PATTERNS AND PROCESSES IN MORPHOSPACE: GEOMETRICMORPHOMETRICS OF THREE-DIMENSIONAL OBJECTS

P. David Polly1 and Gary J. Motz2

1Departments of Geological Sciences, Biology, and Anthropology, Indiana University, 1001 E. 10th Street,Bloomington, Indiana 47405, USA ⟨pdpolly@indiana.edu⟩

2Department of Geological Sciences and Center for Biological Research Collections, Indiana University,1001 E. 10th Street, Bloomington, Indiana 47405, USA ⟨garymotz@indiana.edu⟩

ABSTRACT.—Focusing on geometric morphometrics (GMM), we review methods for acquiringmorphometric data from 3-D objects (including fossils), algorithms for producing shape variablesand morphospaces, the mathematical properties of shape space, especially how they relate to mor-phogenetic and evolutionary factors, and issues posed by working with fossil objects. We usethe Raupian shell-coiling equations to illustrate the complexity of the relationship between suchfactors and GMM morphospaces. The complexity of these issues re-emphasize what are arguablythe two most important recommendations for GMM studies: 1) always use multivariate methodsand all of the morphospace axes in an analysis; and 2) always anticipate the possibility that thefactors of interest can have complex, nonlinear relationships with shape.

INTRODUCTION

Morphometrics literally means ‘the measurement ofmorphology,’ and is a broad topic with tendrils thatcreep into almost every branch of paleontology,biology, statistics, and informatics. In this paper, wereview the acquisition and analysis of morphometricshape data extracted from three-dimensional (3-D)objects and discuss issues related to the analysis ofhigh-dimensional data by comparing theoreticaland empirical morphospaces by analyzing 3-Dbrachiopod valves generated using the shell-coilingequations of Raup (1961, 1966).

Our focus is on geometric morphometric methods(GMM; Table 1), a class of methods in which theanalytical variables are the Cartesian coordinates(XYZ values) of geometric points, usually referred toas landmarks or semilandmarks, which represent thelocations of structures on objects such as fossils(Bookstein, 1991; Dryden and Mardia, 1998). GMMthus differs from other multivariate morphometricmethods that analyze lengths, areas, volumes,angles, and other kinds of measurement data

(e.g., Reyment et al., 1984). One of the primarydifferences is that GMM uses Procrustes super-imposition (Table 1) to rescale objects to unit size,translate them to their geometric centroids, and rotatethem to minimize the sum of squared differencesbetween corresponding points, thereby standardizingthe objects in a common coordinate system,removing the mathematical component of size, andminimizing shape differences between objects tofacilitate hypothesis testing of patterns of shapevariation (Gower, 1975; Rohlf and Slice, 1990; Rohlf,1990). GMM is undoubtedly the most widely usedtype of morphometrics today. Its strengths include:1) the ability to effectively remove size and thus focusthe analysis on pure shape; 2) the comparativesimplicity of collecting coordinate data (althoughdata collection is easier from two-dimensional [2-D]photographs than from 3-D objects), and 3) the ease ofvisualizing results as transformations of the morpho-logy itself rather than as tables of numbers (Bookstein,1989; Klingenberg, 2013).

Procrustes superimposition imposes importantlimits on GMM. Analyses are usually restricted to

The Paleontological Society Papers, 22, 2016, p. 71–99Copyright © 2017, The Paleontological Society1089-3326/17/2399-7575doi: 10.1017/scs.2017.9

71

single rigid objects and a single constellation oflandmark (or semilandmark) points. This restrictionimposes hurdles for studying the relationshipsbetween mobile parts, such as moveable skeletalelements, limb segments, or other unfixed structures(but see Rohlf and Corti, 2000; Bookstein et al.,2003), and for coping with missing data or the evo-lutionary gain and loss of structures. Procrustes alsodistributes variance among all the coordinate points,even if only one of them is truly biologically variable(Chapman, 1990; Richtsmeier and Lele, 1993;Richtsmeier et al., 2002). The strength of GMM,therefore, is making comparison in overall differ-ences in shape, but it cannot be used to preciselylocalize which points are varying.

In this paper, we begin by pointing readers toseminal works and other overviews of GMM meth-ods and applications. We then describe how GMManalysis works, with a focus on the mechanics ofconstructing shape variables and morphospaces. Weinventory currently available software for extractingmorphometric data from 3-D digital objects. Forcompleteness, we also briefly review common sta-tistical questions that can be answered using mor-phometric data and their analytical solutions. We alsodiscuss issues associated with analyzing fossils,including breakage and deformation. However, ourprimary focus is on issues related to identifyingbiological factors from analysis of multivariateshapes. Our discussion includes an exploration of therelationship between theoretical parameter spacesand empirical shape morphospaces using theRaupian shell-coiling equations, an examination ofhow different strategies for quantifying the shapeof 3-D objects can influence the construction of

morphospace, and how phylogenetic covariancesinteract with other biological factors to determinethe orientation of morphospace axes.

All analyses, modeling, and simulations in thispaper were performed in Mathematica (Wolfram,2015), often with assistance of the functions inthe Geometric Morphometrics, Phylogenetics, andQuantitative Paleontology packages (Polly, 2014,2016a, b), unless otherwise noted.

PREVIOUS REVIEWS OFMORPHOMETRICS

Several comprehensive treatments and reviews ofGMM methods already exist, to which we referreaders interested in topics that are not coveredadequately in this review. Two books from the 1990sare the core references for GMM. Bookstein (1991),colloquially known as the ‘Orange Book,’ made theseminal leap into geometric morphometrics, com-bining direct analysis of Cartesian coordinatesinstead of interlandmark distances, and thin-platespline deformations (Table 1) as both an analyticalalgorithm for decomposing shape variation and anillustrative, d’Arcy Thompson-esque method forshowing the transformation of one shape intoanother. Dryden and Mardia (1998) presented GMMfrom a mathematical and statistical perspective.

Many GMM methods were first presented in anunofficial series of edited volumes that, similar to theOrange Book, are known by the colors of theircovers. The ‘Red Book’ (Bookstein et al., 1985) wasthe antecedent of GMM, remembered especiallyfor the presentation of truss analyses, an approachthat focused on landmarks but used standard

TABLE 1.—Definitions of important terms in geometric morphometric methods (GMM). Classic references are providedwhen appropriate.

Geometric morphometrics (GMM) = is a class of shape analyses that use Cartesian coordinates as variables, usually based onProcrustes superimposition to remove variation due to size, translation, and rotation (Bookstein, 1991; Dryden andMardia, 1998).

Ordination = any method for transforming variables to a new set of axes. In GMM, the most common ordination is PrincipalComponents Analysis using the covariance method, which rigidly rotates landmark coordinates to a set of orthogonal axes.

Procrustes superimposition = a least-squares algorithm that rescales, translates, and rotates landmark constellations to minimize thesum of squared distances between corresponding landmarks, a criterion that facilitates hypothesis testing that shape differencesare statistically significant (Gower, 1975; Rohlf and Slice, 1990).

Shape models = landmark configurations constructed from PC eigenvectors for a particular point in shape morphospace (seeEquation 3). Shape models can be constructed for any point in morphospace, regardless of whether it is occupied by a real shape(Rohlf, 1993; Dryden and Mardia, 1998).

Thin-plate splines = a two- or three-dimensional spline algorithm that extrapolates shape deformation to areas between landmarks.TPS is frequently used to illustrate shape differences as a d’Arcy Thompson-like grid and is sometimes used to decompose shapevariation by spatial scale (Bookstein, 1989, 1991).

72 The Paleontological Society Papers

morphometric algorithms to analyze a ‘truss’ ofinterlandmark distances. The highlight of the ‘BlueBook’ (Rohlf and Bookstein, 1990) was a series ofpapers on Procrustes- and thin-plate spline-basedmethods. The power and popularity of GMMmethods quickly took root and were developed bypapers in the ‘Black Book’ (Marcus et al., 1993),including the debut of so-called relative warpsanalysis, a GMM variant of principal componentanalysis (PCA) that is used to produce shape scores(Rohlf, 1993), and the ‘White Book’ (Marcus et al.,1996), which contains a variety of methodologicaland applied papers. A second ‘Blue Book’ (MacLeodand Forey, 2002) focuses specifically on issuesrelated to phylogenetics and morphometrics.A second ‘Black Book’ (Slice, 2005) focuses onanthropological applications, but includes severalimportant methodological papers including a fulltreatment of 3-D sliding semilandmarks (Gunz et al.,2005). Finally, the ‘Yellow Book’ was published as aspecial issue of the journal Hystrix (Cardini and Loy,2013), and contains reviews of GMM, includinganalyses involving phenotypic trajectories, phylo-genetic comparative methods, and semilandmarks.Collectively, these volumes contain much of theGMM canon.

Three important textbooks provide syntheticintroductions to GMM for beginners. GeometricMorphometrics for Biologists (Zelditch et al., 2012)covers the basic theory and methods of analysis(including statistical analysis), and provides appliedexamples in several areas of evolutionary biology.Paleontological Data Analysis (Hammer and Harper,2006) is a general reference for analytical methodsfrequently used by paleontologists, including GMM.It contains concise essays on individual methods thatinclude succinct descriptions of their purpose andtheory as well as clear equations and algorithmsfor implementing them. Morphometrics with R(Claude, 2008) also covers the theory and methods ofGMM, but with a focus on implementing them in theR statistical language.

Finally, three journal article reviews of GMM areespecially noteworthy in the present context. Adamset al. (2004, 2013) have twice critically reviewed thestate of the field, providing readers with succinct andinsightful introductions to methods and prospectusfor future developments. Viscosi and Cardini(2011) reviewed GMM methods with examples of

quantifying and analyzing leaf shapes, and touchedon important issues of measurement error, statisticalanalysis, sample sizes, and visualization. Their paperprovides readers with the knowledge and skillsneeded to competently perform publishable qualityGMM analyses with little other introduction. Finally,Webster and Sheets (2010) introduced GMM topaleontologists in the 2010 Paleontology Societyquantitative methods short course.

QUICK-START TECHNICAL GUIDETO GMM

As explained in more detail below, GMM begins withthe Cartesian coordinates of landmark points placedon the objects of interest, which are then Procrustessuperimposed and converted to shape variables inthe form of principal components scores (Drydenand Mardia, 1998). Statistical tests, phylogeneticcomparative methods, or other analyses are usuallyperformed on the shape variables using the sametechniques applied to any multivariate dataset. Manyof the software packages described below carry outthe computations presented in this section; however,readers can find the following algorithmic detailsuseful for building their own custom analyses inR (R Core Team, 2013), Mathematica (Wolfram,2015), MATLAB (MATLAB, 2015), or in anotherprogramming environment.

Selecting landmarks is the most important step ofany analysis. Typical GMM analyses are based on ahandful of landmarks placed on homologous struc-tures in 2-D digital photographs, but increasingly, itis possible to acquire landmarks from 3-D objects(e.g., meshes or voxel datasets; Fig. 1). Such data arequick, easy, and inexpensive to collect. The term‘landmark,’ although often used generally to refer toany Cartesian point used in GMM, strictly refers to asingle point placed on biologically homologousstructures. The term ‘semilandmark’ refers to pointsplaced algorithmically on a curve (including anoutline) or surface (Bookstein, 1991; Gunz andMitteroecker, 2013). Usually, the curve or surfaceitself is biologically homologous, but the positions ofthe individual semilandmarks are arbitrary. Onceplaced, semilandmarks are analyzed exactly likelandmarks, but their biological interpretationcan differ. Semilandmarks, like landmarks, can be2-D or 3-D.

Polly and Motz—Three-dimensional geometric morphometrics 73

The simplest strategy for applying semiland-marks to curves or outlines is to space a constantnumber of points equidistant from one another inthe same direction from a homologous start point, aprocedure that is also used in elliptical Fourier shapeand eigenshape analysis (Lohmann, 1983; Rohlf,1986; MacLeod and Rose, 1993; Polly, 2008b). Thecurve can also be broken into segments anchored byordinary landmarks if it has homologous waypoints,which tends to improve correspondence betweensimilar shapes (MacLeod, 1999). Because Procrustessuperimposition minimizes distances between corres-ponding points, even segmented semilandmarkcurves can align in counterintuitive ways, dependingon how the semilandmark placement algorithminteracts with variation in the shape of the curve;corresponding semilandmarks can end up in locationsthat do not minimize shape differences. Slidingsemilandmarks were developed to minimize theeffect of semilandmark placement by iterativelymoving them around a curve or across a surface sothat either the thin-plate spline-bending energy orthe Procrustes distance between the shapes isminimized (Bookstein, 1997; Gunz et al., 2005). Aswith Procrustes superimposition itself, slidingsemilandmarks help ensure that shape differencesare minimized, with the tradeoff that their placementis sample dependent. Each strategy quantifiesshape in a slightly different way, so their optimal

superimpositions and ordinations (Table 1) willdiffer. A comparison of three semilandmark place-ment strategies is presented below.

Rohlf and Slice (1990) described how Procrustessuperimposition could be applied to a sample ofshapes, a procedure sometimes known as GeneralizedProcrustes Analysis (GPA). Procrustes super-imposition is a modification of the algorithmproposed by Gower (1975) for comparing multi-variate ordinations of the same data. Briefly, eachobject is centered by subtracting the centroid from thelandmark coordinates and then is scaled to unit sizeby dividing the coordinates by its centroid size (thesquare root of the sum of squared distances betweenthe centroid and several landmarks). The objectsare then iteratively rotated about their centroids tominimize the sum of squared distances betweencorresponding landmarks. These steps begin withcalculating the mean of the sample, which is alsoknown as the consensus shape, translating eachobject to that mean, and then rotating them aroundtheir centroids (center points) until the sum of squaresstops getting smaller. The rotation matrix for eachstep is found from the singular value decompositionof the dot product of the transpose of each object andthe current mean (Gower, 1971).

Procrustes superimposition removes degrees offreedom from the landmarks, which has importantimplications for statistical analysis. One degree offreedom is removed for scaling, two or three degreesare removed for translation of 2-D or 3-D data, andone or three degrees are removed for rotation of2-D or 3-D data. For 2-D data, the total degreesof freedom will be 2k − 4 (where k is the number oflandmarks) or n− 1 (where n is the number ofobjects), whichever is smaller. For 3-D data, the totaldegrees of freedom will be 3k− 7 or n− 1, whicheveris smaller. Note that there will often appear to bevariance in at least one more dimension, which is anartifact of the curvature of shape space (Dryden andMardia, 1998; Rohlf, 1999). To correct for this, shapevariables can be projected to a Euclidean tangentspace (a noncurved approximation of shape space inwhich analyses can be performed using standardalgorithms), although this is seldom necessary forbiological data, including fossils, because shapevariation is usually so small that the curvature of thespace is negligible (Rohlf, 1999). The curvature ofshape space can be an issue for objects that have

FIGURE 1.—Collecting landmarks from a 3-D scan of aDevonian brachiopod: (A) photograph of the Devonianbrachiopod, Spirifer murchisoni Hall, from the IndianaUniversity Paleontology Collection (IUPC #2400); (B) a 3-Dmesh, obtained from a Creaform GoScan20 structured lightscanner, with four landmarks (loosely following Collins,2014) denoted on the specimen by solid black circles,as follows: (i) posteriormost point of pedicle valve;(ii) lowermost point of medial sulcus along commissure;(iii) lateralmost point along the flat portion of the commissureon the hinge; and (iv) the ventralmost point of the parasulcusalong the commissure.

74 The Paleontological Society Papers

shapes that are more variable and random, such asstone tools (e.g., Costa, 2010), sedimentary particles(e.g., MacLeod, 2002), or randomly generated models.

Once the coordinates have been superimposed,their covariance matrix is calculated. First, the con-sensus shape is subtracted from each shape to pro-duce residuals. This step centers the principalcomponent space on the mean shape. The covariancematrix is then calculated from the residuals. Note thatfor 3-D objects, especially ones that use semiland-marks, the covariance matrix can be large and canpresent computational problems. Presuming that thenumber of objects is smaller than the number ofcoordinates, it could be more efficient to perform theprincipal components ordination in Q-mode (basedon a matrix of the objects) instead of the normalR-mode (based on a matrix of the variables)(Legendre and Legendre, 1998).

Singular value decomposition (SVD) of the cov-ariance matrix is used to compute the eigenvectorsthat define the axes of the principal component space:

SVD ½P=UVWT (1)

where P is the covariance matrix of the coordinateresiduals (the symbol P is used in evolutionary geneticsfor the phenotypic covariance matrix); U is the matrixof eigenvectors with one column per axis of the prin-cipal component space; V is a diagonal matrix of sin-gular values (which are the same as eigenvalues whenthematrix being decomposed is square, symmetric, andpositive definite, as it is with covariancematrices);W isidentical to U for square, symmetric, positive definitematrices and can be ignored for our purposes; and Tindicates a transposed matrix. The eigenvalues reportthe variance of the data along each eigenvector, whichmeans along each axis of the principal componentsmorphospace. By convention, these axes are sorted indescending order by their variances. Note that only thefirst 3k− 7 (or 2k− 4) or n− 1 axes will have nonzerovariances. Higher axes can be discarded.

Principal components (PC) scores, which are theshape variables used for further analysis (see below),are obtained by multiplying the residuals by theeigenvector matrix:

S=RU (2)

where S is the matrix of scores, R is the matrix ofresiduals, and U is the eigenvector matrix obtainedfrom SVD. Again, axes that exceed the expected

dimensionality of the Procrustes superimposed datacan be discarded.

LANDMARK DATA ACQUISITION FROM3-D OBJECTS

Digitization of fossil specimens is now progressing ata rapid pace, in large part due to major technologicaladvances in digital photography, CT scanning,structured light and laser-based surface-scanningsystems, and photogrammetry and structure-from-motion programs. However, the acquisition ofdata from digitized specimens that are suitable forgeometric morphometrics provides an increasinglycomplex set of challenges, with an ever-growingnumber of software programs and analytical tools tomeet these challenges. This section focuses on theacquisition of data from 3-D digitized surface scansof fossil specimens. Such datasets can be obtainedfrom any number and combination of the virtualiza-tion methods outlined in the various papers fromthis short course publication. For 2-D objects andvoxel-based 3-D data, unsuperimposed coordinatesare usually in pixel (or voxel) units unless specialeffort is made to scale them to real-world units (e.g.,millimeters). Each unit of a digital image, termeda pixel, has color-value information and a specificlocation. As an extension of this format, the unit ofmany three dimensional images is called a voxel.Each voxel, similar to a pixel in a 2-D array, is a pieceof data that records either color, or grayscale, andlocation information (in three dimensions, X, Y,and Z). Volumetric data (e.g., raw tomographicdatasets or MRI scans; Sutton et al., 2012), can beanalyzed in terms of the Cartesian coordinates of thevoxels within that dataset. However, surface scans(e.g., those generated by laser-based scanning sys-tems, photogrammetry, or structured light scanners)are generally presented as meshes. Meshes are arraysof nodes as 3-D points that are connected by inter-nodes. Typical representations of meshes are inpolygonal faces, usually triangles, in which eachvertex of the polygon gives the specific 3-D coordi-nate information required for geometric morpho-metric analyses, and the faces are the component thatcan record color information.

Geometric morphometric methods, therefore,rely upon the careful selection of precise Cartesiancoordinates for landmark-based analyses. Most GMM

Polly and Motz—Three-dimensional geometric morphometrics 75

programs use a standardized text-file format, includ-ing: 1) .TPS, which is the de facto standard for GMMas popularized by Rohlf (2015) in his TPS suite ofGMM programs, 2) .NTS files (NTSYSpc format ofRohlf, 1988), 3) .CSV (comma-separated values file),or 4) .TXT (standard ASCII or binary text file).The specific format of each of these file types is onethat can cause great consternation in preparingshape data for GMM, and each program treatslandmark data acquisition differently. The mostwidely utilized and standardized format is that of the.TPS file. The ‘thin plate spline’ format of Rohlf(2015) consists of architecture that specifies a numberof properties for each specimen in sequence beforerepeating the pattern for the subsequent specimen inthe text file. The first line of text specifies the numberof landmarks (‘LM = p’ or ‘LM3 = p’ for 3-Dobjects, where p is the number of landmarks), andthe second line begins a per-row listing of theCartesian coordinates for each landmark. Thefollowing lines can contain identifiers, image names,outline specifications, etc., ending with the scalefactor. A sample .TPS file has the following basicformat:

LM3 = 49.124494 10.345037 4.594074

−1.783271 −10.989462 −8.90592510.716694 −3.154946 14.635059−4.783275 −13.257661 4.594074IMAGE = 2400_Spirifer_murchisoni.JPGID = 1SCALE = 0.06664

Although the Windows program tpsUtil is quitehelpful in structuring these .TPS files, there are manyother programs that can produce structured files thatcan be imported for morphometric analysis. Thefollowing section reviews features of a selectedset of programs that can be used for acquiring land-mark coordinates from images or meshes for theiranalysis.

PROGRAMS FORGMMDATAACQUISITIONAND ANALYSIS

This section contains a brief review of software foracquiring landmark coordinates from 3-D digitalobjects and for conducting GMM analyses (Table 2).We focus primarily on open-source packages. TABLE2.

—Sum

marytableof

GMM

softwarepackages.Lin

=Linux;Mac

=Macintosh

OSX;NA

=notapplicable;surf=

surface;

vol=

volume;

Win

=Microsoft

Windows;+=

present;-=

absent.

Program

Nam

eCost

Operating

System

Acquisition

Ana

lysis

3-Dda

tatype

3-Dform

atinpu

tGMM

form

atou

tput

GPA

PCA

TPS

PLS

EVANToolbox

mem

bership

Mac,W

in,L

in+

+surf

.OBJ

.CSV,.TXT

++

++

FIJI

free

Mac,W

in,L

in+

−surf,v

ol.BMP,.PNG,.JPG,.DICOM

.CSV,.TXT

−−

−−

ImageJ

free

Mac,W

in,L

in+

−surf,v

ol.BMP,.PNG,.JPG,.DICOM

.CSV,.TXT

−−

−−

IMP

free

Win

−+

surf

.OBJ,.PLY

.NTS,.TPS,.TXT

++

−+

Landm

ark

free

Win

+−

surf

.PLY,.STL,.PTS

.DTA,.LAND,.PTS

−−

−−

GMM

for

Mathematica

free

(+$140

for

Mathematica)

Mac,W

in,L

in+

+surf

.TXT,.CSV,.TPS

.CSV,.TXT

++

++

MeshL

abfree

Mac,W

in,L

in+

−surf

.PLY,.STL,.OBJ,.QOBJ,.OFF,.PTX,.VMI,.BRE,.DAE,

.CTM,.PTS,.CPTS,.XYZ,.TRI,.ASC,.TXT,.X3D

,.X3D

V,.WRL

.PP

−−

−−

ISE−MeshT

ools

free

Mac,W

in,L

in+

−surf

.STL,.OBJ,.PLY,.VTK

−−

−−

−Morpheusetal.

free

Mac,W

in,L

in−

+surf

.PLY

.LND,.TXT,.RDATA,.RAW,

.TPS,.NTS

−−

−−

MorphoJ

free

Mac,W

in,L

in−

+surf

.TXT,.CSV,.NTS,.TPS

.MORPHOJ,.TXT

++

++

PAST

free

Mac,W

in−

+−

−.TPS,.TXT

−+

−−

R−geom

orph

free

Mac,W

in,L

in+

+surf

.PLY

.CSV,.NTS,.TPS,.TXT

++

++

R−Morpho

free

Mac,W

in,L

in+

+surf

.DTA,.PP,.PTS,.TPS,.CSV

.CSV,.PTS,.TXT

++

++

R−shapes

free

Mac,W

in,L

in−

+surf

−.CSV,.TXT

++

+−

SPIERS

free

Mac,W

in−

−vo

l.BMP,.PNG,.JPG

.PLY,.STL,.VAXML

−−

−−

TIN

Afree

KNOPPIX

++

vol

.AIFF,.ANLZ,.RAD,.NEMA,.PGM,.DICOM

.NTS,.TPS

++

−−

tpsSuite

free

Win

++

NA

NA

.TPS

++

++

76 The Paleontological Society Papers

Landmark 3.6 is a Windows-only free softwarepackage developed for evolutionary morphing(Wiley et al., 2005) that allows users to placesemilandmarks on 3-D mesh surfaces and exportthem for analysis in other programs, but it does notperform geometric morphometric analyses per se.Landmark does provide a GMM context for‘morphing’ one mesh (.PLY, .PTS, and .STL inputs)into another mesh (e.g. along ontogenetic trajectories,evolutionary sequences, etc.). Landmark isavailable from http://graphics.idav.ucdavis.edu/research/EvoMorph.

MeshLab is an open-source program (MacOS,Windows, Linux) for the processing and editing ofsurface meshes (Cignoni et al., 2008), and is anessential tool for the inspection, editing, cleaning, andrepairing of meshes, in addition to the conversion ofmyriad 3-D file formats. MeshLab also contains aPointPicker tool that will allow for a .PP text file to besaved and then formatted and imported into theGMM program of your choice. MeshLab is availablefrom http://meshlab.sourceforge.net/.

ISE-MeshTools, from InteractiveSoftwarE, is anopen-source, 3-D interactive fossil reconstructionfreeware (MacOS, Windows, and Linux; Lebrun,2014), and is a good mesh manipulation software forpreparing data for GMM or acquiring landmarks andexporting for use in other programs. MeshTools isdesigned to maximize the extensibility of generallyrigid constraints of GMM by approximating missinglandmarks, merging complementary datasets (e.g.,landmarks from both dorsal and ventral orientationsof a specimen), and augmenting the general power ofstatistical inference possible for suboptimal datasets.ISE-MeshTools is freely available at http://morphomuseum.com/downloadMeshtools.

ImageJ and FIJI (FIJI Is Just ImageJ) are Java-based open-source programs (MacOS, Windows,Linux) that are powerhouses for image analysis andprocessing of digital images. Many specializedpackages are available for ImageJ (Schneider et al.,2012) and a number of the ‘standard’ packagesfor ImageJ are bundled and distributed as FIJI(Schindelin et al., 2012). There are a number of plu-gins for ImageJ that can be used to acquire 2-Dcoordinates from images or 3-D coordinates fromimage stacks (e.g., PointPicker and 3D Viewer).These plugins are available with ImageJ and FIJIfrom http://imagej.net/Downloads.

The TPS series of software (Windows only;open-source) arguably set the standard for GMM(Rohlf, 2015), in which .TPS file creation and land-mark placement occurs in tpsUtil and tpsDig.Superimposition, image unwarping, and averagingare executed in tpsSuper; tests for a small amountof morphological variation, as an appropriatenessmetric for thin-plate spline methods, are performed intpsSmall; fitting and visualization of thin-platesplines on trees are performed in tpsTree; multi-variate multiple regression of shape onto independentvariables is carried out in tpsRgr; relative warpsanalysis in tpsRelw; and partial least-squares analysisfor shape covariance utilizes tpsPLS. The TPS seriesof software is available from http://life.bio.sunysb.edu/morph/soft-tps.html.

The Integrated Morphometrics Package (IMP) isa suite of open-source GMM programs (Windowsonly; Sheets, 2003) that will take text file inputs(.NTS, .TPS, .TXT) to perform a host of analyses,including generalized Procrustes superimposition,shape-variation visualization, principal componentsanalysis (PCA), quantification of morphological dis-parity, statistical testing of difference in mean shapes,statistical testing of assignments of specimens togroups, canonical variance analysis (CVA), two-block partial least squares, residual analysis, andcomparisons of ontogenetic trajectories. The variousIMP programs are available at http://www3.canisius.edu/~sheets/IMP%208.htm.

A package for the R statistical environment (opensource for MacOS, Windows, Linux), geomorph(Adams and Otarola-Castillo, 2013) is a versatile andeasy-to-use suite of landmark acquisition and GMMdata manipulation functions, with the ability to cap-ture landmark coordinates, import landmark datafrom other programs in 2-D and 3-D, and performnumerous statistical analyses of shape variation andcovariation. Additionally, Adams and Otarola-Castillo (2013) provided a thorough walkthrough ofdata acquisition, 3-D landmark placement, and var-ious analyses performed on sample datasets includedwithin the R package. Although the package isrestricted to .PLY file imports that are strictly of theASCII format, additional 3-D file formats can beutilized, if converted using the Morpho packagefunction ‘file2mesh’ (see below). Analyses includeProcrustes ANOVA and pairwise tests, comparisonof rates of shape evolution on phylogenies, fit of

Polly and Motz—Three-dimensional geometric morphometrics 77

sliding semilandmarks to surfaces and curves,generalized Procrustes analysis (GPA), quantificationof morphological integration between modules,estimates of morphological disparity, two-blockpartial least-squares analysis, and more. R-packagegeomorph provides excellent graphical depictions ofshape evolution and patterns of shape variation and isavailable by using ‘install.packages (‘geomorph’)’ inthe R console or from https://cran.r-project.org/web/packages/geomorph/index.html.

Morpho is an open-source package for theR statistical environment (MacOS, Windows, Linux;Schlager, 2016) is a toolbox for numerous geometricmorphometric methods including sliding operationsfor semilandmarks, importing, exporting and manip-ulating of 3-D-surface meshes, and semiautomatedplacement of surface landmarks. Morpho canperform two-block partial least-squares regression,Riemannian distance calculations, principal compo-nents analysis (PCA), canonical variance analysis(CVA), .TPS grid interpolation, superimposition, andvisualization; it also contains functions for the easyexporting of arrays from R to MorphoJ or the EVANToolkit (formerly Morphologika; Phillips et al.,2010). The Morpho package can batch-process land-mark placement, and automatically superimposessemilandmarks on a folder of meshes usinga template ‘atlas’ and is available by using ‘install.packages (‘Morpho’)’ in the R console or from https://cran.r-project.org/web/packages/Morpho/index.html.

MorphoJ is an open-source, cross-platformprogram (MacOS, Windows, Linux; Klingenberg,2011) developed for the analysis of geometricmorphometric datasets, and will input .CSV, .NTS,.TPS, and .TXT files that record 3-D landmarkcoordinates (captured in other utilities) and outputstandard text files (for use in other GMM utilities)as well as .MORPHOJ project files for easy returnto analyses in MorphoJ. Analyses that can beperformed in this program include principal compo-nents analysis (PCA), matrix correlation, covarianceof shape variables, two-block partial least squares,regression, evaluation of modularity, canonicalvariate analysis (CVA), discriminant analysis,morphology mapping onto phylogenies, and somequantitative genetic capabilities to factor in herit-ability and additive genetic variation. MorphoJis freely available at http://www.flywings.org.uk/MorphoJ_page.htm.

Morpheus et al. is an open-source cross-platforminterface (MacOS, Windows, and Linux; Slice, 2013)for GMM visualization and basic preparation foranalyses to be carried out in other programs, and isessentially a command line interface with an attachedgraphics display pane for data visualization.Although .PLY files can be input, great utility isderived from the numerous export options available.Morpheus et al. will export .LND, .NTS, .RAW,.RDATA, .TPS, and .TXT files so that it plays wellwith other applications and is available from http://morphlab.sc.fsu.edu/software.html.

Geometric Morphometrics for Mathematica(Polly, 2016a) is a free add-on package for the com-mercial Mathematica symbolic computation program(Wolfram, 2015) that performs GMM analyses,including Procrustes, two-block partial least squares,thin-plate splines, principal components of shape,multivariate regression and ANOVA, EuclideanDistance Matrix Analysis, and projections of phylo-genetic trees into morphospace. It is complementedby the Phylogenetics for Mathematica (Polly, 2014)and Quantitative Paleontology for Mathematica(Polly, 2016b) packages. It is available at http://mypage.iu.edu/~pdpolly/Software.html.

PAST, or Paleontological Statistics (MacOS andWindows; Hammer et al., 2001) is an open-source,broad-based GUI program that includes a number ofdata transformation tools, graphical plotting features,numerous univariate and multivariate statisticaloperations, modeling algorithms, geometric mor-phometric operations (including 2-D and 3-D princi-pal component analysis, thin-plate spline for 2-Dlandmarks, linear regression of 2-D landmarks, andelliptic Fourier shape analysis of outline data), andanalyses for diversity, time series, stratigraphy,cladistics, and scripting of the PAST program itself.PAST is a workhorse of paleontological statistics thatis available, with excellent documentation, at http://folk.uio.no/ohammer/past/.

TINA (This Is No Acronym) is a robust machine-vision environment available for the KNOPPIXoperating system, a Linux distribution (Schunkeet al., 2012). The KNOPPIX OS is available as abootable disc image that is burned onto a CD andloads when the boot sequence in the BIOS of amachine is modified to load from the CD drive. TINAis complicated, but is a very powerful and high-levelanalytical environment for estimating anisotropic

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measurement errors on landmarks, and likelihood-based pseudolandmark placement with measurementcovariance estimates (as a favored method to tradi-tional Procrustes superimposition and PCA). TINAalso performs shape alignment by linear modelingand iterative optimization, landmark covarianceestimation, landmark acquisition reproducibilitystudies using anisotropic measurement covarianceto assess statistical equivalence between replicatedstudies, and Monte Carlo tests that use modelparameters with covariance estimates to evaluateexpected/obtained measurement errors in shapeanalyses for both surface and volumetric datasets.TINA and KNOPPIX are available from http://www.tina-vision.net/tina-knoppix/software.html.

ANATOMY OF A PC MORPHOSPACE

Figure 2 illustrates the properties of a typical PCmorphospace. Each point within the shape spacerepresents a unique configuration of the 500semilandmarks used to construct it. The markers A–Fshow the location of six brachiopod valves afterProcrustes superimposition. Only two axes are shownhere, but this morphospace has a total of five mean-ingful axes. The axes are sample limited at n − 1, butlarger samples could have as many as 1493 axes(3k − 7, where k is the number of landmarks). The PCscores record the position of each shell along eachaxis (illustrated by the two arrows pointing tovalve A). There are five scores for each brachiopod inthis example, one for each of the five axes. The scoresare uncorrelated (the product-moment correlationbetween the six scores on any two PCs is 0.0).

Each axis represents a literal spectrum of shape,illustrated here by a series of shape models (seeTable 1 for definitions) constructed for the positionsindicated by the scores under each one. Shape modelscan be constructed for any point in morphospace byreversing the operation in Equation 2 and addingback in the mean shape:

M= S U1 +C (3)

where M is a matrix containing the XYZ coordinatesof the model shape, S is the set of scores for the pointof interest in morphospace, U−1 is the inverse of theeigenvector matrix, and C contains the XYZ coordi-nates of the mean (consensus) shape. Note that foreigenvectors, the matrix transpose of U gives the

same result as the inverse and is computationallymore efficient to calculate. The origin of the axes(score 0,0) represents the shape of the sample mean(consensus shape) and is therefore identical on allindividual PC axes for score = 0.

The units of the axes are ‘Procrustes units,’where1 unit equals the centroid size of each shell aftersuperimposition, the magnitude of which is a func-tion of the number of landmarks. The space is scaledso that the Euclidean distance between two objects(e.g., D and E) is equal to their Procrustes distance(the sum of the distances between correspondinglandmarks, also known as the partial Procrustesdistance). Note that the distance between objectsequals the Procrustes distance only if all the axes areused in its calculations: only lengths a and b areshown in Figure 2, but the Euclidean distance alongc was calculated using all five dimensions.

Because each point in morphospace represents aunique configuration of landmarks, and because thedistances between shapes is preserved, the PC scorescan be used as shape variables for most otheranalyses, including statistical testing (Rohlf, 1993;Dryden and Mardia, 1998). The scores have theconvenient property of being statistically indepen-dent from one another (the correlation between scoreson different axes is 0). Note, however, that subsets ofscores (e.g., scores from one of several groups in thelarger sample) will be correlated. Each axis (eigen-vector) consists of a linear combination of theoriginal variables and thus represents a ‘package’ ofvariation among landmarks that is internallycorrelated but uncorrelated with other axes. The axesthemselves are sample dependent, and the fact thatthey are orthogonal (uncorrelated) is imposed bydesign, not by biology. As with Procrustes distances,however, all of the scores are needed to fully captureeach shape. Thus it is seldom justifiable to discardhigher axes when performing shape analyses.

As illustrated below, the axes of PC space aresample dependent because the first axis lies along themajor axis of variation in the sample. The orientationof the first PC will change (sometimes radically) asobjects are added or removed from the analysis. Thescores and the eigenvectors will therefore change asthe sample changes. However, the changes are alwaysrigid rotations and translations of the axes, andthese transformations preserve interobject distances.Thus, the Procrustes distance (Euclidean distance)

Polly and Motz—Three-dimensional geometric morphometrics 79

between two objects remains the same if calculatedusing all axes, no matter which other objects wereused to construct the morphospace, and nomatter howmuch the relationship between the objects appears tochange in a 2-D scatter plot.

THEORETICAL AND EMPIRICALMORPHOSPACES

The GMM morphospace described above is empiri-cal, derived from Cartesian coordinate sampling ofshapes, not a theoretical morphospace as in theRaup (1961, 1966) shell-coiling space. The axes ofGMM morphospaces are linear combinations of theCartesian coordinate variables representing shapesof objects, whereas the axes of theoretical mor-phospaces are parameters of the processes thatgenerate morphology (Thompson, 1917; Raup, 1966;McGhee, 1999). Those processes can have nonlinearrelationships to shape (and thus with shape space)and nonorthogonal relationships to one another(Mitteroecker et al., 2004; Polly, 2008a, Adams andCollyer, 2009; Viscosi and Cardini, 2011; Bookstein,2013, 2016; Gerber, 2014). Nevertheless, as long asthe same configuration of landmarks is used toquantify shape in both kinds of spaces, there must

exist a one-to-one mapping between them, although itcould be complex and nonlinear.

The complex relationship between theoreticaland empirical morphospaces can be illustrated usingthe Raupian (1961, 1966; Raup and Michelson,1965) shell-coiling equations (Fig. 3). These equa-tions, which were derived from earlier work onlogarithmic spiraling in shell growth (e.g., Thomp-son, 1917), describe the general shapes of mollusksand brachiopods as functions of rotation of the shellaperture around a coiling axis. The parameters in theequations represent the distance of the aperture fromthe axis (D), the rate of whorl expansion (W), and thetranslation of the aperture along the axis (T). Notethat T is always 0 for nonspiraled shells such as thebrachiopod valves in Figure 3. Along with the shapeof the aperture itself (S), these parameters can be usedto generate most of the forms present in bivalve,gastropod, cephalopod, and brachiopod valves.

Here, we used the Raupian equations to createvirtual brachiopod valves, the surfaces of which wereprepopulated with semilandmarks by placing semi-landmarks around the aperture curve (we usedequally spaced points) and rotating it in incrementsaround the coiling axis. Brachiopods are bilaterallysymmetrical, therefore, we held the Raupian transla-tion parameter constant at 0. To more efficiently

A

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2 (

9.0%

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(0,0) mean shape

Shape Spectrum along PC 1S

hape Spectrum

along PC

2

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132

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(0.0, 0.1)

(0.0, 0.05)

(0.0, -0.05)

(-0.15,0.0) (-0.15,0.0)

FIGURE 2.—Anatomy of the first two dimensions of a principal component (PC) morphospace. See text for explanation. Thebrachiopod valves used to construct this morphospace were generated using the Raupian shell coiling equations and areillustrated in Figure 3 (see Theoretical and Empirical Morphospaces section).

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model brachiopods, we rescaled the Raupian originalparameters so that our W parameter is ~1/100th of themagnitude of the Raupian parameter and our Dparameter is scaled with respect to the hinge lineinstead of the aperture center. By experimentation,a range of values consistent with brachiopod shapewas identified for W (1.4 to 2.0) and D (−2.2 to −1.2)Because brachiopod apertures are semicircular, thecenters must be shifted negatively toward the coilingaxis to produce a realistic shape. We added our ownparameters to vary aperture shape (S). A Gaborwavelet function was used to deform an ellipticalsemicircle into a bilaterally symmetrical pattern ofwaves to define a 3-D curve representing the aperture,onto which 20 equally spaced semilandmarks wereplaced. Shape variation in the aperture is controlledby three parameters: S1 is the number of peaks in thewavelet (which varied from 2 to 6 in this example); S2

is the amplitude scaling parameter (0.3–0.5); and S3 isthe arc length of the aperture with respect to half of anellipse (±0.0–0.3 Pi). Samples of brachiopod valveswere then created by simulating tip values for each ofthese five parameters using Brownian motion on aphylogenetic tree (Martins and Hansen, 1997).

We used the virtual clade of 28 brachiopods fromFigure 4 to illustrate how theoretical morphospace asdefined by the generating parameters in the Raupianequations maps onto empirical morphospace asdefined by shape variation in the resulting brachiopodvalves. The valves (numbered dots) are graphed intwo dimensions of the theoretical parameter space inFigure 4B. The axes of this morphospace are theparameters used to generate the valves (W, D, S1, S2),two of which are shown here. Their phylogenetictree was projected into the parameter space byreconstructing the node values of each parameter

A

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A B C D E F

0

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S2

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Node 0

Node 1

Node 2

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FIGURE 3.—Brachiopod valves generated with the Raupian equations and phylogenetic modeling: (A–F) virtual brachiopodvalves used to construct the morphospace in Figure 2 illustrated in lateral, dorsal, and anterior views; the grey points arethe semilandmarks that arise by rotating the semilandmarks of the aperture in 15° intervals once around the coiling axis usingthe Raupian equations; (G) diagram illustrating four of the five Raupian parameters used to generate the valves; (H) a simplephylogenetic tree with six tips. Tip values for each parameter were generated using Brownian motion. See text for details.D = distance of aperture from the coiling axis; S1 = number of waves; S2 = circumference length; (S3, wave amplitude, notshown); W = whorl expansion rate.

Polly and Motz—Three-dimensional geometric morphometrics 81

(Martins and Hansen, 1997; Revell and Collar, 2009)and connecting the branches of the tree (Rohlf, 2002).An empirical morphospace was constructed for thesame valves using Procrustes superimposition and

PCA on the valve shapes as represented by the semi-landmarks generated by the Raupian equations. Thevalves are shown in the first two dimensions of GMMempirical morphospace (Fig. 4C) with their phylogeny

Tim

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1.6 2.0 2.4 2.8-2.2

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root

W

D

A

B C

FIGURE 4.—Twenty-eight virtual brachiopod valves generated using the Raupian equations by simulating tip values for the fiveparameters using Brownian motion: (A) valves on a phylogeny generated using a constant-rate birth-death process; (B) valvesplotted in two dimensions of the parameter space; the phylogeny has been projected into the theoretical based on reconstructedancestral node values for the parameters; shape models along the margins show variation related to each of the two parameters;(C) valves plotted in an empirical morphospace based on GMM Procrustes superimposition and ordination; the phylogeny wasprojected based on node reconstructions of shape scores; each axis represents a linear combination of variance in thesemilandmarks, illustrated with shape models, not a Raupian parameter per se; the relative placement of valves is broadly similarin these two spaces (especially if PC 2, the direction of which is arbitrary, is flipped), but differ in many details (see text fordiscussion).

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projected using the same method, but based on the PCshape scores instead of the Raupian parameters.

The theoretical and empirical morphospaces thusrepresent two different ordinations of the sameobjects, one based on generating parameters and theother based on shape analysis. They have many broadsimilarities. The relative positions of the valves aresimilar in both spaces; the W axis of parameter spaceroughly corresponds to PC 1 of shape space, and theD axis roughly corresponds to PC 2 (if the latter isflipped). However, many small discrepancies can beobserved that are due to the nonlinear mapping ofparameters into shapes that arises from the non-linearities of the Raupian logarithmic equations. Asdiscussed in detail below, this mapping betweentheoretical parameter space and empirical shapemorphospace is very complex.

One of the most intuitive mathematical differ-ences between the two morphospaces is that theiraxes have different units and scaling. The GMMmorphospace is a truly Euclidean morphospace(sensu Mitteroecker and Hutteger, 2009) in whichdistances between objects can be calculated usingEuclidean geometry (e.g., the Pythagoras theorem forcalculating the distance between two points based onthe lengths of sides of a right triangle). This ispossible in the empirical shape morphospace becausethe axes all have the same units (Procrustes distanceunits), thus creating a natural scaling between them.Furthermore, the two PC axes that define the spaceare orthogonal by design. Distances between objectsin the empirical space are therefore also in Procrustesunits, and they equal the square root of the sumof squared differences between corresponding land-marks on the valves. The theoretical parameter spacedoes not have all of these properties, in large partbecause its axes have different units. The units of Ware rates equal to the proportional change in aperturesize per revolution, and the units of D are in distancesof whatever units the aperture size is measured (oftena proportion of its circumference or area). Euclideandistances between points in the theoretical space,although mathematically calculable, cannot beexpressed in common units. Although the distancesand directions in the empirical shape morphospacecan be interpreted in terms of shape differences andtransformations, which in turn allow processes suchas growth to be studied in the shape morphospace, thedistances and directions in theoretical morphospace

are more difficult to interpret. The theoreticalparameter space is a type of affine space (Mitter-oecker and Hutteger, 2009) that preserves someaspects of spatial geometry, including paralleltrajectories, but not others (e.g., direction, anglesbetween trajectories, and distances between points).All of the latter features will vary depending onarbitrary decisions about how to scale one axis toanother (i.e., how to equate a change in W with achange in D). One of the consequences of thesemathematical differences between the two morpho-spaces is that the ancestral shapes reconstructed inthem cannot be expected to match because the algo-rithms used to reconstruct ancestral node values relieson Euclidean mathematical assumptions that are notmet by the theoretical parameter space.

This is not to say that the equations of theoreticalmorphology and their associated parameter spacesare not useful—indeed, theoretical morphospaces arearguably invaluable to the quantitative study of form.Many kinds of operations in parameter space canbe imagined, including likelihood and Bayesianestimations of the parameters that underlie the evo-lution of form in a species or clade. The parameters oftheoretical morphology equations are (as an ideal)linked to the biological processes (evolutionary ordevelopmental) that produce morphology.

STATISTICAL ANALYSES

Statistical analysis of shape, by which we mean test-ing a hypothesis for statistical significance against arandom null model, is a topic too varied to coveradequately in a review such as this (the literaturereviewed above includes many papers on statisticalanalysis). Briefly, however, a large number of statis-tical GMM questions in paleontology fall into one oftwo general types: testing two or more groups fordifference in shape (e.g., samples from differentlocalities or stratigraphic intervals), and testing shapevariation for association with another variable, eithercontinuous or categorical (e.g., body size or dietarytype). Testing shape for association with a continuousvariable is a type of regression question, whereastesting groups for shape differences and testing forassociation with a categorical variable are bothtypes of analysis of variance (ANOVA) questions.Because shape is multivariate, both kinds of testswill be multivariate (Dryden and Mardia, 1998).

Polly and Motz—Three-dimensional geometric morphometrics 83

Frequently, paleontological data points are from dif-ferent species, in which case phylogeny must also betaken into account (Martins and Hansen, 1997).Because shape is a complex variable that is unlikelyto be multivariate normal, permutation or other non-parametric resampling statistical tests are usuallypreferred over parametric tests (Manly, 2004;Kowalewski and Novack-Gottshall, 2010). We willpresent examples of multivariate regression andMANOVA, both with and without phylogeneticcorrection in the phylogenetic section of this paper.

DEALING WITH BREAKAGE ANDDEFORMATION IN FOSSILS

All else being equal, analyzing the shape of fossils isno different than analyzing the morphology of extantorganisms (Goswami and Polly, 2010a). As long asthe same morphological structures are preservedintact, extinct and extant organisms can be combinedinto the same GMM analysis with no specialconsideration. Fossils frequently present specialchallenges, however, because of breakage anddistortion.

Broken fossil specimens can result in missinglandmarks, a problem that is difficult or impossible toovercome for GMM analysis because Procrustesanalysis requires all landmarks to be present tooptimize the superimposition. In ordinary morpho-metrics, various strategies exist for bypassing orimputing missing data, such as substituting thesample mean value for missing data points in PCA,which is possible because only deviations from themean contribute to the ordination (e.g., Reymentet al., 1984; Strauss et al., 2003). These tactics do notwork with GMM because the mean value of eachlandmark coordinate is dependent on the super-imposition of the entire sample, which is optimizedbased on the fit of all landmarks. Imputed landmarksthus have a greater potential to affect the outcome ofGMM analyses than imputed data points do inordinary distance-based statistics. Nevertheless, sev-eral strategies have been proposed to estimate thepositions of missing landmarks. Sometimes they canbe estimated by eye based on the topography of themorphology around the missing point (e.g., theposition of a broken tooth cusp can often be estimatedaccurately based on the slope of the surroundingarea). In symmetrical structures, missing landmarks

can be estimated from their symmetrical homologsby mirroring (Mardia et al., 2000; Webster andSheets, 2010). If the broken fossil is part of alarger sample with complete sets of landmarks, theposition of missing landmarks can be estimatedbased on their covariance with other landmarksin the unbroken sample (Gunz, 2005; Gunz et al.,2009; Mitteroecker and Gunz, 2009). This strategydepends on the covariance structure being similarin the fossil with the missing landmark as in themore complete sample. All else being equal, the morenonmissing landmarks there are and the moreclosely related the sample, the more accurate thereconstruction will be.

Sediment compaction, tectonic stress, and otherprocesses frequently deform fossils. Undeformedfossils should always be given preference for GMMstudies that depend on accurate comparisons of livingmorphology. Techniques exist, however, for retro-deforming fossils (Hughes and Jell, 1992; Zollikoferand de León, 2005; Angielczyk and Sheets, 2007;Gunz et al., 2009). The simplest methods are used onsymmetrical specimens that are intact but plasticallydeformed (plastic deformation changes the shape ofthe object without breaking it), such as equalizingthe distances of symmetric landmarks across areconstructed sagittal plane (Ogihara et al., 2006).Other methods find the minimum amount ofstretch required to make the two halves symmetrical(Zollikofer and de León, 2005), or find the best fit of adistorted specimen to an undeformed one usingalgorithms similar to those used to place slidingsemilandmarks (Gunz et al., 2009). Brittle deforma-tion, which can include both breakage and plasticdeformation that has independently affected thebroken pieces, requires both piecing the object backtogether and finding optimal parameters for plasti-cally retrodeforming it (Zollikofer and de León,2005). This procedure has so many independentparameters that many solutions can be found thatdiffer in shape (Boyd and Motani, 2008).

THE COMPLEX RELATIONSHIP OFMORPHOSPACE AXES TO BIOLOGICAL

FACTORS

A common goal of morphometric analysis is toidentify trajectories in morphospace that are pro-duced by biological factors such as developmental

84 The Paleontological Society Papers

processes, functional relationships to the environ-ment, or phylogenetic history (e.g., Monteiro, 1999;Kim et al., 2002; Mitteroecker et al., 2004; Bastiret al., 2006; Hunt, 2007a, b; Figueirido et al., 2009;Adams and Collyer, 2009; Pierce et al., 2009; Pollyet al., 2013b; Schreiber et al., 2014). GMM is apowerful tool for this kind of study, but it must beremembered that the PC axes of morphospace cannotbe expected to naturally align with the shape variationproduced by individual biological factors; therefore,univariate analysis of a single PC axis is unlikely tofully recover the relationship between it and anindependent variable.

The paths of trajectories through morphospace ofRaupian shell-coiling parameters illustrate why this isso. Because the Raupian equations were used togenerate the brachiopod valves in Figure 4, theycan serve as proxies for morphogenetic processesand their relationship to shape space. The Raupianparameters are analogous to genotypes, and hisequations are analogous to the developmental inter-actions that produce phenotypes. Just as a develop-mental biologist might systematically alter geneticbackgrounds to understand developmental processesfrom the phenotypes that are produced, so, too, canwe systematically change the Raupian parameters totrace the corresponding phenotypes in shape space.Knowing the Raupian equations, we know the truerelationship between parameters, processes, andshape, and can evaluate the extent to which we couldcorrectly recover them if we only knew the para-meters and phenotypes.

Figure 5 shows the trajectories of four of theRaupian parameters (W, D, S1, and S2) projected intothe first three dimensions of the valve morphospace(compare with Fig. 4C). Valves 3, 23, 35, 40, 46, and49 are labeled in Figure 5A to aid in orienting it withrespect to Figure 4C (note that Fig. 5B is based on adifferent set of random valves). Each trajectoryrepresents shapes generated when one parameter isvaried and the others are held constant at the value ofthe mean shape (which causes the four trajectories tointersect at the morphospace origin). In this example,W roughly parallels PC 1, in part because varying therate of whorl expansion affects the positions of nearlyevery semilandmark on a logarithmic scale, thusgenerating a lot of shape variance. W is at a slightangle to the PC axes and is gently curved and thus isnot fully congruent with PC 1. D roughly parallels

PC 2. S1 is more complex in that it is not linear withrespect to shape space, but rather forms an undulatingwave traveling along PC 2 with peaks and troughs inPC 3. S1 is the parameter controlling number ofwaves in the valve margin. The path of its trajectorythrough shape space can be understood by imaginingthe shapes of the valve margins with three and fivewaves respectively: because of bilateral symmetry,three of the five sulci would be in the same positionand thus not contribute to shape difference. Now,imagine margins with three and four waves respec-tively: none of the sulci would line up, which wouldmake valves with four waves more different from onewith three waves than a valve with three sulci is fromone with five. The oscillating trajectory of S1 istherefore caused by alternation between even andodd numbers of waves. S2, the circumference, has aneven more complex trajectory, tracing a stair-steppattern at an angle along both PC 2 and PC 3. Thiserratic path is caused by ranks of semilandmarksalong the radial axis slipping on and off of sulcias wave spacing is changed by changes in thecircumference.

Although the correlation between shape W and Dcan be captured by univariate regression of PC 1 orPC 2 onto them, respectively, the two aperture-shapeparameters produce trajectories that are both multi-variate in their orientations (traveling along morethan one axis) and nonlinear in their paths. Theserelationships are evident in the univariate summarystatistics in Table 3, which reports the amount ofvariance accounted for by each of the first ten PC axes(including as percent of the total variance) and thecoefficient of determination (R 2) of the regression ofthe scores from each axis onto each of the Raupianparameters. R 2 can be interpreted as the proportion ofshape variation along each axis that is explained bythe parameter. As expected, W explains 97% of var-iance along PC 1 and very little on any other axis.Similarly, D explains a high percentage of PC 2variation (79%), although it also explains a sub-stantial proportion of PC 3 (16%). The univariateresults for the two aperture-shape parameters aremore complicated, as one might expect. S1 has thehighest explanatory value for PC 3, but has somevariance scattered across other PCs. Note that for allthree of these parameters, the sums of the R 2 valuesacross the first 10 PCs is nearly 1.0 (100%) becauseall of the shape variance associated with the

Polly and Motz—Three-dimensional geometric morphometrics 85

parameter is distributed somewhere in the morpho-space, most of it in the first ten axes. S2 has someassociation with PC 2 (22%) and PC 3 (17%), but itsvariance is distributed more erratically across manyPCs. In fact, only 67% of S2 shape variance isaccounted for by the first 10 PCs—43% of its

variance is on PCs 11–27, even though they accountfor <0.1% of the total shape variance in the dataset.The reason is that circumference changes create onlysmall differences in the positions of semilandmarkscompared to W, D, or S1, thus appearing insignificantin the total pool of variation in this dataset, and theyproduce shape variation that is almost randomlydistributed.

THE SAMPLE DEPENDENCE OFMORPHOSPACE ORIENTATION

Added complexity comes from the fact that GMMmorphospace axes are sample dependent, but thetrajectories produced by underlying biologicalprocesses are not. PC axes are constructed to satisfy aset of linear mathematical rules for orthogonality(each axis is at right angles to all others) and direction(the first axis is aligned to the major axis of variancein the sample, the second is aligned at right anglesalong the major axis of residual variance, and so onuntil all variance is accounted for). The changedorientation of the PC morphospace is nothing morethan a rigid rotation of the original data: it can beconsidered as a new directional viewpoint of theoriginal Cartesian landmark coordinates in multi-variate space. The orientations of GMM morpho-space axes are therefore driven by the sample onwhich they are based, unlike theoretical morpho-spaces in which axes are firmly tied to parameters.Add or subtract objects from a sample, and theorientation of morphospace axes will change, eventhough the relative positions of the objects within itwill not (the change is a rigid rotation, whichpreserves the distances between them, ensuring theyremain equal to the Procrustes distance).

Sample-dependent changes in morphospaceorientation can have a startling effect on the rela-tionship between individual PC axes and biologicalfactors. Add an outgroup, for example, and whatseemed to be a strongly significant statisticalassociation between PC 1 and body mass based onunivariate tests can suddenly become insignificant.Figure 5B shows a morphospace constructed from adifferent random simulation of brachiopod valvesbased on the same phylogeny with the same para-meters. The Raupian equations that relate shape to theunderlying parameters are identical, but the sample is

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different. The Raupian parameter trajectories appearto be flipped into a new orientation and transformed.W now parallels PC 2 (although not truly parallel),D parallels PC 1 (also oblique), S1 appears to be morehook-shaped than a sine wave, and S2 appears to havea more irregular trajectory, the major direction ofwhich is parallel to PC 3. The switch of W and Dbetween PC 1 and 2 is straightforward in the uni-variate variance explained by them (Table 3), but notso the change in S1 and S2, both of which havecomplex trajectories through shape space so that theirvariance is still distributed across several PC axes, butdifferently than with the original sample. In reality,the trajectories of the four parameters have preciselythe same relationship to each other and to shape; whathas changed is simply the viewpoint in multivariatespace, which is positioned relative to the major axesof variation in our sample (i.e., the orientation of themorphospace relative to the trajectories).

The sample dependency of principal componentsordination means that individual PC axes should not

be expected to be associated exclusively with anyparticular developmental, functional, ecological,temporal, or phylogenetic factors (Monteiro, 1999;Bookstein, 2013, 2016). For a factor to be reliablyassociated with only one PC axis, it would either haveto be responsible for always producing the bulk ofshape variance in every sample and therefore wouldalways dominate variance enough to align with PC 1no matter what the sample, or the factor would haveto be consistently uncorrelated with whatever com-bination of factors determines the orientation of PC 1,which is the only way in which one of the higher PCscould come to lie parallel to it.

Sample-dependent changes in morphospaceorientation will be more pronounced when the sampleconsists of multiple taxa rather than individuals fromthe same biological population. Adding and sub-tracting taxa changes the phylogenetic covariancestructure of the shapes, which is part of the covar-iance structure that drives the principal componentsorientation along with the morphogenetic processes

TABLE 3.—Comparison of statistics for univariate regression of two random shape samples onto the four Raupian parametersused to generate them. The amount of shape variance is explained by each principal component (PC; also expressed aspercent of the total). D = distance from coiling axis; R 2 = coefficient of determination (proportion of variance explained by theregression); S1 = number of waves in the aperture shape; S2 = circumference of the aperture shape; W = whorl expansion rate.

Variance % Explained W (R 2) D (R 2) S1 (R2) S2 (R

2)

Sample 1PC 1 0.01148 69.4% 0.97 0.02 0.01 0.00PC 2 0.00286 17.3% 0.00 0.79 0.02 0.22PC 3 0.00146 8.8% 0.01 0.16 0.82 0.17PC 4 0.00030 1.8% 0.00 0.00 0.02 0.06PC 5 0.00024 1.4% 0.00 0.00 0.05 0.01PC 6 0.00009 0.5% 0.00 0.00 0.06 0.05PC 7 0.00006 0.4% 0.01 0.00 0.01 0.02PC 8 0.00002 0.1% 0.00 0.00 0.00 0.07PC 9 0.00001 0.1% 0.00 0.00 0.01 0.02PC 10 0.00001 0.0% 0.00 0.00 0.00 0.05

Total 0.01653 99.9% 0.99 0.97 1.00 0.67

Sample 2PC 1 0.00412 56.1% 0.05 0.97 0.16 0.35PC 2 0.00270 36.7% 0.95 0.01 0.02 0.00PC 3 0.00036 4.9% 0.00 0.01 0.61 0.00PC 4 0.00012 1.6% 0.00 0.00 0.19 0.08PC 5 0.00002 0.3% 0.00 0.00 0.01 0.40PC 6 0.00001 0.2% 0.00 0.00 0.01 0.12PC 7 0.00001 0.1% 0.00 0.00 0.00 0.00PC 8 0.00001 0.1% 0.00 0.00 0.00 0.02PC 9 0.00000 0.0% 0.00 0.00 0.00 0.01PC 10 0.00000 0.0% 0.00 0.00 0.00 0.00

Total 0.00735 100.0% 1.00 0.99 1.00 0.98

Polly and Motz—Three-dimensional geometric morphometrics 87

(note that these two factors are not the only onescontributing to covariance structure among taxa).Adding or subtracting new individuals to an intra-specific population-level sample should not affect theorientation of the PC axes much, as long as there is asufficiently large number to estimate the covariancestructure reliably (e.g., Cheverud, 1982; Ackermannand Cheverud, 2000; Arnold et al., 2001; Goswamiand Polly, 2010a, b; Marroig et al., 2012). Changingthe number of intraspecific populations representedin a sample might also alter the orientation of PCaxes because phylogeographic and environmentaldifferences in phenotypic covariances occur frequently(e.g., Via and Lande, 1985; Phillips and Arnold, 1999;Polly, 2005). Therefore, multivariate statistics on theentire set of shape variables should always be usedwhen testing for statistical associations between fac-tors such as body size, diet, geographic location, ortaxonomic identity (e.g., Monteiro, 1999; Viscosi andCardini, 2011; Zelditch et al., 2012).

THE DEPENDENCE OF MORPHOSPACEORIENTATION ON LANDMARKING

SCHEMES

As discussed above, many strategies for placinglandmarks and semilandmarks are available. For 3-Dsurfaces, semilandmarks will usually be the obvioussolution, but the choice of placement method willseldom be straightforward and will often be based onwhat aspect of shape a researcher wants to capture andwhat tools are available. These choices will alwayshave an effect because shape distances betweenobjects and the total shape variance in the sample aredirectly related to the interaction between point pla-cements and the underlying shape. However, whetherthat effect is large or small depends on the context.

We illustrate the effects of landmarking inFigure 6 by applying different strategies to the sixbrachiopod valves from Figure 3. The morphospaceordination in Figure 6A is based on the semiland-marks produced by the Raupian equations: semi-landmarks around the aperture are equally spaced andthe ranks from the umbo to the margin are logarith-mically spaced (closer together on the umbo thanclose to the valve margins). One could argue that thisplacement strategy is biologically homologousbecause each rank is analogous to a growth line. Themorphospace in Figure 6B is based simply on an

analysis of aperture shape, with semilandmarksequally spaced around it. The third morphospace inFigure 6C is based on equal spacing of semiland-marks around the aperture and along arcs runningfrom the semilandmarks back to the umbo. Thisscheme is similar to the one in Figure 6A, except thatthe density of landmarks is roughly the sameeverywhere.

The differences between these three semi-landmarking schemes are small in this example.Allowing for the reversal of PC 2, the first twoschemes produce almost identical morphospaceordinations. The similarity might seem surprisinggiven the radical difference in the number of pointsand the complete absence of points that wouldmeasure the dorsal curvature of the valve. However,the similarity in this case is expected because eachrank of points in Figure 6A is itself identical to theaperture shape except in scale and orientation. It isunlikely that ordinations of real brachiopods based onsurface and apertural shape would be as similar asthese hypothetical examples. Although broadlysimilar, the ordination in morphospace Figure 6C ismore different than the other two (e.g., taxon F liesbetween C and D in this ordination instead ofbetween A and D). This difference might also seemsurprising because the placement of semilandmarks isvisually similar to that in Figure 6A, but in Figure 6C,the ranks from umbo to margin are equally spaced,which means that only the final rank has the samecross-sectional shape as the aperture.

The similarity among these three ordinationsshould not be taken for granted. Other publishedexamples demonstrate that the strategy used to placepoints can have a profound effect on the outcome.Salient examples have been provided by Gunz andMitteroecker (2013) that demonstrate how slidingsemilandmark strategies can substantially improvethe fit between shapes (see also Bookstein, 1997;Gunz et al., 2005; and Perez et al., 2006). Polly(2008b) illustrated an example in which the samesemilandmarking strategy (eigensurface, which issimilar to the equal-spaced point scheme in Fig. 6C)works well for one bone (the calcaneum in carnivor-ans), but exaggerates shape differences in another(the astragalus) because the long neck of the latterproduces a sharp boundary that causes substantialchanges in the positions of many semilandmarks inresponse to changes in the proportional length of the

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next, which is only a small part of shape variation inthe astragalus.

LINEAR AND NONLINEAR TRAJECTORIES

Nonlinear relationships between biological factorsof interest and shape space are a different matter.

Although standard multivariate tests might be ade-quate to assess the overall association between shapeand the factor of interest, estimating the factor tra-jectory through morphospace requires more specia-lized techniques. The trajectories can be traced inFigure 5 only because we have access to the equa-tions that were used to generate the shapes. In real

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FIGURE 6.—Differences in morphospace ordinations associated with choice of semilandmarking strategy: (A) semilandmarkplacement is logarithmic, derived directly from the Raupian shell-coiling equations; (B) equally spaced semilandmarks surroundthe aperture of the valve; (C) equally spaced semilandmarks surround the aperture of the valve and other semilandmarks areequally spaced along transects between them and the umbo.

Polly and Motz—Three-dimensional geometric morphometrics 89

datasets, the goal could be to estimate an analogousmorphogenetic process when we know only oneunderlying parameter (e.g., body size, ontogeneticage, whorl expansion rate, or genetic mutation).In this case, we need a nonlinear model that can befit to the data. The issue of nonlinearity is importantbecause many developmental and ecologicalprocesses are known to have nonlinear relationshipsin shape space, e.g., human cranial development(Mitteroecker et al., 2004; Bastir, et al., 2006),ammonite growth (Gerber et al., 2007), trilobiteontogeny (Gerber and Hopkins, 2011), and sexualdimorphism of fish in different environments(Collyer and Adams, 2007).

Trajectories can be traced by connecting points inshape space or by fitting curvilinear (nonlinear)regressions (Mitteroecker et al., 2004). Once trajec-tories have been identified, an approach for comparingthem is phenotypic trajectory analysis (PTA; Collyerand Adams, 2007, 2013; Adams and Collyer, 2009).Any given trajectory can be described by parametersof its own (i.e., size, direction, shape and orientation inmorphospace) that allow it to be analytically comparedto others. Trajectories have a position, direction, andlength (size) in shape space that correspond to thestarting and ending shapes, the change in shape alongthe trajectory, and the amount of change. BecauseGMM morphospace is truly Euclidean, these PTAparameters describe the differences between the kindand magnitude of shape change associated with thebiological factor of interest. Adams and Collyer (2009)measured trajectory size using the path length alongthe segments of the trajectory, which is equivalent tothe total shape change in Procrustes units, overall tra-jectory orientation using the major axis (PC 1) of thetrajectory points (which can be imagined as the straightpath that parallels the trajectory as closely as possible),and trajectory shape using Procrustes analysis of thetrajectory points, which works as comparison ofthe landmark points themselves. Statistical tests(e.g., MANOVA) can then be used to determinewhether differences in these parameters are statisti-cally significant.

SHAPE SPACE AND PHYLOGENETICPATTERNS

Paleontological shape data frequently have aphylogenetic component in the form of lineages of

a single species sampled through a series ofstratigraphic units or a collection of tip species on aphylogenetic tree. Shared ancestry introduces its ownpattern of covariances that not only influence thesimilarities and differences between taxon shapes(i.e., Procrustes distances), but also affect the orien-tation of morphospace (i.e., the shape variables andthe alignment of the principal components relative toother biological factors). These covariances alsocause nonindependence in the data points, whichmust be taken into account for certain kinds ofstatistical analyses (but not for all questions). Theycan also produce sometimes-unexpected distributionsof taxa in shape space.

Methods for phylogenetic comparative analysis oftip taxa are well described elsewhere (e.g.,Felsenstein, 1985, 2012; Graffen, 1989; Harvey andPagel, 1991; Garland et al., 1992; Martins andHansen, 1997; Butler and King, 2004; Revell andCollar, 2009; Revell, 2010), and all involve proce-dures for correcting for phylogenetic covariances aspart of performing statistical analyses such as regres-sions or ANOVAs. The two most common proceduresare: 1) phylogenetic independent contrasts (PIC),which removes the effects of shared common ancestryby reducing the data to distances between pairs of taxaand/or nodes (contrasts) so that the change thattook place along each branch of the phylogeny isrepresented only once (Felsenstein, 1985); and 2)phylogenetic generalized linear models (PGLM),which adjust regression parameters for the expectedcovariance associated with the phylogenetic topology(Martins and Hansen, 1997). Both approaches nor-mally require the assumption that the traits haveevolved by Brownian motion (random walk), butvariants of the PGLM method can accommodateoverdispersion (e.g., disruptive selection) or under-dispersion (e.g., stasis, constraint, Ornstein-Uhlenbeckprocesses). The PGLM and related likelihood methodscan also be used to estimate the most likely ancestralstates for traits, also assuming a Brownian motionmode of evolution (e.g., Martins and Hansen, 1997;Mooers and Schluter, 1999; Pagel, 1999; Polly, 2001;Pagel et al., 2004). Several authors have specificallydiscussed adaptations of these methods forgeometric morphometrics, which generally involvesapplying multivariate versions of these standardmethods to shape scores (Rohlf, 2001; Caumuland Polly, 2005; Klingenberg and Gidaszewski, 2010;

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Gómez-Robles et al., 2013; Klingenberg andMarugán-Lobón, 2013; Monteiro, 2013; Adams,2013, 2014a). Adams (2014b) and Adams andCollyer (2015) proposed a somewhat differentapproach that performs phylogenetic transformationson both the multivariate shape data and the indepen-dent variable before performing statistical tests, whichthey argued improved statistical power. Techniquesused to fit evolutionary models, e.g., Brownian motionor Ornstein-Uhlenbeck to phylogenetic tree data(Mooers and Schulter, 1999; Revell et al., 2008;Harmon et al., 2010; Gómez-Robles and Polly,2012; Slater et al., 2012) and lineage data (Gingerich,1993, 2009; Polly, 2004; Hunt, 2007a, b; Hunt andRabosky, 2010) can also be applied to shape variablesas long as care is taken to use fully multivariatealgorithms.

The reason why phylogenetic correction isneeded is illustrated in Figure 7A–B. We randomlyevolved a univariate trait on the same phylogeny as inFigure 4A to provide an example of a variable thathas no underlying correlation with valve shape.Because of shared phylogenetic history, the randomlyevolved trait appears to have a strong correlation withvalve shape if multivariate regression is appliedwithout correcting for the effect of phylogeneticcovariance (Fig. 7A). Approximately 16% of shapevariance appears to be explained by this random trait(R 2 = 0.16) when all PCs are taken into account,with a significant probability of being greater than 0.0(p = 0.01). This relationship is introduced purely bythe topology of the tree (plus chance): taxa that sharea recent common ancestor are more likely to besimilar in shape and similar in the random trait,therefore a correlation emerges. When the expectedphylogenetic covariance is removed, the correlationdisappears. Figure 7B shows a multivariate regres-sion of phylogenetic independent contrasts of theshape scores onto contrasts of the random trait,with significance measured by randomization test(the GMM method described by Klingenberg andMaurgán-Lobón, 2013). With this correction, theproportion of variance explained by the random traitdrops essentially to zero (R 2 = 0.003) and therelationship is no longer significant (p = 0.25), asshould be the case when the random trait has nounderlying correlation with shape. This exampleworks perfectly because the data fit the assumptions:both shape and trait evolved by Brownian motion

and there is no nonphylogenetic correlation betweenthe two. Real world examples are likely to bemore complicated, if for no other reason thanevolution cannot be expected to follow a Brownianmotion model.

Species lineages sampled through stratigraphicintervals are a special case of phylogeneticallystructured data, one that is simpler to deal with inmany respects. A species lineage is equivalent to asingle branch on a phylogenetic tree—a series ofputatively ancestor-descendant population sampleswith no branching events—such that later samplesshare phylogenetic history with earlier ones. Suchdata constitute a classic time series, from whichspurious correlations can arise between the biologicaldata of interest and independent factors (e.g., Raup,1977; Roopnarine et al., 1999; Gingerich, 2009; Huntand Rabosky, 2010). The correlation is caused by thefact that most time-series data, including Brownianmotion random walks, have general trends over time.Two such trends, even though unrelated, will producea serial correlation, either positive or negative, simplybecause they are measured over the same interval.Arguably the simplest approach to correcting for thisphenomenon is the method of first differencing(McKinney, 1990). Instead of analyzing the valuesof the dependent and independent variables, thedifferences (changes) between points in the timeseries are measured. If the variables are truly corre-lated, they should, on average, change in concert, soan underlying connection between the variablesshould produce a significant correlation between theirfirst differences. This approach is almost identical tophylogenetic independent contrasts in that the timeseries is decomposed into its constituent independentchanges.

Evolutionary time series can produce surprisingpatterns in GMM morphospace (Fig. 7C–D) (Mitter-oecker et al., 2004; Bookstein, 2013). All things beingequal, a sequence of taxa in an evolutionary time seriesforms a shape gradation in which shapes from adjacentintervals are more similar than shapes at the seriesbeginning and end. When data have high dimension-ality, which GMM semilandmarks on 3-D objectsusually have, the first PC of a densely sampled lineagewill usually contrast its beginning and end, much as thetrajectory orientation parameter of Adams and Collyer(2009) is estimated with the first PC of its points. Thesecond PC, which is always at right angles to the first,

Polly and Motz—Three-dimensional geometric morphometrics 91

will usually contrast the middle of the series with itsends, thus producing a ‘horseshoe-shaped’ pattern ofpoints in morphospace. Furthermore, the third PCwill contrast the centers of the regions lying betweenthe middle and the endpoints. The last two panels inFigure 7 illustrate this phenomenon. Ten Brownianmotion random walks with 150 variables (50 land-marks with three dimensions each) were generatedfor 1000 steps, shown with their first two variablesin Figure 7C. Each of these was submitted toprincipal components, all of which are shown toge-ther in a single graph of the first three PC axes inFigure 7D. The same kind of curved pattern canbe expected from any gradational series, such asanalysis of segmental structures such as vertebrae

(e.g., Head and Polly, 2015). Bookstein (2013)described the mathematics behind this phenomenon,the predictability of which increases along with thenumber of variables. This phenomenon is not a pro-blem for analysis per se (other than the issue of serialcorrelation if shape is being regressed onto anothervariable of interest), because the original paths ofeach time series are preserved in the full multi-dimensional shape space. Corrections such asdetrended correspondence analysis should never beapplied to shape data because they distort shape spacein unpredictable ways that can invalidate statisticalconclusions and compromise the ability to maptrajectories in the detrended space back to shape.Instead, all dimensions of the shape variables should

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FIGURE 7.—Phylogenetic effects on morphospace ordinations: (A) multivariate regression of shape scores onto a randomlyevolved variable; shape and the random variable are not correlated, except as an artifact of phylogenetic covariance; theunivariate relationship of PC 1 is shown here; (B) multivariate regression of independent contrasts of shape onto independentcontrasts of the same random variable; phylogenetic covariances no longer influence the regression and the apparent significanceof the relationship has disappeared; (C) ten high-dimensional, Brownian-motion, random-walk lineages illustrated with the firsttwo of 150 variables; (D) the same ten random walks, each ordinated with principal components and plotted together. See textfor discussion.

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be used so that the analysis captures the true pattern inthe data rather than the partial pattern on the first fewaxes. For example, evolutionary model fitting wouldlikely find a directional trend in these data if only thefirst two or three PCs were considered, but Brownianmotion would be the obvious pattern based on theentire dataset.

The curvature of gradational shapes in the firstfew axes of shape space is usually present in anysample with a phylogenetic component, such as isvisible in the W trajectory in Figure 5A, but thepattern is often subtle because sampling of the tree issparse (usually tips only), and because the branchingpattern adds noise to the gradation. However, prin-cipal component plots of tip taxa often have their ownregular patterns: the first PC axis often separates themost distantly related taxa because they are, all otherthings being equal, the most different in shape. Thesecond axis separates the next closest clade, the thirdaxis the next closest, and so on. Differences betweenclosely related tips are often found on the highestaxes. With the caveats already mentioned, this is not aproblem as long as all dimensions of shape space areincluded in analyses, and phylogenetic comparativemethods are applied when necessary. Revell (2009)described an ordination method called phylogeneticprincipal components analysis (pPCA) that removesthe expected covariance due to phylogeny from thecovariance matrix used to calculate the eigenvectors(P in Equation 1). This procedure aligns objects with anaxis that is not affected by phylogenetic covariances,but it does not remove phylogenetic correlation fromthe PC scores (Revell, 2009; Polly et al., 2013a). Theproperties of pPCA complicate many of the ordinaryinterpretations of morphospace (e.g., the first PC ofpPCAmight account for less variance than higher PCs)and should be used with caution. Fully multivariatestatistical results should be identical regardless ofwhether PC or pPC shape scores are used.

CONCLUSIONS

Geometric morphometrics provides a powerful set ofmethods for exploring, analyzing, and modeling 3-Dshape. Practical hurdles for acquiring GMM datafrom 3-D objects are substantial, but are becomingeasier as new software tools are developed. Theore-tical hurdles for how to represent the complexity of3-D objects with landmarks and semilandmarks are

only beginning to be addressed, partly because someof the issues of homology, weighting, and artifactswere not apparent until recent years when 3-D ana-lyses began to be feasible. The complexity of 3-Dshape analysis re-emphasizes what are arguably thetwo most important recommendations for GMMstudies: 1) always use multivariate methods and all ofthe morphospace axes in an analysis; and 2) antici-pate the possibility that the factors of interest have acomplex, nonlinear relationship with shape.

ACKNOWLEDGMENTS

L. Tapanila and I. Rahman generously invited us toparticipate in this short course and were patient withour schedules. A. Reese scanned the brachiopodsused in Figure 1. P. Novack-Gottshall providedinsightful comments on some of our analyses. Specialthanks to S. Marcus for fielding questions, handlingspecial formatting requests, and seeing both thispaper and the entire volume through to publication.Scanners used to acquire data were supported by theIndiana University Center for Biological ResearchCollections. Supercomputing was supported by LillyEndowment through the Indiana University Perva-sive Technology Institute and the Indiana METACytInitiative. This research was partially supportedby National Science Foundation grant NSFEAR-1338298 to PDP.

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