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Comparison of four simple methods for estimating sexual dimorphism in fossils.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 94:465476 (1994)
Comparison of Four Simple Methods for Estimating Sexual
Dimorphism in Fossils
J. MICHAEL PLAVCAN
Department of Biological Sciences, Uniuersity of Cincinnati, Cincinnati,
Ohio 45221
KEY WORDS
Sexual dimorphism, Coefficient of variation,
Finite mixture analysis, Primates
ABSTRACT
Estimating sexual dimorphism in skeletal and dental features of fossil species is difficult when the sex of individuals cannot be reliably
determined. Several different methods of estimating dimorphism in this situation have been suggested: extrapolation from coefficients of variation, division of a sample about the mean or median into two subsamples which are
then treated as males and females, and finite mixture analysis (specifically
for estimating the maximum dimorphism that could be present in a unimodal
distribution). The accuracy of none of these methods has been thoroughly
investigated and compared in a controlled manner. Such analysis is necessary
because the accuracy of all methods is potentially affected by fluctuations in
either sample size, sex ratio, or the magnitude of intrasexual variability.
Computer modeling experiments show that the mean method is the least
sensitive to fluctuations in these parameters and generally provides the best
estimates of dimorphism. However, no method can accurately estimate low to
moderate levels of dimorphism, particularly if intrasexual variability is high
and sex ratios are skewed. o 1994 Wiley-Liss, Inc.
Sexual dimorphism can be a substantial
component of intraspecific morphological
variability, and its recognition is important
in understanding morphological variation
and species recognition in the fossil record
(for recent discussions, see Cope, 1993;
Kelley, 1993; Martin and Andrews, 1993;
Plavcan, 1993; Teaford et al., 1993). Furthermore, among primates sexual dimorphism in canine tooth size and body weight
are associated with variation in mating system and intrasexual competition (CluttonBrock et al., 1977; Harvey et al., 1978; Leutenegger and Kelly, 1977; Kay et al., 1988;
Plavcan and van Schaik, 1992; Greenfield,
1992). Following this, it has been suggested
that the behavior of extinct species can be
inferred on the basis of sexual dimorphism
(Fleagle et al., 1980; Kay, 1982a,b). For example, in the primate fossil record polygyny
has been inferred for Oligocene anthropoids
(Fleagle et al., 1980), Siuupithecus (Andrews, 19831, Chinese hominoids (Wu and
0 1994 WILEY-LISS, INC
Oxnard, 19831, and Eocene omomyids
(Krishtalka et al., 19901, and either monogamy or low-competitionpolygyny in australopithecines (Leutenegger and Shell, 1987;
McHenry, 1991).
Such inferences of the behavior of extinct
species are only as good as the estimate of
dimorphism. Estimating dimorphism is a
trivial task if the sex of individuals can be
identified by discrete, sex-specific morphological characters or if the sample frequency
distribution is clearly bimodal (in all cases,
though, one must have good evidence that
the “dimorphism” is not a product of the
mixing of two similar species or geographic
variants of a single species). However, accurately estimating dimorphism is no simple
Received June 14,1993; accepted February 28,1994.
J. Michael Plavcan’s current address is Department of Anatomy, New York College of Osteopathic Medicine, Old Westbury,
NY 11568. Address reprint requests there.
466
J.M. PLAVCAN
task if males and females cannot be reliably
identified, either because sex-specific morC. torquatus
phological characters are not preserved or
MIF = 1.41
were not present (many skeletal and dental
characters-ven
canine teeth-are often
dimorphic only in size) or because male and
female distributions overlap too much. The
latter problem is exacerbated by small sample sizes, which confound the recognition of
bimodal distributions in dimorphic characC. nictitans
ters (Cope, 1989). Dimorphism of up to 28%
(one sex larger than the other) can be hidden
within unimodal distributions for sample
sizes of up to 100 individuals (Godfrey et al.,
1993) (Fig. 1). While the canine teeth of
many anthropoid primates and their ancestors are considerably more dimorphic than
this (allowing the sexing of individual speciC. cephus
mens as long as the canine teeth are preserved), cranial and skeletal features of dimorphic primates often fall below this
threshold (Godfrey et al., 1993; Godfrey personal communication). Where male and female distributions overlap, some specimens
may be identifiable as belonging to one or
the other sex, but these will tend to lie at the
extremes, producing overestimates of the
C. pogonias
degree of dimorphism in the population
(Kelley, 1993). Such problems with estimatM/F= 1.16
ing sexual dimorphism in extinct taxa due to
uncertain sex assignment have received
particular attention for hominoids and hom0
inids (Kay, 1982a,b; Kelley, 1993, Oxnard,
90
1987; Leutenegger and Shell, 1987) and
2o
Maxillary Canine Buccolingual (mm)
most recently subfossil lemurs (Godfrey et
al., 1993).Since most extinct species are repFig. 1. Frequency histograms of maxillary canine
resented by small samples of fragmentary buccolingual breadth for combined-sex samples of Cerremains, a method of estimating dimor- copithecus pogonias, C . cephus, C. nictitans, and Cercophism that does not rely on visual assess- cebus torquatus illustrating approximate degrees of bimodality associated with different degrees of sexual
ments of frequency distributions andlor the dimorphism (MIF). Units are the same in each histoprior knowledge of the sex of individuals is gram. Data from Plavcan (1990).
clearly desirable. This would both avoid the
problems of uncertain sex assignment and
allow estimation of dimorphism using cra- (Fleagle et al., 1980; Kay 1982a,b; Godfrey
nial and skeletal remains that are not easily et al., 1993). Such techniques rely on the fact
that, as dimorphism increases, combinedsexed.
To get around the problem of estimating sex sample variability increases as a funcsexual dimorphism in fossil samples where tion of the separation between male and fethe sex of individuals cannot be reliably de- male means. Division of a sample into
termined, several studies have suggested hypothetical male and female subsamples
that it is possible to estimate sexual dimor- about either the mean or median of the samphism using techniques that correlate sam- ple (Godfrey et al., 1993) is not often used
ple variability with sexual dimorphism but represents the simplest way of estimat-
i
1
FOUR METHODS FOR ESTIMATING SEXUAL DIMORPHISM
ing dimorphism in a fossil sample. Two more
sophisticated methods are extrapolation of
dimorphism from coefficients of variation
(Fleagle et al., 1980; Kay, 1982a,b),and estimation of maximum dimorphism in unimodal samples with finite mixture analysis
(described in Godfrey et al., 1993). These
methods have been employed to estimate
sexual dimorphism in hominoids (Kay
1982a,b), australopithecines (Leutenegger
and Shell, 19871, and subfossil lemurs (Godfrey et al., 1993), all of which present difficulties for estimating sexual dimorphism.
Division of a sample into two subsamples
by the mean or median (referred to hereafter
as the “mean” and “median” techniques following Godfrey et al., 1993) are the simplest
techniques for estimating dimorphism. The
combined-sex sample is simply divided into
two subsamples either about the sample
mean or the median, and the ratio of the
means of the newly created subsamples represents the estimate of dimorphism. These
methods assume that male and female distributions do not o v e r l a p a situation that
rarely occurs except when dimorphism is extreme. Criticism also may be leveled at these
techniques for arbitrarily creating male and
female means, thereby creating the impression of sexual dimorphism when in fact none
may be present. Nevertheless, Godfrey et al.
(1993)find that the mean method appears to
be quite accurate, especially when dimorphism is substantial.
Using data from extant species, several
studies report that coefficients of variation
(CV) from pooled-sex samples are very
highly correlated with sexual dimorphism
as measured by a ratio of male to female
means (Fleagle et al., 1980; Kay 1982a,b;
Leutenegger and Shell, 1987). These studies
suggest that dimorphism in extinct species
can be easily extrapolated from a simple regression equation between CVs and dimorphism. This relation makes intuitive sense.
The CV is nothing more than the sample
standard deviation divided by the sample
mean (usually multiplied by 100 to express
the ratio as a percentage). With increasing
sexual dimorphism, the difference between
male and female means increases, causing a
proportional increase in the pooled-sex sample standard deviation. However, this tech-
467
nique has been strongly criticized on the
grounds that, for several extant anthropoid
species, CVs of dental dimensions from combined-sex samples are not necessarily
higher than those of single-sex samples
(Martin, 1983; Martin and Andrews, 1984;
Vitzthum, 1990). While most of these observations are based on non-dimorphic measurements of teeth, they severely undermine confidence in the technique.
Finite mixture analysis (FMA) is described in Godfrey et al. (1993). Unlike the
other three techniques, this technique is
designed to demonstrate either the lack of
dimorphism in a sample or the maximum
dimorphism that could be present within a
unimodal distribution. Because dimorphism
of up to 28% can be hidden within a single
unimodal distribution, even for sample sizes
as large as 100 (Godfrey et al., 19931, this
method could prove very useful. FMA is
based on the observation that unimodal distributions can be generated from the mixture of two normal distributions, but only
within certain limits. The method quantifies
the maximum separation of male and female means that can be contained within a
unimodal distribution on the basis of the
combined-sex sample range. To briefly summarize the method presented in Godfrey et
al. (19931, within a given whole-sample
range, a certain number of standard deviation, k,will on average occur. This value will
vary depending on sample size and can be
looked up in a table provided by Pearson
(1932) (also reproduced in Godfrey et al.,
1993). The maximum number of subsample
(male and female) standard deviations that
can be contained in the total sample’s observed range and still produce a unimodal
k. The inverse of this
distribution is
value is equivalent to the percentage of the
observed whole-sample range comprising
the difference between the mean of the
whole sample and either of the subsample
means. Therefore, multiplying this result by
the observed sample range yields the distance of the whole-sample mean from either
subsample mean. This value is added or subtracted from the whole-sample mean to yield
the means of the two subsamples, which are
then used to calculate the maximum dimorphism that could be contained in the sample.
468
J.M.PLAVCAN
All of these methods rely to some extent
on the assumption that as dimorphism increases, the pooled-sex sample variability
increases in proportion to the difference between the male and female means. Because
of this, each method is potentially confounded by fluctuations in sex ratio, small
sample sizes, and fluctuations in intrasexual variability. Deviations from a balanced
sex ratio should lower pooled-sex sample
variability, since the pooled-sex sample
mean will be closer to the mean of one sex,
and therefore more individuals will be closer
to the pooled-sex mean. Small sample sizes
increase the likelihood that pooled-sex variation will be influenced by sampling error
and increases the likelihood that the sex ratio will be imbalanced (or even that a sample
is composed of only one sex!). Finally, since
the pooled-sex sample standard deviation is
a function of both intrasexual variation and
the difference between male and female
means, increased intrasexual variability
can potentially swamp out variation due to a
difference between the male and female
means especially when dimorphism is relatively slight.
While some analysis of the influence of
sample size, imbalance in the sample sex
ratio, and high intersexual variability on
each method has been provided (Kay, 1982a;
Godfrey et al., 19931, these studies have
used data from a limited number of extant
species and have not actually quantified and
compared the error of each method with
variation in each of these three factors.
Without a direct comparison of the methods,
it is difficult to decide under what conditions
which, if any, works best. Computer modeling offers an easy way to examine the influence of confounding factors on the accuracy
of each method. By demonstrating the advantages and disadvantages of each technique using artificial data, the method
likely to yield the best results can then be
selected depending on the actual nature of
the data that one is likely to encounter in the
fossil record. For example, one method may
work better than others a t small sample
sizes, or may be relatively robust to changes
in the ratio of males to females in a sample.
This analysis presents the results of a
computer modeling experiment which com-
TABLE 1. Procedure for generating simulations
Step 1: Set the initial population parameters: 1)male
and female CVs; 2) female mean
Step 2: Set the male mean, based on the female
mean, so that dimorphism
equals a set value
Step 3: Randomly sample a specific number of males
and females from the (infinite)population
with parameters set in steps 1 and 2
Step 4: Calculate the actual sample dimorphism
Step 5: Calculate estimates of dimorphism based on
the CV, FMA, mean, and median methods
Step 6: Repeat steps 3-5 100 times and then calculate
the averages and standard deviations of the
dimorphism estimates
Step 7: Go back to step 2, recalculating the male
mean to achieve a new level of dimorphism,
and repeat steps 3-6; do this for as many
levels of dimomhism as desired
pares the accuracy of these four methods under the influence of variation in sample size,
intrasexual variability, and the ratio of
males to females in a sample.
METHODS
The computer model and
experimental design
Table 1 provides an outline of the analytical procedure. A computer model was used
to randomly generate gaussian samples of
males and females using the algorithm of
Box and Mueller (19581, modified so that
sample means and variances could be selected by the user. Generation of these samples was analogous to randomly selecting a
subsample from an infinite sample of known
mean and variance. Therefore, while each
sample generated had a unique mean and
variance, the average mean and variance of
a large number of such samples were normally distributed about the user-defined
mean and variance.
For each experiment, a series of 100 samples were generated for 10 levels of dimorphism, rangingfrom 1.0 to 1.9 in increments
of 0.1 (expressing the ratio of male to female
means-a convention used throughout the
text). Thus, for an experiment, 100 samples
would be selected from an infinite population with sexual dimorphism of 1.0; then 100
more samples would be selected from an infinite population with sexual dimorphism of
1.1, and so on until a total of 1,000 samples
was generated. Because the samples were
randomly generated, dimorphism was not
FOUR METHODS FOR ESTIMATING SEXUAL DIMORPHISM
469
exactly 1.0, 1.1,etc., but instead the dimor- with this program at different sample sizes
phism of the samples was normally distrib- and levels of intrasexual variation. Sample
uted around the “true” dimorphism (that is, size was not allowed to randomly fluctuate
the dimorphism specified by the user).
because this is a known value in the fossil
While dimorphism of 1.9 will usually pro- samples. Intrasexual variation was not alduce bimodal distributions with little or no lowed to randomly fluctuate because this
overlap between male and female mans (ob- factor is strongly dependent on the variable
viating the use of any of the techniques ex- being measured and is thus highly sampleamined here), this range of dimorphism was dependent. For example, teeth are comnecessary to accurately characterize the be- monly known to be much less variable than
havior of all four methods, especially the CV other skeletal measurements (Yablakov,
method. In fact, interspecific variation in di- 1974; Simpson et al., 1960),while variability
morphism in morphological characters is in the canine teeth fluctuates substantially
highly variable. Thus, canine dimorphism among species (Gingerich, 1974; Plavcan,
often exceeds 1.9 in magnitude, while skull 1990).
lengths rarely exceeds about 1.4 (Godfrey,
Extrapolating dimorphism with CVs
personal communication). For most applications to data in the fossil record, the techBefore performing the simulations, it was
niques analyzed here will be useful when necessary to reformulate the way CVs are
dimorphism is less than about 1.4 in magni- used to estimate sexual dimorphism. In the
tude, though this upper boundary depends past, dimorphism in a fossil sample was estion sample size, sex ratio, and the level of mated by extrapolation from a simple linear
intrasexual variability present in a particu- regression between dimorphism and CVs
calculated from a limited number of species
lar sample.
For each sample the program calculated (Kay, 1982a,b; Leutenegger and Shell,
the actual sexual dimorphism (&al$Xfemale) 1987). Output from initial experiments
and the sexual dimorphism estimated using clearly indicated that the relation between
each of the four methods (CV, FMA, mean, the CV and dimorphism is not linear. Ln
and median). The means and standard devi- transformation of the estimates of dimorations of the dimorphism estimates were phism produced a linear relation with an
then calculated and tabulated for each set of extremely high correlations (Fig. 2a). How100 samples a t each level of true dimor- ever, variation in sex ratio and intrasexual
phism.
variability produced differences in the relaA number of different experiments were tion between the CV and dimorphism. Imperformed for different sample sizes, sex ra- balances in the sex ratio of the sample protios, and intrasexual variation. For each of duce different slopes in the relation between
these three factors, a series of experiments the CV and In-transformed dimorphism
was performed holding the other two factors (Fig. 2b). The magnitude of the change in
constant. For example, four sets of 1,000 slopes depends both on the degree of imbalsamples corresponding to sample sizes of 10, ance of the sex ratio and on whether more
20, 30, and 40 individuals were generated, males or more females are present in the
all with balanced sex ratios and intrasexual sample. With increasing intrasexual variCVs of 5.5. Next, four more such sets were ability, the intercept of the relation between
generated, but this time with intrasexual the CV and In-transformed dimorphism is
CVs of 7.0. This procedure was repeated for reduced (shifting the scatter of points to the
various sex ratios and levels of intrasexual right), and the correlation between the CV
variability. In all experiments, the variabil- and In-transformed dimorphism is reduced
ity of males and females was kept equal.
(increasing the scatter of points) (Fig. 2c).
Finally, the program was modified so that
The equation used to extrapolate dimorthe sex ratio of each population was ran- phism from CVs obviously depends on the
domly selected, mimicking the situation in degree of intrasexual variability and the exthe fossil record where the sex of individuals act sex ratio of the sample. Unfortunately,
is unknown. A series of experiments was run neither of these parameters can be known
470
J.M. PLAVCAN
c
C
b
for fossil samples requiring the use of this
technique to estimate dimorphism. The only
practical approach is to calculate the equation assuming a balanced sex ratio and the
lowest level of intrasexual variability that
can be reasonably expected in the sample.
Since most fossil remains are teeth, this
analysis used the average variability in the
postcanine teeth of primates, which corresponds to a CV or roughly 5.5. With these
assumptions, a regression between CVs and
In-transformed dimorphism from a computer experiment produces the following
equation:
Y = -.047
+
.0214 * X
where Y is In-transformed dimorphism and
X i s the combined-sexCV. This equation was
used t o calculate CV-based estimates of sexual dimorphism in the experiments.
RESULTS
Basic comparison (Balanced sex ratio,
low intrasexual variability, constant
sample size)
-0.2 I
0
3
50
Combined-sex CV
Fig. 2. Bivariate plots of In-transformed dimorphism
vs. combined-sex coefficients of variation (CVs) from
random sampling experiments. All experiments were for
sample sizes of ten. Bottom panel (a) demonstrates the
high correlation and linear relation when sex ratios are
balanced and intrasexual variability is low (male and
female CVs = 5.5).Note that the cluster of points at the
bottom left of each plot is the random scatter generated
when dimorphism is 1.00. Middle panel (b) compares
output from samples with eight males and two females
(higher slope) against samples with two males and eight
females (lower slope). As for panel a, male and female
CVs were both 5.5. Top panel ( c ) shows the effect of
increased sample Variability (male and female CVs =
10.0).Note the increased scatter of points and the larger
scatter of the “bulb”reflecting greater intrasexual variability.
With balanced sex ratios and low intrasexual variability, the mean and median
methods provide the most accurate estimates of sexual dimorphism overall (Table
2). While all four methods tend to overestimate dimorphism when the true dimorphism is low (<1.2), the CV and FMA methods overestimate dimorphism less than
either the mean or median methods. When
true dimorphism is 1.2 or greater, both the
mean and median methods are quite accurate, while the CV method tends to slightly
overestimate dimorphism. Importantly, the
FMA method begins to substantially underestimate dimorphism when the true dimorphism is greater than 1.1 (under these conditions).
At low levels of true dimorphism (less
than 1.41, standard deviations of dimorphism estimates from the mean, median,
and FMA methods are lower than those of
the observed dimorphism, indicating that
the estimates of each method are actually
less variable than the observed dimorphism
(Table 2). At higher levels of true dimorphism (1.4 or greater), the standard devia-
471
FOUR METHODS FOR ESTIMATING SEXUAL. DIMORPHISM
TABLE 2. Comparison of the auerages and standard deviations of the actual dimorphism us. estimated dimorphism
using the four proposed techniques for populations with balanced sex ratios, male and female CVs of 5.5, and
combined-sex sample size of ten
True’
dimorphism
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
cv
Observed
FMA
Mean
Median
Average
sd
Average
sd
Average
sd
Average
sd
Average
sd
1.00
1.10
1.20
1.30
1.40
1.50
1.61
1.70
1.81
1.90
0.035
0.039
0.043
0.044
0.053
0.046
0.055
0.054
0.064
0.058
1.07
1.11
1.20
1.31
1.41
1.51
1.63
1.73
1.85
1.94
0.026
0.039
0.045
0.048
0.057
0.048
0.061
0.057
0.068
0.061
1.08
1.11
1.15
1.20
1.23
1.27
1.32
1.34
1.39
1.41
0.021
0.027
0.028
0.031
0.029
0.028
0.035
0.029
0.037
0.033
1.09
1.13
1.21
1.30
1.40
1.50
1.61
1.70
1.81
1.90
0.021
0.032
0.040
0.043
0.053
0.046
0.055
0.054
0.064
0.058
1.09
1.12
1.20
1.30
1.40
1.50
1.61
1.70
1.81
1.90
0.020
0.031
0.041
0.044
0.053
0.046
0.055
0.054
0.064
0.058
‘“True”is the level of dimorphism used to generate the populationsand is thus not a mean. “Observed”is the actual dimorphism of the randomly
generated populations.All means are based on 100 populationsper level of true dimorphism.
tions of the mean and median methods are
identical to those of the observed dimorphism. This is because, under the parameters defined in this experiment, there is virtually no overlap in male and female
distributions. Finally, standard deviations
of the dimorphism estimates from the CV
method are slightly higher than those of the
observed dimorphism, except when true dimorphism is less than or equal to 1.1.
Sample size
Standard deviations of dimorphism estimates from all methods decrease substantially with increasing sample sizes. However, no method offers any apparent
advantage over the others with increasing
sample sizes.
lntrasexual variation
The magnitude of intrasexual variability
is critical to the accuracy of all methods,
though overall the mean and median methods are least affected by high intrasexual
variability. With low intrasexual variability
(male and female CV = 5.5),standard deviations of dimorphism estimates are relatively
low at all levels of dimorphism, and both the
mean and median methods tend to overestimate dimorphism at low levels of true dimorphism (Table 2). As intrasexual variation increases, each method not only
strongly overestimates dimorphism whenthe true dimorphism is low, but also overestimates dimorphism at progressively higher
levels of true dimorphism. For example,
when male and female population CVs equal
14.0, the mean method overestimates dimorphism even when true dimorphism
equals 1.5 (Table 3).
With increased intrasexual variability,
the CV method also consistently overestimates dimorphism at all levels. This follows
from the particular equation used to estimate dimorphism (see above). Were the
magnitude of intrasexual variation known,
the equation could be adjusted to provide
more accurate estimates.
When intrasexual variability is high, the
FMA method overestimates low levels of dimorphism less than the other methods. At
the level of intrasexual variability presented
in Table 3, the FMA method begins to actually underestimate dimorphism when the
true dimorphism exceeds 1.3. Comparing
this to Table 2 where the FMA method begins to underestimate dimorphism when the
true dimorphism exceeds 1.1,it is clear that
the exact level of dimorphism in which the
FMA method begins to underestimate
rather than overestimate dimorphism is
proportional to the degree of intrasexual
variability.
As for the basic comparison (Table 2),
standard deviations of dimorphism estimates from the mean, median, and FMA
methods are lower than those of the observed dimorphism. However, at the higher
level of intrasexual variability, standard deviations of the mean and median methods
remain lower than those of the observed dimorphism at higher levels of true dimorphism. This is because, with increased intrasexual variability, overlap between male
472
J.M. PLAVCAN
TABLE 3. Comparison of the averages and standard deviations
of the actual dimorphism us. estimated dimorphism for
populations with increased intrasexual variation (Male and female CVs = 14.0, balanced sex ratios, and combined-sex
sample size of ten)
h e '
dimorphism
1.0
Obsenred
Average
sd
1.00
1.10
1.20
1.29
1.42
1.51
1.60
1.71
1.80
1.90
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.093
0.093
0.095
0.130
0.128
0.139
0.134
0.157
0.168
0.167
cv
FMA
Average
sd
Average
sd
1.28
1.28
1.36
1.43
1.55
1.64
1.73
1.85
1.93
2.05
0.094
0.094
0.106
0.127
0.148
0.148
0.153
0.181
0.179
0.188
1.22
1.22
1.27
1.31
1.36
1.40
1.44
1.49
1.52
1.56
0.067
0.067
0.074
0.075
0.091
0.084
0.104
0.106
0.108
0.108
Mean
sd
Average
1.25
1.25
1.30
1.37
1.47
1.53
1.62
1.73
1.80
1.91
0.073
0.076
0.081
0.103
0.118
0.129
0.122
0.152
0.158
0.167
Median
Average
sd
1.24
1.23
1.29
1.35
1.44
1.52
1.60
1.72
1.80
1-90
0.070
0.069
0.081
0.102
0.115
0.133
0.129
0.155
0.164
0.167
'True" is the level of dimorphism used to generate the populationsand is thus not a mean 'Observed" is the actual dimorphism of the randomly
generated populations. All means are based on 100 populationsper level of true dimorphism.
TABLE 4. Comparison
of the averages and standard deviations of the actual dimorphism us. estimated dimorphism for
populations with unbalanced sex ratios (eight males and two females per population, male and female CVs = 5.5)
True1
dimorphism
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Observed
Average
sd
1.01
1.10
1.21
1.30
1.40
1.50
1.60
1.69
1.80
1.92
0.043
0.051
0.060
0.060
0.059
0.065
0.062
0.072
0.078
0.092
cv
FMA
Average
sd
Average
sd
1.07
1.09
1.16
1.22
1.28
1.34
1.40
1.45
1.50
1.56
0.030
0.034
0.041
0.040
0.038
0.038
0.039
0.038
0.040
0.043
1.08
1.10
1.14
1.18
1.21
1.23
1.27
1.29
1.31
1.34
0.022
0.025
0.029
0.027
0.025
0.027
0.030
0.027
0.028
0.028
Mean
Average
sd
1.09
1.11
1.17
1.23
1.31
1.41
1.50
1.59
1.72
1.82
0.025
0.028
0.043
0.062
0.072
0.099
0.114
0.143
0.154
0.176
Median
Average
sd
1.09
1.10
1.14
1.17
1.20
1.23
1.26
1.28
1.30
1.32
0.024
0.025
0.028
0.027
0.025
0.027
0.031
0.027
0.029
0.033
'"True"is the level of dimorphism used to generate the populationsand is thus not a mean. "Observed"is the actual dimorphism of the randomly
generated populations. All means are based on 100 populationsper level of true dimorphism.
phism is low and dramatically underestimate dimorphism with increasing true dimorphism, with the FMA and median
methods performing worse than the CV
method. Interestingly, the CV-based estimates of dimorphism are much more accurate when more females are present in a
Sex ratio
sample than when more males are present
Bias away from a balanced sex ratio, like (Table 5). With more females in a sample,
increased intrasexual variation, greatly af- the CV tends to overestimate dimorphism at
fects estimates of dimorphism using any of all levels of true dimorphism, and the deviathe methods. However, the mean method is tions of the estimates from the true dimormuch less affected than any other. For ex- phism are not as great as when more males
ample, with eight males and two females are in the sample.
With unbalanced sex ratios, the standard
present in a sample of ten individuals, the
deviations
of dimorphism estimates from
mean method slightly overestimates dimorphism when true dimorphism is low ( G 1.1) the CV, FMA, and median methods are conand slightly underestimates dimorphism sistently lower than those of the observed
when true dimorphism is higher (Table 4). dimorphism (Table 4). Standard deviations
The CV, FMA, and median methods slightly of dimorphism estimates from the mean
overestimate dimorphism when true dimor- method are less than those of the observed
and female distributions persists at higher
levels of dimorphism. Finally, the standard
deviations of dimorphism estimates from
the CV method are consistently higher than
those of the observed dimorphism a t all levels of true dimorphism.
FOUR METHODS FOR ESTIMATING SEXUAL DIMORPHISM
TABLE 5. Comparison of the averages of estimates of
sexual dimorphism from the CV method when sex ratio is
biased toward more males (eight males, two females) and
toward more females (eight females, two males)
True
dimorphism
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Eight males
Average
sd
1.07
1.09
1.16
1.22
1.28
1.34
1.40
1.45
1.50
1.56
0.030
0.034
0.041
0.040
0.038
0.038
0.039
0.038
0.040
0.043
Eight females
Average
sd
1.07
1.11
1.17
1.26
1.35
1.46
1.57
1.68
1.78
1.91
0.030
0.039
0.049
0.058
0.053
0.070
0.075
0.080
0.103
0.099
dimorphism a t low levels of true dimorphism (less than 1.3) and greater at higher
levels of true dimorphism.
Random fluctuation in sex ratio
Results from experiments where the sex
ratio of each sample was randomly selected
demonstrate that, overall, the mean method
provides the best estimates of dimorphism
(Table 6). This result is repeated at all levels
of intrasexual variability and at all sample
sizes. Surprisingly, average CV-based estimates of dimorphism are nearly as good as
those from the mean method, though standard deviations of dimorphism estimates
are consistently higher than those from the
mean method. Neither the FMA nor the median method provides better estimates of dimorphism than the CV or mean methods at
any level of true dimorphism.
DISCUSSION
Two features of the various methods need
to be evaluated to determine which is best
for assessing dimorphism in fossils. First,
which method generally provides estimates
of dimorphism that are close to the actual
values of dimorphism present in a sample?
Second, where dimorphism is slight, which
method provides a reliable upper limit to
dimorphism for samples where dimorphism
is relatively low?
The results of this study demonstrate
that, among the techniques examined here,
the best way of estimating dimorphism in a
sample where the sex of individuals is unknown is the simplestdivision of the sam-
473
ple about the mean into two subsamples.
Compared to other methods, the mean
method is the most accurate even when intrasexual variability is high and when sex
ratios are strongly biased. At low levels of
dimorphism (less than about 1.21, the mean
method usually overestimates dimorphism
a bit more than the other three methods and
seems most appropriate for setting a conservative upper limit on the amount of dimorphism that could be present in a sample.
Because such estimates represent upper
limits, care should be taken not to interpret
these estimates as evidence for any particular degree of dimorphism.
The CV method provides estimates of sexual dimorphism that are almost as accurate
as those of the mean method, although it is
more sensitive to departures from a balanced sex ratio and increasing intrasexual
variability. The accuracy of the method is
contingent upon deriving a regression equation that appropriately accounts for variation in sex ratio and intrasexual variability
(assuming that some characters are more
variable than others). Potentially, if the limits of intrasexual variability in a character
can be established in extant species, approximate empirical “confidenceintervals” could
be derived by estimating dimorphism from
regressions derived for the upper and lower
limits of expected variability. The utility of
such a procedure is limited, though, by the
decreasing correlation between the CV and
dimorphism at progressively higher levels of
sexual dimorphism with increasing intrasexual variability. This is clear in Figure 2,
where the scatter of points about the lower
right portion of the plots increases dramatically with increasing intrasexual variability. Furthermore, uncertainty about
whether levels of variability in extant species truly reflect the situation in the fossil
record precludes assigning meaningul probabilities to such confidence intervals.
At first glance, the results for the FMA
method suggest that it is of little utility for
estimating dimorphism in fossil samples.
The results for the FMA method are not surprising, though, since with increasing dimorphism the assumptions of the method
are violated. The FMA method is potentially
useful under the circumstances for which it
474
J.M. PLAVCAN
TABLE 6. Comparison of the averages and standard deviations of the actual dimorphism us. estimated dimorphism for
populations with randomly selected sex ratios (males and females CVs = 5.5, combined-sex sample size = 10)
True’
dimorphism
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
cv
Observed
FMA
Mean
Median
Average
sd
Average
sd
Average
sd
Average
sd
Average
sd
0.99
1.10
1.19
1.31
1.40
1.50
1.61
1.70
1.80
1.90
0.034
0.037
0.049
0.044
0.049
0.053
0.067
0.066
0.065
0.066
1.07
1.11
1.19
1.30
1.39
1.48
1.58
1.68
1.80
1.89
0.030
0.038
0.047
0.052
0.066
0.075
0.106
0.121
0.119
0.145
1.08
1.11
1.15
1.20
1.24
1.27
1.31
1.34
1.39
1.43
0.023
0.027
0.030
0.031
0.027
0.035
0.043
0.046
0.050
0.066
1.09
1.12
1.19
1.30
1.39
1.48
1.57
1.68
1.80
1.90
0.024
0.031
0.045
0.056
0.070
0.074
0.105
0.094
0.069
0.072
1.09
1.12
1.18
1.26
1.33
1.39
1.44
1.53
1.61
1.65
0.023
0.030
0.040
0.056
0.072
0.090
0.123
0.144
0.151
0.169
‘“True”is the level of dimorphism used to generate the populations,and is thus not a mean. “Observed”is the actual dimorphism of the randomly
generatedpopulations.All means are based on 100 populationsper level of true dimorphism.
The median method has little to recomwas designed-estimating the maximum dimorphism that could be present in a unimo- mend it in comparison to the other three
dally distributed population. Unimodal dis- methods. Under the best of circumstances
tributions can be generated from popu- (low intrasexual variability, balanced sex
lations showing dimorphism of approxi- ratio) it performs nearly as well as the mean
mately 1.3 (Godfrey et al., 1993) or more method, but it is much more sensitive to dewhen sample sizes are small. Under all con- partures from a balanced sex ratio and inditions, the mean method provides, on aver- creased intrasexual variability.
It must be noted that with increasing
age, slightly higher estimates of dimorphism when true dimorphism is less than sample sizes, an unbalanced sex ratio be1.3 and so seems to be better suited to the comes less of a problem. The likelihood of
purpose of setting an upper limit on dimor- drawing any particular sex ratio in a sample
phism in a sample. However, all methods is easily calculated from a binomial probaoverestimate dimorphism in these cases, bility distribution. Assuming an equal sex
and the lower estimates of the FMA method ratio in the original population and an equal
in fact are more accurate. The problem is probability of males and females being prethat with decreasing intrasexual variability, served (neither assumption is necessarily
the FMA method actually underestimates true [Oxnard, 198711, one can calculate that
dimorphism at progressively lower levels of at a sample size of ten, there is a 95% probatrue dimorphism. Thus, for characters that bility that the sex ratio should not exceed
typically show reltively low variability, such about 4:l in favor of either sex. However, at
as the teeth, it must be kept in mind that the a sample size of 20, this ratio falls to about
FMA method can potentially underestimate 2.3:l. Conversely, for samples of fewer than
rather than overestimate dimorphism. six individuals, the chances are better than
Since intrasexual variability cannot be 5% that a sample will be composed of only
known where the sex of individuals cannot one sex. This latter point should not be forbe reliably determined, this technique gotten when dealing with small samples,
where estimates of dimorphism could potenshould not be used alone.
As for the CV method, the FMA method tially be calculated on small samples comcan be adjusted to fit different assumptions posed of only one sex!
It is interesting to note that for the mean,
about intrasexual variability and sex ratios
(Godfrey et al., 1993). Potentially, this could median, and FMA methods, standard deviaallow one to set approximate confidence in- tions of dimorphism estimates tend to be
tervals on the estimate of maximum dimor- lower than those of observed dimorphism,
phism in a population, though such a prac- especially when the true dimorphism is low.
tice is potentially subject to the same This means that, even though these methods tend to overestimate the true dimorproblems as the CV technique.
FOUR METHODS FOR ESTIMATING SEXUAL DIMORPHISM
475
phism (at low levels of true dimorphism), the
CONCLUSIONS
estimates themselves are actually less variThe problem of estimating sexual dimorable than estimates generated when the sex phism in small fossil samples where the sex
of individual specimens is known. The only of individuals cannot be reliably determined
exception t o this occurs when the mean has no easy solution. Even though the mean
method is used to estimate dimorphism method is overall the best of the four techwhen the sex ratio is unbalanced, but this is niques investigated here, in fact none of the
only likely to be a substantial problem with methods provides reliable estimates of dismall sample sizes (less than about ten). morphism when true sexual dimorphism is
Thus, even when a sample of fossils contains low. All four methods are confounded by
a few individuals that can be sexed, these fluctuation in sex ratio and intrasexual
techniques are still useful in providing sup- variability. While it is possible to adjust the
plemental upper estimates of dimorphism if FMA and CV techniques to account for fluca large number of specimens that cannot be tuation in these parameters, the very fact
identified by sex can be used. For example, if that the sex of individuals is not known
a collection consists of 20 jaws containing means that neither the sex ratio nor intrathe first molar tooth, but only a few of these sexual variability can be known a priori.
jaws can be sexed, variability-based esti- When sexual dimorphism is low such that
mates could be useful for setting an upper sample distributions appear t o be unimodal,
limit on the amount of sexual dimorphism the best that can be hoped for is to estimate
that could be present in the molar teeth us- the maximum amount of dimorphism that
ing the entire sample rather than just a few could be present in the sample (Godfrey et
individuals. Thus, a variability-based esti- al., 1993).
mate of dimorphism can help control for
sampling error in the estimate of dimorACKNOWLEDGMENTS
phism based only on the sexed individuals.
I thank Richard F. Kay for advice and
Finally, all of the methods examined here
help.
Gene Albrecht, Dana Cope, Rebecca
strongly overestimate sexual dimorphism
German,
Laurie Godfrey, Bill Hylander, Jay
when intrasexual variability is large and the
Kelley,
Rick
Madden, Mike Sutherland, and
actual population dimorphism is low. Under
two
anonymous
reviewers provided helpful
these conditions, it is unlikely that any
comments
and
discussion.
Dave Hertwick
method that uses sample variation can accuprovided
assistance
with
computers.
This
rately estimate low to moderate levels of diresearch
was
supported
by
NSF
dissertation
morphism (from 1.0 to roughly 1.3), since
the total sample variation due to the separa- improvement grant BNS 8814060 and NIDR
tion between male and female means is postdoctoral fellowship 5 F32 DE05605-02.
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