Journal of Archaeological Science (2002) 29, 317–322
doi:10.1006/jasc.2001.0733, available online at http://www.idealibrary.com on
The Modified Triangular Graph: A Refined Method for
Comparing Mortality Profiles in Archaeological Samples
Teresa E. Steele* and Timothy D. Weaver
Department of Anthropological Sciences, 450 Serra Mall, Bldg 360, Stanford University, Stanford,
CA 94305-2117, U.S.A
(Received 21 March 2001, revised manuscript accepted 18 May 2001)
Plotting the percentages of juvenile, prime, and old individuals on a triangular graph has become a popular method for
analysing mortality profiles or age structures found in archaeological faunal samples. This method allows easy
comparisons of multiple samples or multiple species and appears to work even with small sample sizes. However, the
utility of triangular graphs is compromised for two reasons: (1) samples cannot be statistically compared and (2) the
points on the graph are based on percentages, and, therefore, they do not provide information about sample size. The
modified triangular graph described here offers a method for approximating 95% confidence intervals around the data
points by using bootstrapping. Samples with non-overlapping contours are likely to have had different pre- or
post-depositional histories. The 95% density contours reflect sample size since they shrink as samples get larger. Thus,
the modified triangular graph allows more confident comparisons of three age class data.
2002 Published by Elsevier Science Ltd.
Keywords: FAUNAL ANALYSIS, ZOOARCHAEOLOGY, AGE PROFILES, AGE STRUCTURES, AGE
DISTRIBUTIONS, SAMPLE SIZE, AGE AT DEATH, BOOTSTRAPPING.
Mortality profiles found in the fossil record vary
greatly, however, and these models are mainly useful as
baselines for comparison. A full discussion of mortality
profile interpretation is beyond the scope of this paper,
and more information is available in many sources (e.g.
Klein, 1982b; Klein & Cruz-Uribe, 1984: 55–57, 85–92;
Levine, 1983; Lyman, 1994: 114–132; Stiner, 1990,
1991, 1994: 271–315).
There are multiple ways researchers analyse the
mortality profiles found in archaeological assemblages,
and each method has strengths and weaknesses. The
purpose of this paper is to discuss the triangular
graph, one method for analysing age distributions that
has become popular recently. We offer a modified
triangular graph that strengthens this method.
Introduction
aunal analysts have long recognized that the age
distribution of a species in an assemblage provides information about the specimens’ pre- or
post-depositional history. Age structures or mortality
profiles found in fossil samples can inform on the mode
of death or bone accumulation. Age structures are
often compared to two theoretical models that are
based on observations in wildlife biology and characterize stable populations of large mammals that give
birth to only one offspring at a time. The first model
describes the age structure of a live herd on the
landscape, and, therefore, it is often called a ‘‘living’’
structure (e.g. Stiner, 1990). This mortality structure is
also frequently called a ‘‘catastrophic’’ profile (e.g.
Klein, 1982b), because it is the age structure that would
be found in a fossil assemblage if an entire herd was
destroyed by a flash flood, volcanic eruption, or other
disaster. The second model is directly related to the
first, because it corresponds to deaths that occur in
between each age class in the living structure (i.e. the
natural attrition on the herd); thus, this model is
often called an ‘‘attritional’’ profile (e.g. Klein, 1982b).
Attritional mortality affects mainly the youngest and
oldest members of the population, and, therefore,
an attritional age structure is often referred to as a
‘‘U-shaped’’ profile (e.g. Klein, 1982b; Stiner, 1990).
F
Triangular Graphs
A common way of representing age structures in
zooarchaeological publications is to use a triangular
graph, or ternary diagram, as proposed by Stiner
(1990, 1994). In this method, specimens are assigned to
one of three age classes: young, prime and old, and the
proportions of individuals in each class are plotted on
a triangular graph (see Figure 1a). The vertical axis
represents the percentage of old individuals, and the
top corner represents 100% old specimens in the
sample. The right corner represents 100% prime domination, and the left corner represents 100% juvenile
*E-mail: [email protected], [email protected]
317
0305–4403/02/$-see front matter
2002 Published by Elsevier Science Ltd.
318
T. E. Steele and T. D. Weaver
(a)
100% Old
(b)
100% Old
1 = Hypothetical 1
2 = Hypothetical 2
3 = Hypothetical 3
1 = Hypothetical 1
Old dominated
0% Prime
0% Juvenile
0% Prime
0% Juvenile
3
Attritional
Living
2
1
1
Juvenile
dominated
100%
Juvenile
Prime
dominated
0% Old
(c)
100%
Prime
100% Old
100%
Juvenile
0% Old
(d)
100% Old
1 = Hypothetical 1
2 = Hypothetical 2
3 = Hypothetical 3
0% Prime
100%
Prime
= Sample size 100
= Sample size 40
= Sample size 12
0% Juvenile
0% Prime
0% Juvenile
3
2
100%
Juvenile
1
0% Old
100%
Prime
100%
Juvenile
0% Old
100%
Prime
Figure 1. (a) A triangular graph indicating the five zones for different age structures as defined by Stiner (1990: 318). Three hypothetical data
sets plot in different zones, and, therefore, they would be interpreted as representing three different mortality profiles. (b) A modified triangular
graph showing the distribution of the 10,000 re-samples and the 95% density contour for sample 1. Many of the re-samples plot in the same
location, so less than 10,000 points are visible. (c) A modified triangular graph with the 95% density contours for the three hypothetical samples.
Although each sample plots within a different zone on the graph and therefore should have different age structures, not all the samples can be
separated. Only samples 2 and 3 can be confidently differentiated from each other, while neither samples 1 and 2 nor samples 1 and 3 can be
differentiated. (d) A modified triangular graph demonstrating the effects of sample size on the ability to distinguish age structures, because the
95% density contours increase with smaller sample sizes. The percentage of each age class remains the same, and only the total sample size
changes. All data are listed in Table 1.
domination. Zones that represent either the ‘‘living’’
(left of prime-dominated) or ‘‘attritional’’ (right of
juvenile-dominated) age structures are also labelled on
the graph. When a sample plots within one of these five
zones, it is assumed to have the indicated age structure.
If two points plot close to each other and within the
same zone, the samples are said to possess similar
mortality profiles. We created three hypothetical data
sets each with a sample size of 35 individuals, listed in
Table 1 and plotted in Figure 1a. Each of these data
sets plots in a different zone on the graph, and,
therefore, they would be interpreted as representing
three different mortality profiles.
Triangular graphs are widely used by researchers
studying both prehistoric and modern hunting to
describe the mortality patterns in their samples (e.g.
Alvard, 1995, 1998; Dı́ez et al., 1999; Gaudzinski,
1995; Marean, 1997; Speth & Tchernov, 1998; Stiner,
1990, 1991, 1994, 1998). They have gained popularity,
because they are visually appealing, are easy to create,
and allow the simultaneous comparison of multiple
samples or multiple species. Also, since triangular
graphs are based on only three age classes, they can be
used with various age determination methods and
coarse-grained age data, such as data from wildlife
biology. As with any method of age structure analysis,
the boundaries between the age classes must be
well defined and replicable by other researchers. Since
triangular graphs are based on percentages, it appears
as if they can present more information about
smaller samples than other methods can provide.
Stiner (1998: 315) suggests that a sample size of 12
might be adequate for the triangular graph, while
histograms of ten age classes require a minimum of at
The Modified Triangular Graph 319
Table 1. Data used in this paper. The percentage of each age class is listed in parentheses
Sample
Hypothetical 1
Hypothetical 2
Hypothetical 3
Sample size 100
Sample size 40
Sample size 12
Elandsfontein*
Klasies River Mouth*
Kebara Gazella MP**
Kebara Dama MP**
Kebara Gazella UP**
Kebara Dama UP**
n
Juvenile
Prime
Old
35
35
35
100
40
12
95
64
316
114
86
95
12 (34·3)
19 (54·3)
5 (14·3)
33 (33·0)
13 (32·5)
4 (33·3)
20 (21·1)
42 (65·6)
96 (30·4)
43 (37·7)
39 (45·9)
43 (45·3)
17 (48·6)
11 (31·4)
16 (45·7)
50 (50·0)
20 (50·0)
6 (50·0)
54 (56·8)
19 (29·7)
196 (62·0)
53 (46·5)
39 (45·9)
44 (46·3)
6 (17·1)
5 (14·3)
14 (40·0)
17 (17·0)
7 (17·5)
2 (16·7)
21 (22·1)
3 (4·7)
24 (7·6)
18 (15·8)
7 (8·2)
8 (8·4)
*(Klein, 1982a: 155; Lyman, 1994: 130).
**(Speth & Tchernov, 1998: 231).
least 30 or 40 individuals (Klein & Cruz-Uribe, 1984:
59; Lyman, 1987; Shipman, 1981: 157).
The triangular graph method has two related disadvantages. First, percentages of each age class are
plotted, and, therefore, sample sizes are not taken into
account when comparing assemblages. This increases
the tendency to consider small samples as informative
even though they can be heavily influenced by
sampling processes. Second, samples cannot be compared statistically. Graphs are visually inspected, and
patterns may be erroneously identified where none
actually exist. If these two limitations can be addressed,
the triangular graph method will be considerably more
informative.
The Modified Triangular Graph
Bootstrapping the age class data offers a way to
account for sample size and approximate confidence
intervals around age structure data points on the
triangular graph. The bootstrap is a simulation method
for making statistical inferences based solely on the
observed data, and it is usually used when standard
parametric inference techniques are difficult or incorrect (Efron & Tibshirani, 1993; Mooney & Duval,
1993).
As used here, the method works as follows (for
further discussion, see Efron & Tibshirani, 1993;
Mooney & Duval, 1993; Sokal & Rohlf, 1995). A
fictional sample is created by randomly re-sampling
with replacement from the observed age class data.
This process is repeated 10,000 times. For each of these
10,000 re-samples, the percentages of the three age
classes are recalculated and plotted on the triangular
graph. This produces a scatter of points around the
original sample location, since each re-sample potentially can have duplicates or be missing individuals
from the original sample. A 95% density contour is
then calculated and drawn around the bootstrapped
re-samples. This density contour can be considered a
95% confidence interval around the observed age class
frequencies. When multiple samples are compared, and
two density contours do not overlap, then the two age
structures differ at approximately the 0·05 level. If two
contours overlap or touch, then the two samples are
not significantly different at ]0·05. Figure 1b shows
the re-sampled points and the resulting 95% density
contour for Hypothetical sample 1. Note that 10,000
points are not visible, because multiple re-samples
often have the same proportions of age classes.
This procedure works because as the original sample
size increases, the bootstrapped scatter of points
approaches the distribution which would have been
obtained by repeated sampling from the true age
structure or population in a statistical sense (Efron &
Tibshirani, 1993; Mooney & Duval, 1993). The procedure is not as exact with smaller sample sizes, but it
is still usually very good (Mooney & Duval, 1993).
Once the distribution is approximated by the 10,000
re-samples, 95% confidence limits are obtained by
excluding the outlying 5% in a way analogous to using
the normal distribution to obtain confidence limits
around a sample mean.
In the modified triangular plot, 95% confidence
limits are approximated using density contours. These
contours are similar in function to the density ellipses
calculated by many statistical software packages;
however, ellipses should be used only with normally
distributed data, since they are drawn using parametric statistical techniques. The bootstrap points on
the triangular plot are not normally distributed, and
the axes are not standard. Therefore, the 95% density
contour must be calculated non-parametrically, similar
to drawing a topographic contour, except that point
density is being contoured instead of elevation. A
contour is picked such that 95% of the point density
is enclosed. The scatter of points is first smoothed
using a Gaussian kernel smoother (Silverman, 1982;
1986) to make the contour less ragged and more
accurate.
320
T. E. Steele and T. D. Weaver
(a)
100% Old
(b)
100% Old
= Gazella (MP)
= Dama (MP)
= Gazella (MP)
= Dama (UP)
E = Elandsfontein
K = Klasies River
Mouth
0% Prime
0% Juvenile
0% Prime
0% Juvenile
E
100%
Juvenile
K
0% Old
100%
Prime
100%
Juvenile
0% Old
100%
Prime
Figure 2. (a) A modified triangular graph displaying the age structure data for the extinct giant African buffalo found in the South African sites
of Elandsfontein and Klasies River Mouth. The non-overlap of the two density contours indicates that these two samples differ significantly
when complete age profiles are examined. Data are from Klein (1982a: 155) and Lyman (1994: 130). (b) A modified triangular graph showing
the age structure data for gazelles and fallow deer from Kebara Cave, Israel. MP stands for Middle Paleolithic and UP for Upper Paleolithic.
Data are from Speth & Tchernov (1998: 231). All data are listed in Table 1.
Figure 1c illustrates the modified triangular graph
method using the three hypothetical samples from
Table 1 and Figure 1a. The three samples were bootstrapped, and the 95% density contours were calculated. The density contours for samples 1 and 2
overlap, as do those for samples 1 and 3, while those
for samples 2 and 3 do not. This suggests that sample
1 cannot be differentiated from 2 and sample 1 cannot
be differentiated from 3, while samples 2 and 3 probably do represent different age structures. These results
are supported by the Kolmogorov–Smirnov test. This
test compares the cumulative frequency distributions
of two histograms, and the triangular graph is really a
visually appealing way of representing a 3-bar histogram of age class frequencies. Samples 1 and 2
and samples 1 and 3 cannot be differentiated
(Kolmogorov–Smirnov Z=0·84 and 0·96, P=0·49 and
0·32 respectively), while samples 2 and 3 are significantly different (Kolmogorov–Smirnov Z=1·67,
P<0·01). These results suggest that samples cannot be
differentiated simply because they fall within one of the
five designated zones on the triangular graph. The
confidence interval around each point must also be
considered.
An additional advantage to bootstrapping the three
age class data is that the resulting density contours
are sensitive to sample size. Figure 1d depicts three
assemblages where their age class frequencies remain
the same, but sample sizes are 100, 40, and 12 (data are
listed in Table 1). The density contours around the
data points become larger with decreased sample size.
Bootstrapping allows smaller samples to be compared
more informatively. The confidence interval around
the sample of 12 is quite large, but it need not overlap
with another sample if the other sample is large and in
a distinctly different section of the graph. The density
contour around the sample of 100 is still sizable. This
indicates that if data are compared without confidence
intervals, misinterpretations are possible.
Figure 2a shows an example of the modified triangular graph using archaeological data. Lyman (1994:
128–132) uses Klein’s data on the extinct giant African
buffalo (Pelorovis antiquus) from the South African
sites of Elandsfontein and Klasies River Mouth to
compare two methods of analysing age structures: the
triangular graph and histograms of ten age class frequency data (see Klein, 1982a for the original data and
for a discussion of the histogram method and the sites).
Using the Kolmogorov-Smirnov test, both Klein and
Lyman found that the ten age class histograms created
from the assemblages were significantly different.
Lyman also collapsed Klein’s ten age classes into
three and plotted them on a triangular graph. The
Elandsfontein and Klasies River Mouth samples plot
in the zones corresponding to the living and attritional
structures respectively, confirming the histogram
analysis. We use the same three age class data as
Lyman (listed in Table 1) to create a modified triangular graph. The separation of the 95% density contours shown in Figure 2a further confirms that the
Elandsfontein and Klasies River Mouth samples differ
significantly.
This example also raises a concern regarding the
difficulties of studying complete age profiles such as
those depicted in the modified triangular graph. As
noted by many researchers (e.g. Hulbert, 1982; Klein &
Cruz-Uribe, 1983, 1984; Kurtén, 1953: 75; Lyman,
1994), juvenile specimens are likely to be lost in the
archaeological record due to post-depositional biases,
and, therefore, they often should be discounted when
studying mortality profiles. Histogram and modified
triangular graph analyses both show that the major
difference between the Elandsfontein and Klasies
River Mouth samples is in the number of juveniles
The Modified Triangular Graph 321
represented. As discussed in Klein (1982a) and further
shown in Klein & Cruz-Uribe (1996), analyses of both
samples based simply on adults show no difference.
This is not obvious when using the triangular graph,
because it is difficult to visually discount the juvenile
specimens. In these two samples, the difference in the
number of juveniles could distinguish natural attrition
and subsequent scavenging from active hunting (Klein,
1982a), but in other sites the difference could
represent biased preservation. Given the potential
problems of post-depositional loss of juvenile specimens, analysing complete age structures requires
caution. The Elandsfontein and Klasies River Mouth
example suggests that although the modified triangular
graph is an improved method of analysing complete
age structures, methods that compare only adults
should always be used as well.
Speth & Tchernov’s (1998) data for Kebara Cave
(Israel) offer another example to illustrate the modified
triangular graph. They used the triangular graph
method to compare Middle and Upper Paleolithic
hunting of gazelle (Gazella gazella) and fallow deer
(Dama mesopotamica). They concluded that all four
assemblages fall within the living structure zone of
the triangular graph and, therefore, that Paleolithic
accumulators from Kebara hunted as ambush predators and were able to take prime animals. If the points
had fallen in the attritional zone, the hunters would
resemble cursorial predators who primarily hunted the
youngest and oldest animals (see Stiner, 1990, 1994 for
more details). Figure 2b shows Speth & Tchernov’s
data on the modified triangular graph. We agree that
the accumulators of all four assemblages hunted in a
similar fashion, because all four of the 95% density
contours overlap. However, the spread of the confidence contours into the attritional zone, particularly in
the Upper Paleolithic samples, indicates that these
assemblages need not reflect living structures. These
assemblages could represent living or attritional structures or something in between. This example raises
concerns about relying on just the zones for mortality
profile identification. More meaningful interpretations
may be gained by direct comparison of multiple
samples rather than by determining within what zone
the samples fall. If necessary, samples can be directly
compared to the model living and attritional structures
by scaling the model data to reflect similar sample sizes
as the study samples and using the modified triangular
graph to analyse the data.
The Modified Triangular Graph Program
A modified triangular graph program was written for
Macintosh computers and is available by contacting
the authors. The data are entered into a text file so that
the first line is the sample name, the second line
designates the color of the sample on the output graph,
and the third to fifth lines are the raw (not percentages)
numbers of young, prime, and old individuals in the
sample. Any number of samples can be entered sequentially in the file. This file is opened in the Triangle
program by selecting it from the file menu, and the
program plots the samples’ points with their surrounding 95% density contours. A number on the graph
identifies each sample, and a legend lists the numbers
and corresponding sample names. Multiple samples
can be run at the same time either by adding them in
the original data file or by opening multiple files.
Currently the program draws only 95% density contours, but we plan to modify it so that 50%, 90%, and
99% options are available. The output graphic can be
copied and pasted into other graphics programs, such
as Powerpoint or Photoshop, to modify for figures or
presentations. We created all of the figures in
this article by transferring results from the Triangle
program into Powerpoint.
Conclusions
The triangular graph is a popular method for comparing the age structures of multiple samples and species
using multiple types of age determination methods.
However, it cannot be used for statistical inference,
since it does not account for sample size. The modified
triangular graph program described here solves this
problem by bootstrapping the original data to allow
statistical differentiation of samples on a triangular
graph. The program produces 95% density contours
that reflect sample size. Therefore, it allows multiple
assemblages to be compared more confidently. A
Macintosh program to calculate and plot 95% density
contours on a triangular graph is available from the
authors.
Acknowledgements
Richard G. Klein kindly commented on an earlier draft
of this paper. The L.S.B. Leakey Foundation, the
Mellon Foundation, and the Stanford University
Department of Anthropological Sciences generously
provided support for this research.
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