) by going to the File pull-down menu located on the toolbar. These vectors can

be input into the program VecDisplay (described in Chapter 10), which shows the

difference between two vectors or their sum using a variety of display options.

2. SVD Block2, Group1+2 shows SA1 for the second block for both groups simulta-

neously. The plotting styles and editing options are as described for (1). This option

is not available if Block 2 is not homologous between groups.

3. SVD Block1, Group1 shows SA1 for Block 1 of Group 1; the available plotting styles

and editing options are the same as for PLSMaker, described above.

4. SVD Block1, Group2 shows SA1 for Block 1 of Group 2. The graphical options are

as described in (3).

5. SVD Block2, Group1 shows SA1 for Block 2 of Group 1. The graphical options are

as described in (3).

6. SVD Block2, Group2 shows SA1 for Block 2 of Group 2. The graphical options are

as described in (3).

7. Data Block1, Group1+2 shows the landmarks for the first block for both groups;

the data for Group 1 are in blue, those for Group 2 are in red. The Procrustes GLS

superimposition is the only option. Editing options are as given in (1).

8. Data Block2, Group1+2 is the same as (7), except that Block 2 is shown (this option

is not available if the Block 2 is not homologous between groups).

9. PCA Block1, Group1 shows PC1 for Block 1 in Group 1. Display and editing options

are as given in (3).

10. PCA Block2, Group1 shows PC1 for Block 2 in Group 1. Display and editing options

are as given in (3).

11. PCA Block1, Group2 shows PC1 for Block 1 in Group 2. Display and editing options

are as given in (3).

12. PCA Block2, Group2 shows PC1 for Block 2 in Group 2. Display and editing options

are as given in (3).

13. PCA +SVD Block1, Group1 shows PC1 and SA1 for Block 1 of Group 1. SA1 and

PC1 are displayed by vectors of relative landmark displacements (the position of the

landmarks is determined by the coordinates of the reference form). Those for Group 1

are shown in black, those for Group 2 in red. Editing options are as described above

for (1).

14. PCA +SVD Block2, Group 1 is the same as (13), except that the plot shows PC1 and

SA1 for Block 2.

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PARTIAL LEAST SQUARES ANALYSIS 289

15. PCA +SVD Block1, Group2 is the same as (13), except that the plot shows PC1 and

SA1 for Group 2.

16. PCA +SVD Block2, Group2 is the same as (14), except that the plot shows Group 2.

17. −PCA +SVD Block1, Group1 allows you to reverse the direction of the PC. The

signs of PCs and SAs are arbitrary; you may find that PC1 and SA1 look nearly

identical except that the arrows point in opposite directions.

18. −PCA +SVD Block2, Group1 is the same as (17), except that the plot shows Block 2.

19. −PCA +SVD Block1, Group2 is the same as (17), except that the plot shows Group 2.

20. −PCA +SVD Block2, Group2 is the same as (18), except that the plot shows Group 2.

21. PCA Block1, Group 1+2 shows PC1 for Block 1 for both groups simultaneously.

The PCs are displayed by pairs of vectors of relative landmark displacements. Those

for Group 1 are shown in black, those for Group 2 in red. The positions of the land-

marks are determined by the coordinates of the reference form. Editing options are

as given for (1).

22. PCA Block2, Group 1+2 is the same as (21), except that Block 2 replaces Block 1.

23. −PCA Block1, Group 1+2 is the same as (21), but reverses the sign of one of the PCs.

24. −PCA Block2, Group 1+2 is the same as (22), but reverses the sign of one of the PCs.

Comparing correlations between different pairs of blocks of a single group

In this analysis, the two “groups” are competing hypotheses of integration. The same set

of landmarks represents Block 1 for both hypotheses because we are asking if that block

is more highly correlated with one Block 2 than with another Block 2. Therefore, the first

step is to turn off the default option that Block 2 is homologous between groups. The

same file is input as Block 1 of both groups, and two different files are input as the two

Block 2s. To do the analysis, click on Do SVD 2Block to see the preliminary results (the

statistical analysis will be done when you click on Bootstrap SVD Angle). If you want

more than 100 bootstraps, or an α level other than 0.05, type your preferences in the

boxes provided. The results will appear in the Auxiliary Results box (another window).

The first results are the angles between Block 1 and each of the two “groups”. Below that

are the correlations between Block 1 and each Block 2, giving the observed correlation,

its confidence interval, and its standard error (which can be used in analytic tests of the

difference between correlations). The final three lines are the results of the resampling-

based test of the equality of correlations.

The variety of output files that can be saved, and the options for graphical displays, are

detailed above.

References

Bastir, M., Rosas, A. and Sheets, H. D. (2004). The morphological integration of the hominoid

skull: a partial least squares and PC analysis with morphogenetic implications for European Mid-

Pleistocene mandibles. In Developments in Primatology: Progress and Prospects (D. Slice, ed.),

in press. Kluwer Academic/Plenum Press.

Bookstein, F. L. (1982). The geometric meaning of soft modeling, with some generalizations. In

Systems Under Indirect Observation: Causality–Structure–Prediction (K. G. Jöreskog and H.

Wold, eds) pp. 55–74. North Holland Publishing Co.

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290 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Bookstein, F. L., Gunz, P., Ingeborg, H. et al. (2003). Cranial integration in Homo: singular warps

analysis of the midsagittal plane in ontogeny and evolution. Journal of Human Evolution, 44,

167–187.

Corti, M., Fadda, C., Simson, S. and Nevo, E. (1996). Size and shape variation in the mandible of

the fossorial rodent Spalax ehrenbergi.InAdvances in Morphometrics (L. F. Marcus, M. Corti,

A. Loy et al., eds) pp. 303–320. Plenum.

Hingst-Zaher, E., Marcus, L. F. and Cerqueria, R. (2000). Application of geometric morphometrics

to the study of postnatal size and shape changes in the skull of Callomys expulsus. Hystrix, 11,

99–113.

Houle, D., Mezey, J. and Galpern, P. (2002). Interpretation of the results of Common Principal

Components Analysis. Evolution, 56, 433–440.

Jöreskog, K. G. and Wold, H. (eds). Systems Under Indirect Observation: Causality–Structure–

Prediction. North Holland Publishing Co.

Klingenberg, C. P., Badyaev, A. V., Sowry, S. M. and Beckwith, N. J. (2001). Inferring developmental

modularity from morphological integration: analysis of individual variation and asymmetry in

bumblebee wings. American Naturalist, 157, 11–23.

Lowe, A. A., Özbeck, M. M., Miyamoto, K. and Fleetham, J. A. (1997). Cephalometric and demo-

graphic characteristics of obstructive sleep apnea: an evaluation with partial least squares analysis.

The Angle Orthodontist, 67, 143–154.

Lundrigan, B. (1996). Morphology of horns and fighting behavior in the family Bovidae. Journal of

Mammalogy, 77, 462–475.

Rohlf, F. J. and Corti, M. (2000). Use of two-block partial least squares to study covariation in

shape. Systematic Biology, 49, 740–753.

Rüber, L. and Adams, D. C. (2001). Evolutionary convergence of body shape and trophic morpho-

logy in cichlids of Lake Tanganyika. Journal of Evolutionary Biology, 14, 325–332.

Sampson, P. D., Streissguth, A. P., Barr, H. M. and Bookstein, F. L. (1989). Neurobehavioral effects

of prenatal alcohol: part II. Partial least squares analysis. Neurotoxicology and Teratology, 11,

477–491.

Streissguth, A. P., Bookstein, F. L., Sampson, P. D. and Barr, H. M. (1993). The Enduring Effects

of Prenatal Alcohol Exposure on Child Development: Birth through Seven Years, A Partial Least

Squares Solution. International Academy for Research in Learning Disabilities monograph series

no. 8. University of Michigan Press.

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PART

III

Applications of Morphometric

Methods to Complex

Hypotheses

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Disparity and variation

Disparity and variation are closely allied concepts – both refer to the general idea of “vari-

ety.” Disparity usually signifies the variety of a group of species and is the outcome of evo-

lutionary processes; variation, on the other hand, refers to the variety of individuals within

a single (homogeneous) population and is the raw material necessary for evolution. In light

of the theoretical distinction between the two concepts, it may seem difficult to cover both

in a single chapter. However, the distinction between the concepts lies in the processes that

produce them and the theories that predict them. The metric (or formula) for measuring

disparity among species is the same as that used to measure variation within a species.

Because the same metric is used to measure both, we cover them both in the same chapter.

Even so, to avoid confounding concepts that have little in common aside from a metric,

we begin by reviewing their biological meanings, then turn to the issue of measurement.

Disparity

Disparity may be an unfamiliar term to many biologists, but it has emerged as a major

theme in the paleobiological literature. The term was introduced to clarify the distinction

between two notions of diversity that were often confounded: (1) phenotypic variety (often

but not always morphological), and (2) taxonomic richness. Over the past decade, owing

largely to work by Foote (especially Foote, 1990, 1993a, 1993b) the distinction between

them has been clarified – a major step towards increasing both conceptual clarity and

methodological rigor. In the early literature the number of taxa was often used as a measure

of “disparity,” but, as Foote showed (1993b), and as many other studies have confirmed,

the number of taxa increases even as their morphological variety decreases.

To date, most studies of disparity have focused on its temporal dynamics over a geo-

logical time scale. The chief questions addressed by such studies are:

1. What is the temporal pattern of disparity?

2. What evolutionary processes explain those patterns?

Such studies are almost invariably based on fossils because they require sampling disparity

at multiple times in the geological record. Some groups studied in this way include Cam-

brian marine arthropods (Foote and Gould, 1992; Wills et al., 1994), Paleozoic blastozoans

ISBN 0–12–77846–08 All rights of reproduction in any form reserved

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294 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

(e.g. Foote, 1992), stenolaemate bryozoans (Anstey and Pachut, 1995), crinoids (e.g.

Foote, 1994; Ciampaglio, 2002), gastropods (Wagner, 1995) and Ordovician trilobites

(Miller and Foote, 1996). The growing empirical literature on disparity repeatedly doc-

uments a surprising historical pattern: disparity initially increases and then stabilizes or

even decreases while the number of taxa increases.

Efforts to explain this pattern have focused on two classes of hypotheses: ecological

and developmental. Ecological hypotheses postulate that ecological space is initially open

and then becomes saturated; limits on disparity are thought to arise from the structure

of the ecological space. In contrast, developmental hypotheses propose an intrinsic expla-

nation for limits on disparity – the acquisition of developmental constraints that stabilize

morphology (see Wagner, 1995 and Ciampaglio, 2002 for reviews of hypotheses and

approaches to testing them). Whether any explanation is even needed has been questioned

in a profound (if difficult) theoretical analysis (Gavrilets, 1999). At present it is not clear

what we ought to expect from disparity under plausible models; nor is it clear what role

artifacts might play in the patterns detected by empirical analyses. It is also difficult to

isolate causal factors that might explain the temporal dynamics of disparity because of the

multiplicity of uncontrollable factors that can influence those dynamics, including rates of

speciation and extinction, selectivity of extinction or speciation that is non-random with

respect to morphology, the magnitude of change within a lineage, and factors potentially

limiting that magnitude (such as developmental and selective constraints).

Of the various factors that can influence disparity, constraints may be the least under-

stood – partly because they are rarely documented prior to analyzing disparity. Instead,

constraints are inferred from the data, even though it is not clear how either developmental

or selective constraints ought to influence disparity. Both sorts of constraints are thought

to limit disparity, which may seem intuitively obvious; however, like many intuitions, it

may be faulty. We know little about the impact of either sort of constraint on disparity, and

determining their impacts will require studies that document constraints independently of

such supposed effects. We cannot simply infer constraints from decreases in disparity when

we do not know if they generally decrease disparity. Instead, we need to determine whether

development is constrained or not, and then ask how those constraints affect disparity.

In at least one case, developmental constraints are inferred to increase disparity (Zelditch

et al., 2003).

Studies of disparity of living taxa are still relatively rare, but they have been used to

address basic issues in evolutionary biology – such as whether decoupling of integrated

parts increases disparity (Schaefer and Lauder, 1996), whether biomechanical and mor-

phological disparity are related to each other (Hulsey and Wainwright, 2002), and whether

developmental constraints might limit disparity (Zelditch et al., 2003). Surprisingly few

studies have tried to relate ecological heterogeneity and morphological disparity, an

obviously important direction for future research (Roy and Foote, 1997).

Any biological explanation for an empirically documented pattern rests on the assump-

tion that the pattern is real. Whether it is real or an artifact depends partly on how disparity

is measured, and also on the sampling design. Both metrics and sampling designs have been

foci of critical reviews. In particular, a number of critics have taken issue with the phenetic

approach to disparity implicit in the use of a variance as its metric (e.g. Wills et al., 1994).

Alternative metrics, which measure change along branches of a phylogeny, have been rec-

ommended, but they are still in their infancy. Such metrics are difficult to apply when

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DISPARITY AND VARIATION 295

ancestors have not been sampled (or are unknown), and they also pose an interpretative

challenge because they redefine disparity, replacing the idea of variety (around an average)

with that of directed change away from the ancestor (see Wills et al., 1994; Wagner, 1997;

Smith and Lieberman, 1999). A second criticism is that measures of disparity typically

do not consider the biological significance of the contributing variables. It is conceivable

that large morphological changes could have few biological consequences, and some small

changes affecting just a few morphological details could have profound consequences for

function. In that light, weighted measures of disparity that take the biological significance

of the changes into account might seem more justified than measures of disparity per se

(see Wagner, 1995).

For recent reviews of the literature, including critical discussions of metrics and meth-

ods, and summaries of empirical studies, see Foote (1997), Ciampaglio et al. (2001) and

Wills (2001).

Variation

Variation within populations is a major theme in evolutionary biology because it is so fun-

damental to evolution – phenotypic variation provides the opportunity for selection to act,

and genetic variation enables selection to effect change. Variation is the raw material on

which selection acts, and its structure can influence the outcome of selection. Because evo-

lution can be constrained by limited or biased variance, the variance–covariance matrix is

sometimes viewed as an intrinsic constraint on evolution; such limits or biases arising from

developmental processes are developmental constraints (see Maynard Smith et al., 1984).

Although that view of variation emphasizes its role as a potential constraint, the structure

of (co)variation itself may be molded by selection. Theoretical models predict that pheno-

typic and genetic (co)variance structures evolve to match patterns of developmental and

functional integration (e.g. Lande, 1980; Cheverud 1982, 1984; Wagner, 1988; Wagner

and Altenberg, 1996). This matching is expected to result from differential elimination

of pleiotropic effects between members of different functional complexes, combined with

the maintenance (or augmentation) of pleiotropic effects within a complex. There is much

empirical evidence that phenotypic and/or genetic covariances reflect developmental and

functional relationships among traits, a conclusion based on many exploratory studies

(Olson and Miller, 1958; Berg, 1960; Van Valen, 1962, 1970; Gould and Garwood, 1969).

In addition, many studies have deduced the structure of (co)variation among measurements

from developmental and functional theories (e.g. Cheverud, 1982, 1995; Zelditch and

Carmichael, 1989; Kingsolver and Wiernasz, 1991; Marroig and Cheverud, 2001). Most

studies concentrate on a single developmental stage, but a few have examined the onto-

genetic dynamics of variance (e.g. Foote, 1986; Zelditch, 1988; Zelditch and Carmichael,

1989; Zelditch et al., 1993).

The concept of variation is also central to systematic studies, both because systematists

study evolutionary processes and also because the systematic value of a character is partly

a function of its variability. In the systematics literature the term “variation” is sometimes

used very broadly, such as when talking about “ontogenetic variation.” In that context the

“variation” results from the mixture of ages in the sample; because individuals differ in age,

they differ in everything that changes with age. Ontogeny is thus the factor explaining the

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296 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

variation within the sample, but that is not the variance on which selection acts (unless we

seriously entertain the idea that selection favors adults over juveniles, which is unlikely in

the first place and would not have any evolutionary consequences in the second). To study

the variance on which selection could act, we would first need to remove the variation

resulting from the heterogeneity of the sample. Should removing that variation strike you

as an improper manipulation of the data, ask yourself whether it is reasonable to imagine

that selection acts on it.

A classic hypothesis linking variance to disparity is often called the “Kluge–Kerfoot”

phenomenon: traits that vary the most (within populations) are also the ones that most

differentiate populations (Kluge and Kerfoot, 1973). The original empirical support for

the hypothesis was harshly criticized on methodological grounds (e.g. Sokal, 1976; Rohlf

et al., 1983), but the hypothesis has re-emerged in the recent literature with more impressive

empirical support; the dimension of greatest (genetic) variance is sometimes regarded as

the evolutionary line of least resistance (e.g. Schluter, 1996).

Metrics for disparity and variance

As mentioned above, there is no universally accepted metric for disparity (there is for

variation, so we will focus on disparity throughout this section). One major distinction

among available metrics is whether they measure the variety of forms in a sample or

the diversification along branches of a cladogram. The first could be viewed as a static

measure of disparity, the second as a dynamic measure of diversification. We will focus on

the first approach for two reasons: the first is that we define disparity in terms of variety

rather than in terms of magnitudes or rates of diversification; the second is that ancestral

morphologies are rarely observed and known to be ancestral. Without direct observations

of known ancestors, ancestral morphologies must be inferred, and the methods for inferring

ancestral morphologies are still a matter of dispute.

Metrics for the variety of observed forms can be subdivided into two broad classes: (1)

those applied to continuously valued variables (such as size and shape) and (2) those applied

to ordinal or categorical data. The distinction (which is based on the type of data) is impor-

tant, because continuously valued variables are measured on an unambiguous scale, which

is not the case for ordinal or categorical data. For example, if we want to know how dif-

ferent two organisms are, and one is 10 mm while the other is 12 mm, we can say that their

difference is 2 mm. Given a third, which is 14 mm, we would say that the difference between

the first and third is 4 mm, and the difference between the second and third is 2 mm. Because

2 mm is equal to 2 mm, we can say that the difference between the first and second organ-

isms is equal to that between the second and third. We might choose a scale that takes

proportions into account, so that 2 mm counts for more when organisms are near 1 mm

than when they are near 100 mm, but still the scale is unambiguous and measurements are

mathematically commensurable. In contrast, if we classify morphologies into three types –

“one,” “two” and “three” – “one” and “two” are taken to be one unit apart, as are “two”

and “three,” but we cannot say that the difference between “one” and “two” is equal to the

difference between “two” and “three.” Perhaps the first two types differ by the presence

or absence of a notochord, whereas the second two differ by the presence or absence of a

tubercle on the tibia. The problem faced here does not arise when coding discrete classes for

phylogenetic analyses because the characters may be equally informative in that context.

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DISPARITY AND VARIATION 297

However, weighting them equally in studies of disparity implies that they contribute equally

to morphological variety. Fortunately size and shape data are continuously valued vari-

ables, so we will concentrate on metrics of disparity suited to continuously valued variables.

The metrics for continuously valued variables can be either Euclidean or non-Euclidean

distances, although most workers use Euclidean distances. We can also distinguish among

metrics by whether the measures are of: (1) linear distances between forms (correspond-

ing to a standard deviation); (2) squared distances between forms (corresponding to a

variance); or (3) volumes. Measures of volume might seem most desirable because they

could appear to capture the most information about the size of the occupied morphospace.

Unfortunately no satisfactory measure of volumes is available yet, because measuring them

involves multiplication rather than addition. When distances along dimensions are multi-

plied, a trivial distance along one deflates the size of the space. For example, if we multiply

distances along several dimensions, such as 0.4, 0.3 and 0.2, we get a volume of 0.024, and

if we multiply that product by 0.002, we get 0.000048 – therefore, adding information

about that fourth dimension reduces the size of the space to nearly zero. Logically, we

would expect that the additional information would only increase the size of the space.

Another disturbing feature of this volume-based approach to disparity is that the volume of

several slightly disparate variables can be far larger than the volume of three very disparate

variables and one nearly invariant variable. For example, above we considered a case of

three disparate variables and one that is nearly invariant. We might have another case in

which there are also four variables, each with a disparity of 0.1; the product of (0.1)(0.1)

(0.1)(0.1) =0.0001, which is more than twice the volume of the first case (0.000048). In

contrast, if we restrict our analysis to only the first three variables, the disparity would be

(0.1)(0.1)(0.1) =0.001 – substantially less than that of the first case (0.024).

If we had an objective and non-arbitrary method for ignoring some dimensions (so that

their low levels of disparity do not deflate the space), we could circumvent these problems.

However, all methods for deciding whether to exclude a variable depend on subjective

arguments, and the decision about whether to exclude a variable can have an enormous

impact on the results. For that reason, we prefer metrics based on standard deviations and

variances. Both standard deviations and variances are equally useful metrics, and there is

no reason to debate which of them is preferable because one is easily derived from the other.

The major reason for using a variance is that variances are additive. Because of that prop-

erty, we can calculate the overall disparity of a group, then partition it into the contribution

made by each taxon (the partial disparity of that taxon; Foote, 1993a). The additivity of

variances means that the sum of partial disparities equals the overall disparity. However,

it is worth noting that the two measures weigh outliers differently, and consequently their

results can differ. Standard deviations and variances are not linearly related, and a highly

distinctive taxon has a much greater impact on a variance than on a standard deviation.

Measuring disparity

To measure morphological disparity (MD) by a variance, we calculate:

MD =



j=1

(N − 1)

(12.1)