then any use of the Lincoln“Petersen index (or any other use of what are thought to

be bilaterally paired skeletal elements) is likely invalid.

Identifying Bilateral Pairs

Krantz (1968:287) noted early on that “in experienced hands there should be little

doubt as to which mandibles [or any other bilaterally paired elements] pair off and

which do not.” Twenty years later, Todd (1987:180) provided a detailed statement on

the procedure for identifying bilateral pairs:

Initial estimates of possible anatomical re¬ts can be based on metric attributes of the

elements. In the case of bilateral re¬ts, potential candidates for pairs can be further

examined visually. Familiarity with elements from individual carcasses allows for the

recognition of attributes that are individually distinctive and bilaterally uniform. The

patterns of muscle attachment shape and prominence, synovial fovea shape and depth,

and the proportions of components of articular surfaces within paired elements in the

[animal] carcass create an individually distinctive “¬nger print” for element mates.

Bilateral mates are usually mirror images of each other in these attributes.

Two problems accompany anatomical matching. First, matching takes a consid-

erable amount of time (Adams and Konigsberg 2004; White 1953a); more time is

required as the size of the collection (and thus the number of possible pairs) increases.

The other problem is more serious. The analyst can determine the sex of the individ-

ual represented by paired bones in a mammal skeleton from a very limited number

of skeletal elements (innominates, frontals of some antler-bearing ungulates), unless

the taxon under consideration is sexually dimorphic. Exacerbating the dif¬culty of

matching is the fact that the degree to which paired bones (and teeth) display the

same ontogenetic stage simultaneously is unclear. Thus for illustrative purposes the

focus here is bone size. This criterion is used by virtually all who have identi¬ed

bilaterally paired bones (e.g., Allen and Guy 1984; Chase and Hagaman 1987; Enloe

2003a, 2003b; Enloe and David 1992; Fieller and Turner 1982; Morlan 1983; Nichol and

Creak 1979; Todd 1987; Todd and Frison 1992; Turner 1983; Wild and Nichol 1983b).

Chaplin (1971 :74) provides an early example of a matching procedure based on

bone size. He indicates that left and right distal tibiae of sheep (Ovis aries) can be

identi¬ed as bilateral pairs from the same individual if they are identical in “maximum

width,” or if they are within ¤ 0.3 mm of each other in maximum width, but lefts

and rights are not pairs if their maximum widths differ by ≥ 0.4 mm. How he

established the 0.3 vs. 0.4 mm maximum width is not speci¬ed, but it introduces the

quantitative paleozoology

130

matching problem. Identifying bilaterally paired bones in paleozoological collections

rests on the assumption that left and right skeletal elements from the same organism

will be symmetrical. Bones (and teeth) are, however, typically asymmetrical to some

degree. Indeed, temporally ¬‚uctuating asymmetry has in the past two or three decades

become a very important research topic in biology (e.g., Gangestad and Thornhill

1998; Palmer 1986, 1994, 1996; Palmer and Strobeck 1986; Pankakoski 1985). To identify

bilateral pairs among a commingled set of left and right elements from multiple

individuals of a taxon demands that we answer the question “How symmetrical is

symmetrical enough to conclude a left and right humerus (for example) are from the

same individual” (Lyman 2006a)? Answering that question involves determining a

“tolerance” or the maximum allowable asymmetry between bilaterally paired bones

(Nichol and Creak 1979). Determination of the tolerance rests on study of known

pairs.

Adams and Konigsberg (2004) performed a “test for the accuracy of pair-matching”

by choosing skeletons of ¬fteen humans recovered from an early historic burial

ground associated with an archaeological village. Based on morphological indica-

tors such as “robusticity, muscle markings, epiphyseal shape, bilateral expression of

periosteal reaction, and general symmetry” (Adams and Konigsberg 2004:145), they

successfully identi¬ed all pairs of femora, all pairs of tibiae, and all but one pair of

humeri. But they also suggested that the error rate might increase were the sample

size to increase because large samples would obscure between-individual variation

(Adams and Konigsberg 2004:146). As sample size increases (as the number of indi-

viduals increases), the degree of discontinuity between individuals will decrease such

that one individual will, in a sense, blend into another individual. This will occur

with either morphological traits or metric (size) traits.

The bivariate scatterplot in Figure 3.12 shows two measurements of paired distal

humeri from forty-eight museum skeletons of deer. Both white-tailed deer (O. vir-

ginianus, N = 17 pairs) and mule deer (O. hemionus, N = 30 pairs) are included;

the forty-eighth pair of distal humeri is from a hybrid of the two species. Both

species are represented in many paleozoological collections in western North America

(Livingston 1987; Lyman 2006b) and their remains can occasionally be distinguished

based on morphological criteria (Jacobson 2003, 2004); the two species cannot be

distinguished on the basis of the size of the distal humerus. The two measurements

taken on the distal humeri were the anterior breadth of the distal trochlea (DBt) and

the minimum (antero-posterior) diameter of the (latero-medial) center of the distal

trochlea (MNd). Use of two measurements rather than one should make any effort to

identify bilateral mates more accurate because specimens must be symmetrical in

two dimensions rather than one.

estimating taxonomic abundances: other methods 131

figure 3.12. Latero-medial width of the distal condyle and minimum antero-posterior

diameter of the middle groove of the distal condyle of forty-eight pairs of left and right

distal humeri of Odocoileus virginianus and Odocoileus hemionus. Try to determine which

left element is the bilateral match of which right element.

Conceive the difference between the left and right DBt as de¬ning the length of

one side of the right angle of a right triangle (= a in Figure 3.13), and the absolute

difference between the left and right MNd as de¬ning the length of the other side

of the right angle of the right triangle (= b in Figure 3.13). Then, the Pythagorean

theorem (a2 + b2 = c2 ) can be used to ¬nd how asymmetrical the distal humeri are in

terms of the two measurements. If the left and right distal humeri were perfectly sym-

metrical, then the hypotenuse (c-value) of the right triangle would be zero because

a = 0 and b = 0. None of the distal humeri pairs include specimens that are perfectly

symmetrical; the hypotenuse length or c-value of the triangle de¬ned by the left

and right specimens of each pair > 0.0. There is no statistically signi¬cant difference

between the c-values displayed by pairs of distal humeri of the two deer species

(O. virginianus, 0.753 ± 0.81 ; O. hemionus, 0.467 ± 0.282; Student™s t = 1.771 ;

p > 0.08), so the mean of all forty-seven, plus the hybrid (forty-eighth specimen),

was determined; the average c-value = 0.561 ± 0.541. A tolerance level one might

adopt is a c-value ¤ 0.561 ; any set of one left and one right distal humerus of deer the

DBt and MNd measurements of which de¬ned a c-value ¤ 0.561 would be identi¬ed

as a bilateral pair.

quantitative paleozoology

132

figure 3.13. A model of how two dimensions of a bone can be used to determine degree

of (a)symmetry between bilaterally paired left and right elements. Upper, the Pythagorean

theorem describing how the lengths of the three sides of a right triangle are related; lower, a

model of how two measurements of a skeletal part considered together de¬ne a right triangle.

The length of the hypotenuse (c-value in the text) is a measure of how (a)symmetrical two

specimens are; specimens L and R are more symmetrical than specimens l and r.

Were you to try to identify bilateral pairs of one left distal humerus and one

right distal humerus using the tolerance level just identi¬ed, you might get some

of the pairs correct. But you likely could not match them all because of the chosen

tolerance level (some pairs would have c-values > 0.561), and of those that you did

match, some would likely represent incorrectly identi¬ed pairs. With respect to the

latter, consider Figure 3.12 again. Assuming we did not know which lefts went with

which rights in this ¬gure, use of the c-value of 0.561 to identify pairs in this data

set is dif¬cult if not impossible because of the large number of specimens. Earlier

analysts seem to have recognized this problem, although their wording was inexplicit

(e.g., Uerpmann 1973:311). Later commentators were more explicit. Nichol and Wild

(1984:36“37) noted that “it is much harder to identify the extra unmatchable elements

in a collection of bones from a larger number of animals of a species than it is in a

smaller collection. For example, if there are 100 lefts and 100 rights of a bone, it will

be rather dif¬cult to produce an estimate of MNI much greater than 100, whether

estimating taxonomic abundances: other methods 133

100 or 200 animals are represented, while it may be very easy to see that a single left

and a single right come from different individuals.”

The blizzard of points in Figure 3.12 comprising both left and right specimens com-

prises the problem. Within the center of the distribution it is effectively impossible to

determine which lefts go with which rights, and in a paleozoological sample approxi-

mating this size it would be impossible to identify true pairs and to ¬nd truly unpaired

specimens. Only at the peripheries of this point scatter do we ¬nd individual left ele-

ments without multiple possible right mates and individual right elements without

multiple possible left mates. Exacerbating the problem are the facts that populations

of deer killed and deposited in many sites (= thanatocoenose) likely represent sev-

eral dozen individuals, yet archaeologists generally collect only a sample of those

remains “ perhaps as little as 10 percent or even less “ the Whitean MNI of the

population (= thanatocoenose) might be 6“10. This is in part why the Lincoln“

Petersen index seems so attractive “ the manner in which it is calculated explicitly

acknowledges that the materials at hand are a sample of a population (though which

population in Figure 2.1 is unclear). But Figure 3.12 indicates that even were one to

use the tolerance level or c-value of 0.561 for distal humeri based on the model in

Figure 3.13, it is likely that false pairs would be identi¬ed. A more conservative c-value,

such as, say, 0.25, would overlook many true pairs and still result in the identi¬cation

of some false pairs.

Perhaps bilateral pairs could be identi¬ed using morphological criteria such as the

size and rugosity of muscle scars and other traits, but deciding which left elements to

compare with which right elements likely would begin with similarly sized specimens.

Similarly, it might be possible to identify bilateral pairs in samples of fewer than a

dozen lefts and fewer than a dozen rights, but if these specimens comprise but a

sample of the population (and it is likely they will for one or both of two reasons “

only part of the population was recovered because of sampling and recovery, or only

part of the population was recovered because of differential preservation), then who

is to say that the population would not approximate that in Figure 3.12 (see Lyman

2006a for a real example).

In light of this discussion of distal humeri and a similar analysis of deer astragali

discussed elsewhere (Lyman 2006a), the assumption that analysts can accurately

identify bilateral pairs required of quantitative units, such as the Lincoln“Petersen,

index is unwarranted. And there is yet another seldom acknowledged dif¬culty with

quantitative methods aimed at measuring or estimating taxonomic abundances that

require information on the number of bilateral pairs in the assemblage.

The Lincoln“Petersen index is dependent on aggregation. Pairs would not be

sought among a set of left skeletal elements from a stratum deposited about

3,000 years ago and a set of right elements recovered from a stratum deposited

quantitative paleozoology

134

about 2,000 years ago (unless perhaps one sought to measure postdepositional dis-

turbance). This underscores in yet another way the potentially signi¬cant in¬‚uence

of aggregation on derived quantitative measures and estimates. Even if we have a sin-

gle stratum, and we are comfortable with the procedure we use to identify bilateral

pairs, have we gained any resolution with respect to taxonomic abundances? The

data in Table 3.15 suggest it is unlikely that we have.

CO R RE C T I N G F O R V A R I O U S T H I N G S

When Shotwell (1955, 1958) discussed how he estimated taxonomic abundances, he

noted that not only do individuals of different taxa have different numbers of skeletal

elements per individual, individuals of different taxa also have different numbers

of taxonomically identi¬able skeletal elements. (For sake of simplicity, ignore the

potentiality that skeletal elements might be broken and thus constitute specimens

rather than skeletal elements.) Thus he advocated determination of the corrected

number of skeletal elements per taxon, or “the number that would be expected

if all species contributed the same number (the standard number of elements) of

recognizable elements” (Shotwell 1955:331). To calculate the corrected number of

skeletal elements (CNSE), he used the equation:

CNSE = [E (SNE)]/ENE,

where E is the number of elements identi¬ed in a collection, SNE is the standard

number of elements per individual that the analyst can identify, and ENE is the

estimated number of elements that the paleozoologist can identify in one complete

skeleton of the taxon under consideration. SNE is, according to Shotwell (1955), an

arbitrary number chosen so as to reduce the amount of corrective math.

It was argued in Chapter 2 that corrections for variation in the frequency of iden-

ti¬able elements per taxon were unnecessary for several reasons. Nevertheless, such

weighting of skeletal frequencies has been suggested by more than one paleozo-

ologist (e.g., Gilinsky and Bennington 1994; Plug and Badenhorst 2006; Plug and

Sampson 1996). Holtzman (1979), for example, suggests calculation of what he terms

the “weighted abundance of elements” or WAE. This value is calculated as

WAE = NE j /NEi j ,

where NE j is the frequency of skeletal elements of taxon j identi¬ed in an assemblage,

and NEi j is the frequency of skeletal elements of individual i of taxon j that can be

identi¬ed. Because WAE uses the number of skeletal elements rather than the number

of specimens, Holtzman (1979:80) argued that it “is less susceptible to biases arising

estimating taxonomic abundances: other methods 135

Table 3.16. Abundances of beaver and deer remains at

Cathlapotle, and WAE values and ratios of NISP and WAE

values per taxon per assemblage

NISP WAE Ratio, beaver: deer

Beaver, precontact 111 1.247 NISP, precontact

= 0.143

Beaver, postcontact 238 2.674 NISP, postcontact

= 0.177

Deer, precontact 775 14.091 WAE, precontact

= 0.088

Deer, postcontact 1,347 24.491 WAE, postcontact

= 0.109

from variation in degree of fragmentation or number of [identi¬able] elements per

individual,” although he also acknowledged that it worked well only if differential

preservation across taxa was not a problem. The methods used for determining the

number of skeletal elements represented by a collection of anatomically incomplete

specimens (an assemblage of broken bones) are considered in Chapter 4. It suf¬ces

here to note that Holtzman™s WAE can be described as the number of skeletal elements

identi¬ed per taxon, adjusted to account for the number of elements in one com-

plete skeleton that can be identi¬ed by the paleozoologist. An example will illustrate

Shotwell™s and Holtzman™s concerns.

Although the frequencies of deer remains and of beaver remains from Cathlapotle

are NISP (Table 1.3), let us treat them as if they are frequencies of anatomically

complete skeletal elements. Frequencies of the two taxa are given in Table 3.16. If

each tooth is considered as an independent skeletal element, the skull is considered

one element, and vertebrae, carpals, and tarsals are ignored, each individual deer has

the same number of skulls, mandibles, scapulae, humeri, radii, ulnae, innominates,

femora, tibiae, and ¬bulae (distal end only in ungulates) as each individual beaver.

But a beaver has two clavicles whereas a deer has none; a beaver has four incisors

whereas a deer has eight; a beaver has sixteen metapodials whereas a deer has four;

and a beaver has forty-eight phalanges (¬rst, second, and third) whereas a deer has

twenty-four. Thus a single deer can be conceived (for sake of discussion) to have ¬fty-

¬ve identi¬able elements whereas an individual beaver has (for sake of discussion)

eighty-nine identi¬able elements.

If we follow Holtzman™s (1979) procedure, then (NISP) abundances of beaver

and deer at Cathlapotle change to the WAE values indicated in Table 3.16. Notice

that the relative abundances of beaver and deer do not change whether NISP or

quantitative paleozoology

136

WAE values are used. Deer always outnumber beaver. Notice as well that the WAE

values provide a sort of fractional MNI in the sense that there are suf¬cient skeletal

elements of beaver in the precontact assemblage to represent about one and a quarter

individuals, and there are suf¬cient skeletal elements of beaver in the postcontact

assemblage to represent a bit more than two and a half individuals. Calculation of

Shotwell™s CNSE or Holtzman™s WAE does not gain us much accuracy in estimating

taxonomic abundances.

Rogers (2000a) has been concerned with how the differential accumulation of

skeletal parts in conjunction with the differential preservation of those parts might

in¬‚uence estimates of animal abundance (among other things). He notes that empir-

ical work has focused on the two taphonomic processes of accumulation (what he

terms deposition) and attrition, and that focus has resulted in major gains in knowl-

edge. But he was concerned that statistical methods had not developed or improved

at the same pace as empirical knowledge had improved (Rogers 2000b). Rogers there-

fore developed what he termed “analysis of bone counts by maximum likelihood”

(ABCML). ABCML is a complex statistical algorithm into which one plugs various

data such as skeletal part frequencies (see Chapter 6), bulk density per skeletal part (as

a proxy for preservation potential), economic utility per part, and the like. Attrition

of skeletal parts is assumed to be proportional to the bulk density of each part, and

the probability of transport of a part is assumed to be proportional to the utility of a

part. Rogers (2000a:122“123) explicitly acknowledged that ABCML was “incomplete”

because it has empirical weaknesses. Weaknesses include the modeling of attrition

and transport based on density and utility, respectively; actualistic (ethnoarchaeo-

logical) research indicates that both of these processes are in¬‚uenced by more than

just density and utility. Application of the ABCML protocol to a zooarchaeological

collection (Rogers and Broughton 2001) makes these points explicitly clear for two

reasons. First, the assumptions that are required are numerous, and second, the qual-

itative results are characterized as “more trustworthy” and more likely to be correct

than are the quantitative results (Rogers and Broughton 2001 :763, 772).

It is likely that ABCML has not often been used for two reasons. First, many paleo-

zoologists lack the statistical sophistication necessary to comprehend what ABCML

is and how it works (Rogers 2000b). Paleozoologists must learn more about statistical

methods, and they must overcome about seven decades of disciplinary historical iner-

tia that has focused on deterministic questions rather than probabilistic ones (Lyman

1994c). Second, the method requires one to assume much regarding the taphonomic

history of the collection. Has the collection been subjected to differential transport

intrataxonomically or intertaxonomically or both? Has it been subjected to dif-

ferential attrition intrataxonomically or intertaxonomically or both? A colleague

estimating taxonomic abundances: other methods 137

suggested that other methods analytically overlook these possibilities when

estimating taxonomic abundances, but ABCML deals with them. It also provides

con¬dence intervals, allowing one to assess how tight (or loose) an estimate is.

Rogers and Broughton (2001 ) advocate the use of ABCML because of perceived

¬‚aws in more simplistic analyses. In particular, they correctly note that even non-

parametric measures of association such as Spearman™s rho (between, say, NISP and

MNI) assume that NISP counts represent independent tallies. We know NISP counts

do not (or are highly unlikely to) represent tallies of independent specimens (Chap-

ter 2). Correlation coef¬cients also assume that different taxa have equal numbers of

identi¬able skeletal elements (ignoring fragmentation), and we know that they do

not. Finally, Rogers and Broughton (2001 ) correctly argue that different values of a

correlation coef¬cient cannot be related theoretically to the intensity of a taphonomic

process. Correlation coef¬cients are, however, used by paleozoologists to gain insight

to possible associations; none builds an argument on just a correlation coef¬cient

and instead consults other pertinent data to help understand why a correlation exists

(or doesn™t exist). Few paleozoologists infer the intensity or degree of a taphonomic

process simply on the basis of the magnitude of a correlation coef¬cient; rather, a

coef¬cient is usually interpreted in nominal-scale terms. Two variables are corre-

lated, or they are not. What the presence (or absence) of a correlation means is a

taphonomic issue more than a statistical one.

ABCML presents a detailed view of the sorts of variables that one must con-

sider in any analysis of transport and attrition and how those processes are likely

to in¬‚uence taxonomic abundances. In this I (in Lyman 2004c) agree completely

with Rogers™s (2000a:123) observation that detailed consideration of the empirical

requirements of ABCML will “help identify the [taphonomic, biological, archaeo-

logical, etc.] parameters that should be estimated and reported.” Time will tell if the

method gains popularity among paleozoologists.

S I ZE

In a recent discussion, Orchard (2005) argued that the relationship of bone size

to body size can be used to assist with the estimation of taxonomic abundances.

The method is easy to grasp conceptually. Each skeletal element has a particular

statistical relationship to body size that can be established with museum specimens.

That relationship can be described by a regression equation. Once such equations are

established for multiple skeletal elements, different skeletal elements in a prehistoric

collection may be used to estimate the body sizes represented. Let™s say that the most

quantitative paleozoology

138

common skeletal element in a prehistoric collection is right astragali, and the sizes

of those specimens suggest that there are ¬ve individuals (= MNI) of varied sizes. If

distal right tibiae suggest that there are only four individuals (= MNI) but one of those

tibiae indicates an individual body size larger than any of the individual body sizes

indicated by right astragali, then a sixth individual is added to the tally of individuals.

There is a practical problem; “the time and effort involved in gathering compar-

ative data and generating regression formulae, as well as the dif¬culty in obtaining

adequate comparative samples, can be prohibitive” (Orchard 2005:357). Generat-

ing regression formulae is relatively easy. It is the process of acquiring the requisite

data that is dif¬cult. Consider, for example, Emerson™s (1978) data for white-tailed

deer summarized in Table 3.12. He worked the several weeks of an annual hunting

season. I spent 2 weeks visiting eight collections of deer skeletons in various muse-

ums and comparative zooarchaeological collections to generate the data presented

in Figure 3.12. Those collections are housed in widely separated localities (Wyoming,

Montana, Washington, British Columbia). There is a more fundamental problem

with Orchard™s suggested procedure, however, and it can be illustrated with some of

the data collected when the Figure 3.12 data were collected.

Consider Emerson™s (1978) data summarized in Table 3.12. As noted earlier, those

data provide the equation Y = “104.96 + 4.11 X, where Y is the body weight or

individual biomass in kilograms, and X is the maximum lateral length of the astragalus

in millimeters. Applying that equation to the seventeen left and seventeen right

astragali constituting bilaterally paired elements of white-tailed deer that I measured

produces the results summarized in Table 3.17. Variation between the individual body

size estimated by the length of the left astragali and the body size estimated based on

the length of its bilaterally paired right astragali mate ranges from 0.25 kilograms to

2.3 kilograms with an average of 0.94 kilograms. Similar analysis of forty-three pairs

of astragali from mule deer indicates difference in body size estimates provided by left

and right elements averages 1.01 kilograms and ranges from 0.08 to 4.11 kilograms.

The problem that presents itself is precisely the one illustrated in Figure 3.13 but

the variables are different. In the former case the problem concerned the degree of

symmetry of distal left and right humeri in terms of size; now it is symmetry in

estimates of body weight derived from the size of astragali. Recall that only those

specimens that provide asymmetrical results (do not have bilateral mates in the

collection) will also add to the tally of individuals represented by an assemblage of

skeletal elements. What tolerance level should be chosen and why? How symmetrical

should the two estimates of body size be in order to conclude the size of the same

individual has been estimated twice? Orchard (2005) provides no guidance, and he is

estimating taxonomic abundances: other methods 139

Table 3.17. Estimates of individual body size (biomass) of seventeen

white-tailed deer based on the maximum length of right and left astragali.

Estimation equation is Y = “104.96 + 4.11X, where Y is the body weight or

individual biomass in kilograms, and X is the maximum lateral length of the

astragalus in millimeters (after Emerson 1978)

Length of Right Length of

Pair right weight left Left weight Difference