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

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18 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

fitness, or of any other factor of interest. When we do not have any such factors in mind

in advance of a study, we can explore the data algebraically, using the methods of matrix

algebra to determine if any interesting patterns emerge (principal components analysis,

PCA, is an example of this kind of algebraic exploration).

Because many biologists begin a study by exploring patterns in the data, the section on

analytic methods begins with an overview of ordination methods (Chapter 7). These are

useful for extracting simple patterns from complex multidimensional data because they

provide a space of relatively low dimensionality, capturing most of the variation among

specimens (PCA), or most of the differences among groups (canonical variates analysis,

CVA). We explain the algebra underlying these methods, compare them, and discuss when

each is appropriate in light of particular biological questions.

The next three chapters cover methods of statistical analysis. We begin with an overview

of computer-based statistical methods, i.e. computer-intensive methods for constructing

confidence intervals and/or hypothesis testing, such as bootstrapping and Monte Carlo sim-

ulations (Chapter 8). The next two chapters discuss the two broad classes of hypotheses that

are conventionally tested statistically. Chapter 9 addresses hypotheses about the effects of

an independent categorical variable – Hotelling’s T

-test, analysis of variance (ANOVA),

and multivariate analysis of variance (MANOVA); Chapter 10 addresses hypotheses about

the effect of a continuous variable on shape (regression). The final chapter in this sec-

tion, Chapter 11, covers a method new to morphometric studies, one that analyzes the

covariance between two blocks of variables, partial least squares analysis.

The third section covers applications of morphometric methods to realistically complex

biological hypotheses, addressing more than just existential questions and requiring more

of the answers than just descriptions. We begin with hypotheses that are often stated only

in words, discuss framing them in the terms of more precise formal models, and then

reframe these models into terms suited to statistical analysis. Once a hypothesis has been

framed in the last set of terms, data analysis can proceed in a quite straightforward fashion,

combining an array of techniques. As examples of complex biological questions we include

those posed by studies of disparity and variance (Chapter 12), the analysis of relationships

between ontogeny and phylogeny (Chapter 13), and also systematics (Chapter 14). The

latter chapter represents a bridge between complex but tractable questions and subjects in

need of additional tools.

The final chapter of this book (Chapter 15) briefly discusses two important areas in

which a full set of tools have not been developed yet: (1) methods for analyzing three-

dimensional coordinate data, and (2) methods for analyzing shapes of curves where no

discrete anatomical loci can be found (by locating and analyzing points called “semi-

landmarks”). Neither of these subjects is properly part of a primer that focuses on

well-developed, uncontroversial methods, but both are important for biologists, and both

are subjects of intensive ongoing work. In presenting these subjects we concentrate on the

major points of departure (both conceptual and practical) from the primary subject of this

book, the analysis of two-dimensional configurations of landmarks.

The terminology of statistical shape analysis can be daunting – there are many unfamiliar

words and many terms differ by only a single letter or subscript. Thus we conclude this

book with a glossary of terms, including general statistical terms (e.g. population, sample)

and more specialized terms of shape analysis (e.g. Procrustes distance, partial warps).

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INTRODUCTION 19

Software and other resources

Geometric morphometrics studies require fairly specialized software, not so much to ana-

lyze the data as to depict the results graphically. Fortunately, the necessary software is

readily and freely available. As Mac users will soon realize, virtually all the compiled

software runs under Microsoft Windows.

At present, one major source of software is located at the SUNY Stony Brook website:

http://life.bio.sunysb.edu/morph. Follow the link to Software (the rest of the links go to

other valuable resources, including information about meetings, courses, and a directory

of many people interested in morphometrics, with links to their webpages). We recom-

mend that anyone planning a morphometric study downloads the videodigitizing program,

TPSDig. Not only is this a well-designed and extremely useful program, but also many

writers of morphometric software assume that is the one used for data collection, so the

format in which it outputs the data (TPS format) has become the standard input format

for several programs. There are other useful programs in the TPS series, but we generally

do not provide detailed instructions for using them because we can neither anticipate nor

control any changes in them.

Another major source of morphometric software is located at the website:

http://www.canisius.edu/∼sheets/morphsoft.html. This software, called the Integrated

Morphometrics Programs (IMP), is written by one of us (HDS) and every method of

analysis discussed in this book can be implemented by software in this series. There are

three categories of software: (1) General Release; (2) Undocumented Software (which lacks

manuals but the programs run and have been extensively used in research), and (3) Beta-

Software (which has not been used in any serious research project, so may need considerable

reworking before it is fully useful). There are some additional programs available that

have been used in published research and so are made available; these can be found at the

end of the “Update Information.” At the end of most chapters of this book, we provide

instructions for using the relevant software. These instructions are based on versions of the

programs that have been frozen, so that you can run all the programs using these instruc-

tions. We do, however, anticipate upgrading the software; these upgrades will be available

on the website and will (eventually) be documented. Major changes will be detailed in the

“Update Information” on the bottom of the morphsoft webpage.

Running the IMP programs, which are written in Matlab (Mathworks, 2000) and com-

piled to run under Microsoft Windows, requires first installing a large package of software,

mglinstaller (detailed instructions for installing it, and for installing other programs in the

IMP series, are given below). Different versions of Matlab are often incompatible with

each other (both upwardly and downwardly), so programs written in the future, using a

newer version of Matlab, will require installation of a new version of mglinstaller (in a

different directory).

Another important resource is the listserver Morphmet. It is useful to subscribe to this

list, if only to be informed of new software and notified of any mathematical mistakes

or bugs in the programs. Additionally the list is sometimes quite active, discussing top-

ics of general interest, including conceptual issues like the meanings of size and shape,

and practical issues like dealing with preservational artifacts. Some recent posts have

also provided extensive bibliographies of morphometric studies of mollusks and fishes.

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20 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

To subscribe to this list, send an email to majordomo@wfubmc.edu and include the

following single line in the body of the message: subscribe morphmet.

Downloading and installing mglinstaller

Before you use any program in the IMP series, you need to download and run the self-

expanding mglinstaller (megalo-installer). This will create the directories (folders) where

the other IMP programs must be installed. To download mglinstaller, go to the IMP web-

site (http://www2.canisius.edu/∼sheets/morphsoft.html or www.biocollections.org), find

mglinstaller and click on it. This is a very large file, so it may take a while to download.

After the download is complete, you need to create the directory (folder) where you want

mglinstaller to be expanded. We recommend you call this folder Matlab6 so you can keep

track of the version of Matlab used to write the software. Now expand mglinstaller in that

directory. It will create a folder in Matlab6 called bin, and a folder in bin called win32;

it will also unpack a series of files needed to run the other IMP programs. The other pro-

grams are also packaged as self-expanding files. After you download them, they must be

expanded into the folder win32. If they are not installed in win32, they will not run.

References

Albrecht, G. (1978). Some comments on the use of ratios. Systematic Zoology, 27, 67–71.

Atchley, W. R. and Anderson, D. (1978). Ratios and the statistical analysis of biological data.

Systematic Zoology, 27, 71–78.

Atchley, W. R., Gaskins, C. T. and Anderson, D. (1976). Statistical properties of ratios. I. Empirical

results. Systematic Zoology, 25, 137–148.

Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions. Statistical

Science, 1, 181–242.

Bookstein, F. L. (1989). “Size and shape”: A comment on semantics. Systematic Zoology, 38,

173–190.

Bookstein, F. L. (1991). Morphometric Tools for Landmark Data: Geometry and Biology.

Cambridge University Press.

Bookstein, F. L., Chernoff, B., Elder, R. et al. (1985). Morphometrics in Evolutionary Biology. The

Academy of Natural Sciences of Philadelphia.

Corruccini, R. S. (1977). Correlation properties of morphometric ratios. Systematic Zoology, 26,

211–214.

Dodson, P. (1978). On the use of ratios in growth studies. Systematic Zoology, 27, 62–67.

Hills, M. (1978). On ratios – a response to Atchley, Gaskins and Anderson. Systematic Zoology,

27, 61–62.

Kendall, D. (1977). The diffusion of shape. Advances in Applied Probability, 9, 428–430.

Kendall, D. G. and Kendall, W. S. (1980). Alignments in two-dimensional random sets of points.

Advances in Applied Probability, 12, 380–424.

Klingenberg, C. P. (1998). Heterochrony and allometry: the analysis of evolutionary change in

ontogeny. Biological Reviews, 73, 79–123.

Lagler, K. F., Bardach, J. E. and Miller, R. R. (1962). Ichthyology. John Wiley & Sons.

Strauss, R. E. and Bookstein, F. L. (1982). The truss – body form reconstructions in morphometrics.

Systematic Zoology, 31, 113–135.

chap-02 4/6/2004 17: 21 page 21

PART

Basics of Shape Data

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chap-02 4/6/2004 17: 21 page 23

Landmarks

Landmarks are discrete anatomical loci that can be recognized as the same loci in all speci-

mens in the study. Because landmarks play a fundamental role in geometric morphometrics,

it is important to understand their function in a shape analysis. It is equally important

to understand which functions they do not serve, as that understanding also influences

the selection of landmarks. Criteria for selecting landmarks differ from those applied to

choosing traditional morphometric variables, so some rethinking may be required. This

chapter begins with a summary of some of the basic differences between conventional

and landmark-based studies that bear on landmark selection and on how we may need

to change the way we think about selecting variables. Next is a review of the criteria for

choosing landmarks in light of both biological and mathematical considerations, focus-

ing on general criteria and principles. This is followed by three concrete examples, each

explaining why particular landmarks were chosen. The chapter concludes with a practical

guide to collecting landmark data.

Changing the way we think about selecting variables

One major difference between conventional and landmark-based techniques is most impor-

tant for thinking about the selection of variables: conventional morphometric variables

are selected a priori, meaning that we choose variables before we conduct the analysis;

in landmark-based studies, that is not the case. In studies of traditional measurements

only the variables chosen in advance of the analysis are available for analysis (unless we go

back and remeasure the specimens), and for that reason much emphasis has been placed on

choosing “meaningful” variables. If we do not select the meaningful ones we will not have

them to aid our interpretations, and if we include many that have no particular biological

significance, they could complicate our interpretations of those that do. There are many

considerations that might enter into the decision regarding which variables are meaning-

ful, including relevance to understanding (1) biomechanics, (2) developmental processes,

(3) systematics, and (4) evolutionary processes. For example, in a study designed to analyze

the biomechanics of chewing we would conscientiously select variables for their relevance

to chewing, which might not be the same variables as those that capture the information

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

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24 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

relevant for understanding jaw development or relationships among taxa or evolutionary

processes. Consequently we might need as many as four different measurement schemes,

each one designed in light of the substantive biological questions addressed by the study.

In landmark-based studies, variables are not selected a priori (although landmarks are);

the meaningful variables are discovered by the analysis. All the variables that we could have

measured between any pair of landmarks are included in the analysis, so we do not need

to decide among them before we begin. As mentioned in Chapter 1, given 16 landmarks

we would have to measure 120 variables (i.e. all the lengths, depths and widths that could

be measured by distances between pairs of landmarks) to capture as much information

as we have in the coordinates of the 16 landmarks. Having all that information available

to us does not complicate the interpretation of the results, because the variables do not

enter into the interpretation unless they are found to be relevant. For example, if we are

interested in feeding performance and measure 16 landmarks on jaw bones and teeth,

we will discover which are relevant to feeding performance by analyzing the covariance

between the landmarks and measures of performance. We do not need to know which

variables covary with performance when we begin the analysis – the objective is to discover

that at the end.

Landmarks should provide a sufficiently comprehensive sampling of morphology that

the features of biological significance can be discovered. This emphasis on discovery does

not mean that you should avoid thinking about variables that might be important for your

biological questions. The landmarks you select do determine what you may discover. If

you are interested in the biomechanics of lever arms, you should locate landmarks on

those lever arms or else you will not have the data required to analyze them. However,

you will not lose or dilute information about biomechanics of lever arms by including

other landmarks of unknown relevance – if they are not relevant, they will not covary with

measures of performance. If your only question is “What is the mechanical advantage of

this jaw compared to that one?”, then there is no reason to do a shape analysis – the

question you are asking is about mechanical advantage, not shape. As Bookstein (1996)

pointed out, geometric methods might be “overkill” in such purely biomechanical studies.

However, if you want to place those lever arms in a broader morphological context,

geometric morphometrics helps to provide one.

As a general rule, landmarks should be chosen so you can quantify any differences that

you can see. A quantitative study should capture at least as much information as does an

informal, qualitative inspection of specimens. However, the morphologies should also be

sampled more broadly so that you can discover more than is evident by visual inspection,

and, if you want to pin down where the changes occur, you will need even and fairly dense

coverage of the form. Below we discuss the general criteria for selecting landmarks in more

depth, and then we provide examples of three data sets, explaining what the landmarks

are and why they were chosen.

Criteria for choosing landmarks

Ideally, landmarks are (1) homologous anatomical loci that (2) do not alter their topologi-

cal positions relative to other landmarks, (3) provide adequate coverage of the morphology,

(4) can be found repeatedly and reliably, and (5) lie within the same plane.

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LANDMARKS 25

Homology

The concept of homology plays a crucial role in landmark-based morphometrics. Although

many traditional morphometric studies have been concerned with homology, homology

was not a fundamental concern when selecting measurements. If it had been, some stan-

dard variables (such as “greatest skull breadth” or “least interorbital width”) would not

have been measured. Such variables are not necessarily homologous because they may

be measured at very different points on the skull in different organisms, depending on

where the skull is widest or the interorbital region is narrowest. Consequently, we cannot

say how broadening of the skull or narrowing of interorbital width has been affected by

alterations in skull shape. We cannot trace the changes in skull breadth to the changes in

shape of the skull because we are not measuring the same things on all skulls. In contrast,

homology has been stressed above all criteria for selecting landmarks in geometric morpho-

metrics, and it is undeniably the most important one. For both mathematical and biological

reasons, homology is the paramount consideration when it comes to the selection of

landmarks.

Understanding the role that homology plays both mathematically and biologically

requires an intuitive feel for the mathematics as well as for the biological issues. Sometimes

there are reasons for including landmarks in a study, even though their homology is some-

what dubious (in the last chapter of this book we discuss “semi-landmarks,” points that

aid in studies of regions that lack homologous landmarks). However, it is important to

recognize the compromises resulting from using landmarks of doubtful homology. Most

systematists will presume that homology is a central concern regardless of any mathemat-

ical arguments, but functional morphologists and developmental biologists may be less

convinced that homology is actually necessary. However, there are mathematical argu-

ments that reinforce the biological ones, as discussed in depth in Chapter 4. Because you

will select your landmarks before you read that chapter, and you need an intuitive feel

for the primary mathematical issue before choosing them, the basic mathematical ideas

bear mentioning here. The primary mathematical issue is the interpretation of biologi-

cal change as a deformation: a (smooth) mapping of one set of points to corresponding

points in another form. The mapping only makes sense if the points are truly “correspond-

ing,” and that correspondence requires more than that landmarks have the same name.

It requires a careful consideration of what “correspondence” means.

Correspondence need not imply biological homology – we might think of correspon-

dence in functional (or developmental) terms. For example, we might view points as

corresponding to each other because they are located at the end of an input lever arm

in two different organisms, even if those lever arms are in different locations. In a purely

functional sense those points might indeed correspond, but if our aim is to describe the

transformation from one form to another, and we are using a mathematical model of a

deformation, we need a more restrictive view of “correspondence.” The landmark is not

just serving a corresponding function; it must also be the same anatomical locus.

The importance of biological homology to morphometric analysis has been obscured

somewhat by the definition of homology that sometimes appears in the morphometric

literature. The semantic discrepancy between the notion of homology used by some mor-

phometricians and the one favored by biologists is partly historical (workers in the two

fields have traditionally used the term differently), but there is an important conceptual

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26 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

distinction as well because of the conceptual gap between the subject matter of homology

assessments in biological and mathematical contexts.

Biologists usually think about homology in terms of organismal parts or characters,

whereas mathematicians think about homology in terms of the individual loci (i.e. points)

on those parts. As biologists, our objective in choosing landmarks is to permit making

inferences about the regions between them – we are not interested in the landmarks per se,

but in the shapes of the morphological structures on which those landmarks lie. The role of

the landmarks is to pin down those structures at discrete points that we can recognize as the

same on all organisms. However, this means that our data are the landmarks, the individual

loci, and so we also need to think about the homology of those points. Fortunately, this is

not a wild conceptual leap. We recognize structures are homologous as structures because

they are discrete (distinct from other structures) and recognizable in all specimens. We can

apply the same criteria to intersections of structures (as at sutures), or to their centers, or

to their tips (ends). If discrete and recognizable structures are homologous as structures,

then discrete and recognizable locations on them are arguably homologous as points.

The mathematical framework for thinking about homology is the idea of a deformation,

which extends the correspondence of sampled points to unsampled points lying between.

Using a model of a deformation, such as the thin-plate spline (Chapter 6), we can draw a

picture of a change in shape that extends that change over the whole form, even though

we only sampled it at selected points. In that sense, the deformation imputes homology

to intervening points. For that reason, the mathematical models for deformations have

sometimes been termed “homology functions” (see, for example, Bookstein et al., 1985).

To understand this idea more fully, consider a sample of landmarks on a skull (Figure 2.1);

when looking at the results, we can see changes in the relative positions of landmarks that

imply changes in the proportions of structures sampled by them. We can visualize the

impact of those changes for the shape of the skull using the deformed grid that stretches

where regions are relatively enlarged and contracts where regions are relatively reduced.

A highly literal interpretation of that picture could make us uncomfortable because we do

not know where every single point on one skull is located on the other – we cannot read

the intersections of the grid, for example, as if they are at homologous anatomical loci.

However, we are not trying to impute homology to all those points; rather, we are inferring

changes in shape that are implied by the homologous landmarks. If we are willing to

consider that the structures, such as premaxilla and maxilla (and the sutures between them),

are homologous, and that the presphenoid, sphenoid and basisphenoid (and the sutures

between them) are also homologous, and that foramina are also homologous, we are

specifying correspondences among points. The mathematical analysis uses that information

to infer the changes in shape between the landmarks. If our sample of landmarks is sparse,

we have good reason to worry about inferring changes between them – interpolating from

sparse data is always a cause for concern. However, homology of points between the

landmarks we sample is not imputed or determined by the deformation; homology is

established in advance, by biological arguments. For a deformation to make mathematical

sense, the points in one form must correspond to the points in another.

Sometimes it may seem that points cannot be homologous, as in ontogenetic studies,

because bony tissue is added during growth. Thus the cells located at the suture between

bones at one developmental stage are not the ones found at that suture at another develop-

mental stage. Histologically, the points are not homologous, but it is nonetheless important

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LANDMARKS 27

Figure 2.1 Landmarks sampled on the skull to show the interpolation of changes between

landmarks based on the analysis of displacements of landmarks (relative to other landmarks).

that the anatomical parts be so, especially if we hope to compare growth from age to age or

from species to species. We want to compare rates of growth of comparable parts. That the

landmarks are located in different cell populations does not matter, but the comparability

of the parts does.

Consistency of relative position

Morphometric methods cannot be applied properly when shapes are too different. For

example, if bones are so radically altered in their topology that points on one have moved

past other points (e.g. a foramen that was anterior to a tooth is now posterior to that

tooth), the shapes may be too different to analyze geometrically. Also, in some cases

landmarks may disappear altogether – as when a foramen is present in one taxon (or

age) but not in another. Such changes, while undeniably interesting, are not suitable for a

morphometric analysis. They are not matters of changes in shape so much as of changes in