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and then decrease NISP as fragmentation intensity increases and specimens become
unidenti¬able; moderate breakage will not in¬‚uence MNI, but intensive fragmenta-
tion will result in a decrease in MNI as progressively more specimens fail to retain
suf¬cient anatomical landmarks to allow identi¬cation.
Klein (1980) is correct that MNI will not be in¬‚uenced by fragmentation, whereas
NISP will be, but only in the limited case of assemblages with little fragmentation.
Both NISP and MNI values will be in¬‚uenced by intensive fragmentation. We do
not know, however, the degree of fragmentation intensity at which fragmentation
begins to decrease identi¬ability and to reduce values of both NISP and MNI (Lyman
1994b). The dif¬culty of establishing this degree of fragmentation is exacerbated by
the fact that small fragments are unidenti¬able to skeletal element.
MNI overcomes such problems as intertaxonomic variation in the number of
identi¬able elements per individual. But as I noted, in regard to this problem as it
pertains to NISP, it is easy to analytically correct for intertaxonomic variation in the
number of identi¬able skeletal elements per individual. Either normalize all NISP by
dividing those values by a value that accounts for variation in identi¬able elements
per skeleton, or delete from tallies those taxonomically unique elements (such as
upper incisors in equids when comparing their abundance to bovids, who don™t
have upper incisors).
The most important advantage of MNI is that it overcomes possible specimen
interdependence because of how MNI is de¬ned (Table 2.4). As Ringrose (1993:127)
noted, the “basic principle of the MNI is to avoid ˜counting the same animal twice.™”
If MNI is derived according to the de¬nitions in Table 2.4, there is no way for
two or more specimens in one assemblage to be from the same individual. This is
illustrated in Table 2.5. Notice in that table of ¬ctional data that there is a minimum
of seven individuals (= MNI). Notice also that all postcranial elements have been
assigned to an individual already represented by a skull and both mandibles. The
left scapulae do not represent, so far as we can tell in light of modern methods, an
eighth individual, a ninth individual, and so on, nor do the right scapulae, the left
humeri, and so on. Here, NISP = 71, but obviously if we tally NISP, we count the
¬rst individual ¬fteen times (= NISP for that individual), the second individual is
counted fourteen times, and so on. Of course, the right radius assigned to individual
number ¬ve may actually go with individual number four, or with individual number
eight, but we cannot determine that. Thus, the number of individuals represented
by the seventy-one specimens listed in Table 2.5 is a minimum of seven individuals,
but the specimens might represent more.
MNI would seem to avoid the interdependence problem within an assemblage
such as shown in Table 2.5. But note the emphasized phrase “ within an assemblage.
estimating taxonomic abundances: nisp and mni 45


Table 2.5. A ¬ctional sample of seventy-one skeletal
elements representing a minimum of seven individuals (I)

I.1 I.2 I.3 I.4 I.5 I.6 I.7
+—
Skull + + + + + +
L mandible + + + + + + +
R mandible + + + + + + +
L scapula + + + + + +
R scapula + + + + + +
L humerus + + + + + +
R humerus + + + + +
L radius + + + + +
R radius + + + + +
L innominate + + + +
R innominate + + + +
L femur + + +
R femur + + +
L tibia + +
R tibia +
NISP 15 14 13 11 9 6 3


+ denotes a skeletal element is represented.


What about between assemblages? What if the left femur of an individual is in one
assemblage and the matching right femur is in another assemblage? Were we to tally
each as part of an MNI calculation in the two assemblages, we would have counted
that single individual twice. This introduces the most serious problem with MNI “
aggregation “ and there are other, less serious but signi¬cant problems as well.



Problems with MNI

As with NISP, analysts have identi¬ed what they take to be serious problems with
MNI (e.g., Casteel n.d.; Fieller and Turner 1982; Gilbert et al. 1981; Grayson 1973,
1979, 1984; Klein 1980; Klein and Cruz-Uribe 1984; Plug and Plug 1990; Ringrose
1993; Turner 1983, 1984; Turner and Fieller 1985). These include:

1 MNI is dif¬cult to calculate because it is not simply additive;
2 MNI can be derived using different methods, thereby reducing comparability;
3 MNI values do not accurately re¬‚ect the thanatocoenose or the biocoenose;
quantitative paleozoology
46


4 MNI values exaggerate the importance of rarely represented taxa, or taxa
represented by low NISP values;
5 MNI values are minimums and thus ratios of taxonomic abundances cannot be
calculated;
6 MNI is a function of sample size or NISP, such that as NISP increases, so too
does MNI; and
7 different aggregates of specimens comprising a total collection will produce
different MNI values.

The ¬rst problem “ MNI is dif¬cult to determine “ is not worth considering. No one
has ever said research of any kind was, or should be, easy. But related to this problem
is the fact that MNI is not additive like NISP is. Rather, every time a new bag of faunal
remains is opened and specimens identi¬ed, one has to rederive the MNI (assuming
it was derived before). This is so because the most common skeletal part per taxon
may change with the addition of another bag of bones. This problem, too, is trivial
given that research often involves calculation and recalculation, again and again, as
new data are collected or as new insights are gained and adjustments are made to a
data set.
That different researchers use different methods to derive MNI values (the second
problem) is evidenced by variations in the de¬nitions of MNI in Table 2.4. This
problem is akin to the one that different analysts will produce different NISP values
for the same collection of remains given their varied expertise at identi¬cation.
There is no real way to control this problem, so the methods used to derive MNI
values should be stated explicitly. Were remains matched by size? By age? By recovery
context? All of these or other variables? Potential and varied use of these criteria make
MNI a derived measure.
Some have argued that to not attempt to match potentially paired remains such as
left femora with right femora “ to determine if they originated in the same individual “
results in a misleading MNI value (Fieller and Turner 1982). White (1953a:397) thought
that matching would require “the expenditure of a great deal of effort with small
return [to] be sure all of the lefts match all of the rights.” That is, he believed that
checking every possible bilateral pair of bones for matches (based on the notion of
bilateral symmetry “ that a left element is a mirror image of its right element mate),
and assuming that each of those matches derived from the same individual “ would
result in a relatively small increase in the MNI value, and thus not much in the way of
alteration of taxonomic abundances. Whether or not White was correct with respect
to the magnitude of change in MNI values when matching is undertaken is unclear,
but likely is assemblage speci¬c for many reasons (Lyman 2006a). Some argue that
estimating taxonomic abundances: nisp and mni 47


the difference would be considerable whereas others seem to think it would not be
if “a great deal of effort” were expended, whatever the result, it will be a function of
the identi¬ed assemblage at hand and the procedures used to analytically manipulate
the bilateral pair data rather than the act of matching and identifying bilateral pairs
itself (compare Fieller and Turner [1982] with Horton [1984]).
The third problem, too, can be said to characterize both NISP and MNI. NISP
is a count of identi¬able specimens in the assemblage rather than a measure of
the thanatocoenose or the biocoenose. Similarly, given its de¬nition, MNI is the
minimum number of animals necessary to produce the identi¬ed specimens com-
prising the identi¬ed assemblage. Both quantitative units describe the assemblage
using two rather different variables. Whether either NISP or MNI more accurately
re¬‚ects the thanatocoenose or the biocoenose cannot be assumed given the contin-
gent and particularistic nature of the taphonomic history of the assemblage (e.g.,
Gilbert and Singer 1982; Ringrose 1993; Turner 1983). But, as I noted with respect
to NISP, there are ways to test if the identi¬ed assemblage rendered as a set of MNI
values accurately re¬‚ects the biocoenose. Are the taxonomic abundances indicated by
MNI values what would be expected given independent evidence of environmental
conditions? Do MNI abundances match those from contemporaneous nearby faunal
assemblages that experienced independent taphonomic histories? If the answers to
these questions are all “yes,” then it would be reasonable to suppose that the tax-
onomic abundances in the collection under study are fairly accurate re¬‚ections of
those abundances in the thanatocoenose as well as the biocoenose.
The fourth problem can be appreciated by recalling Table 2.3. There, taxon A is
rarely represented (NISP = 1) whereas taxon B is frequently represented (NISP =
10), but both taxa have an MNI of 1. MNI exaggerates the representation of taxon A
relative to taxon B™s representation. Although this observation is true, it is merely the
converse of the related problem with NISP. Thus one might argue that MNI and NISP
are equally ¬‚awed in this respect. If you are uncomfortable with that, you could note
that taxon A in Table 2.3 is represented by one tibia, and tally only tibiae identi¬ed
as taxon B when comparing abundances of these two taxa. It is likely that such a
procedure would decrease the disparity in representation of the two taxa, but it also
demands the assumption that the other specimens of taxon B are interdependent with
that taxon™s tibiae, an assumption that would likely be dif¬cult to warrant empirically
or theoretically. (Some specimens may be interdependent, but it seems improbable
that all would be interdependent.)
Plug and Plug (1990) identi¬ed the ¬fth problem: MNI values are minimums and
thus ratios of MNI values cannot be validly calculated (see also Gilbert et al. 1981).
They note that if the MNI of taxon A is 10 and the MNI of taxon B is 20, we cannot
quantitative paleozoology
48


use simple arithmetic to calculate the A:B ratio because, with respect to the true
number of individuals, it is very likely that A ≥ 10 and B ≥ 20. Thus any ratio A:B
cannot be validly calculated. MNI values are not ratio scale. Instead, they are perhaps
ordinal scale. Thus we can say, in Plug and Plug™s case of A:B, that it is likely that
A < B, but we cannot say by how much given that both A and B are minimum
values, and we don™t know their true values. A similar argument can be made with
respect to NISP values. They are maximum estimates of taxonomic abundances, so
a ratio of NISP values for two taxa, although easily calculated, may not actually be
a ratio scale measure of taxonomic abundances. That MNI values are minimums
is clear from how they are derived (Table 2.4). Paleozoologists have known these
things since the MNI quantitative unit was introduced (e.g., Adams 1949). What is
perhaps less well-known is that recent simulations indicate that MNI often provides
values considerably lower than the actual number of individuals (ANI) present in a
collection (Rogers 2000a). To illustrate this, look at Table 2.5 one more time. Here,
the MNI is seven; the NISP is seventy-one. If each skeletal element is independent
of every other element (each comes from a different organism), then the ANI is
seventy-one, an order of magnitude greater than the MNI value.
The fact that MNI increases as NISP increases (problem 6) has been recognized for
some time in zooarchaeology (Casteel 1977, n.d.; Ducos 1968; Grayson 1978a). Some
have argued that this statistical relationship warrants use of NISP rather than MNI to
measure taxonomic abundances; the reason is that the same information regarding
taxonomic abundances is contained in both quantitative units, so there is no reason
to determine MNI. Others have noted that although this statistical relationship does
indeed exist between the two measures, the precise nature of the relationship depends
on the particular set of remains involved (e.g., Bobrowsky 1982; Grayson 1984; Hesse
1982; Klein and Cruz-Uribe 1984). Some researchers use the last observation “ that the
relationship between NISP and MNI is statistically particularistic “ to argue that one
cannot predict MNI from NISP in a new sample based on the statistical relationship
of the two in previously studied samples, so perhaps MNI should be determined
(Klein and Cruz-Uribe 1984). This is a clever insight, and it is correct, but it does
not mean we must determine MNI values when seeking measures of taxonomic
abundance. We need not determine MNI because of the relationship between the
NISP for a taxon in an assemblage and its attendant MNI.
It is commonsensical that as the NISP of a taxon increases so too should the MNI for
that taxon. This is so because every individual skeleton comprises a limited number
of elements (or what might become identi¬able specimens comprising the paleozo-
ological record). Adding randomly chosen skeletal elements selected from, say, 100
skeletons of an identi¬ed assemblage, the ¬rst element will contribute one individual.
estimating taxonomic abundances: nisp and mni 49




figure 2.3. The theoretical limits of the relationship between NISP and MNI. Modi¬ed
from Grayson (1978a). Line A indicates that every new specimen does not contribute a new
individual. Line B indicates that every new specimen contributes a new individual.



That is, NISP = 1 = MNI. The second element will contribute another NISP
( NISP = 2), but that element might, or might not, contribute another individual
( MNI = 1 or 2). The third element will contribute yet another NISP ( NISP = 3),
and it might contribute another individual or it might not ( MNI = 1, 2, or 3). And
so on until the probability of adding a bone or tooth of an already represented skele-
ton is greater than the probability of adding a bone of an unrepresented skeleton,
at which point the rate of increase in MNI will slow relative to the rate of increase
of NISP. As Grayson (1978a) noted, there are two limits to the possible relation-
ship between NISP and MNI. Either every new skeletal element derives from the
same individual and thus every NISP > 1 contributes nothing to the MNI tally, or
every new skeletal element derives from a different, unique individual and thus every
NISP ≥ 1 contributes another MNI. These relationships express the limits of all
possible relationships between NISP and MNI (Figure 2.3).
The individual limits to the relationship between NISP and MNI (Figure 2.3) are
unlikely to be found in the real world. Unless one is dealing with, say, the moderately
fragmented skeleton of a single individual animal (NISP is several hundred), it is
likely that NISP will increase more rapidly than MNI. In fact, unless one is dealing
with an assemblage of remains of a single taxon that has but one identi¬able skeletal
element (such as the unbroken shells of a gastropod), it is likely that new NISP
will often be added without adding any MNI. The general relationship between
NISP and MNI is described by the line in Figure 2.4. It is relatively easy to show
that this is indeed the relationship that is found in case after case. The relationship
quantitative paleozoology
50




figure 2.4. The theoretically expected relationship between NISP and MNI. Modi¬ed
from Casteel (1977). As NISP increases, it takes progressively more specimens to add new
individuals.


is curvilinear because, given a ¬nite NISP and a ¬nite MNI, specimens from an
individual already represented are progressively more likely to be added as the sample
size ( NISP) increases. Whatever the kind of skeletal part that is the most frequent
or most common and thus de¬nes MNI, that kind of part will become progressively
more dif¬cult to ¬nd (Grayson 1984).
Ducos (1968) found that this curvilinear relationship could be made linear (and
thus perhaps more easily understood when graphed) if both the NISP data and the
MNI data were log transformed (see Box for additional discussion). As Grayson
(1984:52) later noted, untransformed NISP data and untransformed MNI data may
sometimes be related in a linear fashion, but they often are not. The means to tell if
they are not involves examination (either visual or statistical) of the residuals (the
distance above and below the regression line of the plotted points and the pattern of
the distribution of those points). Typically, log transformation reduces the dispersal
of points to a statistically insigni¬cant level. The slope of the best-¬t regression line
summarizes the rate of change in MNI relative to the rate of change in NISP and
is described by a single number representing a power function (or exponent); the
larger the number, the steeper the slope.
Casteel (1977) found the relationship modeled in Figure 2.4 in a series of assem-
blages of zooarchaeological and paleontological materials representing numerous
taxa. His data originally were comprised of 610 paired NISP“MNI values. (A paired
NISP“MNI value is the NISP value and the MNI value for a taxon in an assemblage
of remains.) Casteel subsequently expanded his data set (Casteel n.d.) to include 3,440
estimating taxonomic abundances: nisp and mni 51



BOX 2.1

It is often easier to grasp intuitively a linear relationship between two variables
than a curvilinear one. In many cases log transformation of NISP and MNI data
causes what is otherwise a curvilinear relationship to become linear. The typical
form of a linear relationship can be expressed by the equation Y = a + bX, where
X is the independent variable (in this case, NISP), Y is the dependent variable
(in this case, MNI), a is the Y intercept (where the line describing the linear rela-
tionship intersects the Y axis), and b is the slope of the line (where the slope of
the line describing the linear relationship represents how fast Y changes relative
to change in X). The simple best-¬t regression line in a graph showing the rela-
tionship between log transformed NISP data and log transformed MNI data is
described by the formula Y = aXb , where the variables Y, a, X, and b are as de¬ned
above. This formula describes what is referred to as a power curve; if b is positive
the curve extends upward from the lower left to the upper right of the graph; if b
is negative the curve extends downward from the upper left to the lower right. If
we transform both sides of Y = aXb to logarithms, then we have log Y = log a +
b log X, linear relationship between log Y and log X. In this volume, I present the
relationship between log X or log NISP, and log Y or log MNI, in the form Y = aXb ,
or what is simply a different form of the linear relationship. The Y intercept should
be zero, given that a zero value for NISP must produce a zero value for MNI, but
practice has been to allow the empirical data to identify a Y intercept; I follow
this practice here noting that should the empirically determined Y intercept dif-
fer considerably from zero, the data used should be inspected to determine why.
Variables a and b are constants determined empirically for each data set.


paired NISP“MNI values. (The manuscript in which Casteel used this larger data set
was never published. It was written in 1977, and afterwards cited occasionally by his
colleagues [e.g., Bobrowsky 1982; Grayson 1979]. I obtained a copy of the manuscript
from Grayson in the late 1970s.) In both cases, Casteel found a statistically signi¬cant
relationship between NISP and MNI like that shown in Figure 2.4. Bobrowsky (1982)
found the same relationship between paired NISP“MNI values using much smaller
data sets. Both Casteel (1977, n.d.) and Bobrowsky (1982) graphed the relationship
using untransformed data; Grayson (1984) and Hesse (1982) used log-transformed
data in their graphs of the relationship. Grayson (1984) summarized many cases
that had been reported by others, and reported several new cases to show that the
relationship was essentially ubiquitous. Klein and Cruz-Uribe (1984) found the same
quantitative paleozoology
52




figure 2.5. Relationship between NISP and MNI data pairs for mammal remains from
the Meier site (data in Table 1.3). See Table 2.6 for statistical summaries.



relationship between NISP and MNI using data sets different from those used by
Casteel, Bobrowsky, Grayson, and Hesse.
The large collection from Meier ( NISP = 5939) shows the relationship between
NISP and MNI nicely (Figure 2.5). The NISP“MNI data pairs are strongly correlated
(Pearson™s r = 0.8734, p < 0.0001). The slope of the best-¬t regression line (= 0.487) is
similar to that reported by others for other data sets; Casteel (1977) reported a slope
of 0.52, for example, and Grayson (1984) reported six others that ranged from 0.40 to
0.64. The precontact assemblage from Cathlapotle (Table 1.3) also shows the nature of
the relationship between NISP“MNI data pairs (Figure 2.6), as does the postcontact
assemblage from that site (Figure 2.7). In all three cases, the correlation coef¬cient is
strong (r > 0.87) and signi¬cant (p < 0.0001). The statistical relationships between
the two variables in each of these three assemblages are summarized in Table 2.6.
The relationship between NISP and MNI shown in Figures 2.5, 2.6, and 2.7 is not
unique to the Portland Basin. Recall that Casteel, Grayson, Hesse, Bobrowsky, and
Klein and Cruz-Uribe found exactly the same relationship between the two variables
in collections from all over the world and representing many time periods and taxa.
estimating taxonomic abundances: nisp and mni 53


Table 2.6. Statistical summary of the relationship between NISP and MNI for
mammal assemblages from Meier (Figure 2.5) and Cathlapotle (Figures 2.6 and 2.7)
(see Table 1.3 for data).

N of
Site Regression equation r p NISP taxa
MNI = ’0.06(NISP)0.487 <0.0001
Meier 0.873 5,939 26
MNI = ’0.098(NISP)0.42 <0.0001
Cathlapotle, precontact 0.916 2,372 21
MNI = ’0.0557(NISP)0.44 <0.0001
Cathlapotle, postcontact 0.901 3,834 24



Together, these cases suggest that the relationship is nearly ubiquitous. Table 2.7 sum-
marizes the statistical relationship between NISP and MNI in fourteen assemblages
of mammal remains from fourteen sites in eastern Washington State. I, along with
two fellow graduate students at the time, identi¬ed the taxa in these collections in
the late 1970s. All fourteen collections display the same kind of relationship between
NISP and MNI as is evident for Meier and Cathlapotle.




figure 2.6. Relationship between NISP and MNI data pairs for the precontact mammal
remains from the Cathlapotle site (data in Table 1.3). See Table 2.6 for statistical summaries.
quantitative paleozoology
54




figure 2.7. Relationship between NISP and MNI data pairs for the postcontact mammal
remains from the Cathlapotle site (data in Table 1.3). See Table 2.6 for statistical summaries.


That NISP“MNI data pairs are often tightly related (the correlation coef¬cient is
large) even in nonarchaeological contexts is also easy to show. Consider a sample of
eighty-four pellets likely cast by a barn owl (Tyto alba) in eastern Washington State.
NISP and MNI values for the total assemblage of prey remains in the eighty-four
pellets (Table 2.8) are arrayed in a bivariate scatterplot in Figure 2.8. The relationship
between the values of the two is linear and strong (r = 0.989, p = 0.0002), as it is
among the archaeological samples discussed previously; the regression equation is:
MNI = “1.57(NISP)0.935 . The same relationship holds for paleontological collections
as well, as Grayson (1978a) showed some years ago.
The fact that NISP and MNI are often strongly correlated could be used to argue
that we should use MNI as the quantitative unit for measures of taxonomic abun-
dance because of the potential for the interdependence of specimens in NISP tallies.
Indeed, Hudson (1990) argued on the basis of ethnoarchaeological data, and Bre-
itburg (1991) on the basis of historic-era zooarchaeological data supplemented by
written documents, that MNI provides more accurate measures of taxonomic abun-
dances than NISP. That MNI would indeed sometimes be a more accurate measure of
taxonomic abundances than NISP is to be expected given everything we know about
the two quantitative units and the in¬‚uences of taphonomic processes and recovery
estimating taxonomic abundances: nisp and mni 55


Table 2.7. Statistical summary of the relationship between NISP and MNI for
mammal assemblages from fourteen archaeological sites in eastern Washington State

Site Regression equation r p NISP N of taxa

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