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eighteen mammalian faunas from eastern Washington State. Data from Table 5.1 .

figure 5.4. Relationships between NISP and NTAXA of small mammals per stratum
(Roman numerals) at Homestead Cave, Utah (after Grayson 1998). Dashed, best-¬t regres-
sion line, r = 0.88, p = 0.3; solid, best-¬t regression line, r = 0.92, p = 0.0001. Faunal material
from strata X and XIII“XVI has not been studied. Data from Table 5.2.
measuring the taxonomic structure and composition 183

Table 5.2. NISP and NTAXA for small
mammals at Homestead Cave, Utah. Data
from Grayson (1998). Faunal remains from
strata X and XIII“XVI were not studied

XVIII 1047 9
XVII 15,421 17
XII 22,661 14
XI 9,996 14
IX 18,043 16
VIII 8,215 13
VII 11,038 15
VI 18,661 17
V 5,093 13
IV 26,200 19
III 2,774 17
II 7,756 20
I 9,906 19

taxa were accumulated and deposited when it was arid. Despite apparent sample size
effects, the mammal assemblages from Homestead Cave appear just as they should
in terms of the relationship between NTAXA and climate.
Another example of studying the covariation of sample size and taxonomic rich-
ness comes from the Upper Paleolithic rockshelter of Le Flageolet I, France (Grayson
and Delpech 1998). The ungulate remains at this site were largely introduced by
humans, but interestingly, there is yet another pair of relationships between NTAXA
(of ungulates) and NISP (of ungulates) (Table 5.3). There is no patterned rela-
tionship between the associated archaeological culture and which line a particular
assemblage of ungulate remains helps de¬ne (Figure 5.5). The analysts found no clear
indication that the degree of fragmentation was creating the two relationships, and
no indication that the differential transport of skeletal parts by bone accumulators
had created the two relationships (Grayson and Delpech 1998). They concluded that
the difference involved variation in diet breadth, or the width of the niche exploited
by the humans that created the assemblages.
There are other ways to compare taxonomic richness values of assemblages of
different sizes. Recall from Chapter 4, for example, that the original recognition
of the sample-size effect (the species’area relationship) was based on the amount
quantitative paleozoology

Table 5.3. NISP and NTAXA for
ungulates at Le Flageolet I, France. Data
from Grayson and Delpech (1998). Strata
with NISP < 30 are not included

XI 651 6
IX 681 11
VIII 461 9
VII 1,768 10
VI 376 8
V 1,244 7
IV 145 5

of geographic area sampled. Thus, one might compare taxonomic richness with
the amount excavated, either the area or volume excavated. Wolff (1975) showed
long ago that the greater the volume of sediment searched for faunal remains, the
greater the number of taxa found (see Chapter 4). Taxonomic richness increases
as the amount of sediment examined increases because as the amount of sediment
examined increases, NISP increases (more specimens are recovered, so more taxa

figure 5.5. Relationship between NISP and NTAXA per stratum (Roman numerals) at
Le Flageolet I, France (after Grayson and Delpech 1998). Dashed, best-¬t regression line
r = 0.99, p = 0.06; solid, best-¬t regression line, r = 0.98, p = 0.02. Some strata omitted as
sample sizes are too small for inclusion. Cultural associations for each stratum are indicated.
Data from Table 5.3.
measuring the taxonomic structure and composition 185

are identi¬ed). As the amount of sediment examined increases, the amount of area
examined increases, which brings us back to the original species“area relationship
discovered by botanists.
Regardless of the technique used to gain insight to the structure and composition of
a fauna, taxonomic richness is often strongly correlated with sample size. Therefore,
the analyst must be ever on the alert for differences in sample size measured as
NISP as a variable that potentially contributes to differences in NTAXA. The
analyst should also realize that the possible in¬‚uence of sample size on all measures
of taxonomic diversity (structure and composition) might be disputed, such as when
all of a site deposit (all of a single stratum, or all of a site within horizontal and vertical
boundaries) has been excavated. In such cases one might argue that a 100-percent
sample has been collected and that taxonomic richness cannot be considered to be
a function of sample size. This is in some senses true, but it also overlooks two
fundamental issues. First, a very small number of sites (or strata within sites other
than trivial cases such as the ¬ll of a single intrusive pit) have been totally excavated
(meaning that a 100-percent sample has been generated). Second, even if a site is
completely excavated, it is likely to be but a portion of some larger cultural system than
is found in that single site or only a portion of the taphocoenose, thanatocoenose,
or biocoenose. This brings us back to Kintigh™s (1984) fundamental dilemma of
de¬ning the population one wishes to model with available samples. Only when that
population is de¬ned beforehand will we know if we have a 100-percent sample or
not, and even then preservation variation and recovery procedures may result in less
than complete retrieval (Chapter 4).

Taxonomic Composition

Two faunas can have the same NTAXA, but share anywhere from none to all of
the taxa represented. How do we compare faunas in terms of the taxa they hold in
common and those taxa that are unique to one or the other? How do we determine if
two faunas are similar in taxonomic composition, and how do we determine if fauna
A and fauna B are more similar to one another than either is to fauna C? Indices
have been designed to answer these questions and to measure just these features
(see reviews in Cheetham and Hazel 1969; Henderson and Heron 1977; Janson and
Vegelius 1981 ; Raup and Crick 1979). For unclear reasons these indices have seldom
been used by zooarchaeologists (Styles [1981 ] is a noteworthy exception). It could
be a result of benign neglect, or it could be that the in¬‚uences of varying sample
size are a concern. Before considering the latter issue, however, let™s consider some
quantitative paleozoology

exemplary indices. These are sometimes referred to as binary coef¬cients, because
they summarize and compare presence“absence (nominal scale) data.
One index is the Jaccard index (J) [originally, “coef¬cient of ¬‚oral community”].
It is calculated as

J = 100C/(A + B ’ C ),

where A is the total number of taxa in fauna A, B is the total number of taxa in
fauna B, and C is the number of taxa common to both A and B. Another index is the
Sorenson index (S), calculated as

S = 100(2C )/(A + B),

where the variables are as de¬ned for the Jaccard index. Given how they are calculated,
the Jaccard index emphasizes differences in two faunas, and the Sorenson index
emphasizes similarities. For example, if A = 6, B = 6, and C = 4, then J = 50whereas S =
66.7. Comparing the Meier site mammalian fauna (A) with the complete Cathlapotle
mammalian fauna (B), A = 26, B = 25, and C = 20. Thus, for these two faunas, J = 64.5
and S = 78.4. Given that the two faunas fall within the same time period, are < 10 km
apart, and occur in virtually identical habitats, it may seem that the indices of faunal
similarity should be considerably higher. This is so because, at least with respect to
statistical precision sampling (as opposed to discovery sampling; see Chapter 4), at
least the Meier site sample seems to be representative because signi¬cant increases
in its size over the last several years of excavation failed to produce any previously
unidenti¬ed taxa (Lyman and Ames 2004).
Why the Meier and Cathlapotle assemblages are not more similar and do not share
more mammalian genera is an ultimate question. It may have to do with variation in
which taxa were accumulated despite similarity of the agents of accumulation; at both
sites humans were the most signi¬cant accumulation agent. Or, it may actually have
to do with a fundamental problem of all such binary coef¬cients (Raup and Crick
1979). That problem can be illustrated with a pair of Venn diagrams (Figure 5.6).
Each of these has been drawn with the Meier and Cathlapotle collections in mind. A
total of thirty-one genera are represented by the two collections. One Venn diagram
suggests each collection is a sample of those thirty-one genera. The other Venn
diagram indicates that each collection is a sample of the total forty mammalian
genera (excluding eight genera of bats) that occur in the area today (Johnson and
Cassidy 1997). Given that neither zooarchaeological collection has signi¬cantly more
than two-thirds of those genera, it is perhaps not surprising that the two do not share
more taxa. Each site collection represents but a sample of the local biotic community.
Another way to make the point of the preceding paragraph is this: Based on earlier
discussions, it should be obvious that sample size (= NISP) will in¬‚uence binary
measuring the taxonomic structure and composition 187

figure 5.6. Two Venn diagrams based on the Meier site and Cathlapotle site collections.
Upper diagram suggests each collection is a sample of the thirty-one genera represented by
the combined collections. Lower diagram indicates that each collection is a sample of the
total forty mammalian genera that occur in the area.

coef¬cients such as the Jaccard and Sorenson indices. This has been known for
decades; the more individuals, the taxonomically richer the sample, so when Paul
Jaccard proposed his index he suggested areas of similar size be sampled, but he
should have suggested similar numbers of individuals be inspected (Williams 1949).
Consider the fact that both the Meier and Cathlapotle collections are samples, and
thus even if remains of all forty mammalian genera known in the area today had
been accumulated and deposited in site deposits, it is likely that remains of rarely
represented taxa would not be recovered. If more of each of those sites had been
excavated, and several thousand more NISP had been recovered from each site, it
is probable that several of those as yet unidenti¬ed genera would occur in those
collections. This would not only represent a shift in the sampling design toward a
discovery model, but it would also increase the magnitude of both the Jaccard index
and the Sorenson index.
Neither the Sorenson index nor the Jaccard index takes advantage of the abundance
of taxa. A simple way to assess the similarity of taxonomic abundances of two faunas
is to calculate a χ 2 statistic (e.g., Broughton et al. 2006; Grayson 1991b; Grayson and
Delpech 1994). To illustrate this, the NISP data for the collection of faunal remains
from eighty-four owl pellets (Table 2.9) is summarized as two chronologically distinct
quantitative paleozoology

Table 5.4. NISP per taxon in two chronologically distinct
samples of eighty-four owl pellets

Taxon 1999 sample 2000“2001 sample
Sylvilagus 5 0
Reithrodontomys 0 19
Sorex 40 6
Thomomys 52 16
Microtus 302 403
Peromyscus 1,147 119

samples in Table 5.4. Chronological distinction concerns when the pellets were col-
lected. χ 2 analysis indicates the two samples differ signi¬cantly in terms of taxonomic
abundances (χ 2 = 586.68, p < 0.0001). The two sets of taxonomic abundances are not
correlated (Spearman™s ρ = 0.6, p > 0.2), which also suggests they may have derived
from different populations, but do the abundances of all of the taxa differ signi¬cantly
between the two samples, or the abundances of just a few of the taxa? To answer this
question, adjusted residuals for each cell were calculated (there are six taxa, and 2
years for each, so twelve cells) to determine if any of the observed values were greater,
or less than would be expected were the two temporally distinct samples derived from
different populations. Basically, the adjusted residual provides a way to determine if
the observed and expected values per cell are statistically signi¬cantly different or not
(see Everitt 1977 for discussion of the statistical method). Expected values (compare
with Table 5.4) and interpretations for each cell are given in Table 5.5. Abundances of
four taxa are causing the statistically signi¬cant difference between the two samples;
specimens of Reithrodontomys, Sorex, Microtus, and Peromyscus are not randomly
distributed between the two chronologically distinct samples. Only Sylvilagus and

Table 5.5. Expected values (E) and interpretation (I) of taxonomic abundances in two
temporally distinct assemblages of owl pellets. See Table 5.4 for observed values

Taxon 1999 E 2000’2001 E 1999 I 2000’2001 I
Sylvilagus 3.7 1.3 p > 0.05 p > 0.05
Reithrodontomys 13.9 5.1 p < 0.05, too few p < 0.05, too many
Sorex 33.7 12.3 p < 0.05, too many p < 0.05, too few
Thomomys 49.8 18.2 p > 0.05 p > 0.05
Microtus 516.8 188.2 p < 0.05, too few p < 0.05, too many
Peromyscus 928.0 338.0 p < 0.05, too many p < 0.05, too few
measuring the taxonomic structure and composition 189

Thomomys occur in the two samples in abundances that are not unexpected; abun-
dances of these two taxa suggest the temporally distinct samples were drawn from
the same population.
Some research has suggested that the Sorenson index provides a better estimate
of similarity than Jaccard™s index (Magurran 1988:96). Not surprisingly, ecologists
designed a version of Sorenson™s index to take account of variation in taxonomic
abundances. That index, Sorenson™s quantitative index, is calculated as

Sq = 2 c N /(AN + B N),

where AN is the total frequency of organisms (all taxa summed) in fauna A, BN is
the total frequency of organisms in fauna B, and cN is the sum of the lesser of the
two abundances of taxa shared by the two assemblages. Using the data in Tables 4.2
and 4.3 for Meier and Cathlapotle mammalian genera, AN (Meier) = 6421, BN
(Cathlapotle) = 6,937, and cN = 4,358 (3 Scapanus + 7 Aplodontia + 342 Castor + 5 Peromyscus +
68 Microtus + 106Ondatra + 39Canis + 5 Vulpes + 102 Ursus + 207 Procyon + 2 Martes + 29Mustela +
3 Mephitis + 51 Lutra + 9Felis + 26Lynx + 43 Phoca + 935 Cervus + 2376Odocoileus ). Thus, Sq =
2(4358)/(6421 + 6973) = 8716/13,394 = 0.651, or 65.1. Recall that the (nonquantitative)
Sorenson™s index was 78.4. Thus, regardless of which index of similarity is used, the
faunas seem fairly similar, though less so when the abundances of taxa are included
than when they are ignored.
A simple way to show similarities and differences between two faunas in terms
of shared taxa, unique taxa, and taxonomic abundances, is to generate a bivariate
scatterplot. Figure 5.7 shows relative (percentage) abundances of those taxa from
Meier and Cathlapotle represented by NISP < 200 at both sites. Notice that were the
relative abundances of taxa equivalent at the two sites, the points would fall close to
the diagonal line; the more equal the relative abundances, the closer to the diagonal
the points would fall. Note as well that more of the points fall on the Meier side
of the diagonal. This suggests that those taxa are relatively more abundant at Meier
than they are at Cathlapotle. Such a graph takes advantage of abundance data in a
visual way. Ecologists are working to develop versions of the Jaccard and Sorensen
indices that also take advantage of abundance data (e.g., Chao et al. 2005), but these
are beyond the scope of the discussion here.
That the binary coef¬cients designed to measure taxonomic similarities of faunal
collections have been largely ignored by zooarchaeologists is likely a good thing. Those
coef¬cients are heavily in¬‚uenced by the sample sizes (= NISP) of the compared
collections because taxonomic richness is signi¬cantly in¬‚uenced by sample size
(Chao et al. 2005). Again, one might use rarefaction in an effort to control sample-
size effects, and that is what some paleozoologists have done (e.g., Barnosky et al. 2005;
quantitative paleozoology

figure 5.7. Bivariate scatterplot of relative (percentage) abundances of mammalian genera
at the Meier site and Cathlapotle. Only genera for which NISP < 200 are plotted. Diagonal
line is shown for reference.

Byrd 1997). This could be a good thing, but it is perhaps unwise for the simple reason
that as was noted more than 20 years ago, the rarefaction procedure was designed
to be used with quantitative units that are statistically independent of one another
(Grayson 1984:152). Those units are also ratio scale values. NISP tallies comprise
units that are probably statistically interdependent and that are also typically at best
ordinal scale values. Given these facts, should one choose to perform a rarefaction
analysis using NISP values, the results should be interpreted in at most ordinal scale
terms. An example will make this clear.
A rarefaction curve based on the eighteen assemblages listed in Table 5.1 con-
structed using Holland™s (2005) Analytical Rarefaction is shown in Figure 5.8 and
suggests that, given a total NTAXA of twenty-eight for the area represented by those
eighteen collections, none of the collections contains all twenty-eight taxa, most
collections contain very few taxa, but none contain too few for their size, and four
(45DO214, 45DO326, 45DO211, 45DO285) of the eighteen collections seem to con-
tain more taxa than they should given their size ( NISP). The rarefaction curve
measuring the taxonomic structure and composition 191

figure 5.8. Rarefaction analysis of eighteen assemblages of mammal remains from eastern
Washington State using Holland™s (2005) Analytical Rarefaction. Data from Table 5.1 .

thus reveals something we didn™t know before because it presents the data in a
unique, interpolated way. If I were analyzing these collections, I would try to deter-
mine why four of the collections were unexpectedly taxonomically rich; perhaps they
are temporally unique, functionally/behaviorally unique, or located in a particular
Were one to perform a rarefaction analysis like that shown in Figure 5.8, one should
¬rst determine if those assemblages are nested. Recall from Chapter 4 that in a series
of perfectly nested faunas, successively smaller faunas will have fewer of the taxa
represented in those faunas that are successively larger, and larger faunas will have
all those taxa represented in smaller faunas plus additional taxa. The interpretive
assumption is that nested faunas all derive from the same parent population. There
are ways to test the degree of nestedness of faunas. That has been done with the faunas
in Figure 5.8; consider Figure 4.12, which shows that the nestedness “temperature”
for this set of eighteen faunas is 18.23 —¦ , meaning the faunas are relatively strongly
nested. The rarefaction analysis in Figure 5.8 thus seems reasonable, if one is willing
to allow an unknown degree of skeletal specimen interdependence and, thus, allow
a bit of statistical sloppiness.
quantitative paleozoology

Taxonomic Heterogeneity

Several indices have been developed to measure taxonomic heterogeneity. Paleozool-
ogists have tended to use only two of these, although there are several different ones
that are occasionally mentioned (e.g., Andrews 1996). By far the most popular one
among zooarchaeologists is the Shannon’Wiener index, sometimes referred to as
the Shannon index. It generally varies between 1.5 and 3.5 (Magurran 1988:35); larger
values signify greater heterogeneity. The Shannon index is calculated as:

H =’ Pi (ln Pi ),

where Pi is the proportion (P) of taxon i in the assemblage. The proportion (some-
times referred to as “importance”) of each taxon in the collection is multiplied by
the natural log of that proportion. Because proportions are < 1, transforming those
values to natural logs results in a negative sign. Values of the products of the multi-
plications are summed, and then converted from a negative value to a positive value
by the negative or ““” sign in front of the summation ( ) sign.
Let™s say we want to determine the taxonomic heterogeneity (at the genus level)
of the total Meier site mammal collection (Table 4.2). The data and mathematical
steps for calculating the value of the Shannon’Wiener index are summarized in
Table 5.6. NTAXA for this collection is 26. The heterogeneity index is 1.556, suggesting
the total Meier site mammal collection is somewhat heterogeneous. For comparative
purposes, consider the fact that the Shannon’Wiener heterogeneity index for the
total Cathlapotle collection, without distinction of the Precontact and Postcontact
assemblages (Table 4.3), has a value of 1.487, indicating that the heterogeneity of the
Cathlapotle collection is a bit less than that of the Meier collection. One contributing
factor here is that the Meier collection, with twenty-six taxa, is taxonomically richer
than the Cathlapotle collection, which contains remains of only twenty-four taxa.
Does a difference in the evenness of the two assemblages also contribute to the dif-
ference in heterogeneity? To answer that question requires calculation of an evenness
index for each collection.
Because heterogeneity is a function of taxonomic richness and evenness, it is
possible that heterogeneity will also be a function of sample size (e.g., Grayson 1981b).
Thus, if one wishes to measure heterogeneity and compare that variable across several
different samples, it is advisable to determine if there is any relationship between the
measures of heterogeneity and NISP for a set of samples. Once again, consider
the eighteen assemblages from eastern Washington State (Table 5.1 ). The relationship
between sample size per site and heterogeneity per site is, in the case of these eighteen
measuring the taxonomic structure and composition 193

Table 5.6. Derivation of the Shannon’Wiener index of heterogeneity for the
Meier site (original data from Table 4.2). Logs are natural logarithms

Taxon NISP Proportion (p) Log of p p(log p) Running sum
’5.878 ’0.016 ’0.016
Scapanus 18 0.00280
’5.878 ’0.016 ’0.032
Sylvilagus 18 0.00280
’6.822 ’0.007436 ’0.039
Aplodontia 7 0.00109
’8.740 ’0.001398 ’0.041
Tamias 1 0.00016
’8.079 ’0.002504 ’0.043
Tamiasciurus 2 0.00031
’6.571 ’0.009199 ’0.053
Thomomys 9 0.00140
’2.933 ’0.156 ’0.209
Castor 342 0.05326
’5.212 ’0.028 ’0.237
Peromyscus 35 0.00545
’8.740 ’0.001398 ’0.238

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