Differences between sliding semi-landmark methods in geometric morphometrics, with an application to human craniofacial and dental variation

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Over the last decade, geometric morphometric methods have been applied increasingly to the study of human form. When too few landmarks are available, outlines can be digitized as series of discrete points. The individual points must be slid along a
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    J. Anat.  (2006) 208  , pp769–784© 2006 The Authors Journal compilation © 2006 Anatomical Society of Great Britain and Ireland  BlackwellPublishingLtd  Differences between sliding semi-landmark methods in geometric morphometrics, with an application to human craniofacial and dental variation  S. Ivan Perez, Valeria Bernal and Paula N. Gonzalez  División Antropología, Facultad de Ciencias Naturales y Museo, Universidad Nacional de La Plata, Argentina Abstract  Over the last decade, geometric morphometric methods have been applied increasingly to the study of humanform. When too few landmarks are available, outlines can be digitized as series of discrete points. The individualpoints must be slid along a tangential direction so as to remove tangential variation, because contours should behomologous from subject to subject whereas their individual points need not. This variation can be removed byminimizing either bending energy (BE) or Procrustes distance (D) with respect to a mean reference form. Becausethese two criteria make different assumptions, it becomes necessary to study how these differences modify theresults obtained. We performed bootstrapped-based Goodall’s F   -test, Foote’s measurement, principal component(PC) and discriminant function analyses on human molars and craniometric data to compare the results obtainedby the two criteria. Results show that: (1) F   -scores and P   -values were similar for both criteria; (2) results of Foote’smeasurement show that both criteria yield different estimates of within- and between-sample variation; (3) thereis low correlation between the first PC axes obtained by D and BE; (4) the percentage of correct classification issimilar for BE and D, but the ordination of groups along discriminant scores differs between them. The differencesbetween criteria can alter the results when morphological variation in the sample is small, as in the analysis ofmodern human populations.  Key words  dental and facial data; minimum bending energy; minimum Procrustes distance.   Introduction  The analysis of human morphological variability has along tradition in anthropological studies. Several recentanalyses, usually based on craniometric data, supportthe existence of low morphological variation amongmodern humans (Howells, 1989; Relethford & Harpend-ing, 1994; Relethford, 1994; Hannihara et al. 2003).Howells’s (1989) study of worldwide craniometricvariation established that differences among modernhuman populations are small based on the comparisonof modern crania with several archaic forms. Likewise,Relethford (1994) and Relethford & Harpending (1994)found that the amount of morphological variationamong geographical regions is relatively low withrespect to intrapopulation variation. Such results are inagreement with those based on genetic data (Lewon-tin, 1972; Barbujani et al. 1997; among others). Thesecraniometric studies have been based on traditionalmorphometric methods, which have been used widelyin anthropology. However, given the low levels ofmorphological variation found among human samples,other techniques that allow the capture of subtleshape differences become necessary. Geometric mor-phometric methods (Rohlf & Marcus, 1993; Adamset al. 2004) have been increasingly applied to the studyof human form over the last decade. These approachesfocus on methods that capture the geometry of mor-phological structures and preserve this informationthroughout the analyses. More powerful morpho-metric analyses can be performed using these morecomprehensive data (Rohlf, 1990, 2000a) and very subtle  Correspondence  Dr S. Ivan Perez, División Antropología, Museo de La Plata, Paseo del Bosque s/n, La Plata (1900) Argentina. T: (54) 221 4259819, E: iperez@fcnym.unlp.edu.ar   Accepted for publication 16 February 2006    Differences between sliding semi-landmark methods, S. I. Perez et al.© 2006 The AuthorsJournal compilation © 2006 Anatomical Society of Great Britain and Ireland  770  shape differences can be visualized (Rohlf & Marcus,1993; Baylac et al. 2003). There are also theoretical rea-sons for the use of geometric morphometrics instead oftraditional methods, including the rigorous statisticaltheory developed for shape analysis (see Adams et al.2004). Concurrently, there is a considerable amountof empirical evidence demonstrating the ability oflandmark-based geometric methods to provide newinsights into patterns of biological shape variation thatcould not be evaluated by traditional methods (Mon-teiro & Abe, 1999; Monteiro et al. 2002; Reis et al. 2002;among others; cf. Lynch et al. 1996). For example, insome studies geometric morphometrics methods wereable to discriminate groups (e.g. species, populations)that could not be separated by traditional methods(Adams & Funk, 1997; Monteiro-Filho et al. 2002;Baylac et al. 2003; Perez, 2003; Bernal, 2005). Given the above-mentioned theoretical and empirical reasons,geometric morphometrics techniques are useful for thestudy of intraspecific morphological variation, such asthe variation among human populations.The most widely used geometric morphometricsmethods are landmark-based approaches that use setsof two- or three-dimensional coordinates of biologicallandmarks (Bookstein, 1996a,b, 1998). From a biologi-cal viewpoint, the use of landmarks may not be suffi-cient because they cannot describe some biologicalforms and patterns (Oxnard, 1978). Further informa-tion may be obtained by increasing the number ofpoint coordinates. However, this cannot be done withpoints defined as landmarks because large areas ofmany biological objects, such as the human cranialvault or facial skeleton, have few or no landmarks andtheir structural information is represented only bysurfaces, curves or outlines. Therefore, the slidingsemi-landmark method was proposed, to capture andanalyse outlines (Green, 1996; Bookstein, 1997). This opera-tion extends the standard Procrustes superimpositionprocedure: in addition to translating, scaling, androtating landmarks optimally, the semi-landmarkpoints are slid along the outline curve until they matchas well as possible the positions of correspondingpoints along an outline in a reference configuration(Adams et al. 2004). This is done because the curves orcontours should be homologous from subject to sub- ject, whereas their individual points need not be (Book-stein et al. 2002). Several criteria have been proposedto slide points along an outline. Two of the most widelyused are minimum bending energy (BE; Bookstein,1996c, 1997; Green, 1996; Bookstein et al. 2002) and perpendicular projection or minimum Procrustes dis-tance (D; Sampson et al. 1996; Bookstein et al. 2002;Sheets et al. 2004). According to the first criterion, thepositions of semi-landmarks along the contour of eachspecimen are allowed to slide along the directionlocally parallel to the outline in order to minimize thebending energy necessary to produce the change inthe outline relative to the reference form (Fig. 1a). Therequirement that semi-landmarks of the mean shapedeform smoothly to the shape of a particular specimenin a manner that minimizes bending energy is equiva-lent to the conservative assumption that the contouron a particular specimen is the result of the smoothestpossible deformation of the corresponding contour onthe reference form (Bookstein, 1996c; Sheets et al. 2004). The minimum Procrustes distance criterion removes thedifference along the curve in semi-landmark positions Fig. 1 (a) Minimum bending energy criterion. The figure shows a semi-landmark point slid along an outline tangent (a1) and projected onto the outline after the relaxation step (a2) (modified after Gunz et al. 2005). (b) Minimum Procrustes distance criterion. The figure shows a semi-landmark point before (b1) and after sliding it toward the line that is perpendicular to the edge at the corresponding semi-landmark of the reference (a2) (modified after Zelditch et al. 2004).  Differences between sliding semi-landmark methods, S. I. Perez et al.© 2006 The Authors Journal compilation © 2006 Anatomical Society of Great Britain and Ireland  771  between the reference form and each specimen byestimating the direction tangential to the curve andremoving the component of the difference that liesalong this tangent (Sheets et al. 2004). The semi-landmarks along the curve are aligned so that thesemi-landmarks of each specimen lie along the lines per-pendicular to the curve that passes through the corre-sponding semi-landmarks on the reference form(Fig. 1b; see Sampson et al. 1996). Owing to the factthat semi-landmark methods permit the generation ofa representation of the overall shape of biologicalstructures, they are destined to gain importance asmorphometric tools. Thus, an understanding of theeffects that the choice of sliding landmarks methodshas on empirical studies of different biological prob-lems becomes a fundamental issue for research.In this work we show the differences between theresults obtained using semi-landmarks aligned by thesetwo different criteria in the morphometric study of thebiological relationships among human populations. Ofparticular concern is the possibility that the differentmethods of handling semi-landmarks might influencethe estimated difference between the mean shapes ofseveral different groups of specimens, or the variationwithin each group. These differences in the mean andvariance might result in differences in principal compo-nents analysis (PCA) of patterns of variance or in theability to discriminate among groups using discrimi-nant function analysis (or canonical variates analysis,CVA). We also evaluate the extent to which the numberof semi-landmarks used may influence these results.Finally, we discuss the implications that such differ-ences have for the study of modern human popula-tions characterized by low levels of morphologicalvariation.   Materials and methods  Two structures differing in their intrapopulation varia-tion and complexity were analysed: (a) the upper firstmolar, which varies little and can be described bymeans of a closed outline (Fig. 2a), and (b) the facialskeleton, which presents greater variation and can bedescribed through a combination of landmarks withopen and closed outlines (Fig. 2b). A total sample of 60upper first molars of individuals from several geo-graphical regions of Argentina (Bernal, 2004) and 66male adult human skulls from the Chubut and NegroRiver valleys (Patagonia, Argentina) were analysed; thespecimens are housed at División de Antropología inMuseo de La Plata and at Museo Etnográfico ‘J. B.Ambrosetti’ in Buenos Aires, Argentina. The Chubutsample, which included individuals from Chubut Valley(CH, n  = 20) and the neighbouring area (  n  = 10), corre-sponds to the later late Holocene (  c   . 500–1500 years   BP  ), a subsample from Negro Valley (RNa, n  = 17) corre-sponds to the same period, whereas the second sub-sample from Negro Valley (RNb, n  = 19) corresponds to Fig. 2 Landmarks and semi-landmarks recorded on dental (a) and facial (b) structures. Drawing by Marina Perez.   Differences between sliding semi-landmark methods, S. I. Perez et al.© 2006 The AuthorsJournal compilation © 2006 Anatomical Society of Great Britain and Ireland  772  middle/late Holocene (  c   . 3000–4000 years BP  ; Barrientos& Perez, 2004, 2005).The specimens were photographed with an OlympusCamedia C-3030 digital camera. The upper first molarswere positioned with the occlusal surface parallel tothe camera at 100 mm. The skulls were positioned inthe Frankfurt plane and the camera lens was locatedin the coronal plane (Buikstra & Ubelaker, 1994). Frontalview images were taken at 250 mm from the prosthionpoint. Seventy-nine semi-landmarks and one landmarkwere obtained from the upper first molars (Fig. 2a);eight landmarks and 74 semi-landmarks were obtainedfrom the facial skeleton (Fig. 2b). The semi-landmarkcoordinates corresponding to the upper first molarswere automatically obtained using tpsDIG 1.40 (Rohlf,2004). The application MakeFan6 (Sheets, 2003), whichplaces alignment ‘fans’ at equal angular displacementsalong a curve, was used to ensure consistent placementof the facial semi-landmark coordinates. Both land-marks and semi-landmarks were then digitized usingtpsDIG 1.40 software (Rohlf, 2004).In landmark-based analysis shape can be defined asthe information remaining in a configuration of land-mark points after the differences due to location, scaleand orientation are removed (Bookstein, 1991, 1996a). The effects of location, scaling and orientation aretypically removed using generalized Procrustes analysis(Gower, 1975; Rohlf, 1990; Rohlf & Slice, 1990) to pro-duce a set of specimens in partial Procrustes super-imposition (Rohlf, 1999; Slice, 2001) with respect to acommon reference form. In the partial Procrustessuperimposition approximation, the set of  x   and  y   coordinates of any single specimen’s digitized two-dimensional landmark points are first centred at thesrcin (0,0) by substracting the centroid or mean locationof all landmarks from each (   x   ,   y   ) pair. After the specimenhas been centred, the centroid size of the configura-tion (the square root of the summed square distance ofall landmarks from the centroid) is set to 1 throughdivision of the coordinates by the initial centroid size ofthe specimen. An iterative procedure is used to deter-mine the mean form onto which all specimens arealigned. During this iterative procedure, all specimensare first aligned as a single specimen, and the meanshape of all specimens is calculated. All specimens arethen rotated to minimize the added squared differ-ences of landmark coordinates between each specimenand the estimated mean shape or reference form. Thisprocedure is repeated until the mean shape does notchange substantially after iteration of the orientationprocedure (Rohlf, 1999). At this point, the specimensare said to be in partial Procrustes superimpositiononto the reference form. A full Procrustes superimpo-sition would allow the centroid size to vary from 1 toreduce further the distance between specimens andthe reference, but the partial Procrustes superimposi-tion is typically preferred (Rohlf, 1999; Slice, 2001).When outlines are digitized at discrete points, a stepis added to generalized Procustes analysis to minimizethe variation tangential to the curve. In this case, indi-vidual points are not claimed to be homologous fromspecimen to specimen, and consequently only the co-ordinate normal to the outline bears information aboutdifferences between specimens or groups (Bookstein,1997; Bookstein et al. 2002). There are various versions of the Procrustes method that slide the points alongthe tangential direction so as to remove this variation(Bookstein et al. 2002; Sheets et al. 2004). Here the semi-landmarks were aligned using both the minimumbending energy criterion (Green, 1996; Bookstein,1997) and the minimum Procrustes distance criterion(Andresen et al. 2000; Sheets et al. 2004), the latter performed following Sampson et al. (1996). The algo-rithm for computing a Procrustes mean curve takes oneof the srcinal outlines  X    i   as an initial estimate of themean curve Y   , then translates, scales and rotates eachof the outlines  X    i   into Procrustes superimposition on Y   by the iterative closest point algorithm (Besl & McKay,1992). Subsequently, new corresponding points oneach of the outlines  X    i   are computed by projectingequally spaced points onto Y   along the normals to the  Y   outline. A new estimate of the mean curve Y   is com-puted as the average of these corresponding points onthe superimposed  X    i   . Finally, each step can be iterateduntil Y   has converged (Sampson et al. 1996) or just onecycle of adjustments can be made (F. J. Rohlf, personalcommunication).Bookstein (1997) used the bending energy model ofthe thin-plate spline method to determine the criteriafor sliding semi-landmarks along outlines. Thisapproach places the semi-landmarks along the contourso as to minimize the bending energy required todeform the reference curve Y   to match the target curve   X    i   (Gunz et al. 2004). The minimization of the bendingenergy is equivalent to seeking the smoothest possibledeformation of one curve into the other, using agenerally accepted mathematical definition of smooth-ness. Semiland6 software (Sheets, 2003) was used to   Differences between sliding semi-landmark methods, S. I. Perez et al.© 2006 The Authors Journal compilation © 2006 Anatomical Society of Great Britain and Ireland  773  slide the points by means of the minimum Procrustesdistance criterion and tpsRelw 1.40 (Rohlf, 2004) wasused to align semi-landmarks based on the minimumbending energy criterion.To evaluate the influence of the number of semi-landmarks, 40 semi-landmark coordinates from molarsand 38 semi-landmark coordinates from facial skeletonwere deleted before performing the alignments. Everysecond coordinate was removed in each outline usingthe tpsUtil 1.29 software application (Rohlf, 2004).Mean shapes of different groups of specimens(drawn from different localities aligned using differentmethods) were compared by computing group meansfrom the superimposed coordinates using partial Pro-crustes superimposition. The differences in the relativepositions of landmarks and semi-landmarks were thenplotted and the partial Procrustes distances betweenthe mean specimens of the different groups were cal-culated using the TwoGroup6 program (Sheets, 2003).Resampling-based (Bootstrap) Goodall’s F   -test(Goodall, 1991; Zelditch et al. 2004) was used to testthe significance of the observed differences in meanshape between groups.Morphological diversity or variation within eachsample was evaluated using Foote’s (1993) disparitymeasurement. This is defined as morphological disparityD = Σ  ()/(  n   −  1) where d    i   represents the distance of thespecimens to the group centroid. Disparity was meas-ured using DisparityBox6 software (Sheets, 2003), whichuses the partial Procrustes distance as a measure of d    i   .The partial warp scores plus the uniform compo-nents, derived using thin-plate spline decomposition ofthe bending energy matrix from the partial Procrustes-aligned landmark and semi-landmark coordinates,were used to perform a PCA of the specimens (Rohlf,1993; Bookstein, 1996b; Dryden & Mardia, 1998). Thepartial warp scores are components along the orthog-onal eigenvectors of the bending energy matrix anddescribe non-affine patterns of shape difference(Bookstein, 1989, 1991), whereas the uniform compo- nents describe affine shape differences (Bookstein,1996d; Rohlf & Bookstein, 2003). Thin-plate splinedecomposition has become a standard technique ingeometric morphometrics because it yields a conven-ient set of variables to perform multivariate statisticalanalysis, as the partial warp plus uniform componentscores express shape changes in the same number ofvariables as there are independent measurements.Additionally, the thin-plate spline method allows foruse in the intuitive deformation grid diagrams todepict shape changes. A PCA of the partial warp plusuniform component scores, typically known as relativewarp analysis, was performed using tpsRelw 1.40(Rohlf, 2004). Discriminant function analysis was thenperformed on the first 50 principal component (PC)axes using the application CVAGen6n (Sheets, 2003).The deformation grids along the discriminant scoreswere obtained with tpsRegr 1.28 (Rohlf, 2004).As PCA is often used as an exploratory analysis of thepatterns of variation in a data set, we used Pearson’scorrelation and Procrustes analysis to compare thepatterns of ordination produced by the two alignmentprocedures (Gower, 1971; Digby & Kempton, 1987;Peres-Neto & Jackson, 2001). The correlation of sitescores along the first PC was used as a measure of con-sistency (Digby & Kempton, 1987). The ordinations inseveral dimensions were scaled and rotated in order tofind an optimal superimposition to maximize their fit(see above). The sum of the squared residuals betweenconfigurations in their optimal superimposition canthen be used as a measure of association (Gower,1971). A permutation procedure (PROTEST) imple-mented by Jackson (1995) was then used to assess thestatistical significance of the Procrustean fit (Peres-Neto & Jackson, 2001). Procrustes analysis was madeusing the vegan 1.4.4 package for R 1.9.1 (Ihaka &Gentleman, 1996).   Results  The differences in mean shape estimation and semi-landmark alignment between the two methods forboth the molar and the facial data are shown in Figs 3and 4. Goodall’s F   -test was used to compare the meanshapes of specimens from CH with those from RNa, andto compare RNb with RNa. The F   -scores and P   -valueswere similar for the minimum bending energy andminimum Procrustes distance criteria, whereas, notsurprisingly, the estimated partial Procrustes distancesbetween the means were larger under the minimumbending energy criterion (Table 1, Fig. 5). For compari-son purposes, the same tests were run on the datawithout any semi-landmarks alignment (only partialProcrustes alignment). The F   -scores obtained weresmaller than those for the aligned semi-landmarks, andthe partial Procrustes distances were similar to thoseproduced applying the minimum bending energy crite-rion (Table 1). d  i  2
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