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Geometric morphometric analysis of mandibular ramus flexure.

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Geometric Morphometric Analysis of Mandibular
Ramus Flexure
A.C. Oettlé, E. Pretorius,* and M. Steyn
Department of Anatomy, University of Pretoria, Pretoria 0001, Republic of South Africa
sexual dimorphism; sex assessment; skeletal biology
Many characteristics of the human skeleton can only be assessed morphologically, which may be
problematic due to factors such as interobserver error and
difficulties with standardization. Flexure of the mandibular ramus is one of these traits, and various researchers
found widely differing results using this morphological
feature. The aim of this study was to determine whether
differences between male and female mandibular rami
could be observed using the computerized method of geometric morphometrics, a valuable tool that helps quantify
shape differences. Twenty-eight mandibular rami of black
females and 43 of black males were photographed in a
standard plane and assessed. It was found that the
females were more scattered on the graph (more variable
in shape), while the males clustered more around the center point where the two axes met (shape more constant).
There was, however, considerable overlap between the
sexes. Although different tendencies exist between the
rami of males (being more flexed) and females (tending to
be straight), the extent of these differences is not adequate
to predict the sex of a single individual. Am J Phys
Anthropol 128:623–629, 2005. ' 2005 Wiley-Liss, Inc.
Determination of sex from skeletal remains can be
approached metrically or morphologically. The advantages of using measurements for this purpose are,
among others, that the results are easy to assess and
interpret. However, many characteristics in the human
skeleton are not measurable through conventional methods, and are thus usually assessed morphologically. Unfortunately these types of analyses may sometimes be
dependent on the experience of the observer, and may be
difficult to standardize. It may also be difficult to interpret the results.
The relatively new method of geometric morphometrics provides a mechanism to quantify morphological
characteristics. Moreover, it also allows for an assessment of exactly where or in what aspect the morphology
between various skeletons differs. Although it has been
used to quantify morphology since the late 1980s, it is a
method that just recently started to become popular in
physical anthropology (e.g., Lynch et al., 1996; Wood and
Lynch, 1996; Hennessy and Stringer, 2002; Rosas and
Bastir, 2002), but its potential is enormous.
Geometric morphometrics is a method dealing with
the study of shape using either homologous landmarks
(Rohlf and Marcus, 1993) or outlines of a morphological
structure. Landmarks are the points at which biological
structures are sampled, and these points capture shape
changes between the same morphological structures in
different species. The definition of shape space by Kendall (1981, 1984) forms the basis of geometric morphometrics, but this space is nonlinear and non-Euclidean,
and therefore conventional linear multivariate statistical
methods cannot be applied to it (Slice, 2001). Furthermore, only via Procrustes superimposition does one enter
into the shape space of Kendall (1981, 1984). Shape
space can, however, be approximated by a linear space
tangent to it. This approximated tangent space allows
for the utilization of standard multivariate statistics on
a data set of homologous landmarks (or x, y coordinates)
of n number of specimens (Slice, 2001).
The main purpose of the method is to permit analysis
of the variability of a morphological structure using a
powerful, comprehensive statistical analysis, and the use
of thin-plate splines to describe the results in terms of
deformations. This produces an exact geometric description of the shape differences between the same structure
in different specimens. Authors like Bookstein (1989,
1991), Rohlf and Slice (1990), Slice (1993), and Rohlf
(1995) developed this method. F. James Rohlf also developed the tps series of programs, which performs the statistics and visualizations of geometric morphometrics.
Other researchers like David H. Sheets further developed the statistics by adding the Integrated Morphometric Package (IMP) (Sheets, 2001). This package,
among other things, allows one to calculate P-values,
using the TwoGroup program, as well as do a discriminant function analysis (or canonical variates analysis;
CVA) using the CVAGen6 program. Pretorius and Clarke
(2000, 2001), Pretorius and Scholtz (2001), and Pretorius
et al. (2001) used the tps programs to compare morpholo-
Susan R. Loth passed away on 23 September 2002, and we are
completing this paper in her memory.
Grant sponsor: National Research Foundation; Grant number:
*Correspondence to: Prof. E. Pretorius, Department of Anatomy,
University of Pretoria, PO Box 2034, Pretoria 0001, Republic of
South Africa. E-mail:
Received 6 March 2003; accepted 4 May 2004.
DOI 10.1002/ajpa.20207
Published online 28 April 2005 in Wiley InterScience
gical structures and indicate relationships between taxa.
The dynamic tps series of programs is continuously
upgraded and improved by F. James Rohlf, and was used
as a tool in the analysis of mandibular ramus flexure in
males and females in this study.
The use of mandibular ramus flexure as a method to
distinguish between the sexes was introduced in Loth
and Henneberg (1996), and was tested by others (e.g.,
Koski, 1996; Indryana et al., 1998; Donnely et al., 1998;
Hill, 2000). Loth and Henneberg (1996) found a distinct
angulation to be present at the posterior border of the
mandibular ramus at the level of the occlussal plane in
males. In most females, the ramus retained its straight
juvenile shape. Initial results on South African skeletal
remains yielded results of about 99% correct separation,
which was slightly lower when other skeletal collections
were added (Loth and Henneberg, 1996).
Most other researchers, however, found significantly
lower accuracies, ranging from as low as 63% (Donelly
et al., 1998) to 91.5% (Indryana et al., 1998). Hill (2000)
and Kemkes-Grottenthaler et al. (2002) also claimed that
the method was too inaccurate to be usable, and said it
was unreliable as it was found to be much more accurate
in males, and has a high inter- and intraobserver error.
The aim of this study was to assess ramus flexure by
means of geometric morphometrics, in order to ascertain
whether the said differences exist and are observable
using a different method of assessment. Geometric morphometrics will also elucidate the exact nature and position of the morphological shape differences in the sexes,
if present.
Seventy-one mandibles (43 males and 28 females) were
used in this study. These mandibles all belonged to
known South African black individuals, and were randomly selected from the Pretoria Skeletal Collection at
the Department of Anatomy, University of Pretoria.
Their ages ranged between 20–87 years, and individuals
with obvious bony pathologies and/or excessive tooth loss
were excluded.
The following landmarks were applied blindly (without
knowing the sex of the specimens) on mandibular rami
before they were photographed. Three homologous landmarks on each left ramus representing its shape were
chosen and applied with pencil on each mandible. The
lowest landmark, landmark 1, was positioned just above
the gonion, i.e., where the ramus started to straighten
from the gonial eversion. The upper landmark (landmark 2) was at the level of the mandibular notch, as
measured by a spirit-leveling instrument, with the mandible on a level surface. A third landmark was chosen
where the observed concavity (flexure) was at its maximum. This concavity was at its maximum where the distance between the posterior border of the ramus of the
mandible and a line connecting the two most posterior
points (roughly corresponding to landmarks 1 and 2) of
the mandibular ramus was at its maximum. Even
though the concavity of some mandibles was less obvious, no ramus was completely straight. The position of
each ramus, relative to the camera, was then standardized by rotating it through 458 on the horizontal plane,
away from the vertical plane to which the digital camera
was kept in a fixed position. This provided an oblique
view from the antero-lateral side that best elucidated
the concavity. The focus point was oriented on the center
Fig. 1. Lateral view of female mandibular ramus, indicating
positions of 11 landmarks.
of the mandibular ramus, as measured and calculated
from its four borders. The captured electronic images (in
jpg format) were then entered into the computer. At the
time that they were analyzed digitally, their sex was not
Four more landmarks were placed on the jpg image,
between landmarks 1–3, and also between 3–2, at equal
distances, using the computer program tpsdig (version
1.31, F. James Rohlf). These 11 landmarks represented
points on the mandibular ramus that elucidated the presence or absence of flexure in that area, as seen in
Figures 1 and 2. These landmarks were then further
analyzed by the tps series of programs. From this tps
series of programs, the following analyses were performed: a relative warp analyses (RWA) and thin-plate
spline analysis. Shape trends were further studied using
the TwoGroup program and CoordGen-program from the
Integrated Morphometric Package.
The RWA analysis was only performed in order to determine general trends in shape between the two sexes; in
other words, from this it can be seen whether a definite
separation between the male and female data set exists,
or whether there is no clear distinction between the shape
of the two sexes. In full shape space (where ¼ 0), RWA is
a principal components analysis (PCA) of the covariance
matrix of the partial warp scores, and is performed by the
program tpsRolw (Bookstein, 1991, 1993, 1996). An alternative is to perform a PCA of the aligned specimens; this
will result in an identical ordination. Relative warps are
computed, using tpsRelw (version 1.25, F. James Rohlf).
This program allows for the analysis to be performed in full
space, the uniform subspace, or the nonuniform subspace.
The program allows choosing a value for (bending
energy). Any value of other than zero converts the analysis to a relative principal component analysis of shape covariances with respect to bending energy instead of Procrustes distance (Bookstein, 1996). Furthermore, Bookstein
(1996) suggested that all such computations apply only to
the uniform subspace of shape space, because, according to
Fig. 2. Lateral view of male mandibular ramus, indicating
positions of 11 landmarks.
the author, the bending energy of every change of shape
within the uniform subspace is uniformly zero. According
to Bookstein (1996), for greater than zero, warps of larger
scale (excluding the uniform component) are assigned more
weight; with less than zero, they are more attenuated.
Where ¼ 1 the researcher can, for example, search for
growth gradients, while ¼ 1 could be used if the
researcher is interested in shape phenomena at a small
scale. Furthermore, when is greater than 0, geometrically
small-scale variation is given less weight than large-scale
variation; the result is that of reducing the weight given to
regions having more landmarks relative to the weight
given to regions having fewer and therefore more widely
spaced landmarks (Rohlf et al., 1996).
Each relative warp can be plotted as a deformation of
the space of the reference configuration of landmarks.
The first two relative warps are usually indicative of
most of the variation between members of the data set,
and therefore their scatterplot presents the most visual
information about variation between the sexes. Shape
changes implied by variation along the first two relative
warp axes can be shown as deformations using thinplate splines.
The question that now arises is of which -value is
most appropriate for the current analysis, and whether
different -values will indeed make a difference in the
RWA plot. To answer these questions, the analysis was
performed using ¼ 0, ¼ 1, and ¼ 1, where both
uniform and nonuniform components were included. Singular value decomposition was also determined for the
first time to relative warps.
Thin-plate splines of the variation between specimens
were also drawn, and shape variations were visualized.
TpsSplin (version 1.17, F. James Rohlf) was used in the
analyses of landmark positions of each specimen. This
program allows visual determination of landmarks
responsible for differences in shape. The computations
involve calculations of a reference, using the generalized
least-squares Procrustes superimposition method (also
known as the GLS method) for the data set (including
male and female specimens). The landmark distribution
of the reference is represented by a perpendicular Carte-
sian grid. The shape differences of each specimen are
represented by the deformation of the reference. Shape
differences can therefore be visualized by studying each
thin-plate spline, or the thin-plates splines of the means
of males and females.
In order to determine whether the shape distances
were statistically significant, a P-value was calculated
using Hotelling’s t-test function of the TwoGroup program, after converting the tps data set into IMP data
using the CoordGen program. A discriminant function
analysis or canonical variates analysis (CVA) was then
performed using the CVAGen6 program. A CVA assesses
the ability to assign specimens in a data set to groups
(e.g., male or female), rather than asking if the two
groups have a different shape.
To test for repeatability, all males in the male sample
were reassigned the chosen landmarks. The ‘‘new’’ male
landmark data were statistically compared to the ‘‘old’’
data set using Hotelling’s t-test function of the TwoGroup program. The ‘‘new’’ data set was also marked as
‘‘unknown’’ and compared to the full original data set
using CVA analysis. Results of the first analysis indicated a P-value of 1, while all ‘‘unknown’’ specimens
were correctly assigned as male in the CVA analysis,
indicating that the placement of landmarks was repeatable.
In Loth and Henneberg (1996, 1998), it was found that
only the distinct flexure at the level of the occlusal plane
of the molars is a consistent indicator of sex. We therefore repeated our analysis, excluding those individuals (5
males and 6 females) in whom the maximum curvature
was not exactly on the level of the occlusal plane.
The RWA analyses of the data set using ¼ 0, ¼
1, and ¼ 1 were carried out including both uniform
and nonuniform components. The six analyses all indicated the same broad result, i.e., that the males and
females are scattered in such a way that no definite distinction between the sexes can be made. Only the two
results where ¼ 0 were used will be discussed here.
The first two relative warps (using ¼ 0; excluding
the uniform component) of the data set accounted for
88.18% of the variation among the specimens (computed
by a singular-value decomposition of the weight matrix;
Rohlf, 1993), while when the uniform component was
included in the analysis, the first two relative warps
accounted for 95.14% of the variation. As an example of
the extent of overlap in the scatter, Figure 3 indicates a
two-dimensional ordination where ¼ 0, without the
uniform component function activated in the tps analysis. Considerable overlap is found between the male and
female distribution, as seen in Figure 3 (this was also
the case in all other plots).
Figure 4 represents the thin-plate splines of the males
(arrow points) and females (black squares) in vector
mode. Thin-plate spline analysis presents a visual aid to
pinpoint the differences in shape. By studying the thinplate splines, the similarity between the averages of the
two sexes is demonstrated. The posterior border of the
mandibular ramus below the level of the occlusal plane
of the molars in males (represented by the landmarks)
shows a slightly more concave angulation backwards, as
compared to the shape of the females. The mean shape
for the total sample, represented by a perpendicular Cartesian grid as well as the mean male and female shapes,
Fig. 3. Two-dimensional relative warp ordination of overall male and female diversity in terms of nonaffine components of shape
variation ( ¼ 0 and excluding uniform component).
was calculated. The distortion of the mean male and
female grids represents the amount of bending energy
required to bend male and female shape to fit mean
shape. Figure 5 represents the thin-plate spline grids of
the mean shape for the complete sample (Fig. 5a), males
(Fig. 5b), and females (Fig. 5c).
A statistically significant P-value of 0.014 for malefemale shape difference was obtained using Hotelling’s ttest. Table 1 indicates the CVA analysis results obtained
using the CVAGen6 program. Table 1 shows that only
about 68–70% of individuals were correctly assigned.
All analyses were repeated excluding those individuals
(5 males and 6 females) whose maximum curvature did
not correspond to the occlusal plane, and similar results
were found.
Geometric morphometrics has proved to be a very valuable tool in elucidating morphological variation in human
skeletal remains. It acts as a sort of independent observer,
with less bias and observer error, as would be the case if
the researcher scored the variations. According to Hennessy and Stringer (2002), geometric morphometrics is
very useful in enhancing and understanding the results of
more traditional morphometric studies. In this study, geometric morphometric analysis was used to assess the presence or absence of ramus flexure as an indicator of sex.
Few studies focusing on sexual dimorphism and using
geometric morphometrics are available in literature.
Rosas and Bastir (2002) concentrated on sexual dimorphism, but included the whole craniofacial complex, studied
in lateral views of crania. Their results clearly indicated
that differences between male and female mandibles
exist. However, they used very few landmarks on the
ramus: one landmark each at gonion, ramus, and condylion. They concentrated more on the basal border of the
mandible, but did find that in males, the gonion tends to
be located more anteriorly, as was the condylion, which
influenced the curvature of the posterior border of the
Various studies investigating the sexual dimorphism of
the mandibular ramus using direct visual assessment
were published, e.g., Loth and Henneberg (1996), Don-
Fig. 4. Vector analysis of average measurements. Females,
solid squares; males, arrow points.
nelly et al. (1998), Indrayana et al. (1998), Hill (2000), and
Kemkes-Grottenthaler et al. (2002). The results of these
studies were contradictory and did not always seem to be
repeatable. Even if assessments on the mandibular ramus
are done in a ‘‘blind’’ fashion, bias cannot be excluded, as
the general appearance of the mandible cannot be hidden
and can give some information regarding the gender of the
mandible. Kemkes-Grottenthaler et al. (2002) commented
on intra- as well as interobserver bias when using ramus
flexure as a means to determine sex. To study this para-
Fig. 5. Thin-plate spline grids of mean male and female
shape (A), mean male shape (B), and mean female shape (C).
meter in a more objective manner, geometric morphometrics were used in this study.
Using 11 landmarks, male-female differences relating
to the shape of the posterior border of the ramus could
not clearly be distinguished by studying the thin-plate
splines of each specimen in this study. The RWA analysis
using the first two relative warps also indicated an overlap in clustering of males and females (Fig. 3). It seems
as if male mandibular ramus morphology is less variable
than female morphology, as indicated in the wider distri-
bution of the female specimens on the relative warp analysis plot. This finding fits with the results of Hill (2000),
who found that males could be more accurately sexed.
Hill (2000) suggested that the pattern of flexion was
more consistent in males, while the female shape was
more variable.
In the vector analysis, comparing mean male and
female shape, the slightly more concave angulation backwards in males, as compared to the shape in females,
can be seen (Fig. 4); this is also seen in thin-plate spline
grids of the mean male and female shape (Fig. 5b,c).
TABLE 1. Percentage males and females correctly and wrongly assigned using canonical variates analysis1
CVA assignment based on shape data
Sex as assigned in
data set
Female (28)
Male (43)
Correctly assigned
Wrongly assigned
Percentage males and
females correctly
assigned by CVA
19 correctly assigned as female
30 correctly assigned as male
9 wrongly assigned as male
13 wrongly assigned as female
CVAGen6 program of IMP series of programs.
This could represent the more prominent ramus flexure
that was noted in males previously (Loth and Henneberg, 1996).
A statistical comparison of the male and female data
sets produced a P-value of 0.014, which indicates that
the two data sets are statistically significantly different.
However, the CVA analysis indicated that 9 of 28 females have a male shape, while 13 of 43 males have a
female shape. Therefore, only 67.8% of the females and
69.9% of the males are correctly assigned. We therefore
believe that, although the P-value suggests a significant
difference between the male and female data sets, this
difference is not of such magnitude and clarity that
ramus flexure can be used as a good indicator of sex.
Loth and Henneberg (1996, 1998) found the ramus flexure a good indicator of sex, if the ramus flexure was at the
level of the occlusion plane. In our study, even when we
excluded those individuals whose ramus flexure was not
at the level of the occlusal plane, we could not achieve
results which clearly discriminated between males and
females. Also, we are not convinced that it is a good idea to
exclude certain individuals on the ground of the position
of the ramus flexure, as this is variable and difficult to
assess. Conclusions made on a sample which has certain
exclusions will be more difficult to apply on an unknown
specimen with confidence.
Koski (1996) gave a short overview of the factors influencing the growth and thus shape of the mandible. This
author concluded that the whole soft-tissue environment
and the occluding dentition, together with the bony jaw,
form a functioning entity whose parts are capable of
influencing each other. The relationship between the
ramus and the condylar process appears to depend on
the functional environment of the lower jaw. The direction of the condyle is approximately perpendicular to the
lateral skull base. In varying functional instances, the
ramus must be the adapting part between the articulating condyle and the tooth-bearing body of the jaw. This
leads to flexures of different degrees, having different
relative positions between the lower ramus and the condylar process that are thus created. In the experience of
Koski (1996), the flexures occur in most individuals, be
they male or female, young or adult. Loth and Henneberg (1996), in their review of the biological reasons for
sexual diversity in mandibles, concluded that the mandible is shaped in response to hormonal influences and
both directly and indirectly by the muscle development
thus stimulated. The possible dimorphism seen may
arise in response to sex-specific hormones. Walker and
Kowalski (1972) also linked this sexual dimorphism to
continuing postpubertal mandibular bone growth in
males in response to hormonal influences. The results
from the current study are more in agreement with the
statements by Koski (1996), indicating that the observed
variations in mandibular ramus morphology have a biomechanical rather than hormonal origin.
Our final conclusion is that ramus flexure does not
provide a definite means of sex determination that can
be used as an isolated sex marker on unknown skeletal
remains. Although the ramus flexure seems to be greater
and more constant in males, the overlap between the
shapes of male and female rami is too great to make it
usable. These results should be followed up on a larger
data set.
We sincerely thank David Sheets for his valuable comments and continued help on the use of the IMP programs, and also F.J. Rohlf for his suggestions. The comments of the anonymous reviewers also helped improve
the paper. We also thank the Department of Anatomy,
University of Pretoria, for allowing us access to skeletons from the Pretoria Collection, and A. da Silva, who
helped with testing the repeatability of allocation of
landmarks. The research of M.S. is funded by National
Research Foundation grant 2054279.
Bookstein FL. 1989. Principal warps: thin-plate splines and the
decomposition of deformations. Trans Pattern Anal Mach
Intell 11:567–585.
Bookstein FL. 1991. Morphometric tools for landmark data.
New York: Cambridge University Press.
Bookstein FL. 1996. Combining the tools of geometric morphometrics. In: Marcus LF, editor. Advances in morphometrics.
New York: Plenum Press. p 131–151.
Donnelly S, Hens SM, Rogers NL, Schneider KL. 1998. A blind
test of mandibular ramus flexure as a morphological indicator
of sexual dimorphism in the human skeleton. Am J Phys
Anthropol 107:363–366.
Hennessy RJ, Stringer CB. 2002. Geometric morphometric
study of the regional variation of modern human craniofacial
form. Am J Phys Anthropol 117:37–48.
Hill CA. 2000. Technical note: evaluating mandibular ramus
flexure as a morphological indicator of sex. Am J Phys
Anthropol 111:572–577.
Indryana NS, Glinka J, Mieke S. 1998. Mandibular ramus flexure in an Indonesion population. Am J Phys Anthropol
Kemkes-Grottenthaler A, Lobig F, Stock F. 2002. Mandibular
ramus flexure and gonial eversion as morphological indicators
of sex. Homo 53:97–111.
Kendall DG. 1981. The statistics of shape. In: Barnett V, editor.
Interpreting multivariate data. New York: Wiley-Liss. p 75–80.
Kendall DG. 1984. Shape-manifolds, Procrustean metrics and
complex projective spaces. Bull Lond Math Soc 16:81–121.
Koski K. 1996. Mandibular ramus flexure: indicator of sexual
dimorphism? Am J Phys Anthropol 101:545–546.
Loth SR, Henneberg M. 1996. Mandibular ramus flexure: a new
morphologic indicator of sexual dimorphism in the human
skeleton. Am J Phys Anthropol 99:473–485.
Loth SR, Henneberg M. 1998. Mandibular ramus flexure is a
good indicator of sexual dimorphism. Am J Phys Anthropol
Lynch JM, Wood CG, Luboga S. 1996. Geometric morphometrics
in primatology: craniofacial variation in Homo sapiens and
Pan troglodytes. Folia Primatol (Basel) 67:15–39.
Pretorius E, Clarke FC. 2000. Geometric morphometric analysis
of the male and female body shape of Hyalomma truncatum
and H. marginatum rufipes (Acari: Ixodidae). Int J Acarol
Pretorius E, Clarke FC. 2001. Geometric morphometric analysis
of the male and female body shape of Amblyomma gemma,
A. variegatum and A. hebraeum. Int J Acarol 27:71–279.
Pretorius E, Scholtz CH. 2001. Geometric morphometrics and
the analysis of higher taxa: a case study based on the metendosternite of the Scarabaeoidea (Coleoptera). Biol J Linn Soc
Pretorius E, Philips K, Scholtz CH. 2001. Geometric
morphometrics, the metendosternite and its uses in phylogenetics of the Scarabaeinae (Coleoptera). Elytron 14:119–140.
Rohlf FJ. 1993. Relative-warp analysis and example of its
application to mosquito wings. In: Marcus LF, Bello E,
Garcı́a-Valdecasas A, editors.Contributions to morphometrics,
volume 8. Madrid: Museo Nacional de Ciencias Naturales.
p 131–159.
Rohlf FJ. 1995. Multivariate analysis of shape using partialwarp scores. In: Mardia KV, Gill CA, editors. Proceedings in
current issues in statistical shape analysis. Leeds, UK: Leeds
University Press. p 154–158.
Rohlf FJ. 2002. Geometric morphometrics and phylogeny. In:
Forey P, Macleod N, editors. Morphology, shape and phylogenetics. London: Francis & Taylor. p 175–193.
Rohlf FJ, Marcus LF. 1993. A revolution in morphometrics.
Trends Ecol Evol 8:129–132.
Rohlf FJ, Slice D. 1990. Extensions of the Procrustes method for
the optimal superimposition of landmarks. Syst Zool 39:40–59.
Rohlf FJ, Loy A, Corti M. 1996. Morphometric analysis of Old
World Talpidae (Mammalia, Insectovora) using partial-warp
scores. Syst Biol 45:344–362.
Rosas A, Bastir M. 2002. Thin-plate spline analysis of allometry
and sexual dimorphism in the human craniofacial complex.
Am J Phys Anthropol 117:236–245.
Sheets DH. 2001. IMP, integrated morphometric package. In:
Sheets DH, Slice DE. 1993. The fractal analysis of shape. In:
Marcus LF, Bello E, Garcı́a-Valdecasas A, editors. Contributions to morphometrics, volume 8. Madrid: Museo Nacional de
Ciencias Naturales. p 161–189.
Slice D. 2001. Landmark coordinates aligned by Procrustes analysis do not lie in Kendall’s shape space. Syst Biol 50:141–149.
Walker GF, Kowalski CJ. 1972. On the growth of the mandible.
Am J Phys Anthropol 36:111–118.
Wood CG, Lynch JM. 1996. Sexual dimorphism in the craniofacial skeleton of modern humans. In: Marcus LF, Corti M, Loy
A, Naylor GJP, editors. Advances in morphometrics. New
York: Plenum Press. p 407–414.
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