’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

194

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

196

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

198

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.

Discussion

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

200

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

assemblages.

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

202

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.

T RE N D S I N T A XO N O M I C A B U N D A N CE S

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

204

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