shape, relative size, and size-adjustments in morphometrics

YEARBOOK OF PHYSICAL ANTHROPOLOGY 38:137-161 (1995)

Shape, Relative Size, and Size-Adjustments in Morphometrics

WILLIAM L. JUNGERS, ANTHONY B. FALSETTI, AND CHRISTINE E. WALL Department of Anatomical Sciences, School of Medicine, S.U.N.Y. at Stony %rook, Stony Brook, New York 11 794 (W.L.J.); National Museum of Health and Medicine, Armed Forces Institute of Pathology, Washington, D.C. 20306 (A.B.F.); Department of Biological Anthropology and Anatomy, Duke University Medical Center, Durham, North Carolina 27710 (C.E.W.)

KEY WORDS Size, Shape, Ratios, Residuals, Morphometrics

ABSTRACT Many problems in comparative biology and biological anthropology require meaningful definitions of “relative size” and ‘(shape.’’ Here we review the distinguishing features of ratios and residuals and their relationships to other methods of “size-adjustment’’ for continuous data. Eleven statistical techniques are evaluated in reference to one broadly interspecific data set (craniometrics of adult Old World monkeys) and one narrowly intraspecific data set (anthropometrics of adult Native American males). Three different types of residuals are compared to three versions of shape ratios, and these are contrasted to ‘(cscores,” Penrose shape, and multivariate adjustments based on the first principal component of the logged variance-covariance matrix; all methods are also compared to raw and logged raw data. In order to help us identify appropriate methods for size- adjustment, geometrically similar or “isometric” versions of the male vervet and the Inuit male were created by scalar multiplication of all variables. The geometric mean of all variables is used as overall “size” throughout this investigation, but our conclusions would be the same for most other size variables.

Residual adjustments failed to correctly identify individuals of the same shape in both samples. Like residuals, cscores are also sample-specific and incorrectly attribute different shape values to individuals known to be identical in shape. Multivariate “residuals” (e.g., discarding the first principal component and Burnaby’s method) are plagued by similar problems. If one of the goals of an analysis is to identify individuals (OTUs) of the same shape after accounting for overall size differences, then none of these methods can be recommended. We also reject the assertion that size-adjusted variables should be uncorrelated with size or “size-free”; rather, whether or not shape covaries with size is an important empirical determination in any analysis. Without explicit similarity criteria, “lines of subtraction” can be very misleading.

Only variables in the Mosimann family of shape ratios allowed us to identify different sized individuals of the same shape (“iso-OTUs”). Resid- uals from isometric lines in logarithmic space, projections of logged data onto a plane orthogonal to a n isometric vector, and Penrose shape distance based on logged data are also part of this shape family. Shape defined in this

0 1995 Wiley-Liss, Inc.

138 YEARBOOK O F PHYSICAL ANTHROPOLOGY [Vol. 38, 1995

manner can be significantly correlated with size in allometric data sets (e.g., guenon craniometrics); ratio shape differences may be largely independent of size in narrowly intraspecific or intrasexual data sets (e.g., Na- tive American anthropometrics). Log-transformations of shape variables are not always necessary or desirable. We hope our findings encourage other workers to question the assumptions and utility of residuals as size-adjusted data and to explore shape and relative size within Mosimann’s explicitly geometric framework. o 1995 WiIey-Liss, Inc.

Although size is widely regarded as a fundamental aspect of any organism’s biology that will influence behavior, anatomy and physiology (McMahon and Bon- ner, 1983; Peters, 1983; Calder, 1984; Schmidt-Nielsen, 1984; Jungers, 1985a; Damuth and MacFadden, 1990; McGowan, 19941, the goal of many comparative studies is to assess similarity or differences among taxa after size is taken into account, controlled for, or “factored out.” In morphometric terms, if we were to take a suite of the same measurements on a mouse lemur and on a gorilla and did nothing to adjust for gross differences in scale between the two, we would probably discover little beyond the obvious fact that Gorilla is larger than Microcebus. Recognition of this pervasive problem is intuitive and obvious; the choice of an analytical strategy and statistical methodology is neither. “Size” itself can be defined in many different ways depending on one’s biological and statistical goals (see below), and the task of determining differences in “shape” and relative size among taxa is especially challenging. As Mosimann and Malley (1979: 175) noted in their review of size and shape variables, “there are a number of precise, inter- esting and quite different possible meanings and these need to be distinguished.”

Attempts to provide useful and consistent formulations of size and shape fuel the current “revolution in morphometrics” (Rohlf and Marcus, 1993; see also Rohlf, 1990; Bookstein, 1991; and Richtsmeier et al., 1992). Although 2-D and 3-D coor- dinates of landmark points are championed as the data of choice in geometric morphometrics, profound disagreement exists over their analysis (e.g., Lele, 1991, 1993). Whereas size is defined as an attribute of an organism (e.g., centroid size), most coordinate-based methods of shape analysis focus on shape differences rather than on shape per se (e.g., finite-element scaling; thin plate splines; Procrustes superimposition; Euclidean distance matrix analysis). Following Mosimann (1970), we suggest that shape should also be viewed as an intrinsic property of an organism rather than as a changing function of different comparative sets. More- over, and contrary to repeated predictions of their imminent doom (e.g., Bookstein, 1978, 1991), interpoint distances remain important sources of information in the study of size and shape.

Whether one is dealing with linear dimensions, areas or volumes, it is not un- common for shape to be redefined as “relative size.” Ratios and residuals represent competing expressions of shape within this relative framework, and both formulations have their advocates and detractors; e.g., compare Lemen (1983) and Cor- ruccini (1987,1995) to Reist (1985) and Albrecht et al. (1993,1995). Some workers have approached this problem by partitioning multivariate distance coefficients into separate size and shape distances (Penrose, 1954; Spielman, 1973; Relethford, 1984; Vasulu and Pal, 19891, while others have tried to sweep size from discrim- inant functions (Burnaby, 1966) or principal components (Flessa and Bray, 1977; Humphries et al., 1981; Bookstein et al., 1985; Somers, 1986, 1989). Apparently new solutions to the same problem continue to appear (e.g., Darroch and Mosi- mann, 1985; Kazmierczak, 1985, 1986; Rohlf and Bookstein, 1987; Howells, 1989; Albrecht et al., 1993; Richtsmeier and Lele, 1993).

The goal of this review is to explore further the distinguishing features of ratios and residuals and their possible connections to other methods of “size-adjustment’’ and shape analysis in comparative studies. Eleven alternative techniques are evaluated here in application to two very different data sets, one broadly interspecific

Jungers et al.1 SHAPE AND RELATIVE SIZE 139

(craniometrics of cercopithecine crania), and one narrowly intraspecific (anthropometrics of male Native Americans). As a preliminary to detailing our materials and methods, we offer a brief review and graphical depiction of the concepts geometric similarity and allometry in both raw and logarithmic data space.

GEOMETRIC SIMILARITY AND ALLOMETRY

“Geometric similarity” is usually taken to be synonymous with “isometry” in analyses of scaling. In simplest terms, this implies that shape is preserved among organisms of different sizes. In raw data space, points lying on the same positively directed ray emanating from the origin exhibit this quality (Fig. la). The ratio of Y/X will be the same for all points on the line Y = m,X; other slopes (e.g., m2 and m3 here) imply different isometric shape vectors (Mosimann, 1970; Mosimann and James, 1979; Bookstein, 1989). Isometry in logarithmic space is revealed by lines with a slope of 1.0 (Huxley, 1932; Gould, 1966,1977); again, the ratio of YIX is the same for all points on such a line (Fig. lb). Different isometric shape vectors are parallel but different in elevation.

“Allometry” refers to nonisometric scaling, wherein shape changes as a function of size; geometric similarity is not preserved. Many functions describe allometric relationships in raw data space (Bookstein et al., 1985; Albrecht et al., 1993). For example, any linear function with a non-zero intercept (e.g., Y = m,X + b, in Figure lc) implies that the ratio YIX will change as a function of X (“linear allometry”). Power functions can also describe allometric changes. The “full allometric” data model of Y = mlXkl + b, and the “simple allometric” data model Y = m3Xk3 (Albrecht et al., 1993) describe curvilinear distortions in shape. Simple allometry is frequently described in log-log plots by straight lines with slopes greater than or less than 1 .O (“positive” and “negative” allometry, respectively; Fig. Id). Obviously, curvilinear and even segmented linear relationships in logarithmic space could reflect allometric changes in the ratio Y/X (e.g., Chappell, 1989; Deacon, 1990; Koops and Grossman, 1993). If shape is defined explicitly and in advance, then isometry and allometry can also be disclosed by nonsignificant versus significant correlations between shape and size (Mosimann and James, 1979; Falsetti et al., 1993).

It will be shown below that some size-adjustments can behave very poorly when allometric relationships characterize a given data set. It is recommended that all variables be examined for isometry versus allometry prior to any size-adjustment. The variables in the two data sets used here are tested in both raw and logarithmic spaces, with isometry as the null hypothesis (raw H,: linear intercept = 0; log H,: slope = 1.0).

SAMPLES AND MEASUREMENTS

Our interspecific sample is drawn from the summary data base provided in the comparative study of guenon craniometrics by Verheyen (1962). Males and females of 13 species (i.e., 26 operational taxonomic units or OTUs) ranging in size from talapoin monkeys to patas monkeys are included in the analyses (Fig. 2). Due to significant sexual dimorphism in cranial size in many species, sex-specific means of 16 linear measurements are used to describe aspects of cranial size and shape in this sample of cercopithecines (also see Martin and MacLarnon, 1988). Taxa, sample sizes, and variables are listed in Table 1.

Our intraspecific sample is taken from the extensive Franz Boas data base on Native North Americans curated at the Univeristy of Tennessee-Knoxville (Jantz et al., 1992). Males from 20 different populations from the Pacific Northwest and Western Plateau regions of North America are examined here. Mean values for 12 different anthropometric variables (six body, six head) are included in all analyses. The 20 OTUs or groups (with their “cultural area”), sample sizes and variables are summarized in Table 2 (also see Falsetti, 1989). Figure 3 is a photograph of an Inuit male and female taken from the archives of the Department Library Ser- vices, American Museum of Natural History.

140 I‘EARBOOK OF PHYSICAL ANTHROPOLOGY [Vol. 38, 1995

lsometrv in Raw Date Space Isometry in Logarithmic Space

X V

Allornetrv in Raw Date SDace

X U

LOG X

Allometry in Logarithmic Space I I

logY=k, logX+logm,

logy =k210gX+logm2 (k,cl .O)

/ I d LOG X

Fig. 1. Graphical display of isometric and allometric functions in raw and logarithmic data spaces. Isometry or geometric similarity (ie., equal ratios of YIX) is indicated in raw space by positively directed rays emanating from the origin (0,O). In logarithmic data models, these rays are translated into parallel linear relationships with slopes equal to 1.0. Many linear and nonlinear equations describe allometry (i.e., ratios of YIX change as a function of X) in both raw and logarithmic plots. See text for descriptions of a-d.

CRITERIA FOR A SUCCESSFUL SIZE-ADJUSTMENT

In order to evaluate objectively the efficacy of the 11 methods of size-adjustment (described below) in recovering shape information, we have adopted Lemen’s (1983) strategy of creating hypothetical OTUs known to be different in size but identical in shape to one of the OTUs included in each sample (also see Corruccini, 1987, 1995). The male vervet (Cercopzthecus aethiops) was selected for the interspecific craniometric analyses, and the Inuit male was designated for the intraspecific anthropometric study. Geometrically similar or isometric versions of each reference OTU were created by scalar multiplication of all variables in the sample: 0.67 (small) and 1.33 (large) for “iso-vervets”; 0.98 (small) and 1.04 (large) for “iso-Inuits.”

The objective criterion adopted here for sane shape between any pair of OTUs is a zero average taxonomic (Euclidean) distance (or ATD; Sneath and Sokal, 1973). Only those size-adjustments that produce zero ATDs among the male vervet and

Jungers et al.] SHAPE AND RELATIVE SIZE 141

Miopithecus talupoin 0

Cercopithecus aethiops B

Erythrocebus patas 0

Fig. 2. Frontal and lateral views of crania of a male talapoin monkey (top), a male vervet (middle) and a male patas monkey (bottom). Images are modified from plates in Verheyen (19621, and reduced to approximately similar size. Note that many aspects of cranial shape change as a function of size differences (from the small talapoin monkey to the large patas monkey).

142 YEARBOOK OF PHYSICAL ANTHROPOLOGY [Vol. 38, 1995

TABLE 1 , Interspecific sample of cercopithecine (guenon) craniometrics

SpecieslSex Sample size Measurements (mm) Allenopithecus nigrooirdis F 7 Postorbital constriction A . nigrovirdis M 6 Foramen magnum length Cercopithecus aethiops F 9 Lambda to Opisthion C. aethiops M 10 Bizygomatic breadth Cercopithecus ascanius F 10 Interorbital breadth C. ascanius M 10 Orbit height Cercopithecus cephus F 9 Nasal aperture width C. cephus M 10 Palate breadth Cercopithecus diana F 5 Symphyseal height C. diana M 7 Ascending ramus height Cercopithecus l’hoesti F 8 Mandible length C. l’hoesti M 11 Bieuryal breadth Cercopithecus mitis F 10 Prosthion to Inion C. mitis M 11 Basion to Nasion Cercopithecus neglectus F 10 Staphylion to Basion C. neglectus M 15 Basion to Prosthion Cercopithecus nictitans F 5 C. nictitans M 9 Cercopithecus mona F 10 C. mona M 10 Cercopithecus hamlyni F 4 C. hamlyni M 2 Erythrocebus patas F 7 E. patas M 9 Miopithecus talapoin F 7 M . talapoin M 10

From Verheyen (1962).

TABLE 2. Intraspecific sample of male Native American anthropometrics

Group Cultural area N Measurements (mm) Inuit Arctic (A) 51 Standing height Carrier Subarctic (S) 29 Shoulder height Chilcotin Subarctic (S) 34 Finger height Tahltan Subarctic (S) 18 Finger reach Kwakiutl NW Coast (N) 56 Sitting height Makah NW Coast (N) 46 Shoulder width Stalo NW Coast (N) 31 Head length Tsimshian NW Coast (N) 97 Head width Bannock Great Basin (G) 26 Facial height Cayuse Great Basin (G) 15 Facial width Ute Great Basin (G) 115 Nose height Paiute Great Basin (G) 82 Nose breadth Coahuilla California (C) 46 Conca California (C) 34 Hoopa California (C) 44 San Luis California (C) 75 Lillooet Plateau (PI 83 Shushwap Plateau (P) 152 Thompson Plateau (PI 132 Wasco Plateau (P) 23

From the Franz Boas database at the University of Tennessee at Knoxville

two iso-vervets and among the male Inuit and two iso-Inuits are considered to be successful, appropriate methods for shape analysis. In other words, OTUs of the same shape or relative size should have no measurable dissimilarity following size-adjustment.

We note that this criterion departs from that advocated by some workers who believe that a size-adjustment is successful if and only if the adjusted variables are uncorrelated with the size variable (e.g., Atchley et al., 1976; Hartman, 1983; Reist, 1986; Albrecht et al., 1993, 1995). We will demonstrate that such “size-free” variables usually fail our test of shape similarity (also see Wood, 1978; Bookstein, 1989). We also see no biological rationale that validates the general use of the


Fig. 3. of Natural History (negative no. 1672; courtesy Department Library Services).

Photograph of male and female Inuits from the photographic collection of the American Museum

zero-correlation rule in morphometric studies (also see Sprent, 1972; Hills, 1978; Corruccini, 1987; 19951, but defer that discussion to later.

SIZE-RELATED TRENDS IN THE GUENON AND HUMAN SAMPLES

Size of each OTU in both samples is defined here as the geometric mean of all variables (16 measurements on guenon crania, 12 on the humans). The geometric mean is a member of the Mosimann family of size variables (Mosimann, 1970,


1975a, 197513, 1987; Mosimann and James, 1979; Mosimann and Malley, 1979) that also includes “sum size” (the sum of all variables), the “norm” (the square root of the sum of squared variables), “max X (the largest variable in a suite), and others. The geometric mean (GM) is computed as the nth root of the product of all n variables (also see Falsetti et al., 1993); using logged data, the log(GM) is simply the average of the logged variables. The GM is also the implicit size variable used both in the multivariate generalization of Huxley’s simple allometry equation (Jolicoeur, 1963; Mosimann and James, 1979; Jungers and German, 1981) and in Penrose’s shape distance if logged data are used (Mosimann and Malley, 1979).

Using the GM as the X (= size) variable, each Y variable in the sample was regressed on X, both in raw and logarithmic data spaces. Isometry in raw data space is indicated by the intercept value not significantly different from zero; for log-log regressions, isometry is indicated by an estimated slope value not significantly different from 1.0. Allometry is the best description of a linear relationship when the intercept is not zero (raw data) or when the slope is greater or less than 1.0 (logged data). The same results (isometric or allometric) obtained for both raw and logged versions of the test. In the interspecific case, 11 of the 16 guenon skull variables were found to be significantly allometric; the five isometric variables were bi-zygomatic breadth, nasal aperture width, palate breadth, ascending ramus height, and the distance between basion and nasion. Dramatic differences in shape among guenon crania can be appreciated qualitatively in Figure 2, which compares a male talapoin monkey, a male vervet, and a male patas monkey. The overall situation was very different in the human sample. Only two variables were best described as allometric, shoulder width and finger reach. Three variables (head length, head width, and nose breadth) were not significantly correlated with the size variable; the remaining seven variables can be characterized as isometric with respect to the GM. In other words, shape differences among the male humans tended to be few and largely uncorrelated with overall size. The fundamental differences between the two samples in their scaling patterns impacts strongly on the various methods of size-adjustment (see below).

SHAPE, RELATIVE SIZE, AND SIZE-ADJUSTMENT

Eleven size-adjustments were applied to all variables in both samples. Size was again taken as the geometric mean of all variables for each OTU, but the same general pattern of results will obtain for many other size variables. A brief description of the calculations follows (virtually all calculations can be made using a combination of SAS/STATS (SAS Institute, 1989) and NTSYS-pc (Rohlf, 1992; version 1.7 or higher):

1. RAW-RES refers to vertical residuals from the linear least squares regression in raw data space (Y = mX + b where X is the GM).

2. LOG-RES are residuals from the linear least squares log-log regression (logy = klogX + logm); i.e., departures from the logarithmic version of the simple allometry formula (Y = mXk). The residuals are kept in log values rather than converted back into linear scale (Hartman, 1983; Smith, 1984).

3. FULL-EQ variables are calculated as residuals in raw data space from the least squares “full” equation, an exponential model that permits a nonzero intercept (Y = mXk + b). The best solution to this nonlinear model appears to be indeterminate; many full equations produce approximately equal residual sums of squares (see Albrecht et al., 1993, for additional details of this model). We used the “best” of the “best lo” option in proc nonlin of the mainframe version of SATiSTATS (SAS Institutes, 1985) using a wide range of starting values for the various coefficients.

4. C-SCORES are a type of double-centered size-adjustment formalized recently by Howells (1989). They are calculated for each variable as (zscore - pensize), where variables are first adjusted to standard form or zscores by

Jungers et al.] SHAPE AND RELATNE SIZE 145

mean-centering and division by the standard deviation over all OTUs, fol- lowed by subtracting the average of an OTU’s zscores ( = pensize).

5 . DM-RAW are ratios formed by dividing each variable by the geometric mean (GM) of all variables for that OTU (i.e., Y/GM). DM refers to Darroch and Mosimann (1985). See Mosimann (1970, 1975a, 1975b, 1987) for additional details.

6. DM-LOG are simply the logged version of DM-RAW, also calculated as logy - logGM. The ratios are hopefully transformed to lognormal distributions by this procedure (Darroch and Mosimann, 19851, but the tests of shape similarity used here are nonparametric in any case. It was discovered by chance (Jungers and Cole, 1992) that identically adjusted variables are produced by an “isometric vector” version of the Burnaby method discussed below (see Rohlf and Bookstein, 1987).

7. K-LOG are size-adjusted variables in the Mosimann shape family developed by Kazmierczak (1985, 1986; Berge and Kazmierczak, 1986). The “analyse logarithmique” proceeds in logarithmic space by double mean-centering fol- lowed by addition of the logged grand mean (also see Chamberlain and Wood, 1987).

8. SIZEOUT refers to a procedure wherein the first principal component (PC1) of the logged variance-covariance matrix is discarded in the belief that this axis represents general “size” (Rohlf, 1967; Lemen, 1983; Reist, 1985; Hart- man, 1988). The scores on the remaining principal components are taken as size-adjusted data for further analysis. Components 2 through 5 are used here because the first five principal components account for over 98% (guenons) and 92% (humans) of the total variance, respectively.

9. BURNABY is a is a size-adjustment inspired by the “growth-invariant” dis- criminant functions of Burnaby (1966). Assuming again that the first principal component of the variance-covariance matrix is a size vector, “size” can be swept from the data matrix by projecting the logged data onto a plane orthogonal to that defined by PC1 (Rohlf and Bookstein, 1987). We keep the adjusted data on logarithmic form here, although they could be antilogged back into linear scale if desired.

These nine procedures generate size-adjusted data that are then used to calculate the average taxonomic (Euclidean) distances (ATDs) among all OTUs in each sample. Recall that our criterion of shape identity is a zero ATD between any pair of OTUs. We also use Penrose’s (1954) partition of the squared distances (ATDs) of two data vectors into a “size distance” and a residual “shape distance” (also see Corruccini, 1973; Sneath and Sokal, 1973: 170). This generates our last two size-adjusted distance matrices (Mosimann and Malley, 1979):

10. PEN-RAW is the shape distance matrix based on raw data. 11. PEN-LOG is the shape distance matrix based on log-transformed data. Mo-

simann and Malley (1979) demonstrate that this adjustment in logarithmic space is related closely to the Mosimann family of shape variables in that the log geometric mean is selected implicitly as the size variable.

For comparative purposes, average taxonomic distances were also computed for each sample from unadjusted data (RAW) and log-transformed raw data (LOG-RAW).

Each of the 11 “adjusted” ATD matrices for each sample is examined to evaluate if the iso-vervets and iso-Inuits have been correctly characterized as being identical in shape; i.e., to see if there are zero ATDs among the geometrically similar OTUs. The mean and standard deviation of the distances in the minimum spanning tree are provided as reference values for each type of distance matrix.

In order to compare the various methods, we also compute pair-wise matrix correlations among all 11 ATD adjusted matrices and the two unadjusted ATD matrices for each sample. The resulting 13-by-13 matrix of matrix correlations is


TABLE 3. Average taxonomic distances among Cercopithecus aethiops (male) and “iso-ueruets” using various methods of size adjustment

Distance matrix format Small iso-vervet Male vervet Large iso-vervet

Small is0 0 Male 0 Large is0 0 [ j r , sd of minimum spanning tree]’ RAW-RES FULL-EQ LOG-RES 0 0 n

* * *

2.86 0 5.70 2.84 0 L1.28, 0.481 DM-RAW 0 0 0 0 0 0 r0.04, 0.021 PEN-RAW 0 87.7 0 347 86.2 0 L5.97, 11.971 SIZEOUT 0 0.05 0

2.59 0 5.84 3.37 0 i1.31, 0.501 DM-LOG 0 0 0 0 0 0 r0.05, 0.02) PEN-LOG 0 0 0 0 0 0 10.002, 0.002l BURNABY 0 0.10 0

0.10 0 0.17 0.07 0 10.04, 0.021 K-LOG 0 0 0 0 0 0 r0.05, 0.021 C-SCORE 0 0.46 0 0.92 0.46 0 10.05, 0.021

0.08 0.03 0.17 0.07 0 10.02, 0.011 10.04, 0.021

‘The mean ( 9 ) and standard deviation (sd) of the distances in the minimum spanning tree are provided as reference values. For example, the distances for iso-vervets implied by LOG-RES and SIZEOUT appear to be absolutely small, but they are really relatively very large (i.e., two to four times the average distance).

then summarized in graphical form using the “unweighted pair-group method, arithmetic average” or UPGMA clustering algorithm (also see Sneath and Sokal, 1973; Lemen, 1983; Reist, 1985). Differences in results and possible inferences drawn from them about similarity are also highlighted by comparing selected distance matrices via OTU phenograms (also UPGMA) and their strict consensus clustering.

RESULTS OF ALTERNATIVE SIZE-ADJUSTMENTS Tests of “same shape”

Using the criterion developed above that organisms of identical shape should have no measureable distance between them after differences in scale are elimi- nated, Tables 3 and 4 summarize the observed differences among OTUs known to be geometrically similar, or isometric versions of each other.

Of the 11 size-adjustments applied to the guenon craniometric database, only four are successful by this criterion: DM-RAW, DM-LOG, K-LOG, and PEN-LOG. All pair-wise distances between the male vervet and the two iso- vervets are zero using these methods. Note that all four are based on adjusted data from the Mosimann family of shape variables, and by design and definition must fit our criterion of “same shape” (see below). All three forms of residuals failed the test based on this criterion, as did C-SCORES, PENRAW, SIZEOUT and BURNABY. In comparison to other non-zero distance matrices, the distances between iso-vervets using the LOG-RES, SIZEOUT and BURNABY adjustments appear to be relatively small, but this is because they are computed using logged data. In fact, the average of the minimum spanning tree values for the LOG-RES distance matrix is only 0.04, so values of 0.07 to 0.17 are really quite large. Sim- ilarly, the 0.08 distance between small and large iso-vervets using the SIZEOUT


TABLE 4 . Average taxonomic distances among Inuit males and “iso-Inuits” using various methods of size adjustment

Distance matrix format Small iso-Inuit Male Inuit Large iso-Inuit

Small is0 0 Male 0 Large is0 0 [ic, sd of minimum spanning tree] RAW-RES FULL-EQ LOG-RES 0 0 0

* * *

4.12 0 12.3 8.21 0 18.16, 3.011 DM-RAW 0 0 0 0 0 0 L0.03, 0.011 PEN-RAW 0 142 0 1560 568 0 t95,551 SIZEOUT 0 0.02 0 0.03 0.01 0 10.01. 0.0051

4.11 0 12.3 8.22 0 L8.15, 3.001 DM-LOG 0 0 0 0 0 0 L0.02, 0.011 PEN-LOG 0 0 0 0 0 0 t0.005, 0.0031 BURNABY 0 0.04 0 0.06 0.02 0 r0.02.0.0ii

0.01 0 0.02 0.01 0 L0.03, 0.011 K LOG 0 - 0 0 0 0 0 10.02, 0.011 C SCORE 0- 0.22 0 0.66 0.44 0 10.64, 0.191

2. sd as in Table 3

method is relatively very large, inasmuch as i t is the largest distance observed between any pair of OTUs in the sample.

A similar result obtains in the Native American sample. Of the 11 adjustments, again only the four Mosimann-like techniques correctly (and necessarily) disclose zero distances among the iso-Inuits and the male Inuit. The apparently small distances reported for LOG-RES, SIZEOUT, and BURNABY are again largely an artifact of the logarithmic space in which they were computed (e.g., the average value of the minimum spanning tree for the Native Americans based on LOG-RES is only 0.03).

Similarity among methods Comparisons of the 11 methods are presented as matrix correlations in Tables 5

and 6. Inspection of these correlation coefficients discloses the fact that several methods are remarkably similar in both the interspecific (guenon) and intraspecific (human) samples. For example, despite differences in variable standardization between DM-LOG (no standardization) and K-LOG (standardized by the grand mean), their respective distance matrices are perfectly correlated. Distances computed from residuals in raw data space are also very similar for both the linear (RAW-RES) and curvilinear (FULL-EQ) models in both data sets. The BURN- ABY and SIZEOUT strategies yield very similary results; if all of the principal component scores are used except those associated with the first axis (rather than only two to five used here) in SIZEOUT, the two methods converge upon the same values.

Recognition of patterns in the correlations between methods is greatly facili- tated by examining how they cluster (UPGMA of the 13-by-13 matrix of distance matrix correlations). Figure 4 is a phenogram of methods for the guenon data set. Two distinct clusters are apparent. The three residual based methods are very similar overall to the BURNABY, SIZEOUT, and C-SCORES techniques. The


TABLE 5. Matrix of distance matrix correlations between methods for size-adjustment in guenon craniometrics

RAW-RES FULL-EQ LOG-RES DM-RAW DM-LOG K-LOG PEN-RAW PEN-LOG C-SCORES BURNABY SIZEOUT RAW LOG-RAW

RAW- RES

1.00 0.99 0.76 0.36 0.37 0.37 0.42 0.35 0.88 0.77 0.76 0.46 0.39

FULL- EQ

1.00 0.74 0.37 0.38 0.38 0.45 0.37 0.87 0.74 0.74 0.27 0.35

LOG- RES

1.00 0.23 0.45 0.45 0.19 0.41 0.91 0.99 0.98 0.39 0.39

DM- DM- RAW LOG

1.00 0.90 1.00 0.90 1.00 0.60 0.51 0.90 0.97 0.41 0.54 0.20 0.42 0.19 0.41 0.51 0.59 0.58 0.62

K- LOG __

1.00 0.50 0.97 0.54 0.42 0.41 0.94 0.88 -

PEN- PEN- C- BURN- SIZE- LOG- RAW LOG SCORES ABY OUT RAW RAW

1.00 0.57 1.00 0.33 0.50 1.00 0.18 0.38 0.90 1.00 0.20 0.38 0.88 0.98 1.00 0.54 0.48 0.51 0.26 0.27 1.00 0.62 0.41 0.58 0.34 0.34 0.96 1.00

TABLE 6. Matrix of distance matrix correlations between methods for size-adjustment in human anthropometrics

RAW- FULL- LOG- DM- DM- K- PEN- PEN- C- BURN- SIZE- LOG- RES EQ RES RAW LOG LOG RAW LOG SCORES ABY OUT RAW RAW

RAW-RES 1.00 FULL-EQ 1.00 1.00 LOG-RES 0.51 0.51 1.00 DM-RAW 0.90 0.90 0.49 1.00 DM-LOG 0.48 0.48 0.97 0.54 1.00 K-LOG 0.48 0.48 0.97 0.54 1.00 1.00 PEN-RAW 0.50 0.50 0.31 0.41 0.30 0.30 1.00 PEN-LOG 0.42 0.42 0.96 0.46 0.97 0.97 0.30 1.00 C-SCORE 0.68 0.68 0.89 0.69 0.90 0.90 0.51 0.85 1.00 BURNABY 0.48 0.48 0.32 0.61 0.38 0.38 0.65 0.34 0.53 1.00 SIZEOUT 0.47 0.47 0.29 0.60 0.35 0.35 0.64 0.31 0.50 0.99 1.00 RAW 0.40 0.26 0.44 0.96 0.27 0.33 0.27 0.27 0.40 0.68 0.67 1.00 LOG-RAW 0.43 0.83 0.81 0.56 0.85 0.50 0.85 0.85 0.43 0.70 0.68 0.59 1.00

other cluster includes a tight grouping of DM-LOG, K-LOG, DM-RAW jointed more distantly by PEN-RAW, RAW and LOG-RAW. With respect to the human data set (Figure 5), the methods tend to cluster more on the basis of raw versus logarithmic data models. RAW-RES, FULL-EQ and DM-RAW are quite similar. A second cluster consists of LOG-RES, DM-LOG, K-LOG, and PEN-LOG; C-SCORES and LOG-RAW link with this log-based grouping. BURNABY and SIZEOUT are linked weakly to PEN-RAW and RAW, and this last cluster is quite dissimilar overall to the other nine analytical stategies. The basis of these differences between the guenon and human data sets will be considered shortly, but it has a great deal to do with the prevalence of allometry in one set (guenonsf and little size-correlated shape differences in the other (human).

Inferences about shape similarity among OTUs in each sample also vary as a function of the method of size-adjustment. For example, Figure 6a-c compares the UPGMA phenograms of the FULL-EQ and DM-RAW distance matrices and their strict consensus (or lack thereof?) phenogram for guenons. Note again that the FULL-EQ method (Fig. 6a) fails to identify the identically shaped vervet male and iso-vervets. The small iso-vervet is depicted as most similar to male Allenopithecus whereas the large iso-vervet is grouped with nothing. Using DM-RAW the vervet male and iso-vervets are correctly linked with zero distances among them (Fig. 6b); skull shape of this group is shown to be most similar overall to male C. rnitis and male C . nictitans. The male and female talapoin monkeys link together but are depicted as quite dissimilar from all other guenons. As a general rule, males of different species tend to group together as do females on the basis of skull shape.


I I

L UPGMA of A 1 1 Met hods: Guenons

0.2 0.4 0.6 0.8 1.0 RAU-RES FULL-EQ LOG-RfS BURNABY SIZEOUT C_ SCORf S DM-LOG

FEN-LOG DM-RAW

Fig. 4. UPGMA phenogram that summarizes the correlations among the 11 size-adjustment techniques and raw data applied to guenon craniometrics (Table 5). See text for the abbreviations and discussion of each method.

I

I UPGMA of A 1 1 Met hods: Humans 0.2 0.4 0.6 0.8 1.0

ES

I r D T I \ I D

Fig. 5. and raw data applied to Native American anthropometrics. (Table 6).

UPGMA phenogram that summarizes the correlations among the I1 methods of size-adjustment

The only point of consensus between these two methods of size-adjustment is in their linkages of male and female talapoins and male and female C . ascanius.

The situation is similar in some ways for the human sample if the UPGMA phenogram from FULL-EQ is compared to that for DM-LOG. The small iso-Inuit (a 98% scalar version of the male Inuit) is shown as similar but not identical to the male Inuit (Fig. 7a), whereas the large (104%) iso-Inuit is linked to the Lillooet and Kwakiutl groups in another adjacent cluster. The Hoopas show little afinity to any of the other groups. In contrast, the DM-LOG solution places the Tahltans as the outgroup to all other OTUs (Fig. 7b); the Inuit male and its isometric versions are correctly clustered with zero distances among them. The only element of consensus (Fig. 7c) between the two methods is the clustering of the male Inuit with the slightly smaller iso-Inuit. Clearly, one's inferences about phenetic resemblance and its potential biological significance would be impacted greatly by the choice of size-adjustment.

UPGMA of Residuals fr-om Full Equation: Guenons 1 4 3 2 1 -0 i

LouUer v A.niqr o v i r d15-m C.neqlect us-m C. pat as-m

F_

r

HI qhver v

C.mi t 1s-m C.aethiops-m

A . n i g r o v i r d i s _ f C. aet h i ops-f C.mona-f C.mona_m C.ascanius-f C.ascanius_m

I M.t alapoin-f M.t alapoin-m C. aet hiops-m C.di aria-f C.neglectus_f C. cephus- f C. m i t I 5- f C .n i c t i f ans-f - C.cephus-m C.nict i tans-m C. d i aria-m C.I’hoesr i - f C.l’hoest i -m C.ham1yni-f C.hamlyni_m

~ 4

i

- C.l’hoest i _ m C.hamlyni-m A.n igrov i rd is_m

Hiqhver v

UPGMA of D&M Shape Uauiables: Guenons 0.16 0.12 0.08 0.04 -0.00

I LouUer v

- -

b C.ascanius_f C.ascanius-m C.l’hoest i _ f

Jungers et al.] SHAPE AND R E L A T N E SIZE 151

C--c

C o n s e n s u s bet ween R e s i d u a l s and Rat 10s: Guenons A 0.7 0.6 0.5 0.4 0.3

C.diana_f C.diana_m C. l ’hoes i i - f C.l ’hoesf i - m C.mitis_f C.mit is_m C.neglect us-i C.neglectus-m C.nict it am-f C.nict i tans-m [.paras-f [.pat a5-m

I LouUer v I I L L J I I V C I Y

A.nigr o v i r d i s - f A. n i qr ovi r di 5-m C. aer h i op5-f C.aerhiops-m C. ascani us-f C. ascani us-m C.ceohus f C.cephus:m

n. i a lapo in - f k r a l a o o i n m c.mond_f C.mona_m C. haml y n i _ f C. haml qn i _ m

Fig. 6. a: UPGMA phenogram that summarizes the average taxonomic distances among guenon taxa (and sexes) using residuals from the “full equation,” Y = mXk + b. b: UPGMA phenogram based on Mosimann (D&M) shape variables. c: Strict consensus phenogram of a and b. The isometric version of the male vervet are denoted by LOVERV (the 67% scalar version) and HIVERV (the 133% version).

DISCUSSION AND SOME RECOMMENDATIONS

As Preuschoft (1989: 89) noted in his review of quantitative approaches to primate morphology: “Most methods of size-correction fall into one of two categories: (1) ratios between variables, or (2) deviations from allometric regression lines, based on some variable selected to represent ‘size.”’

Residuals Residuals have a variety of important statistical functions in regression analysis

(Mosteller and Tukey, 1977; Draper and Smith, 1981; Sokal and Rohlf, 1981). They are frequently examined to see if the assumptions of a linear model have been met (e.g., randomly scattered above and below the line as opposed to patterned behavior or residual autocorrelations), and they help to identify outliers or especially influential observations. Residuals are also widely employed as size-adjusted or “corrected” data in many contexts, including their application to the study of life history variables (Pagel and Harvey, 1988; Harvey and Pagel, 1991; Steams, 1992; Allman et al., 1993). Allometric relationships are assumed to be meaningful “criteria of subtraction” (Pilbeam and Gould, 1974; Gould, 1975; Sweet, 1980; Smith, 1984), and residuals “are of interest because they represent variation in a charac- ter away from an expected or detected trend” (Pagel and Harvey, 1988). These critical assumptions are examined in greater detail below. Many morphometric studies also use residuals as adjusted, “corrected’ or “size-free’’ data (e.g., Kay,


fl a--

Fig. 7. a: UPGMA phenogram that summarizes the average taxonomic distances among Native Amer- ican groups using residuals from the full equation. b UPGMA phenogram based on logged Darroch and Mosimann (1985) shape variables. c: Strict consensus phenogram of a and b. Group and cultural area abbreviations follow Table 2 very closely. LOINUIT and HIINUIT refer to the isometric versions of the male Inuit (98% and 1049'0, respectively).


1975, 1978; Atchley, 1978; Manaster, 1979; Susman and Creel, 1979; Healey and Tanner, 1981; Feldesman, 1982; Hartman, 1983; Creel, 1986; Godfrey, 1988; Jungers, 1988; Martin and MacLarnon, 1988; Wilson and Loesch, 1989; Losos, 1990; Ravosa, 1991a, 1991b; Takahashi, 1990; Ruliang et al., 1991; Moore and Cheverud, 1992; Albrecht et al., 1993; Anthony and Kay, 1993). Note that partial correlations applied to morphometric data are “residuals in disguise” inasmuch as a partial correlation between two variables with size held constant or removed (e.g., Harvey and Zammuto, 1985; Losos, 1990; Steudel, 1994) is equivalent to computing a correlation between the residuals of each variable regressed on the size variable (Kerlinger and Pedhazur, 1973237). Residuals are usually defended on the basis of their zero-correlation with the X (size) variable; size is said to be removed or controlled for by this analytical strategy.

The results of this study are consistent with earlier critiques of residuals. Re- siduals are by definition size-free, but they are also frequently shape-free, especially when allometry is present in the data set. Residuals usually do not recognize OTUs of the same shape or relative size (Lemen, 1983; Corruccini, 1987,1995); i.e., OTUs of identical shape can and usually do have different residual values. This undesirable error is evident in Albrecht et al.’s Table 1 (1995: 196); by a residual definition, two squares are characterized as different! A simple “thought experi- ment” makes this point in a slightly different way: two points of an allometric vector differ in their respective ratios of YIX (after all, that is why the relationship is described as allometric), yet because they both fall on the line they have zero residuals and are “equal” in shape or relative size by the residual definition (also see Figure 1 in Corruccini, 1987). The critical and perhaps unconscious assumption being made by some advocates of residuals is that virtually any observed relationship is not merely a description of how things are; they necessarily assume that the relationship is how things must be. Contrary to this Panglossian assumption (Gould and Lewontin, 19791, the many lines we draw through data, whether linear or curvilinear, in raw or logarithmic data space, are merely empirical, quantitative descriptions that require explanation themselves (Giles, 1956; Jungers, 1984); to automatically reify them all into valid “criteria of subtraction” is overly optimistic at best (e.g., Pilbeam and Gould, 1974; Gould, 1975; Page1 and Harvey, 1988; Steams, 1992) and potentially very misleading. The problem is even more vexing in the analysis of life history variables, for which there are usually no explicit similarity criteria.

The statistical requirement of independence between size and shape (i.e., the zero-correlation criterion of residuals) has no biological justification (Sprent, 1972; Hills, 1978; Corruccini, 1987, 1995). Within this size-free framework, shape information is discarded simply because it is correlated with size. The use of such regression residuals therefore “nearly destroys the possibility of size-shape analysis at all” (Bookstein, 1989: 178). As Oxnard (1978: 233) astutely observed some time ago, the “fact that some aspects of shape are correlated with size does not mean they are size.” Without explicit similarity criteria derived from first princi- ples, it is not obvious how shape can be meaningfully partitioned into “allometric” and “nonallometric” components. Residual definitions of shape or relative size also necessarily change as a function of the comparative set or sample used to construct the criterion of subtraction (Corruccini, 1987, 1995); i.e., shape becomes sample- specific rather than an intrinsic property of the organism.

Regression residuals are also strongly influenced by the statistical leverages associated with the smallest and largest individuals in a given sample (Susman and Creel, 1979; Sokal and Rohlf, 1981; Corruccini, 1987, 1995; Hartman, 1988). This also tends to make the smallest and largest individuals unrealistically similar in residual terms, especially when allometry prevails; e.g., gibbons and goril- las become “inappropriately similar” (Hartman, 1988: 13). This problem is demonstrated dramatically in the study by Corruccini et al. (1983, wherein residual- based cranial shape in squirrel monkeys is more similar to that of goats and cows than to chimpanzees! Contrary to the conclusions reached by Reist (1985), Albrecht


et al. (1993) and others, we agree with Wood (1978), Lemen (1983), Corruccini (1987, 1995) and Hartman (1988) that residuals have no necesary or very useful connection to shape analysis, especially when it can be demonstrated that significant shape changes accompany differences in size.

For many of the same reasons, we question the utility of “encephalization quo- tients” in broad-scale comparisons of relative brain size because they are merely residuals expressed in percentage form (Smith, 1984; also see Cartmill, 1990), and will change as a function of the comparative set (Holloway and Post, 1982). Again, without explicit similarity criteria for “equivalence’’ in brain size-body size comparisons, the various competing regressions are merely sample-specific, empirical descriptions of strongly allometric trends. Very small and very large animals will exert the most statistical leverage on each regression, and will appear to be unduly “similar” in degree of encephalization despite enormous differences in their ratios of brain mass to body mass; the primary allometric signal is discarded because it is assumed that the way the brain size scales is the way it must scale. We acknowledge the frequent need to compare relative brain size, and submit that “narrow allometric” (sensu Smith, 1980) contrasts of brain mass-body mass ratios in size- matched taxa can be useful in this regard.

Multivariate residuals Discarding the first principal component (SIZEOUT) or using a Burnaby (1966)

projection onto a plane orthogonal to the first axis (BURNABY) is almost equivalent to using residuals in a multivariate fashion if the first axis explains the overwhelming majority of the variance (Bookstein, 1989). As Jolicoeur (1963) recognized in his multivariate generalization of the “allometry equation,’’ the first principal component of the logged variance-covariance matrix describes “relative growth” or size-related shape changes (also see Shea, 1985; James and McCulloch, 1990). It is not a simple size vector or latent measure of size. Discarding it (e.g., Flessa and Bray, 1977; Froehlich et al., 1991) or using it as “standard measure” of size (e.g., Creighton and Strauss, 1986) appears to be a bad idea and subject to the same caveats discussed above for residuals (Healey and Tanner, 1981; Lemen, 1983; Hartman, 1988). In those data sets where the first component does not explain most of the variance (e.g., Cheverud, 1982; more generally, in static adult, intraspecific data sets such as the Native American sample used here where PC1 accounts for only 36% of the total variance), it is inappropriate to characterize i t as either an allometric or size vector. This difference helps to explain why SIZEOUT and BURNABY techniques cluster with the residual-based methods of adjustment in the guenon sample but not in the intraspecific human sample.

Cscores Although they have been used recently as size-adjusted data (e.g., Howells, 1989;

Brace and Hunt, 1990; Brace and Tracer, 19921, cscores cannot be recommended for some of the same reasons that lead us to discourage the use of residuals. OTUs of the same shape are not recognized as such by the Howells’ (1989) simple procedure. Because the data set is first mean-centered and standardized by variable over all OTUs, zscores, pensize and the resulting cscores will also be sample-specific and change as a function of their comparative set (also see Corruccini, 1973).

Penrose size and shape Our results also support Mosimann and Malley’s (1979) conclusion that Pen-

rose’s (1954) partition of squared Euclidean distance in raw space into size-distance and shape-distance is ill-advised. Using this method, nonzero shape distances among geometrically similar OTUs were computed for both data sets examined here. Mosimann and Malley also noted, however, that when logged data are used, Penrose’s partition of distance falls into the Mosimann family of size and shape variables (e.g., size is taken as the geometric mean). Our results verify this obser- vation; PEN-LOW clustered with DM-LOG and K-LOG, and ordinations of taxa


using the Penrose logged shape distance are quite similar to those based on distance matrices constructed directly from logged Mosimann shape variables when the geometric mean is selected as size.

Mosimann shape ratios By design and simple algebra, the ratio approaches to size-adjustment and shape

analysis (DM-RAW, DM-LOG, and K-LOG) necessarily satisfy our criterion of faithfully recognizing isometric versions of OTUs. Shape variables in the Mosi- mann framework have been applied productively to a variety of problems (e.g., Darroch and Mosimann, 1985; Berge and Kazmierczak, 1986; Chamberlain and Wood, 1987; Jungers et al., 1988, 1991; Berge, 1991; Jungers and Cole, 1992; Kidder et al., 1992; Simmons et al., 1991; Falsetti et al., 1993; Grine et al., 1993; Richmond and Jungers, 1995). Shape or relative size remains a function of the individual OTU rather than an OTUs comparative set.

Albrecht et al. (1993) argue that ratio adjustments are to be recommended only when their correlations with size are zero (i.e., when they behave like residuals). They also believe that the goal of “most workers” who use ratios do so to “control” or “eliminate” all size effects in data sets comprising OTUs of different sizes. We reject the first stipulation as a statistical straw man (see above), and question the accuracy and generality of the second claim (also see Corruccini, 1995:189). Shape ratios are scale-free in the sense that they are dimensionless variables. Such ratios may or may not be correlated with size, but that is ultimately an empirical determination (Mosimann and James, 1979; Corruccini, 1995). For example, most of the shape variables (whether DM-RAW, DM-LOG, or K-LOG) in the guenon data set are correlated with the geometric mean; significant shape differences are obvious between small (usually female) and large (usually male) crania (Fig. 2). It makes no biological or logical sense to discard this important information about size- and sex-related differences simply because there is a non-zero correlation between craniometric size and shape. Shape ratios created the same way in the human anthropometric data set tend not to be correlated with size (geometric mean), but this result is not very surprising in view of the lack of allometry in the data set. As Mosimann (1987) reminds us, “there is certainly no guarantee that division by a size variable produces shape measurements uncorrelated with size, because, in fact, shape and size often are correlated.”

When the total variances associated with raw versus shape data are compared, the reduction associated with shape variables can range from modest (Darroch and Mosimann, 1985) to dramatic (e.g., Jungers et al., 1988). The reduction in variance is in some ways a measure of how much “size” has been removed from the data set (Darroch and Mosimann, 1985). Size extracted by Mosimann’s ratio methods can be called “isometric size” (Oxnard, 1978) or “geometric size” (Bookstein, 1989). Isometric size can be swept from the data set by creating shape variables of the form Y/GM or by projecting the data onto a plane orthogonal to an isometric vector (Rohlf and Bookstein, 1987; Jungers and Cole, 1992); these are equivalent opera- tions when logged data are used. Obviously, one needs to exercise judgment and care in selecting a relevant size variable for ratio adjustments because definitions of allometry and geometric similarity will be specific to this choice (Mosimann, 1970). This recommendation extends to all studies of size and shape, however, including the choice of size in geometric morphometrics (e.g., centroid size, area, median chord length, etc.). Depending on the goals of the study, one might wish to use the cube root of mass as the size variable (e.g., Sneath and Sokal, 1973; Jungers, 1985b; Vogel, 1988). It is also possible to create a Mosimann-like measure of size that is independent of shape for multivariate lognormal samples (Sampson and Siegel, 1985).

Alternative expressions of Mosimann shape Variables created as residuals to isometric base lines are also part of the Mosi-

mann-family of shape variables (Lemen, 1983; Chamberlain, 1987; Jungers, 1988).


5

4

2

1

(where 1 is a constant) b

1 I I I 1 2 3 4 5

LOG X

Fig. 8. Residuals from an isometric line in logarithmic space ke . , slope = l . O ) , whether forced or empirically determined, are equivalent to the ratios of YIX standarized by a constant equal to the intercept of the line.

In forced isometric (log-log) plots with slopes of 1.0, it is apparent that residuals are equivalent to ratios of YIX standardized by a constant equivalent to the intercept (Fig. 8). K-LOG variables are similar in some ways because they also differ from DM-LOG ratios by scalar constants. Zscore standardization of DM-LOG variables, K-LOG variables, log residuals from bivariate isometry, and logged variables created by the isometric version of BURNABY will result in identical shape data matrices. When empirical relationships are essentially isometric (e.g., Kay, 1975, 1978; Covert, 19861, residuals are again conveying information about relative size that is more or less equivalent to Mosimann measures of shape.

Putative problems with ratios Ratios are not infrequently criticized for their supposedly inferior distributional

properties (Atchley et al., 1976; Albrecht, 1978; Sokal and Rohlf, 1981; Atchley, 1983; Hartman, 1983; Deacon, 1990; Albrecht et al., 1993). Suitable transformations (e.g., to lognormality) can often overcome these problems (Mosimann, 1970; 1975a, 1975b, 1987; Dodson, 1978; Hills, 1978; Oxnard, 1978; Mosimann and Mal- ley, 1979; Darroch and Mosimann, 1985; Frampton and Ward, 19901, and standard parametric statistics can then be used for further analysis. Our experience sug- gests that ratio data are frequently no worse than raw data in this regard; all data need to be examined to see if they meet the assumptions of the statistics chosen to characterize them. Distribution-free, nonparametric alternatives are also avail-


able (Singer, 19791, including the use of multivariate distance statistics (e.g., Grine et al., 1993; Richmond and Jungers, 1995). Rank transformations in shape ratios can also serve as a “bridge” between parametric and nonparametric statistics (Conover and Iman, 1981).

It has also been claimed that Mosimann-like shape variables lead to spurious correlation between the shape and size variables because the size variable is in the denominator of the ratio (Atchley et al., 1976; Prothero, 1986; Page1 and Harvey, 1988). This criticism is valid only when the Y and X (size) variables are independent (Pearson, 18971, as in randomly generated data sets (Atchley et al., 1976; Prothero, 1986); in such cases, the ratio of Y/X should be negatively correlated with X. As Mosimann (1975a), Dodson (19781, and Falsetti et al. (1993) note, however, in many comparative biological situations Y increases as does X, and X and Y are therefore positively correlated. In these circumstances, observed negative correlations between a Mosimann shape variable and size are not spurious at all, but instead reflect a iiegatively allometric relationship (Mosimann and James, 1979). Most importantly in the context of shape analysis, “ratios may be the only meaningful way to interpret and understand certain types of biological problems” (Sokal and Rohlf, 1981: 17).

To log or not to log Should shape variables be expressed in logarithmic or antilogged form (Hart-

man, 1983; Smith, 1984)? Logging data is often done to achieve or approximate a lognormal distribution (see above), but such a result cannot be guaranteed. Dis- tance matrices computed in this study differed slightly (guenons) to considerably (humans), depending in part on whether the adjusted variables were logged or not, Not surprisingly, logged distances were absolutely lower than raw distances in both samples (see above). Reyment (1971) pointed out that log transformations remove the investigator one more step from the biological relationships in the data; in fact, such transformations can even “obscure the relationships the investigator wishes to probe” (Reyment, 1971: 367). Corruccini (1987: 301) also noted that log transformations do not necessarily “improve multivariate distributions, results, or interpretability.” Given the viability of alternative, nonparametric statistics in the face of less than optimal distributions, there is probably no compelling need to work exclusively with logged shape variables. At the very least, it seems prudent to see if and how one’s results and inferences about shape are impacted by log transformations.

CODA

We concur with the opinion of James and McCulloch (1990: 158) that “authors who have objected to the direct use of ratios in morphometric studies . . . have been overlooking some powerful techniques for the direct study of shape and its cova- riation with size.” Sprent (1972: 35) anticipated this conclusion and correctly recognized the great promise of this approach over 20 years ago when he concluded that “this [Mosimann] line of research may prove more fruitful than concentrating on conditions for independence between size and shape.” Finally, Corruccini (1987: 301) suggested that “bio-anthropologists may need to give closer attention to the methods and rationale exemplified by J. Mosimann.” We could not agree more and hope our findings encourage others to question the assumptions and utility of residuals as size-adjusted data, and to explore shape and relative size within Mo- simann’s explicitly geometric framework.

ACKNOWLEDGMENTS

We gratefully acknowledge fruitful conversations and correspondence with many of our colleagues about the issues discussed in this review; special thanks go to Richard Smith, Tim Cole, Laurie Godfrey, Jim Cheverud, Steve Leigh, Steve Hartman, Maria Cole, Brian Shea, Susan Ford and Bob Corruccini. We thank Ted Steegmann for inviting us to present our findings here in the Yearbook of Physical


Anthropology, and for his patience with deadlines! We also thank Luci Betti-Nash for her assistance in preparing the artwork.

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