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1984). Here, ubiquity is also signi¬cantly correlated with NISP (r = 0.709, p = 0.001).
The relationship of sample size and ubiquity across analytical units at 45DO189
is illustrated in Figure 3.9; that relationship as it is manifest at 45OK2 is shown in
Figure 3.10. Again, some taxa may indeed be more ubiquitous than others in the sense
of being found associated with more spatiotemporal analytical units. But it is dif¬cult
to make this argument on empirical grounds because the available data suggest that
had more of each analytical unit with low NISP values been excavated, NISP would
have been larger, more taxa would have been found in those units, and the ubiquity
of rarely represented taxa (those with low NISP values) would have increased.
Dean (2005a:416“417) cited an ethnobotany text as indicating that ubiquity mea-
sures minimize in¬‚uences of “random variations [in] NISP counts” when seeking
to measure taxonomic abundances, and those measures also allow comparisons of
assemblages from different habitats, assemblages collected using different recovery
procedures, and assemblages subject to varied preservation. To be sure, sampling
different habitats will in¬‚uence NISP, as will differential preservation and recovery.
But in so far as ubiquity is a function of NISP “ and the examples given above suggest
estimating taxonomic abundances: other methods 119




figure 3.9. Relationship between NISP and ubiquity of ¬fteen taxa in seven analytical
units in site 45DO189. Data from Table 3.14.


that ubiquity will often be a function of NISP “ differential sampling intensity, dif-
ferential preservation, and differential recovery will also in¬‚uence ubiquity because
they in¬‚uence NISP (see also Kadane 1988).
The preceding is not to say that ubiquity measures are valueless. These could be
quite useful if the in¬‚uences of sample size could be controlled. If, say, the NISP values
of several taxa are very similar (say within 5 percent of each other), and one taxon
has a high ubiquity value and another has a low value, then it would be reasonable
to suspect that some mechanism or agent of accumulation had dispersed widely the
remains of the former taxon and perhaps that same mechanism or agent (or another
one) had concentrated (or failed to disperse) the remains of the latter taxon. Given
this possibility, it is dif¬cult to understand why ubiquity has not been measured more
frequently.



M A TCH I N G A N D P A I R I N G


Recall the de¬nitions of MNI provided by Stock, Howard, Adams, and the ornithol-
ogists interested in raptor diets (and see Table 2.4). White (1953a:397) de¬ned MNI
quantitative paleozoology
120




figure 3.10. Relationship between NISP and ubiquity of eighteen taxa in four analytical
units in site 45OK2. Data from Table 3.14.


this way: The “number of individuals represented by the excavation sample [is deter-
mined as follows.] Separate the most abundant element of the species found . . . into
right and left components and use the greater number as the unit of calculation”
(see also White 1953b:61). White went on to note that this procedure would produce
a “slight error on the conservative side because, without the expenditure of a great
deal of time with small returns, we cannot be sure all of the lefts match all of the
rights” (1953a:397).
What White was getting at with the idea of “matching” is this. Recall that when
faced with three left and two right scapulae of a taxon, the number of individuals is
a minimum because at least three individuals had to contribute these bones. Perhaps
as many as ¬ve individuals contributed the scapulae, but unless we can show that
the two rights do not “match” or bilaterally pair with any of the three left scapulae,
then we must conclude that the two rights are from two of the three individuals
represented by the lefts. How do we make a determination of whether any of the
lefts match any of the rights? Usually bilaterally paired bones are compared, when
these are elements with left and right members, such as scapulae, humeri, tibiae, and
so on. Skulls are not paired bones, but mandibles are; vertebrae are not, but ribs
estimating taxonomic abundances: other methods 121


are, although ribs are seldom matched because they tend to not be taxonomically
diagnostic beyond the family level (if that).
To determine if two potentially paired bones indeed “match” or are bilaterally
paired, they must be compared under the assumption of bilateral symmetry. The
two members of the possible pair (obviously from the same species) are compared in
terms of their size, their ontogenetic (growth) development, the sex of the individual
represented, and the like (Chaplin 1971 :70). Because vertebrates, such as mammals
and birds, are more-or-less bilaterally symmetrical, the left bone should closely match
the right bone in terms of all characteristics if they are from the same individual.
White thought that matching would gain little, but provided little data to this effect.
It is legitimate to ask, then: What does identifying bilaterally paired bones gain in
terms of measures of taxonomic abundance? In fact, can bilateral pairs be accurately
identi¬ed? To answer these questions, the basics of the matching procedure and how
it in¬‚uences measures of taxonomic abundance are reviewed ¬rst. Then a study of
how accurate identi¬cations of bilateral mates might be is presented.



More Pairs Means Fewer Individuals

In his seminal example, Chaplin (1971 :71 “75) described a ¬ctional example of how
identifying pairs would provide a different number of individuals than a Whitean
MNI. He did not reference White, but Chaplin (1971 :70) did note that a Whitean
MNI “ one de¬ned by the most common skeletal element or part “ would constitute
a “not necessarily very satisfactory estimate of the true minimum.” If one had, say,
eleven distal left humeri and ¬ve proximal right humeri, and none of these specimens
overlapped anatomically, then the Whitean MNI would be eleven but Chaplin™s point
was that there might actually be twelve, thirteen, fourteen, ¬fteen, or even sixteen
individuals represented if some of the proximal right specimens were not from the
same animals as the left distal specimens. Thus he focused on identifying those
specimens without matches; matches would not increase the MNI, but bones without
matches would increase that number because they would be added to the number
of pairs or matches. Thus, more pairs mean fewer individuals and fewer pairs mean
more individuals. Left specimens without a mate and right specimens without a
mate are added to the number of pairs as independent representatives of individual
organisms. The more pairs of left and right elements, the fewer independent left and
right elements added to the total. White™s method of determining MNI assumes all
lefts pair up with a right element, and all rights pair up with a left element. Chaplin
was unwilling to make this assumption.
quantitative paleozoology
122


Chaplin™s (1971 ) example of how matching in¬‚uences measures of taxonomic
abundance involved domestic sheep (Ovis aries) tibiae. He used ¬ctional data, but
those data illustrate the issues involved nicely. The NISP of left specimens was thirteen;
some specimens were anatomically complete left tibiae, some represented various
portions of left tibiae. The (minimum) number of skeletal elements represented
(based on age, sex, size) was ten, which also represented the MNI for left tibiae. The
NISP of right tibia specimens in Chaplin™s example was seventeen; these represented
¬fteen elements and ¬fteen MNI. Based on age (epiphyseal fusion) and size (see
below), Chaplin identi¬ed bilateral pairs of left and right tibiae. He derived a formula
for calculating what he called the “grand minimum total” or GMT of individuals.
This formula is:

GMT = (C t /2) + D t ,

where C t is the total number of skeletal elements (not specimens) making up bilater-
ally matched pairs and D t is the total number of elements (not specimens) without
bilateral mates. In his example, Chaplin identi¬ed eight pairs comprising sixteen
elements (eight lefts and eight rights), and had two unmatched lefts and seven
unmatched rights. Thus, C t = 16, Dt = 9, and GMT = (16/2) + 9 = 8 + 9 = 17
individuals. Thus, in this example, the Whitean “most common element” MNI was
¬fteen (based on right tibiae), but matching bilateral elements indicated that because
some elements lacked their bilateral mate and represented independent organisms,
a more accurate (but still a minimum) tally of individuals was seventeen.
A Whitean “most common element” MNI assumes that each left humerus, say, has
a bilateral mate among the right humeri, so only the lefts, or only the rights, but not
both lefts and rights contribute to the MNI tally (see Table 2.5). Chaplin™s GMT can
produce larger tallies of individuals because both left humeri and right humeri can
contribute to the tally. Matching establishes that some left humeri are independent
(come from a different individual organism) of all right humeri, and some right
humeri are independent of all left humeri. Therefore, two “most common elements” “
left and right humeri “ are identi¬ed rather than either left or right humeri only.
Calculating GMT for a collection will give a larger number of individuals the fewer
the identi¬ed pairs. In Chaplin™s example, if we reduce the number of pairs to 7
(C t = 14), that increases D t to 11, so GMT = (14/2) + 11 = 7 + 11 = 18, simply
because we have added one more independent element (another left, or right, tibia
in this example) to the tally. This might make GMT attractive, but do not be fooled
by larger numbers. To be sure, GMT can produce larger numbers of individuals than
the Whitean “most common element” MNI. But does that gain us anything with
respect to measuring taxonomic abundances?
estimating taxonomic abundances: other methods 123


First, the absence of pairs means that MNI = NISP. GMT provides measures of
taxonomic abundance between NISP and Whitean MNI values. In Chaplin™s example,
NISP = 30, GMT = 17, and (Whitean) MNI = 15. Given this, and arguments about
the statistical relationship between MNI and NISP, GMT will be a function of NISP.
Taxonomic abundances based on GMT will also at best be ordinal scale. Think of
GMT as simply a different way to aggregate faunal remains to derive MNI values.
GMT values will depend heavily on aggregation, just as MNI does. This is so because
it is unlikely that one would seek to identify bilateral pairs (or a lack thereof) among
assemblages of tibiae from different strata deposited at different times (unless one
was interested in postdepositional disturbance processes that resulted in inter-strata
movement of specimens). Rather one would choose to seek a bilateral pair among
skeletal elements that potentially derive from the same animal. Thus, how one de¬nes
aggregates plagues GMT just as it does MNI.
Another problem with determination of GMT concerns identifying bilateral pairs.
Because this problem af¬‚icts all measures of taxonomic abundance that use bilateral
pairs in the calculation, it is addressed later in this chapter. The bottom line to
Chaplin™s GMT should be clear. Although GMT tries to make quantitative use of
independent left and right skeletal elements, it is plagued by many of the same
dif¬culties as Whitean MNI values. That it produces data with no more quantitative
resolution or validity than NISP with respect to measures of taxonomic abundances
should cause one to pause before calculating GMT, if it is calculated at all.



The Lincoln“Petersen Index

Chaplin™s (1971 ) GMT is not the only quantitative procedure meant to measure
taxonomic abundances that uses bilateral pair data (e.g., Krantz 1968; Lie 1980, 1983;
Wild and Nichol 1983a, 1983b; Winder 1991 ). A couple of these other techniques
have undergone critical evaluation (Allen and Guy 1984; Bokonyi 1970; Casteel 1977;
During 1986; Fieller and Turner 1982; Gautier 1984; Horton 1984), and are seldom used
these days. They are not considered further. But there is one method that requires
comment because it has several supporters, it has been suggested at least twice by
independent workers (Allen and Guy 1984; Fieller and Turner 1982), and it might
well be suggested yet again given that versions of it are used by wildlife biologists
today (e.g., Amstrup et al. 2006; Hopkins and Kennedy 2004; Slade and Blair 2000).
More importantly, biological anthropologists have recently suggested it is useful for
estimating the number of individual humans in commingled assemblages of remains
(Adams and Konigsberg 2004).
quantitative paleozoology
124


The most frequently advocated method of using identi¬ed bilateral pairs in the
service of measuring taxonomic abundances is analogous to (and in fact derived
from) a procedure used by wildlife biologists to estimate taxonomic abundances.
In wildlife biology, it is known as “capture“recapture analysis,” where a sample of
animals is captured, tallied, and each captured individual (n1 ) is marked (m1 ) and
released (Nichols [1992] provides a good introduction). Then a second sample of
animals (n2 ) is captured. The number of previously marked (m1 ) individuals that
are recaptured (m2 ) and the number of unmarked (new) individuals (n2 ) in the
second sample are tallied. These values “ n1 , n2 , m1 , m2 “ are used to estimate the size
of the population from which the two samples were drawn. The quantitative measure
is usually referred to as the Petersen index, and less often as the Lincoln index (Fieller
and Turner 1982; Turner 1980, 1983); it is here referred to as the Lincoln“Petersen
index. Several paleozoologists ¬nd this index to be superior to Whitean MNI values
and also superior to Chaplin™s GMT values (e.g., Allen and Guy 1984; Fieller and
Turner 1982; Turner 1980, 1983; Turner and Fieller 1985; Winder 1991 ).
In the Lincoln“Petersen index, n1 = m1 . After release of the marked individuals
comprising the ¬rst sample, the proportion (P) of marked individuals in the popu-
lation can be symbolized as n1 /Y = P or m1 /Y = P, where Y is the population size.
What we of course do not know but seek to estimate is Y. So, we convert m1 /Y =
P ¬rst to m1 = PY (multiply both sides of the equals sign by Y), then convert the latter
to m1 /P = Y (divide both sides of the equals sign by P). The Lincoln“Petersen index
assumes that the proportion of marked individuals in the second sample effectively
estimates the proportion of marked individuals in the population, or P = (m2 /n2 ).
Substituting (m2 /n2 ) for P into Y = m1 /P, the result is

Y = m1 /(m2 /n2 ).

Because division by a fraction is equivalent to multiplying by the reciprocal of that
fraction,

Y = m1 n2 /m2 .

And because m1 = n1 , the preceding equation can be rewritten as

Y = n1 n2 /m2 .

Either of the last two formulae provides an estimate of the size of the population
from which the individuals comprising the two samples (n1 and n2 ) were drawn. The
reasoning is illustrated in Figure 3.11 .
Zooarchaeologists who advocate use of the Lincoln“Petersen index for estimating
taxonomic abundances observe that bilaterally paired skeletal elements, such as left
estimating taxonomic abundances: other methods 125




figure 3.11. A model of how the Lincoln“Petersen index is calculated. Square on the left is
the population (twenty individuals) divided into four subpopulations (a“d). Square on the
right illustrates the captured and marked ¬rst sample (n1 or m1 ; shaded), and the second
sample (n2 ; cross-hatched) comprising some marked individuals from the ¬rst sample (cell
b = m2 ) and individuals captured for the ¬rst time (cell d).


and right femora, can be used as follows. The lefts can be considered one sample (say,
n1 ), and the rights can be considered the other sample (say, n2 ). Identifying bilateral
pairs of left and right elements is analogous to ¬nding a marked individual in the
second sample. If left elements are treated as the ¬rst sample, L = n1 ; if right elements
are treated as the second sample, R = n2 ; and if the number of paired bones is treated
as the number of marked recaptures, m2 = number of bilateral pairs, then we can
substitute these symbols into the Lincoln“Petersen index to estimate the number
of individuals in the population from which the bones came. The formula can be
written as

LPind = L R/m2 ,

where LPind is the estimated number of individuals given by the Lincoln“Petersen
index, L is the number of left elements, R is the number of right elements, and m2
is the number of bilateral pairs. In wildlife biology, by convention, if no pairs are
found, LPind is estimated as LPind = (n1 + 1)(n2 + 1). And by convention, LPind is
rounded up to the nearest whole number.
LPind provides an estimate of taxonomic abundance that is greater than MNI and
GMT. The calculated value of LPind for a taxon in any given assemblage depends on
the NISP for that taxon and also on the number of bilateral pairs of skeletal elements
of that taxon regardless of NISP. In the ¬ctional data in Table 3.15, for the sake of
simplicity, each specimen is an anatomically complete skeletal element. As NISP
(= L + R) per row increases in Table 3.15, if the number of pairs (m2 ) remains
quantitative paleozoology
126


Table 3.15. Fictional data illustrating in¬‚uences of NISP and
the number of pairs on the Lincoln“Petersen index (LPind )

NISP Left (n1 ) Right (n2 ) Pairs (m2 ) LPind
20 10 10 2 50
20 10 10 3 34
20 10 10 4 25
20 10 10 5 20
20 10 10 6 17
20 10 10 7 15
20 10 10 8 13
40 20 20 3 134
60 30 30 3 300
80 40 40 3 533
100 50 50 3 834



stable, then the associated LPind increases. As NISP increases from 20, to 40, to 60,
to 80, to 100, if m2 = 3, then LPind increases from 34, to, 134, to 300, to 533, to 834,
respectively. And, if the NISP per row remains stable but m2 increases, then LPind
decreases. Consider those rows in which NISP = 20. As m2 increases from 2 to
8 by increments of 1, LPind decreases from 50, to 34, to 25, to 20, to 17, to 15, to 13.
Increase in NISP causes the estimated number of individuals to increase (all else [m2 ]
equal); increase in the number of pairs causes the estimated number of individuals
to decrease (all else [NISP] equal) until all bones are paired in which case m2 = LPind .
Fieller and Turner (1982) identify several properties of LPind that they argue are
bene¬cial to estimates of taxonomic abundance. One is that con¬dence intervals
can be calculated for any value; as they put it, “it is possible to specify a range of
plausible values for the total population size that are ˜reasonably™ consistent with the
observed proportion of tagged animals” or paired bones (Fieller and Turner 1982:53).
Calculation of both LPind and its associated con¬dence interval for a taxon rests on
several assumptions that, if violated, can cause the estimate to be far off the mark.
For example, the capture“recapture-based Lincoln“Petersen index assumes that in
the time between the release of marked individuals (n1 = m1 ) and the collection
of the second sample (n2 ), the marked individuals will be randomly mixed into the
population. That is, it is assumed that the probability of drawing a marked individual
in n2 will not be in¬‚uenced by the fact that that marked individual was also a part
of the ¬rst sample or n1 . Despite more than 25 years of detailed ethnoarchaeological
studies of faunal remains, only a small bit of that research has focused on the spatial
estimating taxonomic abundances: other methods 127


distribution of those remains (e.g., Bartram et al. 1991 ), and only a small fraction of
that research has focused on matching (e.g., Waguespack 2002). We do not yet know
that the requisite assumption is valid, but the limited available data suggest that it
will not always be. Faunal remains are seldom randomly distributed.
Some suggest the bene¬t of LPind is that it “accounts for random data loss” (Adams
and Konigsberg 2004:140). In taphonomic terms, whether or not a left humerus, say, is
accumulated or preserved should not in¬‚uence the accumulation and preservation of
its bilateral (right) mate. Fieller and Turner (1982) argue that calculation of multiple
LPind values and their respective con¬dence intervals based on different skeletal
elements “ humeri, radii, femora, and so on “ should produce similar estimates of a
taxon™s abundance if the assumptions requisite to its calculation are not violated. The
only thing these similar values indicate is that the included paired skeletal elements
have spatial distributions and preservation potentials suf¬ciently similar that each
provides a similar LPind value. Whether left and right femora are randomly distributed
relative to each other and also relative to both left and right humeri is a separate
question. This distinction is con¬rmed by Fieller and Turner™s (1982:55) suggestion
that dissimilarity in LPind values for different elements reveals something about
“differential deposition [that is] of considerable interest.” Do not confuse deposition
in site sediments with deposition in the area excavated. We do not need to calculate
LPind values to determine that the parts of skeletons are differentially represented,
whether due to varied deposition or preservation. Does LPind provide something not
provided by other measures of taxonomic abundances?
Fieller and Turner (1982:54) argue that LPind is the only quantitative unit that
provides an “estimation of the original killed population size.” According to Fieller
and Turner (1982:51), MNI (of either the Whitean most common element type, or
Chaplin™s GMT type) provides a “simple count” of the number of individuals nec-
essary to account for the bones on the lab table. They argue that LPind , in contrast,
provides an estimate of the original killed population because it takes into account
“missing material” and thus it is a “conventional statistical estimate” (Fieller and
Turner 1982:51). Ringrose (1993:129) agrees that the Lincoln“Peterson index provides
estimates of taxonomic abundances within the “death assemblage.” And Adams and
Konigsberg (2004:138) seem to agree but say the method “estimates the original num-
ber of individuals represented by the osteological assemblage” (emphasis in original).
But consider whether the population of twenty individuals shown in Figure 3.11 com-
prises the biocoenose, the thanatocoenose, or the taphocoenose. Recall Figure 2.1 ,
where the differences between a biocoenose (living population on the landscape),
a thanatocoenose (population of dead individuals, or killed population), a tapho-
coenose (what is accumulated, deposited, and preserved in a site), and an identi¬ed
quantitative paleozoology
128


assemblage (what was recovered, identi¬ed, and reported) are shown graphically.
Fieller and Turner (1982) argue that the thanatocoenose from which the identi¬ed
assemblage is derived is the target variable. They suggest that if MNI or GMT is used
the measured variable constitutes the identi¬ed assemblage and if LPind is used then
the estimated variable is the thanatocoenose.
In response to Fieller and Turner™s (1982; see also Turner 1983) arguments regard-
ing measured and target variables, and their attendant advocacy of the Lincoln“
Petersen index, Grayson (1984:88) pointed out that “an unmatched bone whose part-
ner has simply not been collected has a very different meaning from an unmatched
bone whose partner has disappeared.” Winder (1991 :126) implied the “very differ-
ent meaning” was insigni¬cant because both the failure to recover a bone and the
lack of preservation of a bone merely concerned “classical sampling theory.” But,
Turner (1983:319) himself contradicts this when he points out that “if 100 animals
were killed, 60 whole carcasses removed from the site and only a random sample
of the bones from the remaining 40 animals deposited, then estimates based on the
excavated sample can only refer to the 40 individuals which in this case constitute the
killed population.” How can this situation be mathematically different from a case
in which 100 animals are deposited in a site but bones of only forty animals preserve
and are sampled? How can Turner™s example, or the immediately preceding one, be
mathematically different from a case in which 100 animals are deposited, but bones
of only forty are sampled?
There are no mathematical differences between the three possibilities. This is so
because if the drawn sample consists of sixty bones (thirty-¬ve lefts, twenty-¬ve
rights) drawn randomly from forty (out of 100) animals with skeletons compris-
ing only one bilateral pair, and there are twenty pairs, the LPind = 44 regardless of
anything else. Certainly those forty-four individuals originated in the biocoenose,
passed through the thanatocoenose ¬lter, then through the taphocoenose ¬lter, and
¬nally through the ¬lters of recovery, analysis, identi¬cation, and pair matching.
Which of those “populations” do those forty-four individuals estimate the size of?
That is the key question. Perhaps we do not need an answer for one simple reason.
As Fieller and Turner (1982:57) correctly emphasize, “fundamental to the [Lincoln“
Petersen] method is the ability to determine the precise number of matches in the
assemblage of skeletal parts considered. Omission of true matches in¬‚ates the esti-
mate [and] inclusion of false matches has the opposite effect” (see Table 3.15). This,
then, comprises another requisite assumption “ that analysts can accurately identify
bilateral pairs. It is dif¬cult to determine if this assumption is warranted because few
published tests of it exist. It is necessary, then, to consider the validity with which
bilateral pairs might be identi¬ed. If they cannot be consistently validly identi¬ed,
estimating taxonomic abundances: other methods 129

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