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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

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