Morphometry of the teeth of western North American tyrannosaurids and its applicability to quantitative classification

Abstract

of the teeth of western North American tyrannosaurids and its applicability to quantitative classification. Acta Palaeontologica Polonica 50 (4): 757–776. Gross tooth morphology and serration morphology were examined to determine a quantifiable method for classifying tyrannosaurid tooth crowns from western North America. From the examination of teeth in jaws, tyrannosaurid teeth could be qualitatively assigned to one of five types based on the cross−sectional shape of the base of the tooth and charac− teristics of the mesial carina. A principal component analysis (PCA) revealed that much of the variance in tooth shape was a result of isometry, but some gross morphological variables exhibited strong positive allometry. Non−size associated fac− tors were also important in determining tooth shape, particularly when data on denticle dimensions were considered in the analysis. While PCA identified important factors in variation, PCA ordination plots did not cluster the teeth into distinct, separate groupings based on taxon or bone of origin. The group classification functions determined by discriminant anal− ysis, though not universally successful for classifying unidentified isolated teeth of all tyrannosaurids, do identify bone of origin of adult Albertosaurus, Daspletosaurus, and Gorgosaurus teeth at a statistically acceptable level. Tanya Samman [tsamman@ucalgary.ca] and Leonard.V. Hills [lvhills@ucalgary.ca],

Full text

Morphometry of the teeth of western North American
tyrannosaurids and its applicability to quantitative
classification
TANYA SAMMAN, G. LAWRENCE POWELL, PHILIP J. CURRIE, and LEONARD V. HILLS
Samman, T., Powell, G.L., Currie, P.J., and Hills, L.V. 2005. Morphometry of the teeth of western North American
tyrannosaurids and its applicability to quantitative classification. Acta Palaeontologica Polonica 50 (4): 757–776.
Gross tooth morphology and serration morphology were examined to determine a quantifiable method for classifying
tyrannosaurid tooth crowns from western North America. From the examination of teeth in jaws, tyrannosaurid teeth
could be qualitatively assigned to one of five types based on the cross−sectional shape of the base of the tooth and charac−
teristics of the mesial carina. A principal component analysis (PCA) revealed that much of the variance in tooth shape was
a result of isometry, but some gross morphological variables exhibited strong positive allometry. Non−size associated fac−
tors were also important in determining tooth shape, particularly when data on denticle dimensions were considered in the
analysis. While PCA identified important factors in variation, PCA ordination plots did not cluster the teeth into distinct,
separate groupings based on taxon or bone of origin. The group classification functions determined by discriminant anal−
ysis, though not universally successful for classifying unidentified isolated teeth of all tyrannosaurids, do identify bone of
origin of adult Albertosaurus,Daspletosaurus, and Gorgosaurus teeth at a statistically acceptable level.
Key wo r d s : Theropoda, Tyrannosauridae, dentition, classification, quantitative analysis, Cretaceous, North America.
Tanya Samman [tsamman@ucalgary.ca] and Leonard.V. Hills [lvhills@ucalgary.ca], Department of Geology & Geo−
physics, University of Calgary, AB, T2N 1N4 Canada;
G. Lawrence Powell [lpowell@ucalgary.ca], Department of Biological Sciences, University of Calgary, AB, T2N 1N4
Canada;
Philip J. Currie [Philip.Currie@gov.ab.ca], Royal Tyrrell Museum of Palaeontology, P.O. Box 7500, Drumheller, AB,
T0J 0Y0 Canada; present address: [philip.currie@ualberta.ca]Department of Biological Sciences University of Alberta
Edmonton, Alberta, T6G 2E9 Canada.
Introduction
The teeth of theropod dinosaurs are diverse and relatively
abundant in Upper Cretaceous sediments of western North
America. Many features of theropod teeth allow identification
at high taxonomic resolution, sometimes down to species level
(Currie et al. 1990). Diagnostic features of the tooth crown in−
clude cross−sectional shape, presence of mesial and/or distal
denticles and their locations, and the size and shape of each
denticle. These have been described and utilized in classifica−
tion for various North American taxa, including tyranno−
saurids (Currie et al. 1990; Farlow et al. 1991; Fiorillo and
Currie 1994; Baszio 1997; Sankey 2001; Sankey et al. 2002).
These studies demonstrate that isolated tyrannosaurid tooth
crowns can be identified to the family level.
The teeth of most theropod dinosaurs are contained in the
premaxilla, maxilla, and dentary. In most theropods, the larg−
est maxillary teeth are larger than the largest dentary teeth
(Currie 1987). The denticles are complex structures (Abler
1997), aligned on carinae that sometimes bifurcate in tyranno−
saurids (Erickson 1995).
Tyrannosauridae is a lineage of large−bodied Late Creta−
ceous coelurosaurs (Holtz 1994, 1996, 2000); Sereno 1997;
Norell et al. 2001). The generic and specific taxonomy of the
tyrannosaurids is unstable, and the relationships among these
taxa are controversial (Currie 2003b; Currie et al. 2003).
Several genera, including Albertosaurus,Daspletosaurus,
Gorgosaurus, and Tyrannosaurus (Russell 1970; Currie
2003a) are found in the Upper Cretaceous sediments of
southern Alberta and the northwestern United States.
The crown of a tyrannosaurid tooth (Fig. 1) is a semi−con−
ical structure made up of a stack of nested dentine cones with
a thin external layer of enamel on the crown (Abler 1992).
The premaxillary tooth is D−shaped in basal cross section,
with two serrated ridges (carinae) located on the lingual side.
Maxillary and dentary teeth are more laterally compressed,
less recurved than in other theropods, with a round to ovoid
cross−sectional outline and stout saddle− or chisel−shaped
serrations (denticles) (Fig. 1) on both the mesial and distal
(see Smith and Dodson 2003) margins of the tooth crown.
These serrations are sometimes angled towards the apex of
the tooth, and are generally aligned along carinae that curve
lingually. Further, the denticles on the maxillary and dentary
teeth have the following additional characteristics: (1) they
are relatively large (Farlow et al. 1991), (2) they are more
widely spaced than in many other theropods (Sankey et al.
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Acta Palaeontol. Pol. 50 (4): 757–776, 2005

2002), (3) they have mesial and distal serrations that are
equivalent in size (Chandler 1990), (4) they are wider labi−
ally−lingually than they are long proximodistally, (5) they
generally decrease in size towards the base and apex of the
tooth, (6) they possess sharp ridges of enamel along the
midline, and (7) they are smaller relative to tooth length in
larger individuals, but are absolutely larger in basal diameter
and denticle height (Chandler 1990; Currie et al. 1990). The
internal and external morphologies of tyrannosaurid teeth are
described in detail by Abler (1992). The microstructure and
chemical composition of Tyrannosaurus teeth are discussed
by Dauphin et al. (1989).
Continual replacement (polyphyodonty) of teeth occurred
throughout the life of a dinosaur (Edmund 1960; Chandler
1990; Erickson 1996), with one or two replacement teeth de−
veloping at any given time for each tooth position (Edmund
1960). As the growth rate of the animal decreases over time,
its tooth replacement rates also slow down (Erickson 1996),
and the influence of wear and tear on the replacement of teeth
increases.
Abler (1992) described three types of dental wear surfaces
for tyrannosaurids that were characterized by shape and loca−
tion on the tooth crown. Farlow and Brinkman (1994) reported
wear on the lingual sides of dentary teeth, an unusual feature
given the way tyrannosaurid jaws fit together (Schubert and
Ungar 2005). Jacobsen (1996) observed the presence of wear
on the mesial, distal, labial, and lingual sides of teeth, with a
number of teeth exhibiting more than one type of wear. Wear
features have recently been attributed to two different etiolo−
gies, 1) occlusal attrition, resulting in elliptical shaped facets
with microscopic parallel wear striations, and 2) abrasion, of−
ten resulting in conchoidal shaped spalled surfaces with mi−
croscopic striations that are heterogeneous (Schubert and
Ungar 2005). Attritional wear facets are found on only one
side of a given tooth crown, never on the mesial or distal sur−
faces (Schubert and Ungar 2005). Thus, one might expect to
find occlusal wear facets on the lingual side of a maxillary
tooth crown, and on the labial side of a dentary tooth crown
(Lambe 1917), which could potentially serve as an indicator of
bone of origin for isolated teeth (Schubert and Ungar 2005).
Currie (1987) determined that the teeth of a small thero−
pod, Troodon, vary in structure according to position in the
jaw, and can be separated into four types, similar to Hunger−
bühler’s (2000: 33) “dental sets”: (1) premaxillary, (2) maxil−
lary, (3) mesial dentary, and (4) distal dentary. Similarly, a dis−
tinction between the premaxillary and maxillary/dentary teeth
of tyrannosaurids is easily made from the examination of teeth
in jaws (Chandler 1990; Carr 1999; Brochu 2003) but thus far,
the characteristics that separate these types have not been
quantified.
The feasibility of quantifiably distinguishing taxon of ori−
gin below the family level, or bone of origin for tyrannosaurid
teeth, has not been investigated. Quantifiable distinctions
would be useful for identifying isolated teeth, as they are com−
monly found in North American Mesozoic strata, and are use−
ful for tracking geographical and evolutionary occurrences in
paleoecological studies (Brinkman 1990; Farlow and Pianka
2002).
This study is an attempt to quantify tooth variation within
a restricted co−familial group of dinosaurs from ecologically
similar settings. The viability of two multivariate statistical
methods to discriminate between tooth samples is assessed
using a relatively restricted sample. The efficacy of a given
technique for distinguishing subtle variation within a re−
stricted sample would demonstrate its potential as a method
for tracking ecological and evolutionary shifts within any
stratigraphically well−represented clade. Our exploratory sta−
tistical approach will serve as a methodological baseline for
studying ontogenetically, taxonomically, and stratigraphi−
cally more divergent samples.
To quantify the positional and taxonomic variation of
tyrannosaurid teeth (specifically, tooth crowns) we studied
teeth in tyrannosaurid jaws to determine if there were trends
in serration and gross tooth morphology that can be used to
identify isolated tyrannosaurid teeth. We used principal com−
ponent analysis (Pimentel 1979; Chapman et al. 1981; Weis−
hampel and Chapman 1990) to examine the quantitative
morphometry of tyrannosaurid teeth, as an adjunct to their
descriptive morphometry. In addition to quantifying the scal−
ing relationships among the different dimensions of the teeth
and examining the effects of isometry on these, we ordinated
the results of this analysis in order to examine the relation−
ship of tooth morphometry to bone of origin. This last objec−
tive is also pursued through the use of discriminant analysis
based upon tooth morphometry.
Institutional abbreviations.—BHI, Black Hills Institute, Hill
City, South Dakota, USA; MOR, Museum of the Rockies,
Bozeman, Montana, USA; ROM, Royal Ontario Museum,
Toronto, Ontario, Canada; TMP, Royal Tyrrell Museum of
Palaeontology, Drumheller, Alberta, Canada.
Methods
Specimens.—In situ teeth (in the original alveolar position
in the jaw) from various western North American taxa were
examined (Appendix 1). Of the many isolated tyrannosaurid
teeth in the collections of the TMP, the majority are from the
Judith River Group of southern Alberta. These isolated teeth
consist mainly of shed tooth crowns, the most complete of
which were examined for this study (Appendix 2).
Measurements.—For consistency of measurement, the base
of the tooth crown was defined as a plane that extends hori−
zontally at the level of the base of the distal carina when the
tooth is positioned in approximate life orientation (Fig. 2),
which is roughly parallel to the termination of enamel at the
base of the tooth crown. Measurements (in mm) follow the
methods of Farlow et al. (1991) and Sankey et al. (2002):
(1) fore−aft basal length (FABL); (2) tooth crown height
(THEIGHT), measured from the base of the distal carina;
(3) cross−sectional thickness (XSTHICK), measured at the
758 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005

base of the tooth crown; (4) curvature (CURVATUR); and
(5) distance from the base of the mesial carina (end of the
denticles) to the base of the tooth crown (DMCTOB). The
mesial (M) and distal (D) carinae were each divided approxi−
mately into basal (B), middle (M), and apical (A) thirds
(sensu Chandler 1990). As denticles tend to diminish in size
towards the base and apex of the tooth, denticle width (DW)
and denticle height (DH) were measured for average−sized
denticles in the middle of each section (Fig. 2). Tooth mea−
surements 1, 2, 3, and 5 were taken with dial calipers,
whereas tooth measurement 4 (except for exceptionally large
teeth) and the denticle measurements (6 and 7) were taken
with a calibrated ocular micrometer through a binocular
microscope.
For comparative purposes, ratios calculated for in situ teeth
are (Farlow et al. 1991; Carr 1996): (1) XSTHICK/FABL;
(2) XSTHICK/THEIGHT; and (3) DMCTOB/THEIGHT.
A subjective approximation of the last ratio was used to esti−
mate apriorithe bone of origin for the isolated teeth, based on
the relative lengths of the maxillary and dentary mesial carinae
in Dromaeosaurus (Fiorillo and Currie 1994).
Measurement error.—A single−factor (in situ or isolated)
repeated−measures ANOVA was used to examine the contri−
bution of measurement error to the total variance of the sam−
ple. For a subset of the entire sample (12 isolated teeth and 5
in situ teeth [4 Gorgosaurus libratus,1Daspletosaurus
torosus]), each of the measurements was repeated three times
for each tooth; each replicate was considered as a repeated
measurement in the ANOVA. The departure of the vari−
ance−covariance matrix of each set of repeated measure−
ments from sphericity was evaluated by calculation of the
Greenhouse−Geisser e; a value near 1.0 was taken to indicate
little departure, but in all cases the evalue was used to calcu−
late the appropriate approximation of F for that portion of the
ANOVA (Quinn and Keough 2002). An F−max test was used
to examine each series of repeated measurements for homo−
scedasticity (Sokal and Rohlf 1969), and normality within
each series was tested for using a Lilliefors’ modification of
the Kolmogorov−Smirnov one−sample test (Wilkinson 1990);
these tests of the model’s assumptions determined the
acceptance of the results of the ANOVAs.
In addition, in situ teeth are rarely pristine, and the mea−
surement of THEIGHT is sometimes hampered by slight api−
cal chipping or wear at the base of the distal carina. Farlow et
al. (1991: 163) experienced similar difficulties with incom−
pletely preserved teeth, and measured tooth crown height ver−
tically from the outer rim of the tooth socket to the tooth apex,
noting that their method overestimated tooth height in some
cases. In the present study, our measurement for the more
poorly preserved teeth would tend to slightly underestimate
the value of THEIGHT. However, only teeth with very mini−
mal damage were considered, and can therefore be considered
to be reasonable and representative of the height of the tooth.
Statistical analysis.—To assess the sizes and shapes of the
teeth, a sub−sample (n = 35) of isolated teeth as well as a
sub−set of in situ teeth (n = 71) were chosen for statistical
analysis. For the isolated tooth sample, over 1700 specimens
were examined, 196 were measured, but only the 35 used in
the study were pristine and had all mensural characters
(denticles included). The carinae were not always well pre−
served or complete on the examined in situ teeth, and in order
to maximize the sample size for statistical analysis, denticle
data were not included. Of the ~161 measured in situ teeth,
data from only the 71 best−preserved teeth were used.
Simultaneous size−dependent shape changes in two or
more anatomical dimensions are best considered in terms of
multivariate scaling (Jolicoeur 1963a; Cock 1966; Morrison
1976; Pimentel 1979; Shea 1985; Rohlf and Bookstein 1987;
Strauss 1987; Voss 1988; McKinney and McNamara 1991;
Jungers et al. 1995; Klingenberg 1996). The principal compo−
nent generalization of the allometric equation, derived from
log10−transformed mensural data (Jolicoeur 1963b), describes
multivariate scaling (Jolicoeur 1963a; Pimentel 1979; Shea
1985; Strauss 1987; Voss 1988; McKinney and McNamara
1991). When derived from log10−transformed mensural data
from a single taxonomic group, the first component—the
size−determined shape vector (McKinney 1990)—can be
taken to represent the variance incident upon differences in
size (Somers 1986; Tissot 1988; McKinney 1990; McKinney
and McNamara 1991), whether isometric or allometric. Sub−
sequent components thus describe variance in shape that is not
associated with size (Tissot 1988; Marcus 1990; McKinney
1990; McKinney and McNamara 1991).
The logarithmic spiral manifests itself in the shape of the
vertebrate tooth to a greater or lesser degree (Thompson
1961), and the element of tooth shape (particularly THEIGHT
and CURVATUR) incorporating this cannot be expected to
vary with size over a wide size range according to allometric
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SAMMAN ET AL.—MORPHOMETRY OF TYRANNOSAURID TEETH 759
dentine
denticle
enamel
2mm
Fig. 1. Longitudinal section through the crown of a small, broken, tyranno−
saurid tooth (TMP 86.130.214, Judith River Group, Campanian, Alberta,
Canada). The plane of the break resulted in the loss of the mesial carina, so
only the distal carina is preserved on the tooth. A. Enlargement of the area
outlined in B shows the enamel and the denticles. B. Photograh of the whole
specimen showing dentine (light−colored area).

rules (Batschelet 1979). However, the tyrannosaurid tooth de−
scribes a sufficiently short segment of the logarithmic spiral
that the allometric relationship can be assumed without a loss
in explicatory power over the size range of teeth examined.
In situ and isolated teeth were analyzed separately. Further
subdivisions of these sample groups were not done because
identified taxa are represented by unequal numbers of individ−
uals and/or the bone of origin (Appendix 1). In addition, the
isolated teeth could not be assigned with certainty to either
sub−familial level taxon or bone of origin (Appendix 2),
though the familiarity with tyrannosaur teeth of the first au−
thor, gained from examining many hundreds of specimens,
made educated aprioriestimations for bone of origin possi−
ble. All teeth in each sample were grouped initially as tyranno−
saurid dinosaurs, and within each principal component analy−
sis there was no taxonomic subdivision, nor subdivision by
bone of origin. We assume that the allometric vector does not
differ significantly among any such groupings (Klingenberg
1996), and in any case, principal component analysis will seg−
regate variance not directly explained by size, whether due to
systematic or physical position, from that explained by size.
All of the measurements were log10−transformed to minimize
possible inherent heteroscedasticity (Kerfoot and Kluge 1971;
Zar 1984; Tissot 1988), to better approximate the multivariate
normal distribution (Pimentel 1979), and to better approxi−
mate the linear PCA model (Jolicoeur 1963a; Pimentel 1979;
Shea 1985; Strauss 1987; Voss 1988). Biological data gener−
ally conform to the assumption of multivariate normality
(Marcus 1990). PCA is in any case robust to minor deviations
from the multivariate normal distribution, which are of lesser
importance when PCA is being used to ordinate the data
(Reyment 1990).
Principal component analysis was performed on vari−
ance−covariance matrices derived from the log10−trans−
formed data using SYSTAT (Version 5.0, 1994, SYSTAT,
Inc.). The entire suite of components from each analysis,
ranked by eigenvalue, and incrementally−decreasing subsets
of each suite, were tested for isotropicity of length by Bart−
lett’s c2test for sphericity (Morrison 1976). As the sample
sizes were small, the decision of how many components to
retain was based upon the results of this iterative application
of Bartlett’s approximation of c2(Quinn and Keough 2002).
Acceptance levels for each set of tests were determined by
a stepwise Bonferroni adjustment (Rice 1989). The first
component (PC I) of each analysis was interpreted as the
size−determined shape vector, whereas the remainder of
those retained were examined for evidence of morphological
changes not strongly determined by overall size (Tissot
1988; Marcus 1990; McKinney and McNamara 1991; Jun−
gers et al. 1995). The correlations of each of the morphologi−
cal variables with each of the components, and the percent of
the total variation of each morphometric variable explained
by each of the components, were calculated as described by
Pimentel (1979). The first component was tested for iso−
metry (i.e., all possible ratios of PC I loadings approximating
1.0, or an isometric slope for reduced major axis slopes cal−
culated for any pairwise combination of morphometric vari−
ables in the variable suite—Cheverud 1982, Klingenberg
1996) by comparing it to a theoretical “isometric vector”
(Leamy and Bradley 1982; Cheverud 1982; Voss 1988). The
individual loadings of an isometric vector are computed as
(1/ pi) where piequals the number of variables in the analy−
sis (Jolicoeur 1963b). In the case of the present analyses, the
loadings on the theoretical isometric vector equal (1/ 5=
0.447214) (excluding the denticle variables), and (1/ 17 =
0.242536) (including the denticle variables). The difference
between the observed size−dependent shape components and
the appropriate theoretical isometric vectors was tested by
means of Anderson’s (1963) c2approximation (Pimentel
1979). A variable loading that approximates the value of the
isometric loading exhibits isometry—i.e., equality of relative
growth rate (Jolicoeur 1963a)—relative to an overall mea−
sure of size (i.e., when multiplied by 1/ pi, against the
weighted geometric mean of all variables—Klingenberg
1996). Variable loadings considerably greater than the
isometric loading can thus be interpreted to indicate positive
allometry, whereas variable loadings well below the iso−
metric loading indicate negative allometry (Voss 1988).
The size−associated variance predicted to be explained by
isometry was removed from the original data matrix of the in
situ teeth (restricted to the five gross anatomical variables) by
a modification of the Burnaby method for size−removal
(Burnaby 1966; Rohlf and Bookstein 1987; Marcus 1990).
Here we use the theoretical isometric vector (with loadings
760 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005
3
1
4
5
2
mesialdistal
6
7
Fig. 2. Schematic illustration of tooth and denticle measurements obtained
for this study. Diagram is an outline of tooth specimen TMP 86.130.214.
Denticles on tooth outline in lateral (A), basal (B), and enlarged (C) views,
not to scale. For measurements, tooth is aligned so that the basal termination
of enamel (broken line in A) is approximately horizontal. 1, fore−aft basal
length (FABL); 2, tooth crown height (THEIGHT); 3, cross−sectional thick−
ness (XSTHICK); 4, curvature (CURVATUR); 5, distance from the base of
the mesial carina to the base of the tooth (DMCTOB); 6, denticle width
(DW); 7, denticle height (DH).

of 1/ 5= 0.447214) in place of the usual PC I derived from
the data (Burnaby 1966; Rohlf and Bookstein 1987; Marcus
1990). We assumed that any significant intra−group variance
here (due to bone of origin, individual representation, or tax−
onomic grouping below the family level) was additive to
variance due to size. As PC I in the original PCA would ac−
count for the variance produced by common allometry (Klin−
genberg 1996) as well as that due to isometry, it can be re−
placed by the isometric vector. Variance from other sources
would remain in the modified data set and be susceptible to
further analysis. A new data matrix with this variance re−
moved was generated and subjected to a principal compo−
nent analysis. The principal components extracted from this
analysis are interpreted as describing aspects of tooth shape
that are not dependent upon isometric scaling.
To classify the teeth to a bone of origin, the origins of the
isolated teeth were first determined subjectively, based on
the length of the mesial carina relative to the length of the dis−
tal carina. These assignations were first tested by subjecting
the isolated tooth sample to discriminant analysis, a tech−
nique that has been used by other researchers to distinguish
tooth morphs for various animals (e.g., Palmqvist et al. 1999;
Smith 2002). Using SYSTAT (Wilkinson 1990), the log10−
transformed gross mensural data from the isolated tooth sam−
ple were used to derive canonical coefficients for the five
gross morphological variables, and two discriminant func−
tions for the two putative bones of origin, maxillary and den−
tary (Morrison 1976, Pimentel 1979). The Wilks’ land its
associated F from the MANOVA generated in this process
were used post hoc to evaluate the discriminant analysis
(Morrison 1976). The consistency of the author−assigned
groupings with the discriminant scores of all of the cases was
tested by means of a c2test for independence of association of
the discriminant function’s classification results against an ex−
pected homogeneous distribution (arrayed in a 2×2 contingency
table—Steel and Torrie 1980). The values of the canonical co−
efficients were compared to those of the corresponding PCA
factor loadings in order to further clarify the relationships
among the gross mensural characters of the teeth and their mode
of variance depending upon bone of origin.
As an independent and more powerful test of the dis−
criminant function derived from the isolated tooth data, the
gross mensural data from the in situ tooth sample were used
together with the group classification functions to provision−
ally classify the in situ teeth to bone of origin. The differences
between the scores determined for the in situ teeth were com−
pared to the distributions of the differences in group classifica−
tion scores established for the isolated teeth, and predicted
group membership assigned on that basis. The distributions of
predicted versus actual bone of origin (arrayed as a 2×2 con−
tingency table) were tested for independence of association by
means of a c2test for interaction (Steel and Torrie 1980). Cor−
rect classification by the group classification functions of teeth
already unequivocally known to bone of origin would support the
importance of the variables used to describe tooth morphology,
and validate the subjective judgement of the first author.
Results
Empirical description.—There are 4 premaxillary, 11 to 17
maxillary, and 13 to 17 dentary teeth in tyrannosaurids (Cur−
rie et al. 2003). Five qualitative types of teeth can be ob−
served in the tyrannosaurid jaws (Fig. 3). Premaxillary teeth
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SAMMAN ET AL.—MORPHOMETRY OF TYRANNOSAURID TEETH 761
= basal position of carina
Fig. 3. The five major types of teeth found in the tyrannosaurid jaw, Judith River Group, Campanian, Upper Cretaceous, Alberta, Canada. A. Premaxillary,
TMP 65.26.3. B,C. Mesial maxillary, MM1 TMP 98.68.65 (B), MM2 TMP 66.31.36 (C). D. Distal maxillary, TMP 85.36.342. E,F. Mesial dentary, MD1
TMP 79.14.538 (E), MD2 TMP 89.79.4 (F). G. Distal dentary, TMP 97.12.43. Cross−sectional shape and location of carinae are as indicated, and are to
scale relative to each other. Scale bars 10 mm.

are smaller than all but the most distal maxillary and dentary
teeth, and are D−shaped in cross−section. The mesial (first
one or two) maxillary and dentary teeth are more rounded in
cross−section (personal observation). These correspond to
the incisiform and sub−incisiform tooth shapes mentioned by
Carr (1999). Brochu (2003) also noted a difference in the lo−
cation of the carinae on the first few dentary teeth, which
would result from this difference in cross−sectional shape.
The remainder (distal) of the maxillary and dentary teeth are
more ovoid and labio−lingually compressed in cross−section.
Mesial dentary teeth are procumbent, but curve distally. All
maxillary teeth and distal dentary teeth are more vertical in
orientation, but also curve distally. These distinctions are ob−
vious for in situ teeth, but are more difficult to determine for
isolated teeth because their original alveolar positions are
unknown.
Except in Tyrannosaurus rex, the mesial denticles on the
maxillary teeth start closer to the base of the tooth than those of
the dentary, just as in Dromaeosaurus (Fiorillo and Currie
1994). The widths of both the mesial and distal denticles are
approximately equivalent in both the maxillary and dentary
teeth (Chandler 1990 and personal observation). In the dentary
teeth, however, the heights of the distal denticles are generally
slightly larger than those of the mesial denticles (personal ob−
servation); finer mesial than distal denticles are generally con−
sidered typical of tyrannosaurids (Carr 1996). The direction of
the deviation of the mesial carina from the midline can be used
to distinguish teeth from the left or right sides of the jaw in
762 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005
Albertosaurus
Daspletosaurus
Gorgosaurus
Tyrannosaurus rex
maxillary (M)
dentary (D)
premaxillary (PM)
PM M
D
PM M
D
PM
M
D
Principal component II
Principal component I
Principal component III
Principal component III
Principal component I
Princi
p
al com
p
onent II
2
1
0
–1 012
1.0
0.5
0.0
–0.5
–1.0
–1 0 1 2
1.0
0.5
0.0
–0.5
–1.0
–1
–1 012
Fig. 4. A. Ordination plot for PC I versus PC II for the teeth in jaws. B. Ordination plot for PC I versus PC III for the teeth in jaws. C. Ordination plot for PC
II versus PC III for the teeth in jaws.

tyrannosaurids, as the carina always shifts towards the lingual
side of the tooth. If the tooth is viewed mesially with the base
down, a deviation to the left, for example, indicates either a
right maxillary or left dentary tooth. Also, as tyrannosaurid
dentaries are not as curved as the maxillae, the basal deviation
of the mesial carina from the midline is typically greater in the
dentary teeth.
In some isolated tooth specimens (TMP 94.12.313,
94.12.209, 94.12.4), the first two preserved teeth of the right
dentary of TMP 86.144.1 (Gorgosaurus libratus), and the
first five dentary teeth of TMP 94.143.1 (Daspletosaurus
torosus), there are no discernable denticles on the mesial
apex of the tooth. The enamel is smooth and intact, and there
does not appear to be any evidence of wear. The absence of
apical denticles is likely genetic and not taphonomic in these
samples.
Measurement error.—The distributions of none of the re−
peated measures taken from isolated teeth or from in situ
teeth were significantly non−normal in distribution. The three
repetitions for each measurement, isolated and in situ, were
all homoscedastic. For both of these tests the stepwise Bon−
ferroni adjustment acceptance levels were 0.004 and 0.01,
respectively. The Greenhouse−Geisser evaried but was gen−
erally high (THEIGHT: 0.784; FABL: 0.846; XSTHICK:
0.938; CURVATUR: 0.996; DMCTOB: 0.912), indicating
moderate to small departures from compound symmetry for
the data. Assumptions for the repeated−measures ANOVA
were thus supported, or could be adjusted for. At a stepwise
Bonferroni−adjusted acceptance level of 0.01, the ANOVAs
showed no significant differences between measurements at
different times for any of the variables, and no significant in−
teraction effect between time of measurement and situation
of tooth. We can thus dismiss measurement error as a signifi−
cant factor in this analysis. Additionally, the measurements
used are replicable and should be of use in other studies of
tyrannosaurid teeth.
Statistical analysis.—In the interpretation of the morpho−
metric principal component results, the first principal com−
ponent is interpreted as describing size−related shape dif−
ferences, whereas subsequent components describe non−
size−related shape differences (Somers 1986; Tissot 1988;
McKinney 1990; McKinney and McNamara 1991). Addi−
tionally, variables that show maximum variance explained
by, and strong correlations with, the same principal compo−
nent display variance in shape controlled by the same factor,
whether it be size−associated or otherwise.
Interpreting PCA of the in situ tooth sample for the gross
morphological variables requires only the first four compo−
nents, which collectively account for 99.45 percent of the
sample variance (Appendix 3). PC I differs significantly from
the isometric vector. FABL, THEIGHT, and XSTHICK dis−
play negative allometry, DMCTOB positive allometry, and
CURVATUR strong positive allometry. This reflects how the
teeth become relatively thinner and more recurved with in−
creasing size, as the base of the mesial carina moves up the
slope of the tooth. The opposing trends of THEIGHT and
CURVATUR reflect the contribution of the logarithmic spiral
element of tooth shape in the recurved shape of the tooth. The
size−associated shape vector explains almost two−thirds of the
total variance (Appendix 3), but only accounts for the majority
of the variance of CURVATUR. The remaining gross mor−
phological variables have their variances more evenly spread
among the first three components (Appendix 3). Likewise, the
strongest positive correlation is between CURVATUR and
PC I. The other variables, although having positive correla−
tions with PC I, also have strong correlations with PC II and
PC III (Appendix 3). Because PC II and PC III describe shape
variance that cannot be attributed to size, these large correla−
tions and amounts of variation described must be ascribed to
such factors as bone and taxon of origin. It must also be
pointed out that relatively few individual animals contributed
teeth to this sample (Appendix 1), so individual variation will
make a disproportionate contribution to PC II and PC III.
Unfortunately these factors cannot easily be teased apart.
Removing the variance attributable to isometry from
these data (Appendix 4) reduces the sum of the eigenvalues
by approximately 55 percent, indicating that isometry is re−
sponsible for slightly more than half of the total variance in
the unmodified sample. The first two components derived
from the modified data matrix account for 96.78 percent of
its total variance (Appendix 4). The variance due to allo−
metry and non−size−associated factors such as systematic po−
sition, bone of origin, and individual idiosyncrasy will be
distributed through this total variance. Most of the variance
in FABL, THEIGHT, and XSTHICK, variables which were
negatively allometric in the first PCA (Appendix 3), are ex−
plained by PC I in this analysis (Appendix 4). The majority
of the variance of CURVATUR, a strongly positively allo−
metric variable in the first analysis (Appendix 3), is ex−
plained by PC II in this analysis (Appendix 4). DMCTOB
shows a similar distribution of variance across PC I and PC II
in both analyses (Appendices 3, 4). Both CURVATUR and
DMCTOB are negatively correlated with PC I (Appendix 4).
Overall, the PCA from the modified data set suggests that the
remaining allometric variation left after the removal of the
variation due to isometry is distributed across PC I and PC II.
However, this also implies that some aspects of tooth allo−
metry are not correlated with one another in the absence of
size. The sum of the eigenvalues derived from the data set af−
ter the removal of isometric variance (Appendix 4) is not
greatly in excess of the sum of the eigenvectors for PC II–PC
IV in the original PCA (Appendix 3). Therefore, an ad−
ditional implication is that allometry contributes a relatively
small amount of variance to the whole.
The sample of isolated teeth (Appendix 2) is much
smaller than that of in situ teeth (Appendix 1), which may ac−
count for some of the differences in the PCAs of the two sam−
ples (Appendices 3, 5), especially if this entails a difference
in taxonomic representation. In addition, unlike the limited
number of individuals in the in situ sample, the isolated tooth
sample likely represents as many individuals as teeth. Again,
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SAMMAN ET AL.—MORPHOMETRY OF TYRANNOSAURID TEETH 763

the first four components, explaining 99.11 percent of the to−
tal variance, are non−isotropic (Appendix 5). The loading for
THEIGHT is close to isometric in PC I (Appendix 5), as op−
posed to its negatively isometric PC I loading in the in situ
tooth PCA (Appendix 3). The pattern of loadings for this
vector are otherwise fairly similar for both analyses, al−
though FABL, THEIGHT, and XSTHICK have much more
of their total variances explained by PC I in the isolated tooth
PCA (Appendices 3, 5). PC II of this analysis is poorly corre−
lated with these three variables, and explains little of their to−
tal variance. Instead, it is largely defined by the non−size as−
sociated shape variances of CURVATUR and DMCTOB
(Appendix 5). Most of the remnant variance of FABL,
THEIGHT, and XSTHICK is explained by PC III, which
also has fairly strong correlations with this component (Ap−
pendix 5). In contrast to the analysis of in situ teeth (Appen−
dix 3), the analysis of isolated teeth shows non−size−associ−
ated variance in CURVATUR and DMCTOB that is not cor−
related with the non−size−associated variance in the other
gross morphological variables (Appendix 5). For DMCTOB,
this is likely due to the presence of Tyrannosaurus rex teeth
in the in situ tooth sample. In contrast to the teeth of the other
genera examined, the mesial denticles of both examined
specimens of T. rex teeth start farther away from the base of
the teeth in the maxilla. This would introduce variance in this
variable not necessarily associated with size but definitely
attributable to systematic position.
In the PCA of isolated teeth incorporating denticle data
(Appendix 6), the first three components, accounting for 79.46
percent of the total variance of the data, are the most strongly
associated with the gross morphological variables. PC I, the
size−associated shape vector, explains most of the variance in
FABL, THEIGHT, and XSTHICK with the inclusion of den−
ticle data. Significant amounts of non−size−associated variance
in CURVATUR and DMCTOB are still explained by the sub−
sequent two components (Appendix 6). A much larger amount
of variance in DMCTOB is explained here by PC III, and this
variable is also highly correlated with PC II (Appendix 6).
This contrasts with what was found for DMCTOB in the PCA
of this sample using only the gross morphological variables
(Appendix 5). PC I and PC II also generally explain significant
amounts of variance in denticle dimensions (Appendix 6). Al−
though much of this variance is size−associated, some is not,
and is associated instead with the variance of CURVATUR
(and to a lesser degree DMCTOB) explained by PC II (i.e., for
denticle dimensions DDWB and DDHA). Most of the size−as−
sociated change in denticle shape is negatively allometric, or
in the case of MDHB and MDHM, roughly isometric (Appen−
dix 6). The remaining 20.54 percent of the sample variance
explained by the subsequent six components is distributed
largely among the denticle variables. They therefore contain a
significant portion of non−size−associated variance. DDHB,
DDHM, and DDHA have relatively large amounts of variance
explained by PC IV, whereas MDHA has a significant amount
explained by PC V. Generally patterns in the explanation of
variance by components IV–IX are difficult to discern, and the
component correlations do little to elucidate them (Appen−
dix 6).
Ordination diagrams for the principal components that
explain the greatest amount of variance (PC I, II, and III) in
the unmodified data for in situ teeth allow visual evaluation
of tooth distribution by taxon and bone of origin (Fig. 4). Dis−
tribution of teeth along PC I (Fig. 4A, B) can be attributed to
tooth size (and, to a lesser degree, dinosaur size). Neither the
taxa nor bone of origin separate well into clear, distinct
groupings. This quantifiably supports Carr and Williamson’s
(2000) qualitative observation that the teeth of western North
American tyrannosaurids are not diagnostic below the family
level. To a limited degree, the maxillary teeth separate from
the dentary teeth, as do the premaxillary teeth, in the plot of
PC II against PC III (Fig. 4C).
The discriminant analysis of the isolated teeth using gross
morphology indicated a significant difference in mean be−
tween the two groups by subjectively−assigned bone of ori−
gin (Wilks’ l= 0.451; F = 7.051, 29, 5 df, P < 0.001). Upon
examination of the absolute values of the standardized ca−
nonical coefficients (Table 1), it was evident that the most
important variable was DMCTOB, followed in decreasing
order by FABL, THEIGHT, XSTHICK, and CURVATUR.
Table 1. Canonical coefficients for the dependent variables for the iso−
lated teeth from the discriminant analysis. FABL, fore−aft basal length;
THEIGHT, tooth crown height; XSTHICK, cross−sectional thickness;
CURVATUR, curvature; DMCTOB, distance from the base of the
mesial carina to the base of the tooth.
Variable Canonical coefficient
FABL –0.810
THEIGHT –0.810
XSTHICK 0.532
CURVATUR 0.167
DMCTOB 1.160
Two group classification functions were generated from
this analysis, which can be used to identify western North
American tyrannosaurid teeth as either maxillary or dentary:
(1) 136.637(FABL) + 62.049(THEIGHT) – 84.760(XSTHICK)
– 32.940(CURVATUR) – 17.367(DMCTOB) – 76.762
(2) 121.353(FABL) + 50.957(THEIGHT) – 75.817(XSTHICK)
– 31.400(CURVATUR) – 3.829(DMCTOB) – 62.176
Each of the groups has a characteristic distribution for the
difference between the products of these two functions. The
assignations to maxillary or dentary were originally made on
a subjective basis. Still, the majority of the isolated teeth sep−
arate into two relatively distinct groups based on bone of ori−
gin. If the predictions determined by the discriminant analy−
sis are compared to the a priori classification (Appendix 2),
only one (pathologically anomalous) tooth (TMP 88.36.61)
out of the 35 in the data set was misclassified (Table 2). The
a priori assessment classified it as a dentary tooth, but the
discriminant equations predicted it to be a maxillary tooth.
The distribution of predicted against actual bone of origin is
764 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005

significantly different from what would be expected by
chance (Table 2); the discriminatory functions correctly
assign isolated teeth to their bone of origin.
In order to test the classificatory reliability of the group
classification equations, the equations were used to generate
new scores from the gross morphological values of the in situ
teeth, which were classified as either maxillary or dentary on
this basis (Appendix 1). These scores were compared to the
score distributions derived from the isolated teeth and as−
signed to maxillary or dentary on this basis. When all taxa are
included in the test, the group classification functions are not
sufficient to accurately classify all teeth to bone of origin
(Table 3). With the removal of Tyrannosaurus rex, the distri−
bution of tooth assignment is significantly different from
what would be expected by chance, with only one tooth
misclassified (Table 3). This small tooth belonged to a small
juvenile specimen (TMP 94.12.155) of Gorgosaurus libra−
tus (Currie 2003a). The T. rex sample was not classified to a
significantly different degree from what would be expected
by chance (Table 3).
Discussion
Empirical analysis.—It is reasonable to surmise that few in−
dividual animals are represented by more than one tooth in a
random sampling of isolated teeth (Farlow et al. 1991). Tax−
onomic and ontogenetic variation is undoubtedly present in
the sample of isolated teeth examined. These teeth are gener−
ally not identified beyond the family level. The in situ teeth,
however, can be used to assess taxonomic variation.
A comparison of the ratio XSTHICK/FABL and
XSTHICK/THEIGHT for distal tooth alveolar position 8 (or
close if not present) in the maxillae and dentaries of several
taxa revealed slight taxonomic differences, and contrary to
Currie et al. (1990), slight ontogenetic differences (Table 4).
The teeth in jaws of small juvenile specimens (dentaries,
TMP 94.12.155) of Gorgosaurus libratus (Currie 2003a),
were thinner labio−lingually relative to fore−aft basal length
of the tooth crown, but were comparable relative to tooth
crown height in cross−sectional thickness to larger specimens
of the same taxon (Table 4). The values for these ratios gen−
erally showed a similar range among taxa as they did within a
taxon. Thus, the cross−sectional thickness of a tyrannosaurid
tooth does not appear to be a reliable generic identifier.
Farlow et al. (1991) noted that fore−aft basal length is lin−
early related to cross−sectional thickness, as well as tooth
crown height. For theropods in general, tooth serration basal
length increases with increasing tooth size (Farlow et al.
1991). Tyrannosaurid teeth show greater variability in the
range of serration size along the mesial carina than other
theropods (Chandler 1990). Large individuals of Tyranno−
saurus rex have absolutely larger denticles (and thus lower
denticle densities) than other tyrannosaurids (Carr and Wil−
liamson 2000), meaning that the largest tyrannosaurid teeth
may be identifiable as T. rex. The number of denticles is cor−
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SAMMAN ET AL.—MORPHOMETRY OF TYRANNOSAURID TEETH 765
Table 2. Comparison of predicted versus “true” (based on a priori as−
sessment) classification of the isolated teeth from the group classifica−
tion analysis. Only one (anomalous) tooth of the 35 (sample 14) was
misclassified.
Predicted
Maxillary Dentary Total
“True” Maxillary 11 0 11
Dentary 1 23 24
Total 12 23
Test for homogeneity of classification: c2= 26.64, 1 df, P < 0.005.
Table 3. Comparison of predicted versus actual classification of the in situ
teeth as determined by the group classification equations. For taxa other
than Tyrannosaurus rex, the predictions of the equations were fairly accu−
rate. Note that the five premaxillary teeth from T. rex were consistently
classified as maxillary teeth; they are not included in this table.
All taxa
Predicted
Maxillary Dentary Total
True Maxillary 5 12 17
Dentary 6 43 49
Total 11 55
Test for homogeneity of classification: c2= 1.585, 1 df, P > 0.1.
All taxa except T. rex
Predicted
Maxillary Dentary Total
True Maxillary 4 0 4
Dentary 1 33 34
Total 5 33
Test for homogeneity of classification: c2= 21.623, 1 df, P < 0.005.
Only T.rex
Predicted
Maxillary Dentary Total
True Maxillary 1 12 13
Dentary 5 10 15
Total 6 22
Test for homogeneity of classification: c2= 1.410, 1 df, P > 0.1.
Table 4. Comparison of cross−sectional thickness/fore−aft basal length
and cross−sectional thickness/tooth crown height for posterior tooth po−
sition ~8 (TMP specimens/casts), showing slight differences reflective
of ontogeny and taxonomy for tyrannosaur teeth in jaws. Note that tooth
numbering begins at the front of the mouth and proceeds caudally.
Taxon XSTHICK/FABL XSTHICK/THEIGHT
Maxilla Dentary Maxilla Dentary
Gorgosaurus libratus 0.54 ~0.41, 0.57,
0.72, 0.77 0.29 ~0.33, 0.32,
0.34, 0.43
Daspletosaurus sp. 1 0.55 0.56 0.28 0.32
Tyrannosaurus rex ~0.7 0.74, 0.76 ~0.4 0.48

related with tooth size (Chandler 1990; Farlow et al. 1991),
making this a difficult feature upon which to rely for identifi−
cation purposes (Chandler 1990; Baszio 1997). In an attempt
to overcome this problem, Rauhut and Werner (1995) pro−
posed that a measurement of the denticle size difference in−
dex (DSDI, the difference between a set length of the mesial
and distal carinae) could be used as a taxonomic identifier.
However, DSDI varies along the carina, and the range of
DSDIs in Tyrannosaurus rex encapsulates the DSDI range of
Theropoda, excluding the outliers: troodontids and Richar−
doestesia (Carr and Williamson 2004). In different theropod
taxa, the number of denticles is correlated with tooth size in a
variety of ways. The analysis of shape features unrelated to
size makes identification possible (Baszio 1997). The princi−
ple that non−size−related shape features can be used to
distinguish between tooth specimens is integral to the present
study.
Statistical analysis.—The disproportionate contribution of a
small number of individuals to the in situ tooth sample (Ap−
pendix 1), with the resulting lack of independence among
these data, lessens the power of the analyses based upon them
(Appendices 3, 4; Table 3). However, the perspective that they
enabled upon the characteristics of teeth in the context of their
bone of origin justifies their use, and relevant test statistics
have sufficiently low probabilities (Appendices 3, 4; Table 3)
that they can be accepted regardless. If our assumption con−
cerning the homogeneity of any possible intra−group vari−
ance−covariance matrices in any subset of the data is incorrect,
it will render our interpretations of the PCA results suspect, as
we have interpreted PC I to describe size−associated shape
variance and subsequent PCs non−size−associated variance.
This is difficult to test for, given the sizes and natures of our
samples. However, the effect of removing the variance due to
isometry (Appendix 4), and the high degree to which most
morphometric variables are explained by and correlated with
PC I in the other PCAs (Appendices 3, 5, 6), support this as−
sumption. Further work along these lines, however, should re−
strict samples more closely by phylogeny. The differences in
sample size among the taxa examined here (Gorgosaurus,
8 individuals; Albertosaurus, 1 individual; Daspletosaurus,
3 individuals; Tyrannosaurus, 2 individuals; Appendix 1)
precluded any isolation of the variance attributable to phylo−
geny.
The first principal component is the size−determined shape
vector (McKinney 1990). The maximum variance explained
by PC I in any of the sub−sets of the data is 64.6 percent, for the
in situ teeth (Appendix 3). The smallest amount (48.8 percent)
of variance explained by PC I is for the isolated teeth with
denticle data included (Appendix 6), whereas for the isolated
teeth with denticle data excluded (Appendix 5), it explains
59.07 percent of the total variance. Even though the denticles
generally had large amounts of their total variance explained
by PC I, their inclusion introduced large amounts of non−shape
associated variance (Appendix 6). Differences in shape asso−
ciated with size thus contribute less to variance in tooth shape
than in other osteological features (e.g., Chapman et al. 1981),
although when tooth shape is quantified by gross morphology
alone (Appendices 3, 5), it is still the greatest contributor.
The removal of variance that could be explained by a the−
oretical isometric vector (in place of the empirically−derived
size−associated shape vector for the in situ teeth) reduced the
total amount of variance in that data set by roughly 55 per−
cent, indicating a large contribution by isometry to total vari−
ance in tooth shape. Nonetheless, strong positive allometry
in CURVATUR and DMCTOB when only gross morpho−
logical variables are included in the analysis (Appendices 3,
5) show that some aspects of tooth shape change dispropor−
tionately with size; larger teeth are proportionately more
curved, and have a greater distance between the base of the
mesial carina and the base of the tooth, than smaller teeth. All
of the gross morphological variables are positively allo−
metric when the denticle data are included in the analysis
(Appendix 6), due to the influence of the largely negatively
allometric denticles. The changes in gross morphological
shape associated with size are proportionately greater than
those in denticle dimensions, which rapidly become rela−
tively smaller as teeth increase in absolute size.
The contribution to variance of the denticles was only con−
sidered for the isolated teeth, as preservation of denticles on in
situ teeth was poor. With the exception of MDHM, the maxi−
mum loadings of the normalized data for the denticles was
concentrated in principal components IV, V, VI, VII, VIII, and
IX. Cumulatively, these variables account for 17.3 percent of
the total variance in the data. However, the denticle parame−
ters, except for MDHA (PC V), DDHB (PC IV), and DDHA
(PC II), show maximum variation explained by, and correla−
tion with, PC I. Thus, aside from the denticle variance attribut−
able to size (PC I), the contribution of denticle morphometry
to total tooth shape variance is difficult to interpret. Further in−
vestigation of the influence of phylogeny or function on non−
size associated variance in denticle shape may shed light on
the distribution of this portion of the total variance. The fre−
quently incomplete preservation of denticles does not facili−
tate their use in morphometric analysis, especially if the carina
is divided into sections, and equivalent sections compared. As
the size of denticles usually changes along the carina, this divi−
sion is necessary. However, the measurement of the largest
denticle on each tooth could provide better comparative con−
trol, as it would be a consistent and equivalent parameter for
each tooth. Thus, further analysis is recommended in order to
determine the significance of the contribution of the denticles
to the variance in tyrannosaurid teeth.
Examination of the patterns of variance per component
and component correlations revealed some interesting
trends. For all sub−sets of the data, the variable that had the
most variance explained by a particular component generally
had the highest correlation (not necessarily positive) with
that particular component (Appendices 3, 4, 5, 6). For PC I
(with the exception of the analysis for the isometry−removed
data—Appendix 4), this indicates which dimensions of the
teeth were most strongly influenced by the size of the tooth;
766 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005

CURVATUR for the PCA of the in situ teeth (Appendix 3),
THEIGHT for the PCA of the gross measurements of the iso−
lated teeth (Appendix 5), and to a lesser extent, for the PCA
of the isolated teeth including denticle dimensions (Appen−
dix 6). The sampling problems with the in situ tooth sample,
particularly the influence of Tyrannosaurus rex specimens
(Appendix 1), probably account for this switch in the identity
of the most prominent variable, in addition to accounting for
the non−size−associated prominence of DCMTOB in PC II of
the in situ tooth PCA (Appendix 3). CURVATUR and
DCMTOB both have significant non−size−associated influ−
ences on PC II, in the presence of denticles or not, for the iso−
lated teeth, showing strong correlations with, and high vari−
ance explanation by, this component (Appendices 5, 6). We
hypothesize that the variance of these two features can be at−
tributed to phylogeny, but further analysis of a larger sample
of teeth identified to genus, at least, is necessary to test this
hypothesis.
DMCTOB demonstrated the highest variability and stron−
gest correlation for PC III in the PCA for the isolated teeth in−
cluding the denticle variables (Appendix 6). However, in the
PCA using only the gross morphometric variables, the compo−
nent thus influenced became PC II, and the correlation became
negative (Appendix 5). This pattern was the same for PC II in
the PCA for the in situ teeth (Appendix 3), which also did not
include any denticle variables. This suggests a more complex
relationship between DMCTOB and the denticle dimensions;
to be expected as DMCTOB describes an aspect of the mesial
carina, which is composed of denticles (Fig. 2). Again, the
contribution of denticle dimensions to total variance is ob−
scure, although DMCTOB and the denticle dimensions de−
scribe different aspects of carina morphometrics (Fig. 2).
Despite the reduced statistical independence in the data,
the analysis for the in situ teeth allowed for a comparison of
the relative position of data points of known taxon and bone
of origin in an ordination plot. PC I tended to separate the
teeth by size, but also suggested separation of maxillary and
dentary teeth into two overlapping groups (Fig. 4A, B). PC
III spread out the maxillary teeth and the dentary teeth but did
not show any separate clustering of the two types of teeth
(Fig. 4B, C). PC II grouped the maxillary and dentary teeth
together in a tighter cluster (Fig. 4A, C). Distinction between
teeth originating in these bones must then be due to non−
size−associated shape characteristics. For the most part, how−
ever, no clear grouping of tooth type occurred in any of the
plots.
The separation of maxillary and dentary teeth suggested
by PC I was also evident in the results of the discriminant
analysis. By the absolute values of the canonical coefficients,
the most important variables for distinguishing the teeth were
DMCTOB, FABL, and THEIGHT (Table 1). From the re−
sults of the PCA for the in situ teeth (Appendix 3), FABL and
THEIGHT exhibited strong negative allometry, whereas
DMCTOB was positively allometric. The most strongly pos−
itively allometric variable, CURVATUR, also has most of its
variance explained by PC I and is most strongly correlated
with this factor (Appendix 3), and is of relatively little impor−
tance in the discriminant function (Table 1). Evidently non−
size−associated variance was more important in distinguish−
ing tyrannosaurid teeth than size−associated variance, which
is to be expected. In addition, this supports the common
size−associated scaling relationships among sub−groups pos−
ited by us and used as a necessary assumption for the PCA.
The strong contribution of DMCTOB to PC II, and its nega−
tive correlation with this factor (Appendix 3) indicate that
this feature contributes the most important non−size−asso−
ciated variance useful in discriminating among teeth.
The discriminant analysis generated two discriminant
functions that theoretically allow the determination of the
bone of origin of an isolated tooth. From the a priori assign−
ments of the isolated teeth, the discriminant functions cor−
rectly classified 34 of the 35 teeth. The misclassified tooth
(TMP 88.36.61) exhibited a structural abnormality. In most
teeth the enamel ridge of the carina does not extend proxi−
mally beyond the denticles. In this tooth, the ridge of enamel
extended beyond the end of the denticles at the base of the
tooth crown. This discrepancy resulted in the tooth being
classified a priori as a dentary tooth. However, as the mea−
surement was taken from the base of the mesial carina to the
base of the tooth, this affected the value of DMCTOB, caus−
ing it to be classified as a maxillary tooth by the discriminant
functions. As this was an abnormal tooth, the correct classifi−
cation cannot be known, and this tooth should be disre−
garded. However, the discriminant functions produced pre−
dictions of bone of origin for these data acceptable at a signif−
icant statistical level (Table 2), indicating that they would be
useful in classifying a similar sample of teeth with at least the
same acuity as an informed observer.
In order to test the predictive accuracy of the discriminant
functions, they were applied to the data for the in situ teeth,
which had a known bone of origin. For Albertosaurus,
Daspletosaurus, and Gorgosaurus, the predictions of the
discriminant functions were statistically accurate (Table 3).
Only one tooth for these taxa was misclassified. This tooth
came from the dentary of TMP 94.12.155, a small, juvenile
Gorgosaurus (Currie 2003a). Another Gorgosaurus speci−
men (TMP 86.144.1) and a Daspletosaurus specimen (TMP
94.143.1) were larger juveniles (Currie 2003a), and their
teeth were correctly classified. This suggests that the predic−
tive abilities of the discriminant functions derived from our
data (Appendix 2) may only be applicable to larger, rela−
tively more mature specimens of these taxa. However, if
more small juvenile specimens were available for study, the
data set could be expanded and re−analyzed, potentially pro−
ducing discriminant functions with greater discriminatory
power.
The accuracy of the predictions generated by these
discriminant functions for Tyrannosaurus rex was not statis−
tically significant (Table 3). Of the 28 maxillary and dentary
teeth from MOR 555 and TMP 98.86.1 (high quality research
casts of BHI 3033), only 11 were correctly classified. The
poor fit of the discriminant predictions to the T. rex data
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SAMMAN ET AL.—MORPHOMETRY OF TYRANNOSAURID TEETH 767

could be due to either of two factors. First, it was observed
while measuring the teeth that, unlike the other taxa, the
mesial denticles of T. rex teeth start closer to the base of the
teeth in the dentary. This means that the significance of the
DMCTOB would generally be the opposite for T. rex than for
Albertosaurus,Daspletosaurus, and Gorgosaurus. In addi−
tion, T. rex teeth demonstrate a trend towards a greater tooth
cross−sectional thickness relative to both fore−aft basal length
and tooth crown height relative to other tyrannosaurid taxa.
Thus, the discriminant functions derived from taxonomically
mixed data are not suitable tools for accurately identifying
the bone of origin of T. rex teeth. However, as T. rex does not
co−occur stratigraphically with the other taxa (Eberth et al.
2001) this is not a great concern.
In general, it is difficult to quantifiably distinguish the teeth
of tyrannosaurid taxa. Principal component analysis deter−
mined that the denticles do contribute to the morphological
variance of tyrannosaurid teeth. However, the significance of
this contribution, other than that explained by the size−associ−
ated shape vector, requires further study. The removal of the
variance due to isometry revealed that much of the variance
in tooth shape is due to this, although positive allometry is
important in determining the aspects of shape described by
CURVATUR and DMCTOB, and negative allometry strongly
influences denticle shape over much of the two carinae. PCA
ordination plots demonstrated that the teeth did not unambigu−
ously cluster into distinct groups based on either taxon or bone
of origin. More promisingly, discriminant functions were ef−
fective for classifying the bone of origin of teeth of relatively
mature examined specimens of Albertosaurus,Daspletosau−
rus,andGorgosaurus, although less accurate in classifying the
teeth of Tyrannosaurus rex.
Conclusions and future directions
The teeth of tyrannosaurids can be divided into five types
based on cross−sectional shape and the characteristics of the
mesial carina: (1) premaxillary, (2) mesial maxillary, (3) dis−
tal maxillary, (4) mesial dentary, and (5) distal dentary. The
ratio of the distance of the base of the mesial carina to the
base of the tooth crown/tooth crown height can be useful for
distinguishing between maxillary and dentary teeth. The
mesial denticles for relatively mature specimens of Alberto−
saurus,Daspletosaurus, and Gorgosaurus, start closer to the
base of the tooth in the maxilla, which is not true for the teeth
of Tyrannosaurus rex.
PCA and sphericity tests demonstrated the contribution
of the denticles to the morphometric variance of tyranno−
saurid teeth. However, the significance of the denticles’ con−
tribution to this variance requires further study.
PCA determined that most of the variance for any of the
sub−sets of the data analyzed was accounted for in the first
three principal components. Removal of the influence of the
predicted effects of variance associated with isometry in
shape quantified in the first principal component revealed
that much of the variance in tooth shape is due to isometric
growth, but some gross aspects of tooth shape can be attrib−
uted to allometry. PCA ordination plots demonstrated that
the teeth did not cluster unambiguously into distinct group−
ings based on taxon or bone of origin.
Discriminant functions were effective for classifying the
teeth of relatively mature examined specimens of Alberto−
saurus,Daspletosaurus, and Gorgosaurus. This may result
from variation in denticle or carina morphology that would
reflect functional differences in maxillary and dentary teeth.
As it was possible to assign teeth to bones in the animals but
not to discriminate between taxa, Albertosaurus,Daspleto−
saurus, and Gorgosaurus likely shared aspects of tooth func−
tion. Discriminant functions generally did not accurately
classify the teeth of examined specimens of Tyrannosaurus
rex. The functional implications of this difference require
further study. DFA thus suggests similarities and differences
of tooth function in adult tyrannosaurids that are amenable to
future biomechanical testing.
In summary, although the teeth of tyrannosaurid dino−
saurs must belong to one of five types based on position of
teeth in the bones of the jaws, it is difficult to quantifiably
distinguish these teeth reliably by taxon. However, for larger,
relatively mature specimens (apart from T. rex), discriminant
functions can be used to separate maxillary and dentary
teeth. This method proved more effective for assigning teeth
to bones than did PCA, and suggests the informative discrim−
inatory power of DFA for moderate and larger sample sizes.
Acknowledgements
Staff at the Royal Tyrrell Museum of Palaeontology, Drumheller (espe−
cially Jim Gardner, Jackie Wilke, and Vien Lam), Royal Ontario Mu−
seum, Toronto (especially Kevin Seymour), and Museum of the Rock−
ies, Bozeman (especially Jack Horner and Dave Varricchio) are sin−
cerely thanked for allowing and coordinating access to tyrannosaurid
jaw specimens and teeth in their care during data collection for this
study. Don Brinkman (Royal Tyrrell Museum of Palaeontology),
Christine Chandler (Putnam Museum of History & Natural Science),
and Julia Sankey (California State University Stanislaus) are recog−
nized for their tooth measurement protocol discussions. Thanks are ex−
tended to Thomas Carr (Carthage College) for many insightful discus−
sions about tyrannosaurids. Blaine Schubert (East Tennessee State Uni−
versity) and Peter Ungar (University of Arkansas) are thanked for shar−
ing information from their tooth wear study. Thank you to Eric Snively
(University of Calgary, Department of Biological Sciences) for editing
and suggesting improvements for the manuscript, and Anthony Russell
(University of Calgary, Department of Biological Sciences) for many
helpful suggestions. Much gratitude is also due to colleagues in at the
University of Calgary (Departments of Geology & Geophysics and Bi−
ological Sciences), and the Royal Tyrrell Museum of Palaeontology for
their helpful discussions, advice, access to equipment, and transporta−
tion. Comments from Øyvind Hammer (Geological Museum, Oslo),
Blaine Schubert, Thomas Carr, and anonymous reviewers greatly im−
proved this manuscript. This research was supported, in part, by
funding from the University of Calgary, and contributions from an
NSERC research grant awarded to L.V. Hills.
768 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005

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Appendix 1
Log10−transformed data for the gross mensural characters, and known tooth position for specimens of in situ tyrannosaurid teeth
(n = 71). Negative values for CURVATUR indicate teeth that do not recurve. (j) indicates juvenile specimen (see Currie
2003a). All specimens are from the Judith River Group, Alberta, Canada, except those marked with an *: TMP 81.10.01 is from
the Horseshoe Canyon Formation of the Edmonton Group, Alberta, Canada, MOR specimens are from the Two Medicine For−
mation, Montana, U.S.A and the original of TMP 98.86.1 (high quality research casts of BHI 3033) is from the Hell Creek For−
mation, South Dakota, USA.
Specimen Position FABL THEIGHT XSTHICK CURVATUR DMCTOB
Gorgosaurus libratus
TMP 67.9.164 D 1.348 1.613 1.246 0.368 1.124
TMP 83.36.100 M 1.182 1.509 0.959 0.322 0.740
TMP 83.36.134 D 1.396 1.680 1.236 0.491 1.170
TMP 83.36.134 D 1.354 1.609 1.238 0.041 1.100
TMP 86.144.1 (j) D 1.130 1.464 0.929 0.362 0.944
TMP 86.144.1 (j) D 1.146 1.467 0.940 0.380 0.968
TMP 86.144.1 (j) D 1.130 1.375 0.886 0.415 0.863
TMP 94.12.155 (j) D 0.991 1.217 0.633 0.380 0.362
TMP 95.5.1 D 1.045 1.387 1.049 –0.477 0.778
TMP 95.5.1 D 1.305 1.645 1.161 0.766 1.086
TMP 95.5.1 D 1.301 1.625 1.041 0.230 1.025
TMP 95.5.1 D 1.283 1.599 1.134 0.222 0.996
TMP 95.5.1 D 1.286 1.606 1.140 0.368 1.255
TMP 95.5.1 D 1.255 1.526 1.114 0.301 1.068
TMP 95.5.1 D 1.233 1.480 1.104 0.263 0.934
TMP 95.5.1 D 1.137 1.330 0.954 0.114 0.944
TMP 99.55.170 D 1.230 1.628 1.170 0.230 0.892
TMP 99.55.170 D 1.290 1.660 1.072 0.204 1.021
TMP 99.55.170 D 1.250 1.574 1.104 0.079 0.944
TMP 99.55.170 D 1.243 1.531 1.079 0.279 0.886
TMP 99.55.170 D 1.230 1.496 0.991 0.230 0.949
TMP 99.55.170 D 1.176 1.391 1.004 0.230 0.973
TMP 99.55.170 D 1.146 1.350 0.934 –0.398 0.949
ROM 1247 D 1.053 1.435 1.097 –0.155 1.013
ROM 1247 D 1.236 1.627 1.170 0.431 1.134
ROM 1247 D 1.322 1.672 1.053 0.491 1.049
ROM 1247 D 1.326 1.655 1.121 0.556 1.064
ROM 1247 D 1.316 1.632 1.124 0.505 1.068
ROM 1247 D 1.093 1.336 0.903 0.362 1.061
Albertosaurus sarcophagus*
TMP 81.10.1 M 1.493 1.822 1.274 0.690 0.778
Daspletosaurus sp.1 (see Currie 2003a)
TMP 97.12.223 M 1.537 1.895 1.386 1.045 0.845
TMP 97.12.223 M 1.377 1.666 1.288 0.699 0.898
TMP 94.143.1 (j) D 1.114 1.465 1.025 0.204 0.820
TMP 94.143.1 (j) D 1.223 1.507 1.045 0.519 0.914
TMP 94.143.1 (j) D 1.260 1.501 1.009 0.491 0.959
Daspletosaurus sp.2* (see Horner et al. 1992)
MOR 590 D 1.356 1.750 1.270 0.342 1.083
MOR 590 D 1.354 1.755 1.241 0.146 1.061
MOR 590 D 1.342 1.718 1.204 0.041 1.057

772 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005
Specimen Position FABL THEIGHT XSTHICK CURVATUR DMCTOB
Tyrannosaurus rex*
MOR 555 M 1.603 1.857 1.449 0.806 1.260
MOR 555 M 1.558 1.828 1.393 0.653 1.442
MOR 555 M 1.501 1.751 1.375 0.531 1,412
TMP 98.86.1, high quality research cast PM 1.500 1.711 1.312 0 –0.301
TMP 98.86.1, high quality research cast M 1.691 1.980 1.465 1.061 1.417
TMP 98.86.1, high quality research cast M 1.661 1.914 1.491 1.117 1.450
TMP 98.86.1, high quality research cast M 1.592 1.850 1.441 0.898 1.484
TMP 98.86.1, high quality research cast M 1.474 1.746 1.400 0.806 1.301
TMP 98.86.1, high quality research cast M 1.446 1.693 1.358 0.708 1.340
TMP 98.86.1, high quality research cast M 1.316 1.501 1.176 0.176 1.111
TMP 98.86.1, high quality research cast D 1.408 1.601 1.246 0 0.740
TMP 98.86.1, high quality research cast D 1.566 1.829 1.462 –0.301 0.898
TMP 98.86.1, high quality research cast D 1.498 1.754 1.360 0.732 1.152
TMP 98.86.1, high quality research cast D 1.439 1.626 1.310 –0.398 0.991
TMP 98.86.1, high quality research cast D 1.155 1.233 0.978 0 0.708
TMP 98.86.1, high quality research cast PM 1.427 1.562 1.158 0 –0.046
TMP 98.86.1, high quality research cast PM 1.464 1.618 1.262 0 0.505
TMP 98.86.1, high quality research cast PM 1.450 1.657 1.182 0 0.322
TMP 98.86.1, high quality research cast PM 1.490 1.667 1.288 0 0.505
TMP 98.86.1, high quality research cast M 1.595 1.952 1.668 0.851 0.255
TMP 98.86.1, high quality research cast M 1.667 1.998 1.508 1.188 1.401
TMP 98.86.1, high quality research cast M 1.610 1.806 1.373 1.100 1.430
TMP 98.86.1, high quality research cast M 1.524 1.731 1.326 0.806 1.307
TMP 98.86.1, high quality research cast D 1.400 1.609 1.274 0 0.279
TMP 98.86.1, high quality research cast D 1.606 1.863 1.453 0.398 0.845
TMP 98.86.1, high quality research cast D 1.647 1.880 1.522 0.839 1.272
TMP 98.86.1, high quality research cast D 1.636 1.857 1.494 0.778 1.356
TMP 98.86.1, high quality research cast D 1.595 1.782 1.476 0.724 1.253
TMP 98.86.1, high quality research cast D 1.462 1.744 1.377 0.431 1.215
TMP 98.86.1, high quality research cast D 1.507 1.683 1.360 0.447 1.330
TMP 98.86.1, high quality research cast D 1.459 1.658 1.342 0.204 1.021
TMP 98.86.1, high quality research cast D 1.352 1.474 1.238 –0.523 0.756
TMP 98.86.1, high quality research cast D 1.246 1.365 1.137 0 0.964
Appendix 2
Log10−transformed data for the gross mensural characters (denticle morphometric data available as supplementary data online),
and a priori estimated tooth position for specimens of isolated tyrannosaurid teeth (n = 35).
Specimen
(TMP)
a priori
position FABL THEIGHT XSTHICK CURVATUR DMCTOB
66.28.5 D 0.991 1.243 0.875 0.058 0.633
67.16.62 D 1.104 1.511 1.000 0.862 1.121
79.8.97 D 1.283 1.520 1.130 0.109 0.944
80.16.1094 D 0.968 1.134 0.763 0.234 0.643
80.20.313 D 1.029 1.111 0.875 0.196 0.892
81.18.126 M 1.303 1.676 1.134 0.686 0.820
81.19.7 M 1.255 1.520 1.093 0.196 0.785
82.16.170 D 1.134 1.422 0.833 0.301 0.806
82.18.214 D 1.130 1.362 0.924 0.301 0.851

http://app.pan.pl/acta50/app50−xxx.pdf
SAMMAN ET AL.—MORPHOMETRY OF TYRANNOSAURID TEETH 773
Specimen
(TMP)
a priori
position FABL THEIGHT XSTHICK CURVATUR DMCTOB
84.163.35 D 1.057 1.258 0.851 0.109 0.929
85.36.325 D 0.944 1.127 0.763 0.234 0.681
85.36.342 M 1.215 1.489 0.982 0.410 0.462
87.36.325 M 1.127 1.346 0.944 0.196 0.477
88.36.61 D 1.193 1.483 1.072 0.477 0.663
89.18.70 D 1.297 1.584 1.134 0.359 0.903
89.36.151 D 1.037 1.204 0.892 0.000 0.724
89.79.4 D 1.000 1.455 1.117 –0.243 0.820
91.144.7 D 1.384 1.691 1.220 0.570 1.290
91.36.188 D 1.100 1.346 0.898 0.196 0.771
91.36.431 D 0.973 1.124 0.845 0.058 0.591
91.50.32 M 1.100 1.283 0.857 –0.243 0.415
91.61.9 M 1.107 1.413 0.820 0.234 0.653
92.36.233 D 1.225 1.464 1.090 0.109 0.869
92.36.823 M 1.233 1.529 0.982 0.570 0.792
92.36.961 M 1.155 1.456 0.987 0.385 0.602
93.36.437 D 1.340 1.594 1.068 0.699 1.083
93.36.610 M 1.286 1.645 1.064 0.497 0.146
94.12.209 D 1.294 1.610 1.179 0.359 0.987
94.12.313 D 1.230 1.576 1.053 0.234 1.025
94.12.42 M 1.408 1.710 1.281 0.234 0.987
94.12.759 D 1.029 1.233 0.903 0.058 0.826
96.12.111 D 1.111 1.365 0.857 0.570 0.839
96.12.379 M 1.196 1.493 1.021 0.385 0.748
97.12.43 D 1.324 1.639 1.199 0.497 1.170
99.55.202 D 1.167 1.481 0.940 0.535 0.851
Appendix 3
PCA of in situ teeth, using gross morphological variables. FABL, fore−aft basal length; THEIGHT, tooth crown height;
XSTHICK, cross−sectional thickness; CURVATUR, curvature; DMCTOB, distance from the base of the mesial carina to the
base of the tooth. Components subsequent to PC IV isotropic (PC III–PC V: Bartlett’s c2= 476.963, 5 df; P < 0.0001; PC IV–PC
V: Bartlett’s c2= –400.886, 2 df; P = 0.999). Isometric loading for PC I = 0.447214.
Principal component loadings
I II III IV V
FABL 0.241 0.337 0.372 0.193 –0.808
THEIGHT 0.272 0.306 0.315 –0.843 0.152
XSTHICK 0.252 0.333 0.509 0.495 0.567
CURVATUR 0.733 0.222 –0.636 0.087 0.039
DMCTOB 0.517 –0.795 0.314 –0.01 –0.035
Eigenvalues 0.235 0.075 0.048 0.003 0.002
Percent of variance explained 64.759 20.641 13.258 0.795 0.546
Amount of variance of each variable explained by each component
I II III IV V
FABL 0.452 0.282 0.220 0.004 0.043
THEIGHT 0.557 0.224 0.153 0.065 0.001
XSTHICK 0.402 0.225 0.336 0.019 0.017
CURVATUR 0.845 0.025 0.130 0.000 0.000
DMCTOB 0.547 0.412 0.041 0.000 0.000

774 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005
Correlations between each component and each variable
I II III IV V
FABL 0.672 0.531 0.469 0.06 –0.207
THEIGHT 0.746 0.473 0.391 –0.256 0.038
XSTHICK 0.634 0.474 0.58 0.138 0.131
CURVATUR 0.919 0.157 –0.361 0.012 0.005
DMCTOB 0.739 –0.642 0.203 –0.002 –0.005
Appendix 4
PCA of in situ teeth derived from variance−covariance matrix with variance due to isometric PC I removed, using gross mor−
phological variables. FABL, fore−aft basal length; THEIGHT, tooth crown height; XSTHICK, cross−sectional thickness;
CURVATUR, curvature; DMCTOB, distance from the base of the mesial carina to the base of the tooth. No components isotro−
pic (PC II–PC IV: Bartlett’s c2= 750.660, 5 df; P < 0.0001; PC III–PC IV: Bartlett’s c2= 406.610, 2 df; P < 0.0001).
Principal component loadings
I II III IV
FABL 0.377 0.007 0.061 –0.809
THEIGHT 0.309 –0.017 –0.770 0.333
XSTHICK 0.404 0.073 0.630 0.484
CURVATUR –0.502 –0.736 0.080 0.013
DMCTOB –0.589 0.673 –0.001 –0.021
Eigenvalues 0.091 0.066 0.003 0.002
Percent of variance explained 56.020 40.756 1.902 1.322
Amount of variance of each variable explained by each component
I II III IV
FABL 0.901 0.000 0.001 0.098
THEIGHT 0.807 0.002 0.170 0.022
XSTHICK 0.877 0.021 0.072 0.030
CURVATUR 0.390 0.610 0.000 0.000
DMCTOB 0.513 0.487 0.000 0.000
Correlations between each component and each variable
I II III IV
FABL 0.956 0.016 0.029 –0.315
THEIGHT 0.904 –0.043 –0.415 0.149
XSTHICK 0.943 0.146 0.271 0.173
CURVATUR –0.629 –0.787 0.019 0.003
DMCTOB –0.721 0.703 0.000 –0.004
Appendix 5
PCA of isolated teeth, using gross morphological variables. FABL, fore−aft basal length; THEIGHT, tooth crown height;
XSTHICK, cross−sectional thickness; CURVATUR, curvature; DMCTOB, distance from the base of the mesial carina to the
base of the tooth. Components subsequent to PC IV isotropic (PC III–PC V: Bartlett’s c2= 222.400, 5 df, P < 0.0001;
PC IV–PC V: Bartlett’s c2= –189.687, 2 df, P = 0.999). Isometric loading for PC I = 0.447214.
Principal component loadings
I II III IV V
FABL 0.319 0.008 0.362 0.492 0.725
THEIGHT 0.452 0.050 0.510 0.307 –0.662

SAMMAN ET AL.—MORPHOMETRY OF TYRANNOSAURID TEETH 775
XSTHICK 0.308 –0.173 0.456 –0.797 0.179
CURVATUR 0.615 0.625 –0.455 –0.147 0.049
DMCTOB 0.470 –0.760 –0.440 0.086 –0.037
Eigenvalues 0.104 0.039 0.030 0.002 0.002
Percent of variance explained 59.067 21.990 16.933 1.116 0.894
Amount of variance of each variable explained by each component
I II III IV V
FABL 0.670 0.000 0.247 0.030 0.052
THEIGHT 0.709 0.003 0.259 0.006 0.023
XSTHICK 0.532 0.063 0.336 0.067 0.003
CURVATUR 0.649 0.249 0.101 0.001 0.000
DMCTOB 0.449 0.437 0.113 0.000 0.000
Correlations between each component and each variable
I II III IV V
FABL 0.819 0.012 0.497 0.173 0.229
THEIGHT 0.842 0.057 0.508 0.079 –0.152
XSTHICK 0.729 –0.250 0.579 –0.260 0.052
CURVATUR 0.805 0.499 –0.319 –0.027 0.008
DMCTOB 0.670 –0.661 –0.336 0.017 –0.006
Appendix 6
PCA of isolated teeth, incorporating denticle variables. FABL, fore−aft basal length; THEIGHT, tooth crown height;
XSTHICK, cross−sectional thickness; CURVATUR, curvature; DMCTOB, distance from the base of the mesial carina to the
base of the tooth; MDW, mesial denticle width (basal, middle, apical); DDW, distal denticle width (B, basal, M, middle, A, api−
cal); MDH, mesial denticle height (basal, middle, apical); DDH, distal denticle height (basal, middle, apical). Components
subsequent to PC X isotropic (PC IX–PC XVII: Bartlett’s c2= 68.040, 44 df; P = 0.012; PC X–PC XVII: Bartlett’s c2= 48.695,
35 df; P = 0.062). Isometric loading for PC I = 0.242536.
Principal component loadings
I II III IV V VI VII VIII IX X
FABL 0.288 0.031 0.048 –0.234 0.114 –0.293 –0.210 0.093 –0.210 –0.189
THEIGHT 0.417 0.056 0.104 –0.372 0.119 –0.108 –0.167 0.033 0.112 –0.059
XSTHICK 0.305 0.149 –0.114 –0.195 0.244 –0.146 0.182 0.157 –0.137 –0.357
CURVATUR 0.393 –0.690 0.519 0.203 0.060 0.091 0.104 0.063 0.005 0.094
DMCTOB 0.315 –0.405 –0.820 0.196 0.002 –0.032 –0.048 –0.100 0.044 0.038
MDWB 0.231 0.110 –0.019 –0.144 –0.393 0.087 –0.540 0.296 –0.170 0.429
MDWM 0.205 0.088 –0.045 –0.022 –0.124 0.090 0.255 0.107 0.029 0.119
MDWA 0.127 0.060 –0.008 –0.044 0.031 –0.132 0.267 0.122 0.372 0.012
DDWB 0.177 0.228 –0.053 0.032 –0.086 0.301 0.369 0.269 0.158 0.089
DDWM 0.149 0.154 –0.044 0.080 –0.134 0.063 0.299 0.191 –0.291 0.143
DDWA 0.131 0.118 0.012 –0.068 0.101 –0.175 0.196 0.127 0.188 0.186
MDHB 0.237 0.087 0.052 –0.266 –0.231 0.239 –0.083 –0.582 0.379 –0.188
MDHM 0.262 0.166 0.056 0.079 –0.253 0.233 0.217 –0.426 –0.401 0.039
MDHA 0.126 0.211 –0.019 0.144 0.725 0.520 –0.229 –0.057 –0.041 0.177
DDHB 0.196 0.247 0.084 0.614 –0.205 0.043 –0.288 0.228 0.321 –0.400
DDHM 0.125 0.141 0.070 0.315 0.040 –0.242 0.008 –0.162 –0.411 –0.250
DDHA 0.143 0.243 0.080 0.283 0.128 –0.520 –0.005 –0.334 0.179 0.520
Eigenvalues 0.150 0.055 0.040 0.017 0.012 0.009 0.006 0.006 0.004 0.003
Percent of variance explained 48.831 17.790 12.838 5.418 3.759 2.879 2.015 1.888 1.294 1.008

776 ACTA PALAEONTOLOGICA POLONICA 50 (4), 2005
Amount of variance of each variable explained by each component
I II III IV V VI VII VIII IX X
FABL 0.828 0.004 0.006 0.061 0.010 0.051 0.018 0.003 0.012 0.007
THEIGHT 0.885 0.006 0.014 0.078 0.006 0.004 0.006 0.000 0.002 0.000
XSTHICK 0.775 0.067 0.029 0.035 0.038 0.010 0.011 0.008 0.004 0.022
CURVATUR 0.381 0.429 0.175 0.011 0.001 0.001 0.001 0.000 0.000 0.000
DMCTOB 0.291 0.176 0.519 0.012 0.000 0.000 0.000 0.001 0.000 0.000
MDWB 0.577 0.048 0.001 0.025 0.128 0.005 0.130 0.037 0.008 0.041
MDWM 0.832 0.056 0.010 0.001 0.024 0.009 0.053 0.009 0.000 0.006
MDWA 0.621 0.051 0.001 0.008 0.003 0.039 0.113 0.022 0.141 0.000
DDWB 0.472 0.286 0.011 0.002 0.009 0.081 0.085 0.042 0.010 0.002
DDWM 0.534 0.210 0.012 0.017 0.033 0.006 0.089 0.034 0.054 0.010
DDWA 0.588 0.172 0.001 0.018 0.027 0.062 0.054 0.021 0.032 0.024
MDHB 0.604 0.029 0.008 0.085 0.044 0.036 0.003 0.141 0.041 0.008
MDHM 0.675 0.099 0.008 0.007 0.049 0.032 0.019 0.069 0.042 0.000
MDHA 0.168 0.173 0.001 0.024 0.431 0.170 0.023 0.001 0.000 0.007
DDHB 0.322 0.187 0.016 0.351 0.027 0.001 0.029 0.017 0.023 0.028
DDHM 0.341 0.160 0.029 0.242 0.003 0.076 0.000 0.022 0.098 0.028
DDHA 0.254 0.267 0.021 0.110 0.016 0.198 0.000 0.054 0.011 0.069
Correlations between each component and each variable
I II III IV V VI VII VIII IX X
FABL 0.888 0.058 0.075 –0.241 0.097 –0.219 –0.132 0.057 –0.105 –0.084
THEIGHT 0.931 0.075 0.119 –0.277 0.074 –0.059 –0.076 0.014 0.041 –0.019
XSTHICK 0.869 0.256 –0.167 –0.185 0.192 –0.101 0.105 0.088 –0.064 –0.146
CURVATUR 0.617 –0.655 0.418 0.106 0.026 0.035 0.033 0.019 0.001 0.021
DMCTOB 0.540 –0.419 –0.720 0.112 0.001 –0.013 –0.017 –0.034 0.012 0.009
MDWB 0.754 0.217 –0.032 –0.156 –0.355 0.069 –0.358 0.190 –0.090 0.201
MDWM 0.871 0.226 –0.097 –0.031 –0.146 0.093 0.219 0.089 0.020 0.072
MDWA 0.736 0.211 –0.023 –0.084 0.051 –0.185 0.314 0.139 0.351 0.010
DDWB 0.663 0.516 –0.102 0.039 –0.090 0.274 0.281 0.198 0.096 0.048
DDWM 0.704 0.441 –0.107 0.126 –0.176 0.073 0.288 0.178 –0.224 0.097
DDWA 0.721 0.390 0.033 –0.125 0.154 –0.234 0.219 0.137 0.168 0.147
MDHB 0.770 0.170 0.086 –0.288 –0.209 0.189 –0.055 –0.372 0.201 –0.088
MDHM 0.809 0.309 0.089 0.082 –0.217 0.175 0.136 –0.259 –0.202 0.017
MDHA 0.410 0.415 –0.032 0.156 0.656 0.412 –0.152 –0.037 –0.022 0.083
DDHB 0.565 0.431 0.124 0.591 –0.164 0.030 –0.169 0.129 0.151 –0.166
DDHM 0.553 0.378 0.160 0.466 0.049 –0.261 0.007 –0.142 –0.297 –0.159
DDHA 0.501 0.515 0.144 0.331 0.124 –0.443 –0.004 –0.231 0.102 0.262