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Table 2.14. Fictional data showing that identical distributions of most common
skeletal elements of two taxa across different aggregates will not in¬‚uence MNI.
Note that the NISP per skeletal element is the same as in Table 2.13. Note also
that the four distinct skeletal elements have similar frequency distributions
across the three strata, and that the ratio of taxon 1 to taxon 2 is 1:2 in each of
the three strata, and that MNI values determined while ignoring stratigraphic
boundaries also produce a ratio of 1:2

Taxon 1 Taxon 2
Stratum 1 5 R humeri, 4 L humeri, 4 R 10 R humeri, 5 L humerus, 4 R
femora, 3 L femora femur, 8 L femur
(MNImax = 5) (MNImax = 10)
Stratum 2 1 R humerus, 1 L humerus, 1 R 2 R humeri, 1 L humeri, 1 R
femora, 1 L femur femora, 1 L femur
(MNImax = 1) (MNImax = 2)
Stratum 3 1 R humerus, 1 L humerus, 1 R 2 R humeri, 1 L humeri, 1 R
femur, 1 L femora femora, 1 L femora
(MNImax = 1) (MNImax = 2)



A ¬nal point to consider involves Uerpmann™s (1973:311) observation that the “dif-
ference between number of ¬nds [NISP] and ˜minimum number of individuals™
increases as the size of the sample increases” (Uerpmann 1973:311). The difference
between NISP per taxon and MNI per taxon will increase as NISP increases (Fig-
ure 2.4). Because larger sample sizes allow greater differences between values, differ-
ences between MNImin and MNImax will be greatest in large samples and smallest in
small samples. Thus, large sample sizes, which are desired for statistical reasons (large
samples tend to be more representative than small samples of the population from
which they are drawn, and they tend to increase the statistical power of a test), tend to
be the ones in which MNI ¬‚uctuates the most as different aggregates are de¬ned. As
Grayson (1979:210) noted, “This is not the usual behavior of a unit of measurement.”
Restating problem seven, MNI measures not only taxonomic abundances but
aggregation methods as well (Grayson 1984). This can be shown graphically and
statistically by considering the relationship between NISP and MNI as modeled in
Figure 2.4, but with log-transformed data such that the relationship is linear. The
slope of the simple best-¬t regression line describing the relationship between NISP
data and taxonomically corresponding MNI data should be less steep when more
agglomerative (larger aggregates) methods are used to calculate MNI (MNImin)
than when less agglomerative (smaller aggregates) methods are used to calculate
quantitative paleozoology
66




figure 2.11. Relationships between NISP and MNImin, and NISP and MNImax at site
45DO214.


MNI (MNImax). This is so because MNImin involves few most common elements
so it is dif¬cult to ¬nd and thus add a new most common element. MNImax involves
many most common elements so it is easy (relatively speaking) to ¬nd and thus
add a new most common element. Consider site 45DO214 among the collections
from eastern Washington State mentioned previously (Table 2.7). The slope of the
simple best-¬t regression line describing the relationship between NISP and MNImin
is 0.44 (Table 2.7); the slope of the simple best-¬t regression line describing the
relationship between NISP and MNImax is 0.57. Both sets of data points and both
best-¬t regression lines are included in Figure 2.11 , which shows that the slope for
MNImin is less steep than that for MNImax.
The example of 45DO214 indicates that MNI not only measures taxonomic abun-
dances but it also measures aggregation. But we do not want measures of two variables
to obscure each other, particularly when one may approximate the target variable
and the other has nothing whatsoever to do with the target variable. Is there, perhaps,
a logical way to de¬ne aggregates such that the measures of taxonomic abundances
provided by MNI can be treated as if the aggregates do not signi¬cantly in¬‚uence
those abundances?
estimating taxonomic abundances: nisp and mni 67


De¬ning Aggregates

Recall that an assemblage of faunal remains is the set of remains from a horizon-
tally and vertically bounded space, usually a geological space such as a stratum or a
part thereof. Following Grayson (1984), the term aggregate is used as a synonym for
assemblage, and the term aggregation for the process of de¬ning the spatial bound-
aries of a faunal assemblage. Despite Grayson™s (1973) recognition of the aggregation
problem more than 30 years ago, few analysts other than Grayson (1979, 1984) have
subsequently explored its implications with their own data. Thus it is not unusual to
¬nd paleozoologists still calculating MNI values without considering the aggregation
problem (e.g., Trapani et al. 2006). The aggregation problem is not even mentioned
in one textbook on zooarchaeology (Rackham 1994).
Payne (1972) suggests that aggregates of faunal remains should be de¬ned on
the basis of the homogeneity of taxa and their frequencies. Ignore for the moment
the question of how similar is similar enough for two assemblages to be considered
homogeneous (Payne did not address this question), and consider the following three
things. First, this procedure assumes that natural faunal aggregates exist and we have
but to discover them. But whether natural discoverable faunal aggregates exist or
not is unclear. Furthermore, what kinds of faunal aggregates are to be searched for “
those representing a depositional event, a human-behavior, a death event, or . . .
what? Perhaps the research question being asked would help specify the appropriate
aggregate, but other problems attend their de¬nition. The second thing to consider,
then, is that Payne™s protocol precludes the study of stasis because one aggregates,
say, stratigraphically sequent, similar (homogeneous) assemblages. Finally, Payne™s
procedure comprises a circular process: Change in the fauna would be identi¬ed based
on how the aggregates were de¬ned “ the property of differences or nonhomogeneity
interpreted as change “ because aggregates are de¬ned on the basis of similarity and
homogeneity. Based on these three observations, we might use nonfaunal criteria to
de¬ne aggregates.
Ringrose (1993:128) argues that “not all levels of aggregation are likely to be sensible,
so that the problem [of the in¬‚uence of aggregation on MNI] is perhaps less than it
might seem at ¬rst. If it is not possible for specimens from the same individual to be
present in two locations [that is, to be tallied in two distinct aggregates], then it is
nonsensical to calculate the MNI at a level of aggregation where these two locations
are taken together, since specimens will be, implicitly, counted as being possibly
from the same individual when in fact they cannot be.” Implementing this protocol
of de¬ning aggregates demands a great deal of knowledge regarding the taphonomic
history of the materials under study. Some of it might be found by re¬tting studies
quantitative paleozoology
68


(e.g., Rapson and Todd 1992; Todd and Stanford 1992). However, this is again using
faunal data to de¬ne aggregates and thus imparts a degree of circularity to those
de¬nitions. It also can introduce the problem of matching potentially paired (left
and right) skeletal parts (e.g., Todd and Frison 1992).
Some zooarchaeologists indicate that one should de¬ne faunal aggregates based
on “cultural units rather than arbitrary ones related to excavation logistics” (Reitz
and Wing 1999:198). This is all well and good, but does this mean that two pits con-
taining bones and originating in the same stratum (dating to the same time period
and apparently representing the same cultural context given stratigraphic contem-
poraneity) should be considered separately or together? Archaeologist James Ford
(1962) argued long ago that archaeological “cultural units” such as cultures, phases,
periods, and the like were often de¬ned on the basis of stratigraphically bounded
aggregates of artifacts, but that it was unclear why there should be any necessary rela-
tionship between sediment deposition boundaries and boundaries between cultures.
I agree (Lyman and O™Brien 1999). So what are we to do?
Let us begin by glancing at the solution that paleontologists have used. Fagerstrom
(1964:1198), a paleontologist interested in past biological communities, suggested that
a fossil assemblage representing a community was “any group of fossils from a suitably
restricted stratigraphic interval and geographic locality.” What is suitable is not clear,
though it is hinted at in other paleontological concepts. In vertebrate paleontology, a
faunule is an assemblage of associated animal remains from one or several contiguous
strata, dominated by members of one biological community (Tedford 1970:677). And,
a local fauna is a set of remains from one locality or several closely spaced localities
which are stratigraphically equivalent or nearly so, thus it is a set of taxa close in
(geological) time and (geographic) space (Tedford 1970:678). Identifying prehistoric
faunal communities “ or faunules “ was what Shotwell (1955, 1958) was concerned
about, and he emphasized the taphonomic problems with doing so when one used
an aggregate of fossils the boundaries of which were set by excavation strategies and
stratigraphy.
The preceding brief discussion hints at two things. First, paleontologists often
seek, like Chester Stock and Hildegard Howard did, to determine the census of a
paleocommunity, or a biocoenose. That is their target variable, and they acknowledge
the geological mode of occurrence of the faunal materials that they study, and they
use stratigraphic boundaries and extent of exposures to collect a sample of those
materials. The second thing hinted at is an extremely critical detail. Reitz and Wing
(1999:197) mention it when they state that the aggregates of faunal remains de¬ned
“may depend on the research problem.” Any aggregates de¬ned must depend on the
research problem, as well as whatever taphonomic and site-formational information
is available. Thus, on the one hand, human behaviorally signi¬cant assemblages of
estimating taxonomic abundances: nisp and mni 69


remains, such as those in cache pits or in trash middens or on house ¬‚oors are likely
to be important to questions about human interactions with fauna. On the other
hand, questions regarding paleoecology are likely to be phrased in such a manner
as to require temporally and spatially distinct assemblages of remains, perhaps but
not necessarily representing one “biological community” but certainly providing
insights to the nature of biocoenoses. Temporally distinct assemblages but perhaps
not human behaviorally signi¬cant ones would be of interest to paleoecologists.
Valensi (2000:358) noted that aggregation based on excavation levels “gave an over-
estimation of MNI [as a result of specimen] interdependence [across] some levels.”
Interdependence was recognized by re¬tting specimens of both lithic and bone spec-
imens that came from different depositional units. Valensi used archaeostratigraphic
units as the basis for de¬ning aggregates, and found re¬tting specimens that came
from different units. Her analysis suggests a protocol for de¬ning aggregates. Re¬ts of
lithic specimens would provide nonfaunal criteria for de¬ning faunal aggregates. The
paleozoologist could adopt a rule, such as only when re¬ts across aggregates are min-
imal, whereas re¬ts within aggregates are maximized, have appropriate aggregates
for determining MNI been de¬ned. However, not only is the time cost incredibly
high if the assemblage is large “ do the faunal re¬ts, using the same rule, de¬ne the
same aggregates as the lithics (or ceramics)?
Research questions about taphonomic histories likely will require an estimate of
a taphocoenose, those about hunter or predator selectivity will require not only an
estimate of a thanatocoenose but also the biocoenose from which it derived. Explicit
statement of the research problem and research questions should help the paleo-
zoologist de¬ne aggregates that are pertinent. Of course, any available taphonomic
information such as obvious re¬ts should also be consulted to help set geological
spatial boundaries around the aggregate(s). This does not mean that one will auto-
matically have aggregates that do not share specimens from the same individual, but
perhaps those will be so rare as to not signi¬cantly bias any statistical results.


DISCUSSION


Thus far the problems with NISP as a quantitative unit giving valid measures of
taxonomic abundances (even in a taphocoenose, let alone in a thanatocoenose or
biocoenose) have been considered and it has been argued that all but one of those
problems “ that of possible specimen interdependence “ can be fairly easily resolved
analytically. (Some analysts still fail to realize how easily many of the problems with
NISP can be resolved analytically [e.g., O™Connor 2001 ].) Problems with MNI as
a quantitative unit giving valid measures of taxonomic abundances have also been
quantitative paleozoology
70


identi¬ed and discussed, and it has been shown that many of those are also readily
dealt with analytically. The problem that remains with MNI is aggregation. As implied
above, there is no magic algorithm for solving the aggregation problem because each
aggregate speci¬ed by the analyst may, or may not, have a set of faunal remains all
of which are indeed independent of all other faunal remains in all other aggregates.
Earlier I referred to the latter as Adams™s dilemma. It is aptly referred to as a dilemma
because if, say, stratigraphically bounded aggregates are chosen as the assemblages
to be analyzed, one must assume that the faunal remains in each are independent of
all other faunal remains in other aggregates. But, of course, they might not be.
A chosen sampling design may indicate where to excavate and which screen-
mesh size to use, but the faunal specimens recovered are a result of the taphonomic
history of the assemblage “ which bones and teeth were accumulated, deposited, and
still exist, and where they are located, both horizontally and vertically. The existing
remains of a single individual may be in one or more horizontal loci, in one or more
vertical loci (or strata), or both. (No two specimens can, of course, occupy exactly the
same horizontal and vertical position. By same location, I mean a spatially limited,
horizontally and vertically bounded unit.) Even attempting to match and pair all
skeletal specimens from all excavated recovery proveniences, we will likely never
know what the correct aggregates of faunal remains should be. By correct is meant
those that are not only relevant to our research questions, but also ones de¬ned such
that specimens from a single carcass are not distributed across two or more aggregates.
Given that we cannot know this, we either assume Adams™s dilemma does not exist,
or, we do something other than determine MNI values.
There is, in fact, a relatively simple solution to Adams™s dilemma. The solution
rests on the fact that quite often, virtually the same information regarding taxo-
nomic abundances in an assemblage is found in NISP as is found in MNI. This
statistical relationship has been known for some time (Casteel 1977, n.d.; Grayson
1978a, 1979). In short, MNI is redundant with NISP, where “redundant” means that
the two quantitative units produce the same information. The “same information”
can mean identical, or simply statistically indistinguishable. To show that MNI and
NISP provide the same information in both of these senses, consider the owl pellet
data mentioned before. Recall that the sample comprises eighty-four pellets, that the
relationship between NISP and MNImin is linear (Figure 2.8), and that the relation-
ship is strong (r = 0.989, p < 0.0002). For this sample, 97.8 percent of the variation in
MNI values is explained by variation in NISP values. Clearly, MNI is redundant with
NISP. And, the same applies to the fourteen samples of mammal remains from east-
ern Washington State (Table 2.7). For these assemblages, the relationship between
NISP and MNImin is typically strong (r > 0.75 for 13 of the 14) and signi¬cant
(p < 0.01 for all). For thirteen of these fourteen assemblages, NISP accounts for
estimating taxonomic abundances: nisp and mni 71


more than 51 percent of the variation in MNI. MNI provides information about
taxonomic abundances that is redundant with that provided by NISP (Figure 2.4).
But so what? Constructing an answer to this question requires a consideration of the
scale of measurement represented by NISP and by MNI.



W H I CH S C A L E O F M E A S U RE M E N T ?


Some years ago, Grayson (1984:94“96) noted several critical things. First, he noted that
converting from one ratio scale to another ratio scale based on different measurement
units will not alter the value of a ratio of measurements. This is so because both
ratio scales have natural zero points and their respective units of measurement stay
constant in each. Thus, the ratio of the weight of two items will not alter if ¬rst
measured in pounds and then in kilograms. If the two items are 50 pounds and
75 pounds, the ratio of their weights is 1:1.5; the two items weigh 22.68 kilograms
and 34.02 kilograms, respectively, for a ratio of 1:1.5. As noted earlier in this chapter,
aggregation has the unsavory characteristic of altering MNI tallies, thereby causing
ratios of taxa to change as the manner in which faunal remains are aggregated changes.
The second thing Grayson (1984) noted was that MNI values are not ratio scale
values precisely because they are minimum numbers (Table 2.4). The actual number
of animals represented (by the identi¬ed assemblage) could be as great as the NISP,
although it likely will fall somewhere between the MNI and the NISP given the
probability (> 0.0) of some interdependence of specimens. Thus it cannot be argued
that a taxon represented by an MNI of ten is half as abundant as a taxon represented
by an MNI of twenty, nor can it be argued that if two taxa each have MNI values
of ¬fteen they are equally abundant. Figure 2.12 plots ratios of the abundances of
each pair of taxa based on NISP, MNImin, and MNImax measures of the four least
common taxa in the collection of eighty-four owl pellets (Table 2.9). The ratios vary
by greater or lesser amounts across the three quantitative measures. In particular,
note the variation in ratios between the MNImin and MNImax values. There is no
way to determine which set of ratios most closely measures the actual abundances
of taxa. Clearly, it is ill-advised to treat MNI values as ratio scale because there are
many reasons why they likely are not.
Unfortunately, it is unlikely that NISP values are ratio scale. As Grayson (1984)
noted, if MNI provides minimum tallies, NISP provides maximum tallies. Given that
we do not know the nature (extent) of interdependence of the specimens comprising
the NISP for any given taxon in any given collection, and given that intertaxonomic
variation in such interdependence will differentially in¬‚uence how closely NISP
tracks a taxon™s actual abundance, it is unlikely that ratios of taxa based on NISP values
quantitative paleozoology
72


Table 2.15. Ratios of abundances of pairs of taxa in eighty-four
owl pellets. Original data from Table 2.8. Taxon 1, Sylvilagus;
taxon 2, Reithrodontomys; taxon 3, Sorex; taxon 4, Thomomys

Taxon pair NISP MNImin MNImax
1 “2 0.26 0.20 0.40
1 “3 0.11 0.20 0.29
1 “4 0.07 0.12 0.17
2“3 0.41 1.00 0.71
2“4 0.28 0.62 0.42
3“4 0.68 0.62 0.58

are in fact ratio scale. There is no way to know which set of ratios of abundances of
taxa in the owl pellet fauna (Table 2.15), if any, most accurately re¬‚ects the actual ratio
scale abundances of the taxa. As Grayson (1984:96) noted, because “we know nothing
of the nature of the frequency distribution [of taxonomic abundances] that begins
with MNI and ends at NISP for a set of taxa,” knowledge of ratio scale abundances
of taxa is precluded.
If MNI and NISP do not provide ratio scale taxonomic abundance data, do they
perhaps provide ordinal scale abundance data? Again, Grayson (1984:96“99) provided
Ratio




Taxa Pair
figure 2.12. Ratios of abundances of four least common taxa in a collection of eighty-four
owl pellets based on NISP, MNImax, and MNImin. Taxon 1, Sylvilagus; 2, Reithrodontomys;
3, Sorex; 4, Thomomys. Data from Table 2.8.
estimating taxonomic abundances: nisp and mni 73




figure 2.13. Frequency distributions of NISP and MNI taxonomic abundances in the
Cathlapotle fauna. Data from Table 1.3.



a clear answer. The rank order abundances of taxa are often quite similar across NISP
and MNI; if the two sets of values are signi¬cantly correlated on an ordinal scale, then
the included taxonomic abundances are ordinal scale. Why NISP and MNI should
often be correlated comprises the critical insight as to why we can conclude they are
ordinal scale. In most multitaxa faunas, a few taxa are represented by many individu-
als and specimens, and many taxa are represented by few individuals and specimens.
As taxonomic abundances increase (whether NISP or MNI), the magnitude of the
differences between abundances of adjacent taxa increases. Such frequency distribu-
tions increase the probability that taxonomic abundances are ordinal scale because
there is less chance that variation in aggregation (MNI) or specimen interdependence
(NISP) will alter rank order abundances.
Summing the precontact and postcontact assemblages, eighteen taxa in the Cath-
lapotle fauna (Table 1.3) are represented by seven or fewer individuals whereas only
seven taxa are represented by more than ten individuals (Figure 2.13). Similarly, 20
taxa are represented by 100 or fewer specimens whereas only 5 taxa are represented by
quantitative paleozoology
74




figure 2.14. Frequency distributions of NISP and MNI taxonomic abundances in the
45OK258 fauna in eastern Washington State.




more than 100 specimens. One of the mammalian faunas from eastern Washington
State that has been used in other analyses “ site 45OK258 “ has similar frequency
distributions (Figure 2.14). Grayson (1979, 1984) presented numerous other faunas
with right skewed distributions of taxonomic abundances. Lest one think this sort of
frequency distribution is a function of the faunas we examined, two more examples
may convince the skeptic. A right skewed frequency distribution is found in the two
summed late prehistoric assemblages of mammal remains from the western Canadian
Arctic described by Morrison (1979) (Figure 2.15). And such a frequency distribu-
tion is also found among historic era mammalian faunas described by Landon (1996)
(Figure 2.16).
It matters little why many faunas display a right skewed frequency distribution
of taxonomic abundances [Box 2.2]. The important point is, as Grayson (1984:99)
put it, the “degree of rank order stability will be closely related to the degree of
estimating taxonomic abundances: nisp and mni 75




figure 2.15. Frequency distributions of NISP and MNI taxonomic abundances in two
lumped late-prehistoric mammal assemblages from the western Canadian Arctic. Data from
Morrison (1997).



separation of the taxa involved in terms of their MNI- or NISP-based sample sizes.”
The greater the separation between the abundance of taxon A and the abundance
of taxon B, the less likely changes in aggregation “ if abundances are MNI values “
or specimen interdependence “ if abundances are NISP values “ will alter the rank
order abundances of those taxa. The rank order abundances of rarely represented
taxa, because their abundances are not widely separated, likely will shift with changes
in aggregation and specimen interdependence. As Grayson (1984:98) suggests, “it is
questionable whether [rarely represented taxa] should be treated in anything other
than a nominal, presence/absence sense.”
quantitative paleozoology
76




figure 2.16. Frequency distributions of NISP and MNI taxonomic abundances in four
lumped historic era mammalian faunas. Data from Landon (1996).




Box 2.2

This sort of frequency distribution is known as right skewed “ the tail is to the
right. Such a frequency distribution may result from the accumulation agent
focusing on one or a few taxa “ the frequently represented ones “ and the rarely
represented taxa are background or idiosyncratic accumulations. Alternatively,
recovery may create the frequency distribution. This would be suggested by rare
remains of small animals and frequent remains of large animals (see Chapter 4).
Finally, perhaps the frequency distribution represents what is on the landscape
if accumulation, preservation, and recovery were all random with respect to the
biocoenose.
estimating taxonomic abundances: nisp and mni 77

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