results it is concluded that: (1) all types of morphometric procedures are ... data analysis technique coordinate-point eigenshape analysis is described and its ...
LANDMARKS, LOCALIZATION AND THE USE OF MORPHOMETRICS IN PHYLOGENETIC ANALYSIS Norman MacLeod Department of Palaeontology, The Natural History Museum (London), Cromwell Road, London, SW7 5BD.
Abstract The analysis of morphology is crucial to the study of phylogeny in many ancient and modern organismal groups. Recently a number of arguments have been made in favor of regarding certain kinds of morphometric variables as putatively homologous characters and allowing them to participate, along with other non-morphometric variables, in parsimony-based cladistic analyses. These arguments rest on the assumptions that geometric landmarks incorporate the concept of biological homology and that partial warp and principal warp morphometric variables uniquely incorporate and operationalize the concept of spatial localization, thus providing investigators with the ability to assess patterns and directions of geometrical variation in unmeasured regions of organic forms. Literature review, coupled with empirical investigations of similarities and differences among three different morphometric data analysis methods on two different datasets suggest that there is no support for these assertions. Geometric landmarks, which form the basis for all morphometric measurements and latent shape variables, have no necessary correspondence to biological homology. Partial warp variables, coordinate-point eigenshape variables, and inter-landmark distance-based singular vector shape-change variables all express localized (= regionally-weighted) geometric deformations. However, none of these deformation patterns are localized in the sense of being truly independent from globally-distributed aspects of shape change. All three of the morphometric data analysis methods investigated are capable of inferring patterns of shape change in remote regions of the form so long as a spatially adequate array of landmarks is used as the basis for the shape analysis. Based on these results it is concluded that: (1) all types of morphometric procedures are potentially useful in phylogenetic inference to the extent that they are able to summarize patterns of biotic shape change within the Euclidean space of the original measurements; (2) morphometric interpretations are strengthened by using alternative analytic approaches to probe the geometric system in a hypothetico-deductive manner and achieve a robust and consistent interpretation of underlying differences among operational taxonomic units (OTU s); and (3) while deformationbased morphometric variables can be used to analyze the distribution of organic shapes in a variety of measurement and latent variable spaces, they represent abstract investigative tools that are tied irreducibly to particular samples and data. Such variables can be used to study morphological characters, refine character and character state definitions, and aid in the assignment of character states to OTU s. However, they are not characters themselves and may not be suitable for phylogenetic analyses directly. In addition, a new morphometric data analysis technique coordinate-point eigenshape analysis is described and its use illustrated.
1.1
Introduction
Morphometrics is the quantitative study of patterns of covariance with shape (Bookstein 1991). Since many morphological attributes of biological form are obviously reflections of evolutionary processes (e.g., Darwin 1859; Simpson 1944; Mayr 1963), and since assessments of patterns of morphological variation have been used traditionally to infer phylogenetic relations (e.g., Hennig 1966; Eldredge and Cracraft 1980; Mayr and Ashlock 1991) it would seem be natural to assume that morphometric analyses would play a large role in phylogenetic studies. This has not, however, been the case.
Prior to the 1960 s phylogenetic analyses relied on qualitative assessments of morphological variation. At the time this was the only practical procedure because the imaging, computational tools, and data analysis techniques required for rapid and efficient morphometric analysis were not available. A large number of experiments in using morphometric analyses to constrain phylogenetic hypotheses were conducted under the aegis of the numerical taxonomy/phenetics movement of the 1960 s and 1970 s (e.g., Sokal and Sneath 1963; Sneath and Sokal 1973). But, the significance of many of these experiments was diminished by the separate debate between phenetics and phylogenetic systematics over questions of phylogenetic inference
2 methodology (see Hull 1988 for a history of this controversy). Because most modern taxonomists have adopted a Hennigian perspective on these methodological approaches to phylogenetic studies and because many systematists have associated morphometrics with the more-or-less rejected philosophy of phenetics morphometrics has not been well integrated into contemporary systematic practice. Instead it exists today as a somewhat anachronistic, techniques-driven discipline with a reputation for the complexity of its methods. In addition to suspicions with regard to morphometrics previous association with phenetics (e.g., its reliance on average forms, see below), contemporary systematic theory has had difficulty dealing with the fact that morphometric variables are continuous variables; real numbers as opposed to discontinuous integer-based state codes. Continuously-distributed morphological features have traditionally been regarded as irrelevant to phylogenetic inference because they represent transformational rather then taxic homologies (sensu Rieppel 1980, Patterson 1982) and so are not subject to the tests of conjunction, similarity, and congruence necessary for confirmation as true homologues. The use of continuous variables, such as morphometric variables, has been questioned in phylogenetic contexts because they represent features of aggregate samples (as opposed to individualized features or organisms), and because there is no way of objectively subdividing a continuum (see Pimentel and Riggins 1987). These objections suggest at least a certain measure of confusion between the concepts of a continuous variable and continuous variation. Continuous patterns of variation among OTU s are problematic for simple parsimony-based phylogenetic inference methods (but see Felsenstein 1981; Maddison 1991). Continuous variables are not, so long as the patterns of OTU distribution along the continuously varying axis are discontinuous (e.g., Swiderski et al. 1998). In addition, examples of mean values, variances, and kurtoses of continuous biometric variables have been shown to be under direct genetic or epigenetic control (e.g., Falconer 1981), suggesting that, in principle, these types of variables may carry important phylogenetic signals. Indeed, it is commonplace for blatantly continuous, morphometric variables to be included in phylogenetic analyses of fossil and/or Recent taxa (see Thiele 1993; Rae 1998 and references therein), albeit after qualitative redescription as binary or multi-state variables. Clearly the challenge to systematics presented by continuous variables cannot be met by condemning their use in theory while, at the same time, condoning their de facto use in practice. Recently, Zelditch et al. (1992, 1993, 1995), Swiderski (1993), Fink and Zelditch (1995), and
Zelditch and Fink (1995) have attempted to reintroduce morphometrics into phylogenetic analysis by arguing that particular morphometric methods are compatible with the taxic homology concept. In particular these authors advocate latent geometric morphometric variables (e.g., partial warp and/or principle warp decompositions of superposition-registered landmark constellations) as the only morphometric variables useful in phylogenetic contexts. According to this argument these particular variables (1) are based on the homology of parts rather than linear combinations of variables that optimize discrimination or variance of scores ( Zeldtich et al. 1995, p. 82), (2) can express spatially localized (as opposed to generalized) morphological differences, and (3) facilitate the manipulation of variables chosen during or after (as opposed to in advance of) the analysis thereby allowing the systematist to discover a posteriori, the most significant or useful shape contrasts. Bookstein (1994), Lynch et al. (1996), Naylor (1996) and Rohlf (1998) have challenged the use of geometric morphometric variables in the context of phylogenetic analysis on a variety of methodspecific and generalized grounds. In this essay I seek to examine the issues of (1) the concept and use of landmarks and homology in morphometrics, (2) notions of geometric localization in the interpretation of morphometric variables, and (3) distinctions between the shape variables determined by geometric morphometrics and those based on other multivariate techniques that have been (or might be) applied to analysis of organic geometries. While my discussion of these issues will focus on the application of morphometrics to problems of phylogenetic analysis, it is my hope that clarification of several basic morphometric concepts will contribute to a greater appreciation of the interrelation of morphometric methods, the power of morphometrics as a generalized approach to all types of morphology-based analyses, and the logicallyconsistent use of morphometrics in systematics.
1. 2 The Geometric Basis of Morphometric Observations All morphometric measurements are based on landmarks. Landmarks may be defined operationally as relocatable coordinate positions on an object in a two-dimensional or three-dimensional Euclidean measurement space. This definition is identical in concept to though more specific than that offered by Bookstein (1991, p. 2). The Euclidean distances of multivariate morphometrics (e.g., Blackith and Reyment 1971; Reyment et al. 1984) are distances between landmarks. The boundary coordinates of Fourier (e.g., Lestrel 1997) and eigenshape (e.g., MacLeod 1999) analysis are landmarks. And the
3 landmarks of geometric morphometrics (e.g., Bookstein 1991) are landmarks. Bookstein (1991) identified three classes of biological landmarks: discrete juxtapositions of structures or tissues (Type 1), maxima of curvature (Type 2), or extrema (Type 3). This classification focuses attention on the amount of information necessary to identify or relocate the landmark. Type 1 landmarks require the most biological information to identify and may occur at any point on or within a form so long as that form is composed of different structures or tissue types. While these landmarks are constrained to exist on the boundaries (= outlines) of these structural components or tissue-defined regions, their locations are not determined by any characteristics of the overall boundary or outline. Type 2 landmarks are constrained to lie on the boundaries of single structures or regions and are defined by the nature of the curving surface of that boundary. This definition constrains Type 2 landmarks to be located relative to the distribution of adjacent boundary coordinates. Type 3 landmarks represent those coordinate locations on single structures (irrespective of whether the structure is composed of various substructures or regions) that represent the extremes of the structure s boundaries. Like Type 2 landmarks these points are constrained to lie on the object s outline. No consideration has been traditionally given to the nature of any substructure or tissue when locating Type 3 landmarks. Their definition is dependent entirely on the nature of the outline (= by the distribution of adjacent boundary coordinates), on the orientation of the object, and on the number of axes one wishes to locate extrema along. Because the nature of Type 3 landmarks is so variable and dependent of such a wide variety of conditions Bookstein (1997) has recently revised his 1991 classification and termed this class of landmarks semilan dmarks. The category semi-landmarks includes the former Type 3 landmarks of Bookstein (1991) as well as the boundary coordinates used in outline morphometrics (e.g., Fourier analysis, eigenshape analysis, edgels). Since Bookstein s (1997) revised landmark classification describes the range of morphometrical observations more comprehensively and recognises fundamental similarities between these observation types more consistently, this notation is preferred. Consequently, the former (1991) distinction between landmarks and boundary coordinates has been superseded by a more inclusive and more practical categorization that unifies the artificial distinction recognized by Zelditch and co-workers. Bookstein (1991) explains the rationale behind the term geometric morphometrics as stemming from a desire to preserve the original measurement space of landmarks that previous morphometric data analysis procedures (e.g., distance-based multivariate morphometrics and outline-based
methods like Fourier analysis and eigenshape) had either failed to achieve or recognize as need necessary. This was because it was thought that these latter methods represented transformations of the observed data into an abstract variable space that was incommensurate with the Euclidean space of the original observations. Once the original data had been transformed into the latent variable space it was thought that no meaningful return to the measurement space was possible. It is a basic contention of this essay that, although coordinatedelimited landmark methods represent a particularly elegant and powerful strategy for exploring geometric relations between forms, these methods also share fundamental similarities with other, older morphometric methods.
1.3 The Biological Basis of Morphometric Observations As can be seen from the landmark definitions discussed above, no morphometric observation (= landmark) has anything necessarily to do with the biological concept of homology. Biological homology (taxic and transformational) is based on the identity of morphological structures (e.g., bones, organs, appendages) between taxa such that the structure in question is present, identifiable in the common ancestor of the taxa in question, and unique to the clade containing those taxa (Reippel 1980; Patterson 1982). The aspects of historical identity and hierarchy embodied by the evolutionary concept of ancestry and descent. Geometric (or morphometric) homology (sensu Bookstein 1991), however, is an ahistoric concept. Defined, in the biological sense, as a correspo ndence between parts (p. 56), Bookstein correctly points out that Morphometricians qua morphometricians have nothing much to say about right or wrong notions of homology. (ibid). Geometric homology contains aspects of the criteria systematists use to recognize or postulate biological homology, but it is not synonymous with that concept. More importantly, biological homology refers to unitary structures; not to the infinitesimal coordinate locations on structures that can be defined under the concept of geometric homology. Thus, while it is perfectly acceptable to describe the alternative forms of a radius bone or a canine tooth as long and short because any reasonable set of length measures derived from sets of these objects exhibit non-overlapping distributions, the homology criterion is referenced to, and dependent upon, the evolutionary unity of the entire structure. It is quite a different matter to claim that the geometrical midpoints of the proximal and distal terminus of differently shaped bones (or teeth, or shells) are biologically homologous with the geometrical midpoints of the bones (or teeth, or shells) of other
4 specimens. Even though a wide variety of alternative landmark pairs (in this example) might pass the conjunction, similarity, and congruence tests of biological homologues, there can only be one pair of landmark points that define a length on any biological structure that are true homologues of another pair of landmark points on another (homologous) structure. If true homology cannot be separated from false homology at the level of the mathematical point by the tests available to systematists, the entire question of biologically homologous landmarks is moot. Wagner (1994) points out that in some cases (e.g., pointintersections between three skull bones representing a Type 1 landmark) the concepts of biological and geometrical homology may coincide. However, such situations are comparatively rare and, in the absence of highly-detailed developmental and phylogenetic evidence will always remain either assumptions or assertions. Bookstein (1991) also describes another type of relation that bears on the issue of homology in morphometrics; deformational homology. Tracing the origins of this concept to D Arcy Thompson (1917), deformational homology begins with a series of geometrically-homologous point-to-point mappings on two forms and postulates that the point deformations implied by a comparison of the forms can be described by a single, or a series of, generalized deformational types (e.g., pure inhomogeneous, quadratic, rigid motion involving several landmarks, spiral deformation; see Bookstein 1991 for examples). Unfortunately, Bookstein (1991) did not specify whether the homology he was referring to in advancing this concept of Thompsonian deformational homology was of the geometric or biological variety. Thompson believed that his deformational types (even though he never referred to them in those terms) resulted from the operation of basic physical laws. However, Thompson refrained from discussing his concept in terms of evolutionary theory because he rejected that theory (see Mayr 1982). Regardless of Thompson s opinions on biological theory his concept of deformational homology underpins the morphometric synthesis (Bookstein 1996) because it recognizes that in order to facilitate biologically and geometrical interpretation morphometric data analyses should be portrayed in the language of deformations. Kendall (1984) showed that this deformation space (= shape manifold) exhibited an internal Reimannian geometry whose dimensionality was determined by the number of landmarks participating in the deformation. The linear spaces within which ordinations of objects in empirical shape spaces have been traditionally portrayed and in which the various statistical tests for shape similarity operate are oriented tangent to the hyperdimensional shape manifold with the tangent
point usually occupying the position of the mean shape. In order to be useful in inferring phylogenetic relationship descriptors, whether qualitative or quantitative, must be independent features of individual organisms and able to pass Patterson s (1982) tests of congruence, similarity, and conjunction. Zelditch et al. (1995) based their argument favoring use of principal warp and partial warp-based morphological deformation variables in phylogenetic analysis on a belief that that such variables uniquely meet the criteria stated above. Accordingly, this controversy is grounded on the questions of (1) whether these morphometric variables are different from the types of variables rejected as useful in phylogenetic analysis by Pimentel and Riggins (1987) and Cranston and Humphries (1988), and (2) if different whether deformational homologies can be identified with the concept of biological homology. If there are no substantive differences between inter-landmark distance-based shape variables, boundary/surface coordinate-based shape variables, and landmarkbased shape in terms of their relation to the concept of deformational homology or with the practices employed for using qualitative variables in phylogenetic inference, the argument that geometric morphometric variables are to be preferred for phylogenetic analysis because they are different from other types of phylogenetically-informative variables collapses. Similarly, if it can be shown that the concept of deformational homology refers to geometric, as opposed to biological, homology, the case for using any deformation-based morphometric variable in phylogenetic analysis also collapses. But, if it can be shown that deformation-based morphometric methods can be used to study the geometries of organic morphology in a manner that is more objective, more consistent, and more nuanced than current (largely qualitative and/or datafree) approaches, a path might be opened for the long sought integration of morphometrics into phylogenetic analysis. 1.4 Materials and Methods In order to determine whether different morphometric variables provide intrinsically different basic information types a set of parallel analyses on two different morphometric datasets was conducted and the results summarized using comparable data visualization methods. The two example datasets come from invertebrate paleontology (trilobites) and micropaleontology (radiolaria), but represent typical examples of organic morphologies that might be obtained from a wide variety of Recent and fossil groups. Both datasets contain single type rep resentatives of a number of species belonging to monophyletic
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Figure 1. Representative trilobite cranidia from species in the genera Struszia, Mackenziurus, Avalanchurus, and Fammia. These cranidial images were used to obtain landmark distributions for the trilobite dataset. Original images from Adrain and Edgecombe (1997).
groups. These two groups exhibit radically different morphologies, but were compared in order to prevent over-interpretation of results that might be idiosyncratic to a particular dataset. In addition, both datasets are real (in the sense that they are not simulations) and contain levels of morphological complexity comparable to those in other real systematic datasets. Simulated data can be used to reveal additional aspects of the alternative morphometric methods and their utility in phylogenetic inference. Such investigations, however, are beyond the scope of the present study. The trilobite dataset (Fig. 1) was taken from 12 Struszia, Mackenziurus, Avalanchurus, and Frammia species illustrated in Adrain and Edgecombe (1997). Patterns of two-dimensional morphological variation in this set of trilobites cranidia were quantified using 15 landmarks (Figure 2) and 33 inter-landmark distances (Figure 3) arranged to form a series of box trusses (Strauss and Bookstein 1982; Bookstein et al. 1985). The radiolarian dataset (Fig. 4) was taken from 24 Perispyridium morphotypes illustrated in Pessagno and Blome (1982) and MacLeod (1988). Patterns of two-dimensional morphological variation in this set of radiolarian tests (= skeletons) were quantified using 16 landmarks (Fig. 5) and 24 interlandmark distances (Fig. 6) arranged to form a series of box trusses. The morphometric data described above were used to construct two different data sets for morphometric analyses. Geometric morphometric analysis was represented by a partial warp analysis of the coordinate data (Bookstein 1991; Rohlf 1993; Zelditch et al. 1995). A new technique that applies
Figure 2. Encrinurine trilobite cranidial morphologies (A) and landmarks used to quantify trilobite cranidial shape (B). Black symbols represent Type 1 landmarks. Grey symbols represent Type 2 landmarks. Subscripts refer to left (l) or right (r) and upper (u) or lower (l) relative positions. Struszia illustration from Adrain and Edgecombe (1997).
singular value decomposition (SVD, the basis for all forms of eigenshape analysis) to Procrustesregistered landmark data coordinate-point (C-P) eigenshape analysis is introduced below to represent the eigenshape family of procedures. And multivariate morphometrics is represented by a singular value decomposition (SVD, J reskog et al. 1976) of the raw inter-landmark distances. Presentation of results derived from these alternative procedures using data derived from the same samples and using identical visualization procedures will allow readers to make detailed comparisons between them. Coordinate-point eigenshape analysis was first discussed by MacLeod (1995) and alluded to by MacLeod (1999). It consists of collecting landmark data from a series of forms, constructing a series of shape functions from Procrustes or Booksteinregistered shape coordinates, collecting these shape functions into a data matrix, and submitting this data matrix to a covariance-based SVD. This method can be regarded as a variant of relative warp analysis (Bookstein (1991). Results mirror those of standard and extended eigenshape analyses (see MacLeod 1999) and support linear ordination, shape modelling, morphing, and identification-comparative ranking of regional shape variation modes. The advantages of the coordinate-point eigenshape method over standard and extended eigenshape
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Figure 3. Truss network of trilobite inter-landmark distances used to quantify shape changes in the distance-based multivariate morphometric portion of the study. Symbol conventions as in Figure 2.
analyses include: (1) complete relaxation of the equal spacing between boundary/surface coordinates within corresponding outline segments convention, and (2) ability to incorporate non-outline landmarks within the analysis.
Partial Warp Analysis
Figure 4. Representative radiolarian tests from species assigned to the genus Perispyridium. These images were used to obtain landmark distributions for the radiolarian dataset. Original images from Pessagno and Blome (1982) and MacLeod (1988).
Partial warp analysis of the 15-landmark, Procrustessuperimposed (generalized least-squares) trilobite dataset decomposed the measured landmark displacement into a set of 26 independent shape axes including both affine (2 axes) and non-affine (24 axes) deformational modes. [Note: results obtained for other superposition methods (e.g., resistant fit) are comparable to those reported for the generalized least-squares method.] These axes can be organized to represent the paired (x , y) shape change components of 13 two-dimensional deformations (= the partial warps) that can be portrayed graphically and used to interpret the regionally-weighted foci of shape variation that define the deformations. Figure 7 shows a series of warps from this analysis illustrating the affine and non-affine ends of this shape deformation spectrum. The non-affine warps are further represented by examples from the loworder (= large eigenvalue) and high-order (small eigenvalue) ranges. Conceptually identical, but geometrically variant, patterns of shape deformation were recovered from the partial warp analysis of the 16-landmark, Procrustes-registered radiolarian dataset (Fig. 8). As can be seen from both sets of illustrations the affine warp represents that component of observed shape variation that can be expressed as a simple shear. In other datasets the affine component can also represent a uniform dilation or a combination of shear and dilation. The non-affine modes defined under this analysis constitute a series of either symmetrically or asymmetrically, spatiallyregionalized deformations; hence the name warps.
These warp sets result from an eigenanalysis of a matrix of coordinate-pairwise bending energies In this context bending energy refers to the calculated amount of energy required to bend a hypothetical, flat, 2-dimensional plate defined by a reference configuration of landmarks. Since increases in bending energy are associated with greater degrees of overall shape difference, the bending energy matrix constitutes an R-mode shape dissimilarity matrix somewhat analogous to the R-mode covariance matrix of multivariate analysis. (see J reskog et al. 1976; Chatfield and Collins 1980) The reference shape is usually taken to be the mean shape for the sample (see Rohlf 1993, 1998) whose form is dependent upon the characteristics of the sample. The resulting eigenanalysis-based decomposition yields a series of shape-change vectors within an [(n —3) x 2]—dimensional linear hyperspace (where n = the number of landmarks) ordered such that the higher-order vectors express more generalized and so less energetic deformations and the lower-order vectors express more regionalized and so more ener getic deformations. In understanding the nature and appropriate interpretation of these shape-change warps it is important to note that while different landmarks or landmark subsets may participate differentially in defining a deformation (by virtue of the loading coefficient assigned to the x or y coordinate of the landmark by the eigenanalysis), all landmarks assigned non-zero loadings participate in the
1.5 Results
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Figure 6. Truss network of radiolarian (Perispyridium) interlandmark distances used to quantify shape changes in the distance-based multivariate morphometric portion of the study.
Figure 5. Radiolarian (Perispyridium )morphologies (A) and landmarks used to quantify radiolarian test shape (B). Grey symbols represent Type 2 landmarks. White symbol represents a semi-landmark. Subscripts refer to left (l) or right (r) and upper (u) or lower (l) relative positions. Perispyridium illustration from MacLeod (1988).
definition of each warp. This property of warp definition is identical to the assignment of loading coefficients via eigenanalysis in a variety of multivariate data analysis procedures (e.g., principal components analysis, factor analysis, singular value decomposition, canonical variate analysis, canonical correlation analysis). Accordingly, while it is appropriate to discuss or interpret these warps as expressing deformational modes that accentuate shape changes within particular regions of the landmark constellation under a Thompsonian deformational shape-change model, it is inappropriate to portray such warps as representing logically localized aspects of shape change if that term is understood implicitly to mean independent of other regions (= landmark locations) within the same warp. Moreover, within the shapedeformation hyperspace and the entire sequence of vectors is dependent upon (1) the composition of the sample that determined the reference shape (2) the number, definition, and placement of the landmarks, and (3) the orientation of the landmark constellation. This latter point can be illustrated by removing a landmark from the dataset, recomputing the analysis, and then comparing corresponding warps with one another. If the warp representations were truly
localized, the removal of unimportant landmarks within particular warps should have no effect on their pattern. However, removal any landmark profoundly alters the patterns of all warps by changing the relative patterning of the bendingenergy matrix and distributing the resultant shape variation over a smaller number of eigenvectorbased shape axes (Bookstein 1991). In a similar manner, addition or deletion of taxa from the sample will also change the configuration of the reference shape (the mean shape of the sample) and, in so doing, may dramatically alter the bending energy matrix, principle and partial warp decompositions, and principle and partial warp scores for all taxa. Consequently, it is inappropriate to claim that a partial or principle warp analysis is invariant with respect to sample composition. Partial and principle warp analysis can be made invariant to sample composition by selecting a reference shape a priori (see Zelditch et al. 1992). However, such an a priori designation is logically arbitrary, demonstrably suboptimal from both biological and geometrical points of view (Rohlf 1998), and no different in principle from similar strategies that could be used to force sample composition invariance on a principal components analysis (PCA), SVD ( see MacLeod and Rose 1993), or any other multivariate procedure. Ordinations of objects in the vector spaces defined by these partial warps are presented in figures 9 and 10 for the trilobite and radiolarian samples respectively. The trilobite uniform warps (Fig. 9A) contrast species exhibiting relatively compressed central glabellar regions (e.g., S. dimitrovi) with those characterized by a relatively expanded central glabellar region (e.g., S. onoae, S. petebesti) on the x-axis, and species exhibiting relatively extended
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Figure 7. Affine (A) and non-affine (B-D) partial warps based on the trilobite dataset. Symbol conventions as in Figure 2.
glabella (e.g., S. epsteini) to those characterized by a relatively low glabella (e.g., M. ceejayi, M. deedeei) on the y-axis. No obvious subsidiary groupings of taxa are evident within this slice through the empirical shape space. The most generalized non-affine warp (Partial Warp 12x vs. 12y, Fig. 9B) expresses the degree of transverse compression (x -axis) and anterior shearing of the fixed cheeks (y -axis) that characterize various species. This deformational mode identifies S. dimitrovi and A. simoni as exhibiting relatively atypical, laterally expanded cranidia and S. epsteini as exhibiting an atypical forward placement of the fixed cheeks. Both of the higher-order partial warps (figures 9C and 9D) represent shape deformations that are regionally weighted toward opposite sides of the cranidium, but have no obvious biological interpretation other than expressions of asymmetry (possibly accentuated by inconsistent specimen orientation, spatially-variable preservation, subtle deformation due to geological factors, etc.). Distributions of species within these shape spaces provide further support for the morphological distinctiveness of A. simoni. Uniform partial warps for the radiolarian dataset (Fig. 10A) contrast forms exhibiting a large peripheral shell and relatively small spines (e.g., P. packardi, P. slaughteri) with those characterized by a larger peripheral shell and relatively large spines (e.g., P. mayri, P. ordinarium) on the x-axis and between species exhibiting a relatively small peripheral shell and relatively narrow spines (e.g., P. dobzhanski, P. gemmatum) with those characterized by a small peripheral shell and relatively wide spines (e.g., P. alichakense, P. elegans) on the y-axis. No subsidiary groupings of taxa are supported by these deformational modes. Similarly, the most general non-affine shape deformations (Fig. 10B) suggest no subsidiary groupings, but contrast forms exhibiting relatively small peripheral shells and large spines (e.g., P. darwini, P. schopfi) with those characterized by relatively large peripheral shells and small spines (e.g., P. slaughteri, P. facetum).
Figure 8. Affine (A) and non-affine (B-D) partial warps based on the radiolarian dataset. Symbol conventions as in Figure 5.
Subsidiary groupings of taxa are suggested by the lower-order radiolarian partial warps. However, the geometries of the highly asymmetric deformations represented by these axes make them much more difficult to interpret biologically. Partial Warp 2 subdivides the objects into two groups based on the contrast between forms whose peripheral shell is relatively wide in the region near the primary lateral spines and narrow in the region near the apical spine (e.g., P. facetum, P. foremanae) and those whose peripheral shells are relatively narrow in these areas (e.g., P. darwini, P. gouldi). Partial Warp 1 inverts the shape trends present in Partial Warp 2 (compare figures 10C and 10D) and contrasts species whose peripheral shells are transversely narrow relative to the character of the peripheral shell in the apical region (e.g., P. facetum, P. foremanae) with those whose morphologies are reversed for this shape deformation mode (e.g., P. darwini, P. gouldi). With respect to these two partial warps it is interesting to note that main subsidiary groupings are almost identical, though inverted. This suggests that these lower-order partial warp axes have subdivided a single morphological distinction between subsets of taxa into a series of more-or-less equivalent, but geometrically distinct, shape-contrast axes.
Coordinate Point Eigenshape Analyses. Precisely the same data as were used in the partial warp analysis formed the basis of the c-p eigenshape analysis. However, in this case the inter-object shape distances were represented as a pairwise variancecovariance matrix rather than a bending energy matrix. Despite this difference, the overall result is similar. A dissimilarity matrix in this case a landmark-based variance-covariance matrix was submitted to eigenanalysis via, in this case, SVD resulting in the specification of a series of orthogonal linear shape-change vectors. These
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Figure 9. Distributions of trilobite species within shape planes defined by the following trilobite dataset partial warps: Uniform x and Uniform y (A), Partial Warp 12x and Partial Warp 12y (B), Partial Warp 2x and Partial Warp 2y (C), Partial Warp 1x and Partial Warp 1y (D). Symbols denote genera.
represent a hierarchically-ordered series of shape trends that together account for all aspects of observed shape variation. Whereas the partial warp analysis required the determination of [(n-3) x 2] different mean-form-referenced axes to represent the sum total of observed shape variation (where n = the number of landmarks), c-p eigenshape analysis requires the specification of either k (= the number of objects in the sample) or (n x 2) samplereferenced axes; whichever is smaller. Because of the difference in the number of shape axes determined for the c-p eigenshape and relative warp analysis, some differences in the range of deformation modes used to describe the observed shape changes the objects ordinations within the various shape spaces are expected. But, since the question under examination here is whether shape vectors and shape spaces calculated using nongeometric methods differ in principle from their geometric counterparts in terms of spatial localization, these differences are irrelevant. Regardless, it will be interesting to note any correspondences between the partial warp and c-p eigenshape deformational modes and/or ordinations. Following the methods described in MacLeod and Rose (1993) and MacLeod (1999), shape models based on the eigenvectors of the variance-covariance matrix decompositions can be used to graphically express the patterns of shape deformation these axes represent. While any coordinate location within this shape space may be modelled, sequences of shape models lying along the various shape-change axes are the ones most commonly determined. In the past these have been represented as sequences or overlays of shape models (see MacLeod and Rose 1993; MacLeod 1999). However, it is equally possible to employ the thin-plate spline technique of geometric
Figure 10. Distributions of radiolarian species within shape planes defined by the following radiolarian dataset partial warps: Uniform x and Uniform y (A), Partial Warp 12x and Partial Warp 12y (B), Partial Warp 2x and Partial Warp 2y (C), Partial Warp
morphometrics as a graphical device to represent the shape change trends encoded within the eigenshape axes as Thompsonian deformations. By utilizing this graphing convention one may more readily visualize the nature of the shape-change trends resulting from the c-p eigenshape analysis and compare these to the deformational modes determined as a result of other morphometric data analysis procedures. Figures 11 and 12 illustrate c-p eigenshape models for the first two shape dissimilarity axes of the trilobite and radiolarian datasets respectively. Any generalized shape distinctions among objects within the sample should reveal themselves in the ordination of objects along these two shape axes. Subsidiary groupings of objects may be present on higher-order c-p eigenshape axes, but those axes are rarely used for interpretative purposes owing to wellknow problems with the stability of higher-order eigenvectors and the large influence atypical single observations can have on their placement (see Reyment and J reskog1993 for a discussion). In both datasets the individual dissimilarity eigenshapes correspond to non-affine aspects of the overall shape deformation due to their interaction with the first eigenshape axis during the modelling calculations. Although the modes of shape deformation represented in a c-p eigenshape analysis may differ from those that result from a partial warp analysis, these differences appear to be matters of computational detail (e.g., reference used for the eigenanalysis, the number of eigenvectors calculated), not fundamental differences in the kind of spatially localized information summarized. The second trilobite eigenshape axis (Fig. 11A) expresses contrasts in the relative size of the central glabellar mass and the fixed cheeks. The third eigenshape axis (Fig. 11B) represents a regionalized deformation that
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Figure 11. Thin-plate spline deformation models for the two most important shape dissimilarity axes from the trilobite c-p eigenshape analysis. Note the non-affine character of these deformations and their general similarity to the partial warp deformations illustrated in figures 6 and 7. These eigenshape variable-based patterns have all the “localizability” and “shape contrast” characteristics of partial warps as well as of variety of “geometric morphometric” shape variables. Symbol conventions as in Figure 2.
appears to be more-or-less confined to a medial transverse expansion and distal transverse contraction of the central glabellar mass along with a relatively small distally-directed glabellar translation. These shape deformations appear to be quite distinct from those recovered as a part of the partial warp analysis (compare figures 7 and 11). While the first two radiolarian dissimilarity eigenshape axes express a comparable diversity of deformational modes, the comparison of these to their partial warp results is even more instructive. Radiolarian c-p eigenshapes 2 and 3 (figures 12A and 12B) appear to be more regular variants of the general deformational modes recovered by partial warps 2 and 1 (figures 8C and 8D) respectively, but without the asymmetrical elements that make biological interpretation of the latter somewhat difficult. These deformations express an overall apical-abapical elongation with slight transverse compression of the peripheral shell with reduction in apical and primary-lateral spine length (Eigenshape 2, Fig. 12A), and an abapical compression of the central portion of the peripheral shell with overall radial expansion of the apical region (Eigenshape 3, Fig. 12B). Comparison of the object ordinations specified by partial warp (figures 9 and 10) and c-p eigenshape analysis (Fig. 13) is also instructive. Whereas the non-affine partial warp axes shown above suggest that the trilobite species S. epsteini (Partial Warp 12y, Fig. 9B) and S. dimitrovi, along with A. simoni, represent extreme morphologies (see partial warps 12x and 2y, figs. 9B and 9D), c-p eigenshape analysis indicates a much more diverse array of morphological subgroups. Both S. epsteini and S. dimitrovi appear to form a subsidiary grouping along the c-p eigenshape 2 axis that reflects the anteriorlyexpanded and broadly-rounded glabellar vault of these taxa. At the same time S. epsteini, S. onoae, and S. petebesti appear to form a subgroup along c-p eigenshape 3 that is unified by the more robust character of their pre-occipital glabellar regions. These shape contrasts were not suggested by either
Figure 12. Thin-plate spline deformation models for the two most important shape dissimilarity axes from the radiolarian c-p eigenshape analysis. Note the non-affine character of these deformations and their general similarity to the partial warp deformations illustrated in figures 6 and 7. These eigenshape variable-based patterns have all the “localizability” and “shape contrast” characteristics of partial warps as well as of variety of “geometric morphometric” shape variables. Symbol conventions
the extreme high-order or low-order partial warp ordinations (and they are not present in ordinations based on intermediate partial warps). For the radiolarian dataset the broad shape contrasts present within the most generalized partial warp (Partial Warp 13, Fig. 10B) are not consistent with the two most important shape dissimilarity ordinations of the c-p eigenshape analysis. In addition, the shape subgroups evident on the higherorder partial warps (figures 10C and 10D) are similarly absent from these radiolarian c-p eigenshape shape dissimilarity axes. These results are interesting especially given the geometric similarity of deformational mode that exists between c-p eigenshape 2 and Partial Warp 2 (compare figures 12A and 8C) and between c-p eigenshape 3 and Partial Warp 1 (compare figures 12B and 8D). [Note: the deformational mode represented by c-p eigenshape 1 is strikingly similar to an inverse Partial Warp 2-style deformation. Since the direction of eigenvector axes is determined by the interactions between various sample-specific and algorithmspecific factors, 180¡ rotations of eigenvector orientations are not uncommon in multivariate analysis. Accordingly, inverse deformations (= variables) should be regarded as equivalent deformations (= variables) in morphometric results in the same way that inverse eigenvectors are regarded as equivalent.] These results suggest that the partial warp-based subgroupings of taxa may not be unified by the generalized deformations (that are more definitively represented by c-p eigenshape axes), but by the much more difficult-to-interpret asymmetrical aspects of these partial warps. They also raise the possibility that the lower-order partial warps may not be stable in the sense of reflecting deep-seated shape contrasts existing with the dataset that are relatively immune to compositional peculiarities. Although there is an extensive literature documenting the instability of higher-order principal components,
11 along with new techniques to improve eigenvector stability (see Reyment and J reskog 1993 and referenced therein), there has been relatively little research into the effects of eigenvector instability in the context of geometric morphometrics. The foregoing description of deformational modes and object ordinations resulting from the partial warp and coordinate-point eigenshape analyses have led to different summaries of the predominant morphological signals within these data. One way to resolve such interpretational discrepancies is to employ additional analyses of alternative data or use a completely different analytical approach (e.g., Distance-based multivariate morphometrics [Reyment and J reskog 1993], outline-based techniques such as standard or extended eigenshape [MacLeod 1999]) as additional probes to explore the nature of shape relations within the dataset.
Multivariate Morphometric Analysis Use of Euclidean distance-based multivariate methods represents one candidate for an alternative morphometric data analytic approach because of (1) the long history of the use of these methods in morphometric contexts, (2) the similar (eigenanalytic) basis for the shape decomposition, (3) the ability to use all the landmarks employed in the partial warp and c-p eigenshape analyses to construct a network of inter-landmark distances, and (4) inherent differences between coordinate-based and distance-based representations of the shapes especially if the distance networks are incomplete to assess the major, stable directions of landmark displacement. The truss measurement selection protocol (Strauss and Bookstein 1985) represents a particularly useful procedure for the determination of a landmark-based distance network that meets the third and the fourth criteria specified above. In addition, use of truss networks support the reconstruction of shapes from inter-landmark distance data and along-axis, eigenvector-based shape modelling via a method analogous to that used in eigenshape modelling (e.g., MacLeod and Rose 1993; MacLeod 1999). This means that thin-plate spline visualizations of multivariate morphometric results can be constructed and compared visually to previous shape summaries. At this point let me stress that I am not advocating either the replacement of geometric morphometrics by (or its formal equivalence with) PCA or SVDbased truss analysis. Truss analysis is used in this context only to supply a quasi-independent assessment of the various deformational modes geometries and object ordinations as robust features of the example trilobite and radiolarian landmark samples. That having been said, truss analysis represents an attractive and easy-to-use method for determining inter-landmark distance networks. Its
Figure 13. Distributions of species-specific trilobite (A) and Perispyridium (B) landmark constellations within the planes define by the two most important shape dissimilarity variables in the coordinate-point eigenshape analyses. Note the lack of correspondence between these distributional patterns and those defined by the affine and non-affine partial warps (figures 8 and 9). Symbol conventions as in figures 2 and 5.
basic deficiency, in my view, is that it usually results in so underspecified a distance network that it is difficult or impossible to fully automate the shape modelling procedure required to determine the along-axis deformational modes. While this situation could be improved by increasing the number of distances calculated, that would also require increasing the sample size to avoid underdetermining the measurement matrix. In order to maintain comparability across alternative analyses SVD was applied to the 33 trilobite and 21 radiolarian inter-landmark truss distances in order to extract predominant modes of shape variation and ordinate samples along shape vector axes. As before, only the first three shape vectors resulting from these decompositions (= the major axes of shape similarity and dissimilarity) are examined in detail. Alternative shape decomposition procedures (e.g., PCA) could be employed, but these impose different constraints on the results of those analysis (e.g., number of eigenvectors determined). These differences would then be intermingled with differences that derive from the geometric representation of the landmark data. Since the latter is the topic under consideration it makes sense to minimize the influence of the former by maintaining as much methodological consistency as possible.
Figure 14. Thin-plate spline deformation models for the two most important shape dissimilarity axes from the trilobite inter-landmark distance-based SVD. Note the non-affine character of these deformations and their general similarity to the partial warp deformations illustrated in figures 6 and 7. These eigenshape variable-based patterns have all the “localizability” and “shape contrast” characteristics of partial warps as well as of variety of “geometric morphometric” shape variables. Symbol conventions as in Figure 2.
12
Figure 15. Distributions of species-specific trilobite (A) and Perispyridium (B) landmark constellations within the planes define by the two most important shape dissimilarity variables in the inter-landmark distance-based SVD analyses. Note the strong correspondence between these patterns and those defined by the c-p eigenshape analysis (Fig. 12) and the much weaker correspondence between these distributional patterns and those defined by the affine and non-affine partial warps (figures 8 and 9). Symbol conventions as in figures 2 and 5. See text for discussion.
Inspection of distance-SVD-based along-axis shape deformations for the trilobite (Fig. 14) and radiolarian (Fig. 15) datasets reveals interesting correspondences to previous results. DistanceSingular Vector 2 (Fig. 14A) represents the inverse of the c-p eigenshape Axis 2 deformation (Fig 13A; its geometric equivalent) with the fixed cheek regions undergoing relative expansion at the expense of the glabellum. Distance-Singular Vector 3 (Fig. 14B) represents a less-accentuated and more asymmetric, but modally comparable deformation to that c-p eigenshape Axis 2 Fig. 13B) with expansion of both the free cheek and pre-occipital glabellar region. None of the non-affine deformational modes specified in the partial warp analysis were reproduced by the distance-SVD analysis. Similarities between the distance-based SVD and c-p eigenshape analysis are also reflected in the ordinations of species within the shape space defined by the first two dissimilarity-based distance-singular vectors (figs. 15A), with S. dimitrovi, S. epsteini, S. onoae, S. petebesti, and A. simoni occupying the periphery of the shape distribution within the various along-axis contrasts. Not as many species subgroups are suggested by these distance-based results as by the c-p eigenshape results, perhaps signifying the relative inferiority of distance networks relative to coordinate locations for representing shape change. But those that are present (e.g., S. petebesti, A. s i m o n i ) are consistent not only with the c-p eigenshape-based subgroupings (see figures 13A and 13B), but also with those suggested by the nonaffine partial warps (see figures 9B and 9D). The radiolarian results present a similar picture of methodological-deformational correspondences and differences. The second distance-singular vector axis represents the inverse (= equivalent) of the c-p eigenshape Axis 2 deformational mode (compare Fig. 16A and Fig. 12A) with the apical and primary lateral spine bases-peripheral shell shoulders
Figure 16. Thin-plate spline deformation models for the two most important shape dissimilarity axes from the radiolarian inter-landmark distance-based SVD analysis. Note the non-affine character of these deformations and their general similarity to the partial warp deformations illustrated in figures 6 and 7. These eigenshape variablebased patterns have all the “localizability” and “shape contrast” characteristics of partial warps as well as of variety of “geometric morphometric” shape variables. Symbol conventions as in Figure 5.
contracting radially into the central region while the apical an primary lateral spines simultaneously increase their relative lengths. In terms of methodological comparisons the deformation encoded in Distance-Singular Vector 3 (Fig. 16B), with its sense of lateral expansion in the apical region along with pronounced lateral compression and apically-directed translation of the cephalis and peripheral spines, is interesting particularly in that it appears to generally correspond to aspects of both cp eigenshape Axis 3 (Fig. 12B) and to the non-affine Partial Warp 1 (Fig. 8D). Species ordinations based on these distance-SVD deformations are also similar to those obtained for the trilobite data. There is a strong correspondence between the extreme morphologies identified by both c-p eigenshape and distance-based SVD on these first two Distance-Singular Vectors with little support for subsidiary shape groupings (compare figures 15B and 13B). As before, a few similarities between ordination patterns specified by the Distance-Singular Vector and Partial Warp analysis emerge (e.g., note the consistently extreme position of P. hennigi in figures 10C, 10D, and 13B, and 15B). Overall, though, these comparisons appear typified by a lack of consistency. Once again, this lack of consistency between ordinations of species along deformational axes whose general deformational mode is quite similar especially in contrast to the clear and compelling correspondences between the c-p eigenshape and Distance-Singular Vector results. This seems to suggest that higherlevel partial warp ordinations are strongly effected by the clear, biologically difficult to interpret, and potentially unstable asymmetries in those warps rather than by the more generalized aspects of those deformational modes that lend themselves to straight-forward, stable, and convincing biological interpretation.
13 1.7 Discussion Comparisons Between Methods Very few comparisons between morphometric data analysis methods exist (e.g., Rohlf 1986; Lele and Richtsmeier 1991) and, to my knowledge, none goes into the level of detail afforded by this analysis. The similarities and differences discussed above allow for some generalizations to be made that should clarify some aspects of the relation of these morphometric methods to one another. Even more interestingly, they raise a number of issues that bear on the utility of these methods in both principle and practice, as well as suggesting several areas for future morphometric research. Among the generalizations suggested by this study the two most important are that (1) all morphometrics proceeds from the specification of a shape function for each object in the dataset and the quantification of differences among these shape functions by a pairwise dissimilarity matrix, and (2) the results of any eigenanalysis-based decomposition of a dissimilarity matrix can be represented by a set of deformations that may be parsed into affine or non-affine components. Several pertinent insights derive from these general conclusions. First, that morphometrics is based on the abstraction of biological shape variation into a series of shape variables that together define a shape function should come as no surprise. Shape functions have long been recognized in the outline analysis literature where they are often given specific names (e.g., the Zahn and Roskies shape function of standard eigenshape analysis [see Lohmann 1983], the polar coordinate shape function of radial Fourier analysis [see Christopher and Waters 1974]). These shape functions, in which the order of the variables is determined by convention and their magnitude by measurement, are equivalent to the single column vectors used to describe objects in terms of the inter-landmark distances of traditional multivariate morphometrics or the coordinate locations of geometric morphometrics. All are shape functions and all shape functions represent gross abstractions of the much more complex organic morphologies. Morphometrics analyzes patterns among shape functions, not patterns among shapes, much less differences among biological species. Second, in order to use morphometrics to address biological questions in a hypothetico-deductive context one must be able to formulate at least some predictions as to the patterns of similaritydissimilarity that exist between these shape functions prior to their analysis. Ideally one should have some idea of the morphological implications of various process-level hypotheses and be able to use these to predict the deformational modes that define the various eigenvectors or the object ordinations within the various shape spaces. Perhaps more practically
one can postulate a null model of no significant or consistent pattern of shape deformation, or morphology-based subsidiary shape groupings, and then use morphometric analysis to test this null hypothesis. The danger here is that rejection of the null hypothesis based on morphometric results does not necessarily support any particular process-level explanation for the recovered patterns. Moreover, given the variability in the deformational modes and ordinations illustrated above for the same set of morphometric data, failure to recover an ordination containing subsidiary object groupings using one or another analytic approach does not preclude the possibility of obtaining such groupings using other analyses. The lesson of the examples discussed above is that different analytic procedures are designed to accentuate (or suppress) different aspects of any geometric system defined by shape functions. The goal of biological morphometrics must be to provide answers to biological questions through an understanding of this geometric system rather than through the routine submission of particular types of data to particular analysis methods and then developing untested or untestable process-level explanations (= stories) to account for the result. Third, under this view of the nature of morphometric data and the place of different data analytic methods it can be suggested that morphometric explanations for biological patterns become far richer and more convincing if they can account for the results produced by multiple data analytic methods applied to the same dataset and (if possible) the alternative geometric representations that might be contributed by different shape functions (e.g. analysis of inter-landmark distances, outline analysis). The subsidiary species groupings suggested by the low-order partial warps 1 and 2 appeared suggestive when viewed in isolation. However, the failure of comparable patterns to be recovered by the c-p eigenshape and distance-SVD results especially given the gross similarity the deformations the axes in all three analysis suggest caution. The next logical step would be to perform further experiments designed to explore the underlying reasons for the groupings recovered by the partial warps and interpret the result in light of those findings (= based on a more complete understanding of the geometric system). Similarly, the fact that comparable deformations and ordinations were obtained by c-p eigenshape and distance-based SVD provides a measure of certainty that these results are consistent features of the geometric system that could, in principle, be used to inform a biological character analysis with some degree of confidence. In addition, the deformation data presented above appear inconsistent with the argument that geometric morphometric methods are unique in providing summaries of localized morphological
14 deformations. Using the modelling techniques discussed herein and thin-plate spline graphical techniques (which are fully generalizable to a variety of other morphometric procedures), there appears to be no substantive difference in the type of deformation-based shape variables specified by partial warp, c-p eigenshape, or distance-based SVD analyses. This result removes another of the Zelditch et al. (1995) reasons for preferring geometric morphometric procedures to (at least some types of) outline analysis and multivariate morphometric procedures for the purpose of defining phylogenetic characters/character states and assigning these states to operational taxonomic units (OTU s). [Note: see MacLeod 1999 for comments on the registration problem previously mentioned by Zelditch et al. 1995 and others as a limitation of outline-based morphometric analyses.]
Morphometrics and Phylogenetic Inference As discussed above, in order to be useful in phylogenetic inference characters must represent taxic homologies. Zeldtich et al. (1995) argue that individual partial warps (and no other type of morphometric variable) correspond to the concept of taxic homologies because (1) they explicitly incorporate the concept of a homology of parts, (2) are localizable to particular regions of the form, and (3) allow relevant contrasts between regions of the form to be discovered as a result of the analysis rather than specified a priori. While I agree with Zelditch et al. (1995) that morphometrics can be useful in indeed, is crucial to a properly formulated morphology-based character analysis, I also believe that a case can be made for admitting morphometric methods other than partial warps to the set of phylogenetically-informative morphometric techniques. The relation between landmarks geometric/topological correspondence (= geometric homology) and biological homology has been discussed above. Landmarks do not fit into the concept of biological homology at the level of the individual coordinate point because that concept was never intended to specify intra-structure correspondence to such a refined level. Type 1 landmarks might represent valid homologues under certain conditions, but these types of landmarks are relatively rare in any particular analysis and these is usually no data that can decide the point-based homology question one way or the other. Type 2 landmarks and semi-landmarks correspond to general notions of part matching within biological structures. While these landmark concepts represent generalizations of the form-matching criterion in biological homology, defensible at the level of the overall structure (= the level of the biological homologue), they are not defensible at the level of
the mathematical point. Landmarks, with very few exceptions, are not candidates for homology. In practice all assessments of biological form are tied to landmarks, none of these landmark classes exhibit a detailed or necessary correspondence to the concept of biological homology, but all represent generalized expressions of intra-structure correspondence sufficient for quantitative part matching and are demonstrably superior to the rhetorical or qualitative alternatives. Pimentel and Riggins (1987) criterion that characters useful for phylogenetic inference must be features of individual organisms (see also Cranston and Humphries 1988) has been interpreted to exclude ordinations defined by sample-based properties (e.g., means, variances, principal components) because these abstractions represent sample-based not individual-based observations. As a systematic principle, there can be no argument that sample-based parameters such as these cannot themselves be used (or discontinuously scored) in a character matrix intended for phylogenetic analysis. However, does this also mean that clear discontinuities in the distribution of attribute size or shape must also be so excluded? I do not believe this extrapolation is either warranted or logicallyconsistent with current phylogenetic practice. The problem lies in the confusion of discontinuous patterns of morphological variation between taxa with the data analytic devices that have been developed to reveal and study those patterns. The former is a fact of nature, fully individuated and the focus of character-state analysis. The latter are manmade tools that provide systematists access to aspects of nature that they cannot sense in any other way. To make an approximate physical analogy, every modern systematist would recognize the value of qualitative characters determined through use of a microscope as indicators of phylogenetic relationship. But most would rightly question the practice of including the character able to be observed with a microscope in a phylogenetic analysis. The distinction between the phenomenon under study and the tool used to study the phenomenon is obvious. Devices such as means, variances, principle warps, eigenshape axes, etc. are not as obvious tools as microscopes, but they represent their mathematical equivalent. By using tools such as these it is possible to sense discontinuities in systematic datasets that have phylogenetic significance. However, as Pimentel and Riggins (1987) understood, but expressed badly, no one should not confuse the tool with the target of one s investigation. Failure to understand this distinction has led to the present situation in which it seems perfectly acceptable to qualitatively describe the claws at the ends of aphid legs as simple or complex and code these states discontinuously, but is neither acceptable nor required to employ
15 quantitative methods that might demonstrate whether the claws in question can be objectively subdivided into these categories or where the division should be drawn because to do so would destroy this character s utility in a phylogenetic context. Note also that the qualitative character states aphid leg claw shape: simple, complex are implicitly tied to the sample. Can an aphid claw really be described as complex ? Complex relative to what if not the other members of the sample? This common practice represents a clear violation of Pimentel and Riggins (1987) recommendations regarding independence from sample composition. The routine manner in which obviously sample-referenced character-state designations are used in morphology-based phylogenetic analyses introduces serious questions as to the practicality of this principle, as well as to its routine citation as a justification for excluding morphometric variables from application in phylogenetic studies. A more quantitative assessment would at least affix a sampleindependent scale to this shape distinction. In addition to these heuristic considerations, it must be noted that the shape deformations specified by partial warp analysis or any other eigenanalysisbased procedure cannot be regarded as homologous characters in and of themselves for purely practical reasons. In order to be used in a parsimony-based phylogenetic procedure characters must be able to be assessed for similarity and congruence (Patterson 1982). Shape variables in general and shape deformations in particular fail both tests in important ways. Patterson s (1982) similarity test requires that given three taxa and one character exhibiting at least two different states it must be possible to construct a hypothesis that two taxa are more similar to each other than either is to the third. If two of the taxa have exactly the same shape characteristic the shape character will pass Patterson s test. However, if all three exhibit different states it is impossible to construct an unambiguous hypothesis of hierarchical relationship. This is a consequence of Bookstein s Shape Inhomogeneity Theorem (Bookstein 1980, 1991, 1994) which states that it is impossible to objectively order the states of a shape deformation. Patterson s (1982) congruence test is violated by shape deformations in an even more basic manner. Implicit in the logic of parsimony-based phylogenetic analysis is a commutativity principle that requires all character states to represent noninteractive additions, deletions, or changes to the suite of available characteristics. Shape transformations such as those specified by partial and principal warps require an explicit interaction with a basis shape. In a morphometric analysis this basis shape is the sample mean shape. But, if the warp (= deformational homology) is allowed to participate in a cladistic analysis its place on the resultant tree would imply that the shape
transformation was applied to the shape of the (hypothetical) taxon specified at the proximal node of the branch along which the deformational state change takes place. Even more problematically, shape transformations, unlike traditional cladistic character transformations (that represent unitary structures), exhibit a non-commutativity that is entirely alien to phylogenetic systematics (see Bookstein 1994 for an example). Consequently, and the shape deformation transformations A B C and B A C (where A, B, and C represent different geometric shape transformations) will not produce an equivalent geometric result. The methods advocated by Zelditch et al. (1995) ignore these logical implications that are hidden by the artifice of discontinuous character-state coding procedures. Morphometrics is useful in phylogenetic analysis not because of morphometricians new appreciation for the power of analyzing landmark coordinates (there is no fundamental difference between alternative ways of representing what has always been essentially landmark data), a new ability to analyze these data using eigenanalysisbased decompositions of a shape function dissimilarity matrix (this is how most morphometric data have been analyzed for the last 30 years), or representing shape axes as regionally-weighted deformations (this concept has been implicit in all morphometric procedures, but the tools required to construct such visualization have only recently become available). Rather, morphometrics is useful in phylogenetic analysis because systematists have a basic need to describe morphologies in quantitative terms, to assess variations in the distribution of these morphologies both within and between OTU s and, based on the results of those analyses, formulate character-state descriptions and assign character states to OTU s in a consistent and reproducible manner. Deformation-based shape variables like all shape variables can be used to probe the biological geometries and help point the systematist s eye toward patterns among variables that can be used to discover phylogenetically-useful contrasts. But these multivariate variables are ultimately man-made mathematical abstractions, not biological attributes of organisms. They are inextricably embedded in hierarchical sequences that were created by algorithms to optimize and abstract, non-biological quantities like bending energy or covariance and they cannot exist outside of that context. They are tools, not morphologies (= characters) or distributional ranges (= states). Morphometrics can be instrumental in discovering new characters and new character states to the extent that the use of these tools can inform systematists about morphological discontinuities that exist within nature. To accomplish this task morphometricians and morphometrically-informed systematists must move away from the methodological rivalries that have marred the field s recent history and embrace
16 the morphometric synthesis (which focuses on important underlying similarities among morphometric analysis methods), use alternative procedures to probe and develop a detailed understanding of geometric systems of shape functions, integrate formal, deductive, hypothesistesting procedures into their analytic designs, and apply these results to the interpretation (phylogenetic and otherwise) of organic morphologies. Only by achieving these goals will morphometrics change from being a techniques-driven discipline on the margins of biological thought to occupy its rightful place at the center of morphology-based systematic analysis.
1.8 Summary Currently phylogenetics requires that complex biological morphologies be atomized into characters and then coded into discontinuous character states to participate in a phylogenetic analysis. Although many morphological characters are defined on intrinsically quantitative bases, morphometrics (the study of covariances with biological shape) has played little role in phylogenetic analysis because of its phenetic roots and because of well-established objections to the coding of continuously-varying characters into discrete states. However, since (1) continuously varying characteristics can exhibit discontinuous distributions and (2) many qualitatively-assessed morphometric characters and character states are routinely used in phylogenetic inference, there can be no objection in principle to employing morphometric methods character/character state definitions and assignment of character states to OTU s. Zelditch et al (1995), among others, have recently advocated the use of partial warps as phylogenetic characters on the basis of (1) the correspondence between the concept of a landmark (= the basis for partial warp analysis) and the concept of a biological homologue, (2) the ability of partial warps to quantify regionally-localized aspects of shape variability, and (3) the ability of partial warps to suggest shape contrasts between regions of the measured morphology. Zelditch at al. (1995) state that partial warps (and similar landmark-based geometric variables) are to be preferred in this context because they are the only morphometric variables that exhibit these attributes. Test analyses using partial warp analysis of landmarks, coordinate-point Eigenshape analysis, and singular value decomposition of inter-landmark distance covariance matrices for trilobite and radiolarian datasets, plus a review of the concepts of morphometric landmarks and biological homologues, fail to support these assertions. All morphometric measurements (e.g., inter-landmark distances, boundary/surface coordinates, extremal
points or other semi-landmarks, structure locations/tissue juxtapositions, maxima of curvature) are based on landmarks. Thus, all morphometric variables, observed or latent, are based ultimately on landmarks. In addition, while the concept of biological homology is defined at the level of the biological structure and extends to regional geometric correspondences, it cannot be used in practice to identify homologues uniquely at the level of the mathematical point except by recourse to arbitrary landmark placement conventions. All eigenanalysis-based decompositions of shape functions defined by dissimilarity matrices incorporate the concept of shape deformations as regionally-weighted shape change foci. However, none of these geometric abstractions corresponds to a biological character. Rather, shape deformations should be used as probes to analyze morphological characteristics of organisms that may be later recognized as potential homologues. Finally, because all forms of morphometric data can be used to reverse-specify landmark locations, the graphical tools of deformation-based morphometrics can be used to identify regions of differentiallyconcentrated shape variation regardless of whether those regions fall on outlines or are specified by inter-landmark distances. Since all morphometric data analytic procedures are essentially the same with respect to the attributes cited by Zelditch et al. (1995) as desirable in the application of morphometrics to phylogenetic inference, there seems no reason not to admit that any and all morphometric techniques might be useful in this context. In particular, it should be noted that the use of multiple approaches to the analysis of the same morphometric data, as well as alternative sets of morphometric observations, can contribute to the development of more complete and nuanced analyses of morphological patterns of variation, resulting in improvements in morphology-based phylogenetic results. If morphometricians can set aside their traditional methodological disagreements and get on with the task of employing morphometric analyses to inform phylogenetic analysis through hypothesis testing and refinement of character descriptions/character-state assignments, morphometrics can realize its potential and make unique contributions to biology in general and systematics in particular.
1.9 Acknowledgements This essay benefited from conversations and correspondences I ve had with numerous morphometricians and systematists I ve had over the last 20 years, including F. L. Bookstein, Timothy Carr, Peter Forey, Chris Humphries, Pat Lohmann, David Polly, Richard A. Reyment, F. James Rohlf, Peter Schweitzer, Donald Swiderski, and Andrew
17 Smith. Richard Fortey, Adrian Rushton, and Jon Adrain also patiently explained the intricacies of trilobite cranidial morphological nomenclature to me. In addition, this article greatly benefited from reviews by Joan Richtsmeier, Gill Klapper, and Jon Adrain. Few of these individuals (likely none) will agree with all of the arguments presented herein. Some might disagree with them all. While these contributions were of great help to me in putting this essay together the aforementioned are responsible for none of the data, analyses, or conclusions. That responsibility is mine. I d also like to acknowledge the TpsRelw and TpsSplin programs for the wintel platform that were written by F. James Rohlf and which I used to perform the partial warp analysis and graphically portray the partial warp, c-p eigenshape and distance-SVD deformations. All other data analysis software was written by the author and is available from the PaleoNet Software archive at: http://www.nhm.ac.uk/hosted_sites/paleonet/ftp/ftp.h tml (PaleoNet East) and http://www.ucmp.berkeley.edu/Paleonet//ftp/ftp.html (PaleoNet West).
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