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Rattus 1 0.00016
’8.740 ’0.001398 ’0.239
Neotoma 1 0.00016
’4.162 ’0.065 ’0.304
Microtus 100 0.01557
’2.843 ’0.166 ’0.470
Ondatra 374 0.05825
’8.740 ’0.001398 ’0.472
Erethizon 1 0.00016
’4.058 ’0.070 ’0.542
Canis 111 0.01729
’7.156 ’0.00582 ’0.547
Vulpes 5 0.00078
’4.142 ’0.066 ’0.613
Ursus 102 0.01589
’3.108 ’0.139 ’0.752
Procyon 287 0.04470
’5.773 ’0.018 ’0.770
Martes 20 0.00311
’3.869 ’0.081 ’0.851
Mustela 134 0.02087
’7.386 ’0.004579 ’0.856
Mephitis 4 0.00062
’4.836 ’0.038 ’0.894
Lutra 51 0.00794
’6.571 ’0.009199 ’0.903
Puma 9 0.00140
’5.333 ’0.026 ’0.929
Lynx 31 0.00483
’5.006 ’0.034 ’0.963
Phoca 43 0.00670
’1.927 ’0.281 ’1.244
Cervus 935 0.14562
’0.530 ’0.312 ’1.556
Odocoileus 3,780 0.58869
= ’ = 1.556
6,421 1.0 “ “

assemblages, not statistically signi¬cant, suggesting that variation in heterogeneity
is not being driven by variation in sample size (Figure 5.9). Given the insigni¬cant
correlation between sample size and heterogeneity, we might feel safe in comparing
heterogeneities across these eighteen assemblages were it not for the fact that sample
size and richness are strongly correlated among them (Figure 5.3), and the fact that as
richness increases so too does heterogeneity. Only in those instances in which there
is no such correlation (between sample size and richness and between sample size
and heterogeneity) might one make statements about variation between assemblages
quantitative paleozoology

figure 5.9. The relationship between taxonomic heterogeneity (H) and sample size (NISP)
in eighteen assemblages of mammal remains from eastern Washington State. Best-¬t regres-
sion line is insigni¬cant (r = “0.30, p > 0.2). Data from Table 5.1 .

and taxonomic heterogeneity. Otherwise, variation in heterogeneity may be a result
of variation in sample size across the compared collections.
Notice that I said “might” and “may” with respect to the possible in¬‚uence of
sample size on heterogeneity. Heterogeneity is a summary measure of the relative
(proportional) abundances of taxa. There is, therefore, a potential problem with the
analysis attending Figure 5.9. Speci¬cally, the problem concerns the fact that relative
abundances are involved. That problem can be circumvented by using a statistical test
other than a form of correlation analysis. That test will be introduced after evenness
is discussed.

Taxonomic Evenness

How are individuals distributed across taxa? Are all taxa represented by about the
same number of individuals (an even fauna), or do some taxa contain many individ-
uals whereas other taxa contain very few individuals (an uneven fauna)? To answer
such questions beyond comparing histograms showing the distribution of individ-
uals across taxa in each fauna (the histograms may not be visually distinguishable;
measuring the taxonomic structure and composition 195

figure 5.10. Frequency distribution of NISP values across six mammalian genera in a
collection of owl pellets. Data from Table 2.9.

Figure 5.2), an index of evenness can be calculated. One index that is regularly used to
measure evenness quantitatively requires the Shannon’Wiener heterogeneity index,
which is divided by the log of NTAXA or richness (Magurran 1988). Thus, evenness
is calculated as:

e = H/ lnS,

where H is the Shannon’Wiener heterogeneity index and S is taxonomic richness.
The lower the value of e, known as the Shannon index of evenness (Magurran 1988),
the less even the assemblage. The Shannon index, e, is constrained to fall between 0
and 1, with a value of 1 indicating an even fauna or that all taxa are equally abundant
(Magurran 1988).
Consider the assemblage of mammals represented in the previously mentioned
owl pellet collection and described in Table 2.8. NTAXA (S) is 6, using the NISP
values for this collection the Shannon’Wiener heterogeneity index (H) is 0.922, and
the Shannon index of evenness (e) is (0.922/1.792 =) 0.515. Inspection of Figure 5.10,
which shows the frequency distribution of NISP across the taxa, suggests that indeed,
this fauna is not very even, just as its calculated e value suggests.
Recall the question posed above after the Shannon’Wiener heterogeneity indices
for the Meier collection and for the Cathlapotle collection were calculated “ the Meier
collection was more heterogeneous in part because it was taxonomically richer than
the Cathlapotle collection, but did a difference in the evenness of the two faunas also
quantitative paleozoology

contribute to that difference in heterogeneity? For Meier, e = 1.556/3.258 (where 3.258
is the natural log of 26), so e = 0.4776; for Cathlapotle, e = 1.487/3.178 (where 3.178
is the natural log of 24), so e = 0.4679. The evenness index e varies between 0 and 1 ;
if e = 1, then all taxa are equally abundant (specimens are equably distributed across
taxa). So, the Meier collection seems to be slightly more even than the Cathlapotle
collection “ identi¬ed specimens are a bit more equably distributed across the twenty-
six taxa of the Meier collection than identi¬ed specimens are distributed across
the twenty-four taxa of the Cathlapotle collection. Thus, not only does the greater
richness of the Meier collection contribute to greater heterogeneity, so too does the
greater evenness of the Meier collection contribute to its heterogeneity being greater
than that evident in the Cathlapotle collection.
As with richness and heterogeneity, there are cases when taxonomic evenness
seems to be driven by sample size, though there are also cases where there is no such
relationship (e.g., Grayson and Delpech 1998; Grayson et al. 2001 ; Nagaoka 2001 ,
2002). Until a few years ago, to determine whether or not evenness was driven by
sample size, one measured the strength of the relationship between the two variables.
Consider the eighteen assemblages of mammal remains from eastern Washington
State (Table 5.1 ). Sample size and evenness values among these eighteen assemblages
are strongly correlated (Figure 5.11 ); about 53 percent of the variation in evenness is
explained by variation in NISP (r = 0.73, p < 0.002). On the basis of that correlation,
the paleozoologist would have previously concluded that it would be ill-advised to
suggest variation in evenness across these assemblages was due to some ecological,
environmental, or human behavioral variable rather than simple difference in sample
size. There is now a better way to search for sample size effects on measures of relative
abundance that will be described shortly. First, another measure of evenness needs
to be introduced.
Another index of evenness occasionally used by zooarchaeologists is the reciprocal
of Simpson™s index (Grayson and Delpech 2002; James 1990; Jones 2004; Schmitt and
Lupo 1995; Wolverton 2005). If one assumes that the population is in¬nitely large,
the index is calculated as

1/ pi2 ,

where pi is the proportional abundance of taxon i in the total collection (Magurran
1988:39). However, because the population of organisms in a community, and the
sample of remains in a deposit, are ¬nite, it is appropriate to calculate Simpson™s
index as:

ni [ni ’ 1 ]/N[N ’ 1 ],
measuring the taxonomic structure and composition 197

figure 5.11. Relationship between taxonomic evenness (e) and sample size (NISP) in
eighteen assemblages of mammal remains from eastern Washington State. Best-¬t regression
line is signi¬cant (r = “0.73, p = 0.0005). Data from Table 5.1 .

where ni is the number of specimens of the ith taxon, and N is the total number
of specimens of all taxa (Magurran 1988). The more evenly distributed individuals
(or specimens) are across taxa, the larger the value of this index. Simpson™s index is
known as D because it is more sensitive to the dominance of an assemblage by a single
taxon than is e and D is also less sensitive to taxonomic richness (Magurran 1988).
The reciprocal or inverse of Simpson™s index is used by ecologists and paleozoologists
because the lower the value of the reciprocal, the more the assemblage is dominated
by a single taxon, or the less evenly individuals are distributed across all taxa in the
assemblage. Thus, as 1 /D decreases, the more an assemblage is dominated by a single
taxon. The reciprocal of Simpson™s index for the owl pellet assemblage (Table 2.9) is
1.314, suggesting that the assemblage is fairly strongly dominated by a single taxon (in
this case, Peromyscus), just as we would conclude by simple inspection of Figure 5.10.
The inverse of Simpson™s index of dominance, or 1 /D, for each of the eighteen
assemblages of mammalian genera from eastern Washington State is given in Table 5.1 .
Those index values are only weakly correlated with the NISP per sample size (r =
’0.43, 0.1 > p > 0.05), as might be surmised from the wide scatter of points around
quantitative paleozoology

figure 5.12. Relationship between sample size and the reciprocal of Simpson™s dominance
index (1 /D) in eighteen assemblages of mammal remains from eastern Washington State.
Best-¬t regression line is weakly signi¬cant (r = “0.43, 0.1 > p > 0.05). Data from Table 5.1 .

the simple best-¬t regression line (Figure 5.12). Thus, one would likely be tempted to
conclude that taxonomic evenness or dominance in these eighteen assemblages does
not seem to be signi¬cantly in¬‚uenced by sample size and thus one could discuss
differences in evenness or dominance in terms of ecological variation in site settings
or variation among the taphonomic agents that accumulated the remains or the
like. But again, a correlation has been sought between sample size ( NISP) per
assemblage and a measure of relative abundance (proportions). It is now time to
discuss that issue.


Simple regression analysis such as is illustrated in Figure 5.9 is a reasonable way to
search for sample size effects if absolute measures of sample size are used, such as when
one seeks to determine if there is a relationship between NTAXA and NISP. What
measuring the taxonomic structure and composition 199

Table 5.7. Total NISP of mammals, NISP of deer
(Odocoileus spp.), and relative (%) abundance of deer in
eighteen assemblages from eastern Washington State

Site NISP Deer NISP % Deer
45OK18 31 0 0
45DO204 48 5 10.4
45DO273 84 6 7.1
45DO243 157 34 21.7
45OK2A 366 87 23.8
45DO189 415 251 51.8
45DO282 426 4 0.9
45DO211 474 45 9.5
45DO285 491 33 6.7
45DO214 536 184 34.3
45DO326 640 120 18.8
45DO242 673 368 54.7
45OK287 807 252 31.2
45OK250 1,077 738 68.5
45OK4 1,108 803 72.5
45OK2 2,574 2,021 78.5
45OK11 3,549 1,668 47.0
45OK258 4,433 3,458 78.0

about those cases when relative abundances (percentages or proportions) of one or
more taxa are of interest? Perhaps the paleozoologist wishes to know if the percentages
of a taxon in a series of assemblages increase or decrease in a patterned manner (over
time if the assemblages are chronologically distinct, or over space if the assemblages
are arrayed over an expanse of geographic space). Consider the relative abundance of
deer in the eighteen assemblages from eastern Washington State (Table 5.7). The per-
centage of each assemblage™s NISP representing deer is strongly correlated with
the NISP per assemblage (Spearman™s ρ = 0.763, p = 0.0002), suggesting there
may be a sample-size effect driving the trend. Grayson (1984:126“127) suggested an
empirical way to contend with cases when one ¬nds a correlation between the rel-
ative abundance of one or more taxa and NISP across multiple samples: “Remove
assemblages in order of increasing sample size until the correlation between sam-
ple size and relative abundance is no longer signi¬cant.” Doing so with the eigh-
teen eastern Washington State assemblages results in elimination of the ten smallest
quantitative paleozoology

assemblages “ those with NISP < 600. When those ten assemblages are eliminated,
the correlation between the relative abundance of deer and total assemblage size is
no longer signi¬cant (ρ = 0.667, p = 0.08).
There is a way to evaluate whether relative abundances are in¬‚uenced by sample size
without eliminating assemblages. And it is more statistically sensitive to detecting true
sample size effects than the correlation technique when percentages or proportions
are used as measures of abundance. Cannon (2000, 2001 ) pointed out that correlating
(spatial or temporal) trends in relative abundances of taxa with sample size ( NISP)
is statistically not the best way to search for sample size effects. This is so because
relative abundances do not register whether sample sizes are ¬ve or ¬ve thousand.
Statistically, noting the difference between an absolute tally of ¬ve and an absolute
tally of ¬ve thousand is quite different than saying each comprises 5 percent of the
total collection (100 and 100,000, respectively). Small samples make ruling out sample
size effects dif¬cult; their effect on correlations may simply be due to sampling error
rather than any accurate re¬‚ection of abundance. Rare phenomena are particularly
dif¬cult to inventory “ they will be absent from collections “ unless the samples are
large. If the abundances of several rare taxa are not quite equal in the target population,
but samples are small, the true relative abundances of those rare taxa likely will not
be accurately re¬‚ected by small samples (Grayson 1978a, 1979, 1984). A correlation
between the relative abundance of a taxon and NISP across multiple assemblages
may be driven by small samples simply because of sampling error inherent in those
Using simulated samples drawn from ¬ctional but known populations, Cannon
(2001 ) showed that absolutely small samples of populations that have trends in the
relative abundance of a taxon often display no trends, and also that absolutely small
samples of populations that have no trends in the relative abundance of a taxon
often display trends. Thus, using the regression approach to search for sample size
effects (as with how the data in Table 5.7 were examined) may lead to commission
of a Type I error (rejecting a true null hypothesis that there is no true trend in
relative abundances when in fact there is no trend) or commission of a Type II
error (accepting a false null hypothesis that there is no true trend in abundances
when in fact there is a trend). In both cases, the null hypothesis is that no trend
is present, but sampling error has produced samples that are not representative of
the population. Furthermore, the presence of a signi¬cant correlation coef¬cient is
interpreted as indicating that sample size is the source of the correlation, and the
absence of a signi¬cant correlation coef¬cient is interpreted as indicating that sample
size is not the source of the correlation. As Cannon (2001 :185) astutely observed, the
measuring the taxonomic structure and composition 201

¬rst interpretation at best rests on an incomplete understanding of the relationship
between relative abundances and sample size; the second interpretation presumes
sample sizes are suf¬ciently large to warrant con¬dence but in fact may be too small.
Cannon (2000, 2001 ) suggested that rather than regression analysis or calculation
of a correlation coef¬cient, a different statistical test be used to ascertain if sample
size effects plague an analysis of relative abundances. Cochran™s test of linear trends
is a form of χ 2 analysis that tests for trends among multiple rank-ordered samples
(Zar 1996:562“565). As Cannon (2000:332) notes, Cochran™s test is constructed such
that “signi¬cant trends will not be found when samples are so small that random
error cannot be ruled out at a speci¬ed con¬dence level as the cause of differences
in relative abundance between samples.” Cochran™s linear test seeks trends (either
across space or over time) in relative abundance in such a way as to more directly
take absolute sample size into account than correlation-based analyses. One ¬rst
calculates a standard χ 2 statistic, and then determines how much of that statistic is
the result of a linear trend; if the latter is suf¬ciently (statistically signi¬cant) large,
then one concludes that there is indeed a linear trend in the data independent of
sample size.
Let us return yet again to the eighteen assemblages from eastern Washington State
and the relative abundance of deer remains to determine if there is a linear trend
in the relative abundance of deer or not (Table 5.7). The overall χ 2 statistic is large
and signi¬cant (χ 2 = 4196.6, p < 0.0001), suggesting there is a signi¬cant association
between sample size and frequency of deer remains. The χ 2 statistic for a linear
trend is also signi¬cant (χ 2 = 2239.3, p < 0.0001), suggesting there is signi¬cant
trend in the abundance of deer remains across the eighteen assemblages regardless
of the sizes of those assemblages. Figure 5.13 suggests that there is indeed a trend in
relative abundance of deer across the eighteen assemblages (r = 0.79, p < 0.001). The
problem is that Figure 5.13 gives no indication of possible sample size in¬‚uences on
those relative abundances. Cochran™s test for linear trends does just that, though due
to its relatively recent introduction to paleozoology (Cannon 2000, 2001 ), it has as
yet seldom been used (e.g., Cannon 2003; Nagaoka 2005a; Wolverton 2005).
There are several other, nonstatistical points to keep in mind regarding whether or
not sample size in¬‚uences seem to exist. One is that taxonomic richness, heterogene-
ity, and evenness (regardless of the exact index calculated) were designed for extant
ecological communities (e.g., Jones 2004), and they were designed to use tallies of
individual (statistically independent) organisms as the quantitative units (Grayson
1984). Because it is likely that NISP values are not statistically independent tallies per
taxon, cautious interpretation of paleozoological values for richness, heterogeneity,
quantitative paleozoology

figure 5.13. Percentage abundance of deer in eighteen assemblages from eastern Wash-
ington State.

and evenness should be foremost in one™s mind. Do not be misled into thinking
about these variables as if they are ratio scale variables; chances are good that they
are not, and chances are fair that they may not even be ordinal scale variables. If it can
be shown that taxonomic abundance data based on NISP are ordinal scale variables
(see Chapter 2 for description of a method), then it can be argued that the index
values are also ordinal scale values.
Another point made by Nagaoka (2001 , 2002) about evenness extends to hetero-
geneity. She noted that evenness does not take into account the position of taxa in
a rank ordered (based on abundance) set of taxa. Using her example, taxon A may
comprise 80 percent and taxon B 20 percent of assemblage I, but taxon B comprises
20 percent and taxon B 80 percent of assemblage II. Evenness and heterogeneity
indices will not register these obvious differences, so inspection of how much each
taxon is contributing may be required if something other than simply an index of
faunal structure (is the fauna even or uneven, heterogeneous or homogeneous) is
desired. Jones (2004) adds that evenness will be in¬‚uenced by NTAXA (as will hetero-
geneity). The important point therefore is that evenness and heterogeneity cannot
be considered independently of richness or of each other. One might choose to focus
analysis on one variable, but the other variables (with the possible exception of rich-
ness) will likely require examination in order to correctly interpret the target variable.
measuring the taxonomic structure and composition 203

figure 5.14. Abundance of bison remains relative to abundance of all ungulate remains
over the past 10,500 (C14) years in eastern Washington State.


For nearly as long as paleozoologists have measured taxonomic abundances evident
in the collections of remains they study, they have also tracked those abundances
through time and across space. Sometimes that tracking has involved tables of num-
bers, each column representing a taxon, each row a stratum if the analyst was inter-
ested in temporal variation in abundances. If the analyst was interested in variation
in taxonomic abundances across space, each row could be a geographically distinct
collection locality whereas each column was a distinct taxon. Tables of numbers pre-
sented data, but if more than a few rows and columns were included, such tables are
dif¬cult to interpret visually. A reasonable alternative, then, was to construct a graph
displaying taxonomic abundances across time or space, although raw data might not
be included.
Both tables of numbers and graphs of taxonomic abundances are still constructed
by paleozoologists because they are useful analytical techniques. The graph in Fig-
ure 5.14 is based on data in Table 5.8 (from Lyman 2004b). The graph shows the relative
abundance of bison (Bison spp.) remains in eastern Washington State more or less
by 500-year bins since the terminal Pleistocene; several bins are lumped to simplify
the graph. The graph is signi¬cant for several reasons. First, it shows the abundance
of bison remains relative to the abundance of remains of all other ungulates in the
paleozoological record. There is no correlation between the relative abundance of
quantitative paleozoology

Table 5.8. Frequencies (NISP) of bison and nonbison ungulates per time
period in ninety-one assemblages from eastern Washington State

Time period (yr BP) Bison (Bison sp.) Nonbison ungulates Total
100“500 4 4,196 4,200
501 “1,000 107 3,838 3,945
1,001 “1,500 131 1,512 1,643
1,501 “2,000 73 1,182 1,255
2,001 “2,500 375 218 593
2,501 “3,000 4 3,851 3,855
3,001 “3,500 5 1,639 1,644
3,501 “4,000 1 414 415
4,001 “5,000 1 776 777
5,001 “6,000 10 2,111 2,121

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