Cranial integration in Homo: singular warps analysis of themidsagittal plane in ontogeny and evolution

Fred L. Booksteina,b, Philipp Gunza, Philipp Mitterœckera,Hermann Prossingera, Katrin Schæfera*, Horst Seidlera

aInstitute for Anthropology, University of Vienna, Althanstrasse 14, 1091 Vienna, AustriabInstitute for Gerontology, University of Michigan, Ann Arbor, MI USA

Received 15 March 2002; accepted 4 November 2002

Abstract

This study addresses some enduring issues of ontogenetic and evolutionary integration in the form of the hominidcranium. Our sample consists of 38 crania: 20 modern adult Homo sapiens, 14 subadult H. sapiens, and four archaicHomo. All specimens were CT-scanned except for two infant H. sapiens, who were imaged by MR instead. For eachspecimen 84 landmarks and semilandmarks were located on the midsagittal plane and converted to Procrustes shapecoordinates. Integration was quantified by the method of singular warps, a new geometric-statistical approach tovisualizing correlations among regions.

The two classic patterns of integration, evolutionary and ontogenetic, were jointly explored by comparing analysesof overlapping subsamples that span ranges of different hypothetical factors. Evolutionary integration is expressed inthe subsample of 24 adult Homo, and ontogenetic integration in the subsample of 34 H. sapiens. In this data set, vault,cranial base, and face show striking and localized patterns of covariation over ontogeny, similar but not identical to thepatterns seen over evolution. The principal differences between ontogeny and phylogeny pertain to the cranial base.There is also a component of cranial length to height ratio not reducible to either process. Our methodology allows aseparation of these independent processes (and their impact on cranial shape) that conventional methods have notfound.� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Cranial integration; Geometric morphometrics; Singular warps; Relative warps; Human ontogeny; Human phylogeny

Introduction

In the enduring debate about the tempo andmode of hominid evolution, one of the mosthelpful points of view is to emphasize integration,

the interplay of trends and variability in the formsof components. Whereas biological hypotheses arestatements of cause, findings of morphologicalintegration pertain to the effects of those causes:the correspondence of patterns of covariationamong traits to a posteriori or a priori hypothesesregarding factors such as evolution, development,* Corresponding author

Journal of Human Evolution 44 (2003) 167–187

0047-2484/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0047-2484(02)00201-4

or function (Chernoff and Magwene, 1999). In thisapproach, which is embraced in the present paperas well, integration is assessed by the patterns ofphenotypic statistical association within carefullydesigned samples intended to highlight the effectsof the factor or factors postulated. The logichere corresponds to one version of Olson andMiller’s (1951) “�F-model”, the comparison ofempirically derived clusters of trait-measurements(�-groups) against sets of traits derived a priorifrom preconceptions or exogenous experimentalobservations about development and function(F-groups).

Olson and Miller (1958), in their classic bookon morphological integration, eliminated theF-groups from their formal model; along withmost other modern workers, we prefer the earlierformulation, in which biological theory takes pri-ority over statistics. In our application the �’s willbe covariances rather than correlations and theF-sets will comprise coordinates of landmarks andsemilandmarks in specific regions of the cranium.The use of regions as F-sets in this way goes backat least as far as Sewall Wright’s original biometricfactor analysis (reviewed in Bookstein et al., 1985).

In typical earlier applications, beginning withOlson and Miller’s (1951) work, correlationsbetween scalars (such as distances, ratios or angles)were used to challenge hypotheses about integra-tion. These features needed to be listed in advanceof any analysis, along with the specific patternsexpected that would count as evidence for integra-tive factors of various types. Nowadays, we uselandmark coordinates, not separately measuredfeatures, and hypotheses about integration areexpressed as lists of landmarks, not lists ofmeasured variables. When data are available aslandmark coordinates over two or more regions,geometric morphometrics offers a multivariatetechnique that covers all possible shape measure-ments of the regions separately, exploring thepatterns of their correlations with all possibleshape measures of other regions in one singlecomputation. The analysis results in lists ofmeasures within a region that are optimallyassociated across regions. The lists emerge (some-what as principal components do) in one singledescending order of statistical explanatory power.

The corresponding biological explanatory powerarises from the geometrical or functional separa-tions among the regions sent for analysis. Becausethe data set is one of landmark coordinates, theintegrated features produced are interpretable ascoordinated deformations over multiple regions, amore powerful diagrammatic tool, in general, thananything available in Olson and Miller’s time.

Within any such explanatory setting, theinvestigator may be interested in the regionaliz-ations claimed or in the explanatory factors. Ourpurpose in the present study is a hybrid of thesetwo: to compare two distinct integrative factors,ontogeny and phylogeny, as they apply to a sharedregionalization within one single sample ofhominid crania. Studies of mammalian cranialdevelopment (Cheverud, 1982, 1995; Hanken andHall, 1993; Moore, 1981) indicate that the skull iscomposed of several semiautonomous functional/developmental complexes. At the broadest scale,the skull is composed of three parts, the cranialvault, cranial base, and face (de Beer, 1937;Cheverud, 1996). Although they originate inembryologically distinct regions, they apparentlygrow in a morphologically integrated mannerthrough numerous developmental and functionalinteractions (Lieberman et al., 2000a).

Particular attention has recently been paid tothe role of the cranial base. The cranial base ofanatomically modern Homo differs markedly fromthat of other primates and also is different fromfossil Homo. Among the multitude of hypothesesthat have been proposed to explain the high degreeof midsagittal craniobasal flexion of anatomicallymodern H. sapiens, encephalization and facialorthognathism are the ones most often studied(Biegert, 1963; Dean, 1988; DuBrul, 1977; Gould,1977; Jeffery and Spoor, 2002; Liebermann andMcCarthy, 1999; Ross and Henneberg 1995; Rossand Ravosa, 1993; Strait, 1999). But such expla-nations are considerably weakened if the samemethodology applies without alteration to both astudy of ontogeny and a study of evolutionarychange. As many measures of the hominid skullshow the same trends over ontogeny as they doover phylogeny read backward, the differentiationof these two hypotheses is a matter of somedelicacy. The methodology introduced here may

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187168

be construed as one that draws the greatest con-trast between these two partial summaries. Thecontrast, in turn, may be interpreted as a hetero-chrony or heterotopy, or instead as a replacementof one ontogenetic pattern by another; we cannotpursue these further interpretative problems in thepresent data set.

Our specific purpose in this paper is to contrastthe reports of one particular formal computation,singular warps analysis (see below), between twooverlapping subsets of this sample, one filtered tohighlight effects of an ontogenetic factor, and theother, those of a phylogenetic factor. We will showthat these two factors, while quite similar inrespect of the vault and the face, can be distin-guished by close examination of their effects onthe cranial base. The approach here differs inseveral particulars from that recently proposed byLieberman et al. (2000b) for exploration of asimilar but more focused question, the role of thecranial base in craniofacial integration. Our dis-cussion below will begin with an exploration of therelation between their approach and ours. Butfirst, of course, we introduce our methods, ourdata set, and our findings.

Material and methods

Specimens

We use a diverse cross sectional sample ofmodern and archaic Homo to study the large-scaleeffects of individual-level morphological integra-tion. The sample was selected to cover a widerange of human shape variability. Our sample of34 H. sapiens specimens consists of 20 adults ofboth sexes (nine from Central Europe, includingMladec I (Szombathy, 1925), three San and twoBantu, three Chinese, two Australian aboriginalsand one Papuan) and a cross sectional sample offourteen European subadults ranging in age from3 months to 17 years. The two youngest specimens(3 months and 9 months) were living children,measured from midsagittal slices of an MRIimage. (CT scans of dried infant material areextremely difficult to quantify accurately, owing totheir fragility and to the thinness of the bones even

when imaged unbroken.) All other specimens wereimaged using CT scans. In addition to these 34H. sapiens, the sample included three MiddlePleistocene fossil Homo crania: Kabwe 1 (BrokenHill) (Woodward 1921), Petralona (Kokkorosand Kanellis, 1960), Atapuerca SH5 (Arsuagaet al., 1993), and one H. neanderthalensis cranium,Guattari 1 (Blanc, 1939).

Landmarks and preprocessing

Traditional landmarks were marked on eachcranial CT in 3D, using the software package3DViewnix (Udupa, 1999) in order to identify amidsagittal plane. The 18 unpaired landmarks(Table 1) were then projected onto this plane foreach specimen and the corresponding slicesexported as images. For the MRI scans the mid-sagittal slice was selected according to its uniqueanatomical properties (e.g. crista galli, aquae-ductus mesencephali, septum pellucidum). Digitiz-ation of the landmark locations and capturingof the midsagittal outlines was done usingtpsDig32 (Rohlf, 1998). All the coordinates wereexported and processed with programs written inMathematica�.

In addition to the anatomical landmarks(Table 1), 66 outline points were extracted alongthe inner and outer outlines of the vault and themidline of the maxilla. The outline points (includ-ing the landmarks glabella, bregma and lambda)were considered as semilandmarks free to slidealong their curve-thus acquiring geometric homol-ogy within the sample (Bookstein et al., 1999). The38 configurations of 84 points that resulted wereaveraged and represented as shape coordinatesby standard Procrustes methods (Dryden andMardia, 1998; Bookstein, 1997). Fig. 1 shows atypical midsagittal image with its 84 landmarksand semilandmarks.

A few landmarks are missing in some of thefossil specimens; these are indicated in Table 1.While the modern literature of missing landmarkestimation (Gunz et al., 2002) emphasizes thestatistical use of information from other forms,such procedures would obviously assume the factof integration in essentially the same form in whichwe are testing it here; to avoid bias we therefore

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could not use any of these novel tools. Missinglandmarks were estimated with reference only tothe single form upon which they were missing, byextrapolation of outline curves on midsagittalslices.

All our subsequent analyses require that thecomplete set of 84 points be divided into threesubsets corresponding to the components theintegration of which is to be discussed: cranialbase, vault and face. These are indicated by colorin Fig. 1. The neurocranium is traced up to a pointthat geometrically corresponds to Kabwe’s occipi-tal end (slightly above inion). Where landmarks lieon the interface between two functional com-ponents, we found (by experimenting) that it willnot matter much to which they are assigned forthe analyses to follow; we do not describe thesealternative computations here.

Statistics

Multivariate analyses of this vector of 168 shapecoordinates (two for each of the 84 landmarksand semilandmarks of the representation) werecomputed in two different approaches: relative

warp analysis and singular warp analysis. As anydiscussion of integration must begin with aregional description, both computations do so; butwhereas relative warp analyses refer to predictionof the shape of the craniofacial complex as a whole,the singular warp analyses refer to predictions ofany of its components from the other components.

Relative warp analysisRelative warp analysis, a modification of prin-

cipal component analysis for shape coordinatedata, is perhaps the most commonly encounteredtechnique in the entire modern morphometrictoolkit (Marcus et al., 1986; Bookstein, 1998; Cortiet al., 2000; O’Higgins, 2000; Ponce de Leon andZollikofer, 2001). A relative warp is an eigenvectorof the matrix of variances and covariances ofthe Procrustes shape coordinates. When principalcomponents are computed using covariancesin this way, sums of squared differences of scorespreserve the underlying original geometry ofProcrustes distance.

Landmarks and semilandmarks can be mixed inthis analysis, but the result depends on the relativedensity of landmark and semilandmark spacing (as

Table 1Definitions of landmarks (Martin and Saller, 1957; White and Folkens, 1991)

+ rhinion (anterior free end of the internasal suture)+ nasion (the highest point on the nasal bones in the midsagittal)+ glabella (the most anterior point of the frontal in the midsagittal)+ bregma (the external intersection point of the coronal and sagittal sutures)+ ‘internal bregma’ (the point corresponding to bregma on the inside of the braincase)+ lambda (intersection of sagittal and lambdoidal sutures)+ ‘internal lambda’ (corresponding to lambda on the inside of the braincase)+ opisthion (midsagittal point on the posterior margin of the foramen magnum, G)+ basion (midsagittal point on the anterior margin of the foramen magnum, G)+ sella turcica (top of dorsum sellae, P G)+ canalis opticus intersection (intersection point of a chord connecting the two canalis opticus landmarks with the midsagittalplane, P)+ ‘crista galli’ (point at the posterior base of the crista galli)+ ‘foramen caecum’ (anterior margin of foramen caecum in the midsagittal plane)+ ‘vomer’ (sphenobasilar suture in the midsagittal plane)+ PNS ‘posterior nasal spine’ (most posterior point of the spina nasalis, G)+ ‘fossa incisiva’ (midsagittal point on the posterior margin of the fossa incisiva)+ ‘alveolare’ (inferior tip of the bony septum between the two maxillary central incisors)+ akanthion (top of the spina nasalis anterior)

Landmarks that were estimated in fossil specimens are indicated by initials.G: Guattari, P: Petralona. The estimated landmark coordinates are available from the authors upon request.

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187170

it varies over the cranium). Because this spacingwas arbitrary (see Fig. 1) any report needs toimpose some sort of standardization. From avariety of approaches to this problem we havechosen the simplest, a reweighting of the data sothat vault, cranial base, and face appear to havethe same net Procrustes leverage (point count).Algebraically, we have divided all the variances ofthe vault shape coordinates by 63, the count ofvault points; all the variances of the cranial baseshape coordinates by 6, the count of cranial basepoints; and all the variances of the facial shapecoordinates by 15, the count of facial points.Vault-face covariances are thus divided by �63.15,vault-base covariances by �63.6, and face-base covariances by �15.6. After eigenextrac-tion, the relative warps produced in this wise are“reinflated” by the appropriate multipliers beforebeing visualized as transformation grids.

Singular warp analysisSingular warp analysis is the name given to

applications of Partial Least Squares (PLS) (cf.Bookstein et al., 1996) within morphometrics. Thisgeneral class of techniques (Bookstein, 1998; Rohlfand Corti, 2000) involves a N�2k matrix of shapecoordinates X, and a N�m matrix of other vari-ables Y, that may also be shape coordinates butneed not be. A PLS analysis computes two unitvectors Ux

1, of length 2k, and Uy1, of length m, such

that the covariance of XUx1 and YUy

1 is a maximum.In words, these are the normalized compositescores (linear combinations), one from theX-variables and one from the Y-variables, thathave the greatest mutual linear predictive power.Often a PLS analysis will go on to computeadditional pairs of vectors Ux

2, Uy2 for which Ux

2 isperpendicular to Ux

1 and Uy2 is perpendicular to Uy

1

and the covariance of XUx2 and YUy

2 is a maximum,

Fig. 1. Exemplary midsagittal image with 18 landmarks (open circles) and 66 semilandmarks (solid circles) assigned to the three definedregions: vault (blue), face (green), and cranial base (red). Semilandmarks and landmarks labelled in italics were free to slide along theircurve. For definitions of the landmarks, see Table 1.

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187 171

and so on. In general such multivariate vectors arecalled singular vectors; but, because when the Xsare shape coordinates, each Ux can be visualized asa deformed or warped grid, they are usuallyreferred to more specifically as singular warps.The adjective “singular” here is taken in the math-ematical sense. [The reference is to the standardsingular-value decomposition of the cross-blockcovariance matrix by which all of these compositesmay be computed at once (see Appendix and, forthe algebraic theory, Mardia et al., 1979).]

For the analysis of integration reported here wehave modified this “standard” singular-warp algo-rithm in two different ways. First, just as for therelative warp analysis, we have deflated the vari-ance of each shape coordinate by the count ofshape coordinate pairs in its anatomical com-ponent: a factor of 63 for the vault points, 6 for thebase points, and 15 for the facial points. Afterdeflation, these three candidate componentsaffect one another, in principle, with equal weight,regardless of the differences in their point counts.Second, following the generalized algorithm ofStreissguth et al. (1993), we have extended theanalysis from two blocks to three blocks of shapecoordinates, as indicated in the Appendix. Theanalysis set out in detail there results not in pairsbut in triples of singular warps: one each for thevault, the base, and the face. Just as relative warpsare supplied both as deformations (coefficients ofhow the shape coordinates jointly shift) and asscores, likewise singular warps are supplied both asdeformations and as scores. Figs. 4–9 below showthem in both aspects. Furthermore, because theseare taken as linear combinations of the sameunitary set of 168 Procrustes shape coordinates,they can be visualized as grids in two differentways: either separately, component by component,or jointly. The latter version incorporates the‘spaces between the components’ as well as theirdeformations considered separately. They can bedrawn as a composite spline because the com-ponents stay within the same Procrustes spacethroughout the whole analysis. Singular warpswith negative covariances are reversed beforedrawing those summary figures. As far as weknow, this visualization of the three singularwarps as if they were one single relative warp has

not been published previously. The diagramsgenerated by this device are remarkably suggestive.

Results

Relative warps

The relative warps were computed with � = 0,i.e. with the uniform component included and noweighting by bending energy (Bookstein, 1996). Arelative warps analysis is reported by the jointdistribution of weighted scores together with thediagrams of the grid deformations correspondingto the eigenvectors that generated those scores.Fig. 2 shows the first two relative warps for the fulldataset: all modern adults and children as well asthe archaic ones, N = 38. (The Mladec I cranium iswithin the range of modern H. sapiens shapevariation throughout all these analyses, so forconvenience it is not referred to as a fossil in thispaper.)

Archaic Homo is separated from the moderns inthe first relative warp (Fig. 2a), with the H. sapienschildren at one extreme and the fossil specimens atthe other. The grids of this first relative warpcontrast the tall and large midface and thickvault-bones with large sinuses of the archaicspecimens with the relatively small face of themodern children with thin vault bones and analmost spherical neurocranium (Fig. 2b). The gridsvisualizing the second relative warp contrast aprognathic and relatively elongated cranial shapewith a retrognathic and tall cranial shape (Fig. 2b).

Singular warps

Even in its weighted version, the relative warpsanalysis is aimed at the aspects of these semi-landmark configurations that are jointly most vari-able, while for interpretative purposes we areinterested instead in the separate descriptions ofvault, cranial base, and face and the relationsbetween them. We therefore turn to the method ofsingular warps, and reanalyze two subsets of ourfull sample using this more pertinent method. Withthe one subset of all H. sapiens children and adultswe analyze integration over an individual time

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187172

Fig. 2. Relative warps 1 and 2 for the full data set of 38 specimens. (a) Scatters of the sample keyed to the legend in the frame.Subadults are symbolized by age, fossils are indicated by initials, and the adult modern humans are unmarked. M: Mladec, G:Guattari, P: Petralona, A: Atapuerca, K: Kabwe/Broken Hill. (b) Grids for the first two relative first warps (RW 1, RW 2) shown asdeformations of the average in both directions. In this figure, correlated changes in geometric scale (Centroid Size) are not shown, andthe magnitude of the contrasts on the RW’s is arbitrary.

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scale (ontogeny), while using the other subset ofadult modern and archaic Homo we analyze inte-gration over evolutionary time. The subsets over-lap by 20 adult H. sapiens. But these specimens areostensibly unvarying on both of the explanatoryfactors, ontogeny and phylogeny, which we intendto unravel. We will see, in fact, that they aresubject to an unexpected factor of integra-tion orthogonal to both. The analysis is notconfounded, but rather enriched, by this overlap.

The singular warps can be visualized eitherusing separate thin-plate-splines for vault, baseand face or as one composite spline. The splinesvisualize the patterns of shape difference (the scalesof the grids are arbitrary), while the plots of thesingular warp scores show the contributions ofeach dimension to each actual shape.

Evolutionary integration

Figs. 3–6 deal with the singular warp analysis ofthe adult Homo subset (N = 24). Scores on the firstdimension (Fig. 3) clearly separate archaic frommodern Homo, but show a single linear trend inthe covariation between vault and face, and asimilar trend in the interaction of vault-base(Fig. 3a) and base-face (Fig. 3c). The compositespline (Fig. 4a) shows that the first singular warpof the adult Homo, like the first relative warp for

the full sample, represents the shape difference ofindividuals with thin vault bones and rela-tively small faces versus robust individuals withrelatively larger faces and sinuses and thick vaultbones. The anterior cranial base is inclineddownwards in the moderns (Fig. 4a,c) butupwards in the archaic shapes. Correlations of thesingular warp scores: face with vault, 0.96; vaultwith cranial base, 0.68; face with cranial base, 0.71.

The second dimension of shape covariationamong vault, base and face, Figs. 5 and 6, is byconstruction uncorrelated with the first dimensionblock by block (see Appendix). The grids (Fig. 5)show how this second singular warp correspondsto relatively tall versus relatively elongated crania,with the tall cranium incorporating the less prog-nathic face. There is a cranial base effect entailedalso, as tall crania have a greater relative clivuslength than elongated crania. In keeping with theadjustment of phylogeny (the first singular warp),the archaic specimens cannot be discriminatedfrom the moderns in Fig. 6.

Developmental integration

We study ontogenetic effects in the H. sapienssubset (N = 34). The first singular warp scores(Fig. 7) exhibit a linear trend, children at one endand adults at the other. The corresponding splines

Fig. 3. First singular warp scores for the set of all adult Homo specimens (N = 24). (a) vault-cranial base, (b) vault-face, (c) cranialbase-face. For legend, see Fig. 2. Archaic and modern Homo are clearly separated.

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Fig. 4. First singular warp for the subset of all adult Homo specimens (N = 24): shown as a composite spline (a) and for each of thethree blocks separately: (b) vault (c) cranial base and (d) face. Each of the pairs of grids represents both directions of deformation forthe block involved. The three deformations of the left column are positively correlated, and likewise the right. Deformation toward thearchaic (right column) represents the robust extreme of the forms with relatively larger faces and thicker vault bones. Deformationtoward the modern specimens (left column) with a comparatively smaller face, relatively larger neurocranium, and thinner vault. Hereand in all following figures, the magnitude of the contrasts on the singular warps that are shown as deformations is arbitrary.

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187 175

Fig. 5. Second singular warp for the subset of all adult Homo specimens (N = 24): composite (a) and separate splines (b)–(d). Leftcolumn: Deformation of the average toward a longer cranial shape. Right column: Deformation toward a taller cranial shape.

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187176

in Fig. 8 show relative enlargement of the face andthickening of the vault during growth. The gridsclosely resemble the first singular warp of the adultHomo subset (Fig. 4), except in basicranial shapeand in the angulation of the segment prosthion–anterior nasal spine. The ontogenetic shapechanges of the cranial base differ markedly fromthe shape differences between modern and archaicHomo: The orientation of the anterior cranialbase stays constant while vault and face altertheir shape during ontogenetic development ofH. sapiens.

Again, the second singular warp analysisexpresses elongation or compression of thecranium (Fig. 9), and, as in the 2nd singular warpof the archaic-modern subsample; there is no cor-relation between age and singular warp scores inthe ontogenetic subsample (Fig. 10). The shape ofthe vault is highly correlated with the shape of theface (rz0.9). Crania with a tall vault also haverelatively taller, more retrognathic faces thanelongated crania. The relation of cranial length toheight also influences the shape of the cranial base:The clivus is steeper in short cranial shapes.

Fig. 6. Second singular warp scores for the set of all adult Homo specimens (N = 24). (a) vault–cranial base, (b) vault–face and (c)cranial base–face. Adult moderns and archaics do not separate. (For legend, see Fig. 2.)

Fig. 7. First singular warp scores for the set of subadult and adult H. sapiens specimens (N = 34). (a) vault–cranial base, (b) vault–faceand (c) cranial base–face. We find a linear trend for ontogeny. For legend, see Fig. 2.

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187 177

Discussion

The grids illustrating shape differences of thefirst relative warp, which separates archaicand modern Homo (Fig. 2b), show an almost

spherical braincase of modern H. sapiens com-pared to the elliptical vault of the fossil speci-mens (what Lieberman et al. (2002) term‘neurocranial globularity’), an overall decreaseof bone thickness and facial reduction. This

Fig. 8. First singular warp for set the of subadult and adult H. sapiens specimens (N = 34): composite (a) and separate splines (b)–(d).Left column: Deformation from the mean form toward the infants. Right column: Deformation toward the adults.

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187178

description has evidently omitted mention of thecranial base. It is the purpose of the singularwarps analysis to partition the covariance in

such a way that its changes are integratedwith those of face and vault over two differ-ent subsets of the sample, the one emphasizing

Fig. 9. Second singular warp for the set the of subadult and adult H. sapiens specimens (N = 34): composite (a) and separate splines(b)–(d). Left column: Deformation toward a relatively elongated cranial shape; right column: Deformation toward a taller cranialshape. Similar deformations for the second singular warp were already identified in the subset of adult Homo (Fig. 5).

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187 179

ontogenetic development, the other evolutionarychange.

Methodology for studies of integration

The operationalization of integration by singu-lar warps in this way differs somewhat from that ofother recent publications. For instance, the usefulreview article by Lieberman et al. (2000a,b) notes,with ample citations, that “[integration is mostbasically revealed by complex patterns of correla-tion and covariation which indicate a lack ofindependence among variables . and which canbe recognized a posteriori by comparing theo-retically and empirically derived correlationmatrices.” [p. 161; citations omitted]

This paper goes on to set out, among the crucialissues on which “further research” is warranted,such concerns as the needs to “isolate and definethe actual morphogenetic units which interact, toidentify and quantify their direct and indirectinteractions, and to understand the processes bywhich they interact” [p. 163]. In support of thesegoals, Lieberman et al. call for more data inthree-dimensional form, on developmental mech-anisms and especially the genetic basis of theirvariability, and on evolutionary implicationsof hypotheses like theirs. Along these lines, forinstance Jeffery and Spoor (2002) study phylo-genetic hypotheses of brain-cranial base inter-

action, where “interaction” is operationalized byway of correlations among size measures orbetween size measures and shape measures overontogenetic series.

Our analysis has taken a somewhat differentapproach, in which, so to speak, the roles of theoryand empirical data are interchanged. The “vari-ables” that are taken for granted in the “basic”study of integration excerpted in the quotationfrom Lieberman et al. are replaced in our work bythe configuration of landmark coordinates, withinwhich the search for the specifically informativevariables operationalized as the singular warps isin fact the main point of the investigation. Andwhat Lieberman et al. treat as a comparison ofone “empirically derived” covariance structure toanother derived (somehow) from theory is herereplaced by the comparison of two differentempirical covariance structures, one expressingontogeny, the other, evolution, each one opera-tionalized by a structured subsampling of our dataset as described earlier.

Furthermore, we accept the “actual morpho-genetic units” in this study: the subdivision of thelandmarks and semilandmarks in these three re-gions. In short, our work seems to take for grantednearly all the terms that are contested elsewhere inthis literature, while concentrating on a straight-forward methodological issue that seems to havebeen elsewhere overlooked: which measures of ‘the

Fig. 10. Second singular warp scores for the set the of subadult and adult H. sapiens specimens (N = 34). (a) vault–cranial base,(b) vault–face, (c) cranial base–face. (For legend, see Fig. 2.) Adult and infant forms do not separate.

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187180

vault’, ‘the face’, and ‘the cranial base’ should betaken to indicate their integration—each with theother components—and how the choice of thesemeasures affects empirical claims that the findingof a correlation is pertinent to ontogeny, phylog-eny, or something else. In terms of the quote fromLieberman et al., our concern is not with “lack ofindependence among variables” as pertinent totheory expressed a priori, but instead with the waysin which details of those dependences might shapeinterpretation a posteriori. In particular, processesthat take place before the division into threecomponents is established, or that take place ex-plicitly upon the boundaries among these compo-nents, are not represented by the statisticalmethods introduced here. One version of the argu-ment of this paper is that the same multivariatestatistical methods support studies of integrationas support studies of interaction, so that the dis-tinction among these interpretations (betweenontogenetic cause and evolutionary effect, as itwere) needs to be carried out by a more carefuldifferentiation of the pertinent data resources.

Besides this substitution of shape coordinatesfor variables, there is another redirection embed-ded in the approach we have taken: our implicitemphasis on the cognitive problem of what itmeans to “compare” one correlation matrix, per-haps “theoretical”, to another, or to “quantify .direct and indirect interactions” among morpho-genetic units. The method of singular warps is, inour view, a direct methodological construction thatseems to carry out both of these descriptive pur-poses at the same time. The explicit quantificationof “interactions” among vault, face, and cranialbase in this midplane, carried out once to highlightphylogenetic variation and a second time forontogenetic variation, serves simultaneously forthe comparison of the corresponding covariancematrices to the extent they concern the integrationof these three units.

It follows, we suggest, that the differencesbetween these apparently distinct lines of research,one dealing with “integration” and the other with“interaction, ” are not as central as hinted in therecent literature. If, in a morphometric context,the methods for the study of “interaction” and themethods for the study of “integration” are as

overlapping as we are arguing them to be, thenperhaps a different distinction is justified, betweenthe design of the samples that permit empiricalstudy of the integrated aspects of familiar biologi-cal processes (here, ontogeny and evolution) andthe graphical language by which we can extractsuggestions of the conventional descriptors (e.g.,“globularity” or “length-to-height ratio”) bymeans of which the necessary theoretical discus-sions of genetic and epigenetic origins can begin tobe pursued. In other words, the method we havedemonstrated here might serve as a bridge betweendata sets and the theoretical claims against whichthey have been previously tested only weakly, bystatistical significance procedures. We believe thatthe graphical tests here are stronger, as we willshow in the remainder of this discussion.

Such a use of multivariate analysis, which isgenerally referred to as “exploratory” (theory-generating rather than theory-confirming), hascharacterized biometry for most of the lastcentury. We need not take the space to defend it,as it is so standard a complement to verificationistmodes of reasoning in the same biometric toolkit.The differences we have displayed between picturesof ontogeny and phylogeny in the selfsame data setwill be of great use in the further pursuit ofLieberman et al.’s goals for future work, by distin-guishing what is to be explained by one theoryfrom what is to be explained by others.

First singular warps: evolution and ontogeny

In the first singular warp, which expresses thepatterns having the greatest covariation across thethree components, the integration of vault withface over either evolution (Fig. 4) or ontogeny(Fig. 8) resembles the familiar combination ofglobularity with facial reduction in the relativewarps of Fig. 2. Also, in both analyses the relativelength of the clivus is increased. However, duringontogeny the Procrustes orientation of the anteriorcranial base remains fixed (Fig. 8) whereas archaicand modern Homo differ by an obvious rotation ofthis component (Fig. 4).

The grids in Figs. 4 and 8 show the differencebetween integration over ontogeny and integrationover phylogeny: features of face and vault that

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187 181

have the highest covariance are nearly the same inthe two patterns, but the manner in which thiscommon factor embraces the cranial base is dis-tinctive between the two explanations. The firstsingular warp of the adult Homo (Fig. 4) is corre-lated with facial reduction and neurocranial globu-larity as well as craniobasal flexion. In interpretingthese grids, it is important to understand why thesepatterns do not translate correctly into restate-ments using conventional measurements such as“the cranial base angle”. In both Figs. 4 and 8, thegrid is actually quite inhomogeneous in the vicinityof the dorsum, so that the more appropriatedescription is that of a spatially localized change ofthe transformation, not the change of any singleshape variable. The change in cranial base form inFig. 4 (the singular warp for evolution, read in thedirection of forward time, archaic to modern)incorporates an overall increase of vertical-to-horizontal ratio with a relative rotation “counter-clockwise” of the anterior base from dorsum tocrista galli and a relative repositioning of canalisopticus between dorsum and crista. The change inFig. 8 (ontogeny, read from infant to adult) com-bines a similar increase of vertical-to-horizontalratio and the same relative repositioning of canalisopticus, but incorporates no equivalent of therelative rotation of the anterior cranial base aboutcanalis. There is also a local shift of nasion that isquite opposite between the two transformations.With these cranial base changes is associated anincrease of neurocranial globularity over thearchaic-to-modern transformation, but a decreaseover the infant-to-adult transformation. The rela-tive length of the clivus is increased in modernHomo (Fig. 4), when compared to the archaicspecimens. This morphology emerges during post-natal ontogenetic development within the investi-gated time-range, as the children of our samplehave a relatively shorter clivus than adults (Fig. 8).

Thus neither singular warp can be reportedaccurately as “change in the cranial base angle” orany other single familiar aspect of shape variation.Rather, they combine global and local aspects ofshape change in partially similar and partiallycontrasting ways. Furthermore, changes in faceand vault are much more strongly correlated withone another than either are with the cranial base

feature changes, so that for statistical explanationof the evolutionary shape change no quantificationof the cranial base alone could suffice. (Asexplained in Bookstein et al., 1985, in a single-factor model for three variables, the variableleast correlated with the other two is also leastcorrelated with the common factor.)

As one of us has noted elsewhere (Bookstein,1994; Bookstein (2002)), in geometric morphomet-rics, shape variables come in a complete space of2k�4 dimensions (where k is the number of land-marks in two dimensions), within which it makeslittle sense to guess at the identity of useful vari-ables a priori. Rather, the task of describing ahierarchy of contrasts of shape, such as a clado-gram, lies primarily in the formulation of thevariables in some principled manner. Shape vari-ables that emerge from the contrast of two distinctintegration patterns, as in this paper, are then to betested as potential characters by the methods ofBookstein (2002) that take potential bifurcationsof a cladogram explicitly into account in studiesinvolving a larger subsample of archaic Homo.

That the singular warps call the observer’sattention away from the cranial base angle per sedoes not mean that it might not prove importanton other grounds arising, perhaps, from observa-tions of earlier ontogenetic processes. The cranialbase in H. sapiens flexes in a rapid trajectory thatis mostly complete by about two years of age(Lieberman and McCarthy, 1999). Jeffery andSpoor (2002) have shown the cranial base toundergo several complex shape changes duringprenatal development. As we have only two infantspecimens, we cannot say more here about thecomplex integrative processes during gestation andthe first postnatal years.

Second singular warps

The patterns described so far only constitute thefirst dimensions of singular warps (which meansthey account for the largest covariances betweenthe three coordinate blocks). The second singularwarps are the largest covariances between vault,base and face that are uncorrelated with the firstsingular warps. As we were able to interpret thefirst singular warps as evolution and ontogeny,

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187182

respectively, statistical independence means thatthe shape covariations of the second singularwarps are related neither to ontogeny nor evolu-tion, as can be seen in the overlap of archaic andmodern specimens in the second singular warpfor phylogeny (Fig. 6), or between young andadult specimens in the second singular warp forontogeny (Fig. 10).

The grids of the second singular warp in bothsubsets (Figs. 5 and 9) look almost identical; bothrepresent large scale variation in the height-to-length ratio of these crania. Accordingly, anelongated neurocranium is associated with a rela-tively short clivus and an obtuse basicranial angle,and a short, tall braincase with a relatively longclivus and an acute basicranial angle. Becausearchaic Homo, the modern adult and the modernsubadult H. sapiens specimens completely overlapin the scatters of this second dimension, we con-sider this pattern to specify its own separate pro-cess of Homo cranial variation, orthogonal (byconstruction) to both ontogeny and phylogeny andalso, apparently, unrelated to sexual dimorphismin this sample. It may be associated with heritableintrapopulation variation, epigenetic conditions,or some other morphogenetic explanation.

Although the second singular warp is uncorre-lated with the first singular warp in the specificontogenetic and phylogenetic samples segregatedhere, in any other sample there could be a com-ponent of this elongation to some variable extent,together with a tipping of the clivus associatedneither with orthogenesis nor with growth butinstead with the accidents of sampling on theelongation factor. Mixed samples combining thesetwo processes will generally yield diagrams thatcombine the effects of all the processes contribut-ing to the comparisons. To partial out this con-founding, sample sizes would have to be so large asto be infeasible when dealing with fossil materials.Thus in comparisons across hominid evolutionarygrades, changes in the cranial base angle per se asa single simple measurement should not be con-sidered as separately interpretable aspects of eitherontogenetic or phylogenetic integration. A reportin terms of a cranial base angle, that is, some anglehaving dorsum as its vertex, would be appropriateonly if the corresponding singular warp were fairly

homogeneous in the vicinity of dorsum withprincipal strains bisecting that angle at dorsum(Bookstein, 1991, pp. 220–221). The necessaryhomogeneity does not obtain in either Fig. 4 orFig. 8.

The overlap of archaic and modern specimensin the plots of the second singular warps scoresbears implications for pro/retrognathism as a sys-tematic character. Retrognathism is associatedwith both singular warps for both ontogeny andphylogeny: In ontogeny and evolution the size ofthe face changes relative to the neurocranium(SW 1) and taller crania have less prognathic faces(SW 2). Retrognathy is thus just as problematic aphylogenetic entity as cranial base angulation.Measures like these, so evidently expressing someprocess that is neither ontogeny nor phylogeny,should not be used for species discriminationwithin the genus Homo, as they tap some processof integrated variability that is unrelated to vaultthickness, cranial or neurocranial volume, fossilage, or biological age. By contrast, a measurementlike maxillary height loads on singular warp 1 butnot on singular warp 2, and thus is a morepromising candidate for a phylogenetically usefulquantification.

Neurocranial globularity is a shape featurealong the principal axis of variability and integra-tion all across Homo: the feature expressed by thisgrid actually separates H. sapiens children fromadults better than it separates H. sapiens adultsfrom the fossil specimens. Hence we advise cautionwhen considering this globularity as a systematiccharacter.

Estimated landmarks

As noted in Table 1, two of the fossils in thispaper incorporate a total of six missing landmarkpoints. There is no evidence in either Fig. 3 orFig. 6 that these two forms unduly affect thesingular warp analysis of either vault or cranialbase, but nevertheless the praxis by which thesemissing data were estimated deserves commentary.In general statistical practice, missing data is ordi-narily estimated by some variant of the so-called‘EM algorithm’. Here ‘EM’ stands for “estimation/maximisation”, a computation alternating between

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187 183

(i) prediction of the individual missing data by aregression or analogous procedure based in theactual observations and (ii) fitting of the completeddata set, estimated together with observed, to somepopulation model such as a multivariate normaldistribution. In morphometrics, special concernssuch as symmetry or the need for smoothness ofshape change factors add nuances to algorithmslike these. When the scientific issue at hand is itselfa study of integration, however, the treatment ofmissing data needs to be carefully adjusted so asnot to presume what is actually to be demonstratedfrom the data. In the present study, the concernis with the large-scale covariances among vault,cranial base, and face; and so missing data neededto be estimated by reference to aspects of form thatwere not included in the covariance structurethemselves and that were at a geometric scalesmaller than that of the processes we were actuallystudying. The visual extrapolation of mesoscalecurvatures in the midsagittal bony surface is onesuitable praxis for this purpose. We have notestimated any large-scale aspects of these fossilforms, which are nearly complete as they stand; inparticular, we have not extrapolated outside theboundary of the actual locations observed. Ourmethod of missing data estimation thus permits usto include both Petralona and Guattari in theanalysis, giving the findings somewhat greaterscope without biasing them in any way.

Note that the computation here ordinates thePetralona specimen at one end of an imputeddimension-it is ‘the most archaic’ of the fourcrania. Furthermore, its singular warp scores areequally extreme in terms of vault and face, neitherof which was missing any landmarks, nor in termsof the base, on which both landmarks from thesphenoid are missing. There is thus evidence thatthe estimated positions of those missing landmarksare consistent with the rearrangements of vaultand face shape that are present in the actual data,even though neither vault nor face was involved inthe estimation of these missing points. RegardingGuattari, which is missing three landmarks on thebase, our single-image estimate leaves it locatedquite similarly in all three of the singular warpscore scatters, Fig. 3. There is no evidence that themissing-data estimation has biased our results,

although of course more complete specimens,and also more numerous fossils, would surelystrengthen the argument.

Summary

Integration of the hominoid skull has long beena subject of active research in paleoanthropologyand auxology. This paper adapts one of the classicapproaches to integration, Olson and Miller’s�F-sets, to the modern context of geometric mor-phometrics. Analyses of the integration of cranialvault, base, and face over our mixed ontogenetic/phylogenetic sample of 38 hominid crania in mid-sagittal section show that these two processes,while equivalent in their morphological expres-sions in vault and face, can be distinguished by thepatterns of their correlated effects on the cranialbase. The description of this distinction involvesrelative clivus length and anterior cranial baseorientation. Traditional measurements of thecranial base angle do not distinguish effectivelybetween these ontogenetic and phylogenetic con-texts of integration, nor between either of theseand a third integrative process, involving relativeelongation of the cranium across all three com-ponents, that is orthogonal to both ontogeny andphylogeny in this sample. The posterior cranialbase is strongly influenced by the overall shape ofthe cranium. These independent sources of varia-tion cannot be teased apart by one single angularmeasurement; furthermore, as vault and facesummary scores correlate better with each otherthan either does with the cranial base, no set ofcranial base features is likely, by itself, to suffice toexplain evolutionary shape changes. We recom-mend that future analyses of integration use singu-lar warps in preference to traditional measures ofdistance or angle so as to circumvent these andsimilar potential interpretative pitfalls.

Acknowledgements

We thank the curators of the fossil Homocrania, the subadult H. sapiens crania, and theadult H. sapiens crania for permission to CT-scan

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187184

them. Likewise we thank Wolfgang Recheis (Dept.of Radiology II, University Hospital Innsbruck,Austria) for CT-scanning support and GerhardWeber (Institute for Anthropology, University ofVienna, Austria) for the installation and manage-ment of the digital database. We also thankIngeborg Hirsch, Christine Unteregger andBarbara Wimmer for help with the data collectionand Bence Viola and Dennis Slice for valuablediscussions. This paper benefited greatly from thecritical comments of three anonymous reviewers.This research is supported by the Austrian FederalMinistry of Education, Science and Culture andthe Austrian Council for Science and Technology,Project Number: GZ 200.049/3-VI/I/2001.

Appendix. Algebra of two- and three-componentsingular warp analysis

The two-block PLS analysis reviewed in themain text is usually generated from the singular-value decomposition (SVD) of the cross-covariance matrix ��(1/N)XtY for two matrices ofvariables X: N�p and Y: N�q on the samesample. The SVD expresses the matrix � uniquelyas UxDUy

t , where Ux(p�p) and Uy(q�q) areorthogonal matrices and D(p�q) is diagonal withelements in descending order. Then the first pair ofsingular vectors referred to in the text are the firstcolumn of Ux and the first column of Uy, and thecovariance of XUx

1 and YUy1 is the first element of

D. The second pair of singular vectors combinesthe second column of Ux with the second columnof Uy; the covariance of the corresponding scoresis the second entry of D; and so on.

For the extension to more than two blocks thereis no such standardization of the problem descrip-tion. Rather, there arise a variety of differentalgorithms, each corresponding to one of the theo-rems that characterize the SVD in the classicaltwo-block case. In addition to the maximum-covariance criterion cited in the main text, fourother criteria are often encountered.

1. The singular vectors Ux are the principal com-ponents of the matrix ��t, and the singularvectors Uy the principal components of �t�.

The elements of D are the square roots ofthe eigenvalues of either of those symmetricmatrices.

2. The dyadic product D1Ux1(Uy

1)t is the bestrank-1 least-squares fit to the matrix �; thesum, D1Ux

1(Uy1)t�D2Ux

2(Uy2)t,the best rank-2

approximation, and so on.3. The vectors Ux

1, Uy1, arise upon indefinite

iteration of the sequence of matrix multiplica-tions Ux��Uy, Uy��tUx, starting from almostany arbitrary Ux or Uy.

4. The coefficients of each singular vector, say Ux,are proportional to the regression coefficientsof all the variables of the X-block on the scoreYUy representing the singular vector of theY-block and vice-versa.

To generalize all of this to the case of three ormore blocks of variables, for instance the threecranial components of the application here, it isnecessary to select one of these characterizationsfor extension; they no longer specify the identi-cal computation. We have chosen, followingStreissguth et al. (1993), to extend algorithm 3, theiterative matrix multiplication (but the otherapproaches lead to very similar findings). Write Xv,Xb, Xf for the centered deflated shape coordinatedata of vault, cranial base, and face.

Let Uv, Ub, and Uf be any three arbitraryguesses at the singular warps of these blocks. Thealgorithm that produced the findings here thenloops over the following chain of computations:

Interpret Uv, Ub, and Uf as linear combinations;that is, compute scores

sv = XvUv

sb = XbUb

sf = XfUf

and normalize each s to sample variance 1.Compute the correlations among these scores in

pairs:

rvb = corr(sv, sb)rvf = corr(sv, sf)rbf = corr(sb, sf)

Finally, compute updated estimates of the U’sby summing predictions of the individual variables

F.L. Bookstein et al. / Journal of Human Evolution 44 (2003) 167–187 185

(shape coordinates) using combinations of thescores s with the correlations r as weights:

Uv+Xvt(rvbsb�rvfsf)

Ub+Xbt (rvbsv�rbfsf)

Uf+Xft(rvfsv�rbfsb)

and return to the top of the loop until the iterationhas converged to the level of accuracy youdemand.

At convergence, each singular warp is thepattern of deformation (that is, the profile ofsimultaneous shape coordinate shifts) predictedby the weighted combination of the other twosingular warp scores for the joint shift of eachshape coordinate simultaneously. (This generalizesthe fourth in the list of characterizations of theordinary two-block SVD above.)

We also modified the definition of subsequentsingular warps. In the usual approach, the secondand higher singular warps are orthogonal to thefirst as vectors. In the approach here, the secondand higher singular warp scores are uncorrelatedwith the first as scalars. This permits the extractionof the truly uncorrelated dimensions of integration(statistically independent processes). The secondSW triple is computed by applying the samealgorithm to the residuals after the first SW tripleis regressed out of each shape coordinate.

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