Palcobiolo~~y,
20(2), 1994, pp. 245-258
Estimating numbers of whole individuals from
collections of body parts: a taphonomic
limitation of the paleontological record
Norman L. Gilinsky and J Bret Bennington
1
Abstract.-Paleoecologists have long sought to obtain estimates of the sizes of extinct populations.
However, even in ideal cases, accurate counts of individuals have been hampered by the fact that
many organisms disarticulate after death and leave their remains in the form of multiple, separated
parts. We here analyze the problem of estimating numbers of individuals from collections of parts
by developing a general counting theory that elucidates the major contributing variables. We
discover that the number of unique individuals of a particular species that are represented in a
fossil collection can be described by an intricate set of relationships among (1) the number of body
parts that were recovered, (2) the number of body parts that were possessed by organisms belonging
to that species, and (3) the number of individuals of that species that served as the source of the
parts from which the paleontological sample was obtained (the size of the "sampling domain").
The "minimum number of individuals" and "maximum number of individuals" methods currently
used by paleontologists to count individuals emerge as end members in our more general counting
theory. The theory shows that the numbers of individuals of a species that are represented in a
sample of body parts is fully tractable, at least in a theoretical sense, in terms of the variables just
mentioned. The bad news is that the size of the "sampling domain" for a species can never be
known exactly, thus placing a very real limit o n our ability to count individuals rigorously. The
good news is that one can often make a reasonable guess regarding the size of the sampling domain,
and can therefore make a more thoroughly informed choice regarding how to estimate numbers
of individuals. By isolating the variables involved in determining the numbers of individuals in
paleontological samples, we are led to a better appreciation of the limits, and the possibilities, that
are inherent in the fossil record.
Norman L. Gilinsky and \ Bret Benningtotl.' Department of Geological Sciences, Virginia Polytechnic Institute
and State University, Blacksburg, Virginia 24061
'Present address:
Department of Geology, 114 Hofstra University, Hempstead, New York 11550
Accepted: June 24, 1993
Introduction
One of paleoecology's most enduring goals
has been to use the fossil record to obtain
"snapshots" of populations and communities
from the geological past. Given these snapshots, quantitative ecological principlesgarnered from the study of modern communities-could be applied to give us an understanding of the nature of communities in
the past, the frequency distributions of species that lived in those communities, the trophic relations among ancient organisms, and
the spatial distributions of past populations.
Furthermore, by examining sequences of such
biological snapshots, the paleoecologist could,
ideally, create a kind of ecological motion picture, frame by frame, that would lead to a
rich understanding of community evolution
over a time scale that could not be approached
(a1994 The Paleontological Society. All rights resewed.
by ecologists working with modern-day organisms. Of course, certain kinds of organisms would not be preserved, but the relative
coarseness of paleontological sampling as
compared to what is possible in the Recent
would be more than compensated for by the
vastly greater temporal scale that could be
examined by paleontologists. Parsons and
Brett (1991: p. 22) expressed this ideal vividly
by describing paleontologists as yearning "to
restore the flesh and blood of fossil remains,
and thereby to depict in detail the appearance, and analyze the interactions and dynamics, of ancient communities."
A means for counting unique individuals
would seem to be a minimal prerequisite for
obtaining paleontological snapshots, because
only with such counts would it be possible
to study such phenomena as biomass, popu0094-8373/94/2002-0010/$1 .00
246
NORMAN L. GILINSKY AND J BRET BENNINGTON
lation dynamics, predator-prey ratios, among main, for instance, a very reasonable estimate
others, quantitatively in the fossil record. A of the number of unique individuals can be
big problem in obtaining counts of individ- obtained simply by counting each body part
uals from an assemblage of fossils is the fact as a unique individual. Fortunately, given that
that most fossils are preserved as disarticu- many samples of invertebrates are collected
lated body parts. Do 28 clam valves from a from highly fossiliferous beds, this "maxisingle species represent 14 clams, or 28, or mum number of individuals method" will apsome number in between, for instance? In an ply frequently in paleoecological work.
effort to see if an accurate means for counting
Most fossil assemblages obtained even from
unique individuals could be devised, and thin beds are both time-averaged and locally
eventually applied, we used computer sim- transported (see Kidwell and Bosence [I9911
ulations and mathematical analysis to devel- for an extensive review). The theory develop what is effectively a counting theory that oped here applies to these cases and, in fact,
isolates the relevant variables and elucidates to all cases where the aim is to estimate the
their intricate relationships. To foreshadow number of unique individuals in an assemthe conclusions below, we discover that the blage. But when time-averaging or local
number of unique individuals (of a species) transport are significant, we caution that the
in an assemblage is a function of three basic counts will not be estimates of the original
variables: (1) the number of body parts of that population sizes, only estimates of the numspecies that are preserved in the assemblage, bers of unique individuals represented in the
(2) the number of body parts that comprised collections. These numbers, nonetheless, may
that organism in life, and (3) the number of be of interest in some kinds of studies. The
organisms of that species that were poten- theory also applies to assemblages that were
tially available to be sampled by the pale- produced during a catastrophic burial, where
ontologist (what we define below as the "sam- the number of fossils that are actually prepling domain"). Given values for these served in the sediment may be isomorphous
variables, the theory tells us that the statis- with the number of organisms in the living
tically expected number of unique individ- population. Such assemblages might be very
uals can be determined unambiguously.
rare indeed but, if they exist, the theory deIn practice, however, it will be effectively veloped here may help to estimate the numimpossible to obtain a rigorous estimate of bers of organisms in once-living populations.
the size of the sampling domain and, there- Hallam (1972) has termed these kinds of asfore, it will be impossible to get a rigorous semblages "census populations."
estimate of the number of unique individuals
Perhaps our most important conclusion,
in an assemblage (unless, of course, all of the however, is that there is nothing more pracindividuals were preserved in an articulated tical than theory. For the theory developed
state). So the bad news, as we shall show, is here isolates the important variables related
a rigid limit to the resolving power of the to counting unique individuals, locates the
fossil record. We regard this limit as a kind one variable (the size of the sampling doof taphonomic law, in the sense of Efremov main) that prohibits rigorous estimation of
(1940). This "law of numerical indeterminan- the numbers of unique individuals in assemcy" also explains why previous efforts at blages, pinpoints a very real taphonomic limcounting unique individuals have delimited it to the resolving power of the fossil record,
only the extremes (maximum and minimum and hints at conditions under which we can
numbers of individuals) in the continuum of do the counting anyway.
possibilities. The good news is that the theory
The Problem
also tells us that we nonetheless may be able
to estimate the number of unique individuals
Vertebrate paleontologists, including stuso long as the size of the sample relative to dents of human evolution, have frequently
the size of the sampling domain is known at run up against the problem of counting disleast roughly. When the size of the sample is articulated individuals in their efforts to obsmall relative to the size of the sampling do- tain approximate shapshots of ancient terres-
COUNTING UNIQUE INDIVIDUALS
.
trial communities. A specific instance is
exemplified by Klein's (1980) work on mammalian faunas from Upper Pleistocene and
Holocene archaeological sites in South Africa, where one of the pieces of information
desired for each site is the frequency of occurrence of each mammalian species. In fact,
Klein (1980: p. 226) points out that estimates
of species frequencies are "[tlhe most widely
used numerical data in archaeological faunal
analysis." The problem, even when one can
assume that the fossil deposit is distorted neither by time- nor space-averaging, is that individual mammals are preserved as disarticulated body parts. If the body parts of
individuals are scattered, as they normally are
to some extent (even if transport into or out
of the area of the site is unimportant), then
counting the number of unique individuals
becomes a non-trivial matter. It is unlikely
that all body parts from all individuals represented in the sample will be recovered and,
furthermore, it is unlikely that the parts that
are recovered can all be assigned to particular
individuals.
Two basic approaches to the problem have
been tried. The first has been to sort the bones
by species, and then to assume that each bone
in the sample for a species represents a unique
individual of that species. By this method,
one hundred recovered bones for a species
means one hundred unique individuals. Since
this approach maximizes the number of possible individuals, we call it the XNI approach
(for "maximum Number of Individuals"). As
Klein mentions, the method would almost
certainly overestimate the actual number of
individuals in the assemblage because some
individuals surely contributed more than a
single bone. The second approach, sometimes
called the "Minimum Number of Individuals" method, or MNI, first groups the bones
by species, then by bone type, and then estimates the number of individuals for a species as the number needed to account for the
most common type of bone. For instance, if
the most common bone in the sample for a
species is the left femur, the MNI is defined
as being equal to the number of left femurs.
This method will tend to underestimate the
actual number of individuals due (in this case)
to unrecovered left femurs. A modification of
247
this approach would be to estimate the number of individuals as the MNI plus an additional number of individuals to account for
those that contributed some bones to the sample, but no femurs. However this modification assumes that bone elements can be confidently matched, an assumption that is
usually not warranted.
With a little reflection more complications
come to mind, but all of the approaches that
have been tried to compensate for these complications really amount to attempts to patch
up estimates based on the two main approaches mentioned above. Klein (1980) gives a more
complete treatment of the practical aspects of
the problem than we do here, and Badgely
(1986), Behrensmeyer (1991), and Shipman
(1981) discuss the problem as well. Holtzman
(1979) published a thoughtful analysis of several approaches to counting both vertebrate
and invertebrate fossils, including the MNI,
and his analysis is relevant to the question at
hand, but the emphasis was on estimating
relative abundance and not the number of
individuals per se. Since the XNI and MNI
approaches give us only the extremes in the
spectrum of possible counting conventions
for estimating numbers of individuals, the
obvious question that arises is whether we
can do better than simply to isolate the extremes. Can we develop a method that will
give us the "right" answer or, at least, a rigorously defensible best-estimate?
We have thus far described the problem
from the standpoint of vertebrates because
vertebrate paleontologists have frequently
encountered it in actual empirical work, and
because the problem is so readily brought to
light given that, with so many bones for each
individual, the XNI and MNI methods give
such radically different estimates for the same
data. Invertebrate paleontologists have been
less mindful of the problem, but some sense
of concern is hinted at by the fact that invertebrate paleontologists, too, apply counting conventions in their own work. Many
research workers when counting clams, for
instance, do so by estimating the number of
individuals as the number of left, or right,
valves in the collection, whichever is larger
(e.g., Miller et al. 1992). This procedure is
precisely the MNI approach, but this time
248
NORMAN L. GILINSKY AND J BRET BENNINGTON
applied to organisms with only two body
parts. Other workers count all of the valves
in the collection (for a particular species), a
procedure that is exactly analogous to counting up all of the bones in the vertebrate case
(the XNI approach). (A third method is to
count half the number of valves, but this
number, ironically, is usually smaller than
the MNI!) Despite the different organisms involved, the basic problem of counting individuals is the same. For organisms with multiple body parts, the best that has been
achieved thus far has been minimum and
maximum estimates of the numbers of individuals, even though it is obvious that both
estimates are almost always wrong.
We attempt here to analyze the problem of
counting individuals more abstractly than has
been done before. By doing so, we can attack
the basic question perhaps more rigorously,
from first principles. Accordingly, the work
is mathematical, but the mathematics are not
difficult. We begin with the problem of counting organisms with two body parts (we use
clams as our focal organisms), but we do so
only to keep things simple. We emphasize
that the theory generalizes to organisms with
larger numbers of parts, including vertebrates. The conclusions are given in detail
later, but we mention at the outset that by
thinking of the problem abstractly, as a problem in counting theory, we are able to determine precisely why it is that paleontologists
have thus far been limited to estimating only
the extremes of the possible numbers of individuals sampled from a population, and why
only in rare cases will it be possible to do
much better. We regard explicit awareness of
this limitation of the fossil record, rigorously
defined, as a practical contribution in itself,
a kind of taphonomic "law." On a more hopeful note, however, there may be some circumstances under which this limitation might be
breached, and we shall discuss this aspect as
well.
A Simulation Approach
Motizlation and Example.-If we have a collection of clam valves on hand, and we are
interested in estimating rigorously the number of unique individuals that are represent-
ed by the collection of valves, we need to
know the mathematical relationship between
the number of valves in a collection and the
number of unique individuals. If we had that
relationship, we could calculate the estimated
number of unique individuals in our collection, or we could obtain the number of unique
individuals graphically. (We assume in this
case that it is impossible or impractical to try
to match all of the valves, and that we need
to approach the problem from a mathematical
standpoint. We mention later that it might be
possible to test the model proposed here by
actually collecting specimens and trying to
match the valves.) As mentioned earlier, some
paleontologists have applied the MNI approach to this problem, thereby estimating
the number of unique individuals as the
number of left or right valves, whichever is
larger. Others have used the XNI approach,
simply counting each valve as an individual.
So previous research workers /lave assumed
specific mathematical relationships (as have
vertebrate paleontologists), but those relationships delimit only the extremes. Our aim
is a more sophisticated mathematical relationship, one that accounts for the entire range
of possible estimates, not just the extremes.
One way to explore the unknown relationship is to create a hypothetical population of
clams, to obtain variously-sized samples of
valves from this population, and to see how
many unique individuals are represented in
the samples. This approach is illustrated in
figure 1. It amounts to establishing a population of articulated clams (100 in this case),
labeling both valves of each clam with the
same number (both valves of clam #l are
labeled 1; both valves of clam #2 are labeled
2, etc.), disarticulating the valves, throwing
them in a barrel, . . . stirring well, . . . and then
sampling. To find out how many unique clams
are represented by 20 valves, for instance, 20
valves are sampled randomly from the barrel,
and the number of unique numbers is counted. The number of unique numbers is the
number of unique clams. In performing the
experiment repeatedly, one would immediately encounter the problem of statistical
sampling error, in the sense that different
samples of 20 valves would give different
COUNTING UNIQUE INDIVIDUALS
Take a population
of 1 0 0 clams w i t h
numbered l e f t and
r i g h t valves ...
... disarticulate
and m i x we1 I...
repeatedly sample 20 valves and count
the number of unlque clams...
P
.......
20 valves. 19 unlque clams
2 9 valves, 20 unrque clams
20 valves, I 8 unlque clams
I
LV
1
I
J~IVC;I,
--
I 7 ~ I I I ~ U~C
101113
Mean
... and construct a
nistogram o f the
results.
Number of unique clams
FIGURE 1. Cartoon illustration of a sampling experiment to determine the expected number of unique individuals
that are represented in a collection of clam valves. In the experiment, both valves of each of 100 clams are labeled
with the same number. The clams are then disarticulated, and the valves are mixed well in a barrel by one of the
authors. Twenty valves are then sampled from the barrel randomly, and the number of unique clams represented
in the sample is then determined. By performing this experiment many times, the distribution of numbers of unique
clams can be obtained, and the expected number of unique clams for 20 valves can be estimated as the mean of the
distribution. In the cartoon, about 19 unique individuals are expected when 20 valves are sampled from a collection
of valves from 100 clams.
.
numbers of individuals. But the numbers obtained from repeated samples would define
a distribution (illustrated at the bottom of fig.
I), from which an expected (or mean) value
of the number of unique clams could be obtained, along with confidence intervals about
that expectation.
The expected number of individuals for 20
valves can also be obtained with high accuracy using a computer simulation. The set of
clams that are to serve as the source of the
sample of valves we call the sampling domain. This number is equivalent to the number of clams in the barrel of figure 1, and is
250
NORMAN L. GILINSKY AND J BRET BENNlNGTON
BEK>RE
CLAM
L R
AFTER
L R
1
+
2
&
3
TABLE1. Relationship between the number of valves in
a collection and the expected number of unique clams
represented. Expected number of unique clams was obtained by simulating the process of sampling 0, 10, 20,
. . . valves from a collection of 200 valves (100 complete
individuals). The simulation for each sample size was
run 1000 times. SE is the standard error of the expected
number of unique clams.
4
Number
ol
valves
FIGURE
2. A simulation approach to the problem of determining the relationship between number of valves in
a collection and the number of unique individuals. The
100 by 2 array on the left (labeled "BEFORE), filled with
Os, represents the unsampled left and right valves of 100
clams. Twenty valves are then sampled randomly, with
the randomly sampled valves being represented on the
right side of the diagram (labeled "AFTER") by Is in the
array. The number of unique clams is then counted simply by counting the number of rows that contain one or
more Is (the arrows).
analogous to the number of individual clams
that are represented in the volume of rock
(or sediment) from which a real paleontological collection is made. In the computer simulation the sampling domain is represented
as an array, as shown on the left side of figure
2, and n = 20 valves are selected randomly
from the array (selected valves are depicted
as Is). For the set of valves that are sampled,
the number of unique clams in the collection
can be easily obtained (the arrows in the figure). Repeating the simulation N times (for
N reasonably large) allows the mean (or expected) number of individuals to be calculated for a sample size of n = 20 valves. For
sample sizes other than 20 values, the simulation can be run again for different valves
of n.
Table 1 gives the results of a simulation run
under the protocol described above, for a
range of sample sizes n. Since the number of
Ex ected number
ofunique clams
Variance
SE
clams in the sampling domain was 100, up to
200 valves were available for sampling. For
each sample size (we chose n = 0, 10, 20,. . .
200), the sampling procedure outlined above
was performed N = 1000 times. So the expected number of unique clams for a given
sample size n of valves is equal to the mean
number of clams that were identified in the
N = 1000 iterations of sampling (i.e., the 1000
"collections").
For a collection size of n = 20 valves (the
original problem posed above), and with the
size of the sampling domain set at 100 clams
(200 valves), the collection size n is fairly small
relative to the size of the sampling domain,
so each new valve collected has a high probability of representing a new, unique clam.
This probability continually decreases as additional valves are removed, but it nonetheless remains quite high as long as only a small
sample is being extracted from the whole (in
this case, 10% of the valves). As a result, the
expected number of unique clams is about 19
for a sample size of 20 valves (table 1). At
large collection size, after some 190 valves
25 1
COUNTING UNIQUE INDIVIDUALS
.
have been collected, for instance, it is very
urrlikely that a new valve will represent a
unique new clam. This is because the ten
valves remaining in the barrel will almost
always be counterparts of valves that have
already been collected. That is why, for a collection size of 190 valves, we expect to have
encountered virtually all of the unique clams
in the entire sampling domain (99.7830 of
100, as shown in the table). With an intermediate collection size of 100 valves, we expect to encounter about 75 different individuals.
A plot of the number of valves versus the
number of unique individuals is shown in
figure 3. The relationship is curved, implying
that the ratio between the number of unique
of collected
individuals and the
valves changes as the number of valves in the
collection increases. The curved nature of the
relationship explains why the MNI and XNI
approaches are inadequate. The MNI approach, for example, is the exact estimate only
when the number of valves in the collection
equals 199 or 200, for only in these cases is
the expected number of individuals correctly
calculated as the number of left or right valves,
whichever is larger (the answer is 100). And
the XNI approach gives the expected number
of unique individuals correctly only when
the number of valves in the collection is 0 or
1. The XNI approach does, however, give a
fairly good estimate (with a sampling domain
of 200 valves) even when the collection contains as many as 20 valves (where XNI = 20,
and the simulation-obtained estimate-the
"right answern-is about 19), but it is only
strictly correct at a sample size of 0 or 1 valves.
It is in this sense that the previous counting
conventions represent the extremes. They
give the correct expectations only within one
valve of the minimum and maximum possible
sample sizes. And, you have to use the right
method even at these extremes. It would not
be correct, for instance, to settle on the MNI
or the XNI method, and then to apply it routinely under all circumstances as a matter of
convention.
The width of the band encompassing 95%
of the simulated numbers of unique clams,
peaks at a collection size of 100 valves (one
*
?5
0
20
40
60
80
1 0 0 1 2 0 140
160 180 200
NUMBER OF VALVES IN THE SAMPLE
FIGURE3. Plot of the expected number of unique clams
as a function of the number of valves in a sample for the
range 0 to 200 valves. The size of the sampling domain
is 100 clams. For each sample size of valves 10, 20, . . .
190, 200, the number of unique clams was determined
1000 times. The dotted curves span approximately 95%
of the numbers obtained in these determinations. The
accuracy of the expected number of clams (the solid curve),
obtained from the standard error of the mean, is high
enough that the error bars about those numbers would
be nearly invisible at this scale.
half the number of valves in the sampling
domain) because it is here that the range of
possible outcomes is at a maximum. At a collection size of 100 valves, there can be as few
as 50 unique clams represented or as many as
100. At a collection size of 10 valves, on the
other hand, the entire range of possibilities
spans only from 5 clams to 10 and, at 190
valves, only from 95 to 100. The graph shows
this ballooning of the 95% band near the middle.
Summarizing to this point, the simulation
demonstrates that (1) the expected number of
unique individuals is an inherently statistical
quantity, (2) the expected number of unique
individuals is a nonlinear function of the
number of valves, (3) the MNI and XNI approaches (which assume a linear relationship)
give correct expectations only near the minimum and maximum possible sample sizes,
and (4) the width of the 95% band varies with
the number of valves in the collection. The
curve in figure 3 is precisely the kind of mathematical relationship we are seeking; it gives
expected numbers of unique individuals
252
NORMAN L. GILINSKY AND J BRET BENNINGTON
A: CLAMS
0 ,
0
. . . , . . . , . . . , . . . , . . .
400
800
1200
1600
2000
NUMBER OF VALVES IN SAMPLE
loo0 -
13:CHITINS
m
1000
800-
SW
0
400
800
1200
1600
2000
NUMBER OF PLATES IN SAMPLE
FIGURE4. Plots of the relationship between the number
of valves and the expected number of unique individuals
for sampling domains of several sizes for both clams (A)
and chitins (B).
across the entire range of possible sample
sizes.
The influence of the Size of the Sampling Domain.-The graph in figure 3 can be used to
obtain best-estimates (expectations) of the
numbers of unique individuals for the range
of collection sizes drawn from a sampling domain of 100 clams (200 valves). Unfortunately, however, the curve shown in the figure is
not the only one that is possible. In fact, the
shape of the curve depends strongly on the
size of the sampling domain, that is, the number of clams in the barrel. This can be appreciated readily by imagining an infinitely large
sampling domain. With an infinitely large
sampling domain, the probability of any valve
in a sample being the counterpart of any other valve in the sample approaches zero, no
matter how many valves are collected. In that
case, the curve relating the number of individuals to the number of valves will be linear,
with a slope of 1, and with each valve representing a new individual. At the other extreme, with very small sampling domains, the
probability of a new valve being the counterpart of one that has already been sampled
quickly becomes large as new valves are sampled, and the curve relating the number of
individuals to the number of valves rapidly
becomes strongly concave down. The implication is that there is not one curve describing
the relationship between the number of valves
and the expected number of unique individuals; rather, there must be a family of curves,
one for each sampling domain.
Figure 4A shows the curves for sampling
domains of several sizes. The simulation succeeds in establishing a graphical means of
specifying rigorously the expected number of
unique individuals that are represented in a
collection of valves. But, unfortunately, that
number is a function not only of the number
of valves in the collection, but also of the size
of the sampling domain. As we shall discuss
below, practical applications of the theory are
only possible when the size of the sampling
domain is known, or when certain assumptions can be made regarding the relation between the size of the sample and the size of
the sampling domain.
A Generalization to More Than Two Body
Parts.-Although we motivated the discussion using vertebrates, we have thus far restricted the mathematical analysis to the case
of bivalved organisms, specifically clams. Indeed, the above analysis applies to any organism that has two body parts. But the
counting theory also generalizes easily to organisms with more than two parts by extending the array in figure 2 out to 3, 4, or
more, columns, each column representing an
additional body part. By selecting specimens
(say, of bones) from this larger array, one can
generate curves relating the number of bones
in a collection to the expected number of
COUNTING UNIQUE INDIVIDUALS
unique individuals in much the same way as
with the case of bivalved organisms discussed
previously. To be strictly applicable without
modification, the different body parts would
need to have equal probabilities of being recovered. While probably true for typical bivalves, this is certainly not true for dinosaur
bones! Actual practical applications would
have to adjust the simulation algorithm to
account for the different recovery probabilities.
Figure 4B illustrates an example of a multipart generalization to chitins where the recovery probability of the eight plates can be
regarded as roughly the same. For comparative purposes, we made the axes of figures 4A
and 4B the same. For clams (fig. 4A), a collection of 300 valves from a sampling domain
of 500 clams is expected to yield about 255
unique individuals. But for chitins, a collection of 300 plates from a sampling domain of
500 chitins is expected to yield only 232 individuals. We shall return to this graph below.
253
The equation for calculating the expected
number of species E(S,) for a specified number
n of organisms is:
where Z is the total number of organisms in
the actual collection, Z, is the number of organisms observed to belong to species i, and
S is the total number of species in the actual
collection (Hurlbert 1971; Simberloff 1974;
Heck et al. 1975). The variance (Var [S,]) of
the expected number of species (Heck et al.
1975) is given by:
An Analytical Approach:
The Rarefaction Connection
The simulation approach described above
generates curves relating number of individuals to number of body parts in a manner that
is intuitively appealing. After all, sampling
valves from a barrel is readily seen as analogous to collecting fossils from an outcrop.
But, in fact, the very same curves can also be
regarded as specific instances of rarefaction
curves. Rarefaction analysis is usually applied
to counts of organisms and species, rather than
to valves and clams. It begins with the observed number of organisms and the observed number of species. A graph is then
constructed to express the relationship between the number of organisms and the number of species for hypothetical sample sizes
smaller than the observed sample size, under
the assumption that the observed sample
served as the source of the hypothetical smaller samples. This curve, which is called a rarefaction curve, allows species richnesses for
different collecting sites or habitats to be compared even though one sample might be
smaller than the other (Sanders 1968).
The same theory can also be applied to the
expected number of individuals in a sample
obtained from a larger sampling domain. Let
us suppose that the source of our sample contains S clams (rather than S species), each of
which has Z, = 2 valves. This would be precisely analogous to a case where there were
S species each with Z, = 2 organisms (unusual,
but conceivable). Therefore, the rarefaction
equation applied to clams with two valves
apiece must give the curve for the expected
number of unique clams for various collec-
254
NORMAN L. ClLlNSKY AND J BRET BENNINGTON
tion sizes of tr valves, just as the equation
applied to counts of species and organisms
would give the expected number of species
for various sample sizes of n fossil organisms.
That is, the problem of counting unique clams
is a special case of rarefaction analysis. Furthermore, the size S of the source population
of clams is what we defined earlier as the size
of the sampling domain. The mathematical
theory of rarefaction, with all Z, = 2 and with
S being the size of the sampling domain,
therefore provides the same curves as those
obtained above using simulation.
A straightforward analytical generalization
to organisms with more than two body parts
is possible with the rarefaction approach. For
clams, Z, = 2 because each clam i had two
body parts. For organisms with more than
two body parts, all that is needed is to set Z,
equal to the new number of body parts (for
chitins, Z, = 8), and to apply equations (1) and
(2) above. But, as before, the theory is only
strictly correct when the different body parts
have identical preservation probabilities.
Discussion
The Bad Nezcrs: The Problem of the Samplirlg
Domain.-An ideal solution to the problem of
the number of unique individuals would have
been a single curve relating that number to
the observed number of body parts. This curve
would have provided a means to estimate rigorously the numbers of individuals in a way
that the XNI and MNI methods could not.
Unfortunately, the solution is not nearly so
simple because a third variable, the size of
the sampling domain, needs to be taken into
account. The size of the sampling domain, as
mentioned before, is the number of individuals represented in the statistical universe
from which the actual collection was drawn.
In developing the above counting theory, that
number was defined a priori. But in actual
practice, all we really know is how many
specimens Z we have in our collection; we
don't know how many specimens comprised
the pool from which the collection was drawn.
That is, we don't know the size of the sampling domain.
The following hypothetical case illustrates
the problem. Suppose that one were to iden-
tify a thin layer of rock (or sediment) in the
field and, from this layer, collect 80 clam
valves. The largest number of unique individuals that could be represented is 80, and
the smallest is 40. But, all other things being
equal, whether the best estimate is closer to
80 or to 40 depends on the size of the sampling domain, that is, the total number of
clams that are represented in the volume of
rock from which our sample was collected.
(As usual, we assume that all of the valves are
disarticulated and that we are not willing or
able to match them.) If the sampling domain
included 50 individuals, or 100 individuals,
or 1000 individuals, then the corresponding
best estimates of the number of unique individuals for 80 valves would be 48.081,
64.120, and 77.857, respectively. Clearly the
best estimate is crucially dependent upon the
size of the sampling domain, but we simply
do not know what that size is, so we do not
know which curve to use. This indeterminancy is the bad news; the inability to specify
the size of the sampling domain is what prevents us from estimating rigorously the number of unique individuals in a sample.
Some Good Neuis: We Might Be Able to Guess
When the XNI is Appropriate.-Even though it
is not possible to determine exactly which of
the family of curves to use, it may be possible
to make certain assumptions that can lead to
an informed choice. For instance, the curves
in figure 4A (the clam case) show that the
relationship between the number of valves
and the number of individuals is approximately linear at first and then becomes concave down as the number of valves in the
collection approaches the total number in the
sampling domain. This pattern occurs because the first few valves collected from even
a moderately fossiliferous bed will only rarely be counterparts of valves that have already
been collected (assuming a well-mixed sample). Only after many valves have been sampled, relative to the total in the sampling domain, will valve counterparts start to appear
with high frequency. This suggests that it
might be appropriate to assume a near-linear
relation between number of valves and number of individuals, at least when the number
of valves in the collection is small relative to
255
COUNTING UNIQUE INDIVIDUALS
the total number of clams remaining in the
rock. This assumption would imply that each
valve in the collection can be counted as a
unique individual; that is, it implies that the
XNI approach would be valid.
As can be seen from the curves on figure
4A, the assumption of linearity is good for
any sized sampling domain, provided that the
number of valves in the collection relative to
the size of the sampling domain remains
small. But the length of the near-linear phase
of each curve (and thus the region where the
XNI applies) decreases markedly as the sampling domain shrinks. For this reason, we have
estimated (using simulation) the sample size
at which the assumption of linearity breaks
down for a two-valved organism (clam) and
for an eight-valved organism (chitin) for each
of several sampling-domain sizes. In figure 5,
the line labeled "clams" is the approximate
locus of all points at which the estimate of
the expected number of individuals obtained
by assuming linearity (that is, by counting
each valve as a unique individual) reaches a
maximum error of 10%. For instance, if the
sampling domain contains 600 clams (the vertical dotted line), then one can count each
valve as a unique individual (the XNI approach) for sample sizes of up to as many as
204 valves (the horizontal dotted line that
intersects the solid line for clams) and be assured that the error in the estimate does not
exceed 10%.For chitins (the lower line), if the
sampling domain contains 600 valves, one can
count each plate as a unique individual for
sample sizes of up to 132 plates (the horizontal dotted line that intersects the solid line
for chitins) and be assured that the error in
the estimate does not exceed 10%.Considered
from another standpoint, if one has 100 clam
valves in hand, the number of unique clams
can be estimated as 100 (the horizontal line
of alternating dashes and dots), with 10% error at most, so long as the sampling domain
is at least 294 (where the horizontal line intersects the solid line for clams). But if one
has 100 chitin plates in hand, the number of
unique chitins can be estimated as 100 (the
horizontal line of alternating dashes and dots),
with 10% error at most, only if the sampling
domain is at least as large as 455.
350
-
300
-
250
-
220-
150
-
-
SLOPE .22
0
200
400
600
800
1000
NUMaER OF INDIVIDUALS IN SAMPLING DOMAIN
FIGURE5. Plots, for bivalves and chitins, of the lines
defining the approximate 10% error limits for applying
the XNI method to a collection. For clams, estimating the
number of individuals as being approximately equal to
the number of valves will result in 10%error at most, so
long as the number of valves does not exceed 34% (the
slope) of the total number of clams in the sampling domain. For chitins, estimating the number of individuals
as being approximately equal to the number of plates
will result in 10% error at most, so long as the number
of valves does not exceed 22% of the total number of
chitins in the sampling domain. For example, with a
sampling domain of 600, one may count up to 204 valves
(clams) or 132 plates (chitins) and the error will not exceed 10%. On the other hand, for a collection of 100
valves, one can estimate 100 clams (i.e., use the XNI
method) so long as the sampling domain 2294; for 100
chitin plates the sampling domain would have to equal
at least 455 for the same to hold true.
The slopes of the lines in figure 5 give the
general rule for the two types of organisms.
One can estimate the number of unique clams
as being equal to the number of valves in the
collection (the XNI approach), with no greater than 10% error, as long as the number of
valves in the collection is smaller than about
34% of the total number of clams in the sampling domain. Likewise, one can estimate the
number of unique chitins as being equal to
the number of plates in the collection, with
no greater than 10%error, as long as the number of plates in the collection is smaller than
about 22% of the total number of chitins in
the sampling domain. Of course, this general
relationship could also be determined for organisms with other numbers of body parts,
although we have not done so here.
In general, the 10% error lines have lesser
slopes for organisms with greater numbers of
256
NORMAN L. GILINSKY AND J BRET BENNINGTON
body parts, meaning that the curves relating
numbers of parts sampled versus number of
individuals (such as those in fig. 4) recline
more and more for multiple-part organisms.
This fact has an important implication from
the standpoint of obtaining relative abundances of unique individuals from the fossil
record. As already mentioned, given 100 clam
valves, one can apply the XNI approach to
clam valves without more than 10% error so
long as the size of the sampling domain is at
least as large as 294. A sampling domain of
this size (or larger) is probably reasonable
much of the time for clams. But for the same
number of chitin plates, the XNI approach is
within 10% error only if the size of the sampling domain is at least as large as 455. Since
many cases can be imagined where the sampling domain requirements for applying the
XNI might be met for clams but not for chitins, it follows that the method used for estimating the numbers of clams and chitins
(and other organisms as well, of course) must
depend on the kind of organism being considered. That is, using the XNI (for instance)
for all of the different kinds of organisms in
an assemblage might result in invalid assessments of relative frequencies of unique individuals. We do not attempt a detailed recipe
for making unbiased assessments of relative
frequencies here, but we do wish to mention
that such assessments are not straightforward. Holtzman (1979) deals with the problem of estimating relative frequencies from
another perspective.
In actual practice, one might be able to guess
whether the sample size is small relative to
the number of individuals in the sampling
domain by considering the approximate ratio
of parts sampled to the total number of parts
that could have been sampled. If the recovery
of identifiable parts is modest relative to the
density of a fossiliferous bed, then one can
usually assume that the sampling domain is
large relative to the size of the sample, and
estimate the number of unique individuals as
being approximately equal to the number of
parts.
More Good News: The Relationship to Disturbance.-Another hint to the effective size of
the sampling domain and, therefore, which
curve in figure 4A to use (for clams), comes
from a consideration of the amount of disturbance that is in evidence in the sediment
from which the fossils are obtained. Consider
a case where the valves have been disturbed
and where considerable local transport has
occurred. Some of the valves preserved in the
sample were probably transported into the
collection site from elsewhere within the
habitat, other valves that originated within
the collection site were probably transported
out. In a case such as this, if the collection
site was small relative to the average distance
of transport, then the chances are small that
any valve would remain near its counterpart,
so the size of the sampling domain would
almost certainly be approximately equal to
the total number of valves preserved at the
collection site. It follows, therefore, that a reasonable inference would be simply to count
each valve as a single individual. In general,
the more disturbance there is, the higher the
probability that each valve will represent a
unique individual at a given collection site.
However we caution, as we did earlier, that
the number of individuals of a species that
are represented in an assemblage is not the
same thing as the number of individuals in
the once-living community, except under
special circumstances.
In contrast, now consider a case where the
fossils were very little disturbed. In this case,
one can reasonably assume that most valves
and valve counterparts are preserved somewhere at the collection site being examined
and that the size of the sampling domain is
smaller than in the disturbed case. If the sampling domain is small, it follows that one of
the lower curves in figure 4A apply, and that
only the first few valves retrieved from the
rocks can be regarded as unique individuals.
With a small domain, it may not take very
many valves to begin to move into the concave down region of the curve (see the curves
for 50 and 100 in fig. 4A, for instance) where,
in the limit of all valves in the sampling domain becoming part of the collected sample,
the number of individuals will approach the
MNI. An extreme case would be where a living community were suddenly buried by a
single event of sedimentation, and where all
COUNTING UNIQUE INDIVIDUALS
,
of the clams were preserved in an articulated
state. Here, the number of individuals in the
collection is the MNI, as the theory predicts.
Again, while it is not possible to choose the
appropriate curve from figure 4A exactly, it
is possible to make an informed choice based
upon independent criteria (in this case, an
assessment of the amount of disturbance to
which the fossils have been subject), and to
choose which approach-the MNI or XNIwould be the more appropriate. To obtain an
assessment of disturbance one can use the
many criteria outlined, for example, in Brett
and Baird (1986).
Although it is beyond the scope of the present paper, we mention that an empirical test
of some of the principles developed here
might be practical. For instance, one could
characterize the amount of disturbance to
which a particular assemblage was subjected
by using the criteria described in Brett and
Baird (1986), avoiding the use of the test organisms (say, the clams) in the characterization. Then the clams could be exhaustively
sampled from the sediment, and the number
of unique individuals that are actually there
could be determined by going through the
laborious process of comparing each valve to
every other valve. If the amount of disturbance determined by criteria independent of
the test organisms (the clams) was great, the
prediction would be that the size of the sampling domain was large, and that the number
of unique individuals would best be estimated by making use of one of the higher curves
in figure 4A. With considerable disturbance,
the relationship between number of individuals and number of valves might be linear,
or nearly so. Conversely, low-disturbance assemblages would predict that the size of the
sampling domain was small, and that the
number of unique individuals would best be
estimated using the lower curves in figure 4A,
where the number of unique individuals rapidly approaches the MNI value with increasing sample size. No-disturbance environments (which may not exist in a strict sense)
should find many bivalves still articulated.
Clearly, in this last case, the number of unique
individuals is exactly equal to the MNI. In
effect, by designing the experiment thought-
257
fully, using a set of assemblages characterized
by magnitude of disturbance, in each case
sampling exhaustively (from a specified volume of sediment or rock) and matching valves,
one might be able to develop general, disturbance-related criteria for choosing the right
curve. Of course this test would strictly apply
only to the test organism (bivalves in the case
described), but it would at least give a hint
regarding how far one might be able to go in
applying the theory developed here.
Summary and Conclusions
Paleoecology has long sought to apply ecological concepts to fossil populations, an aim
that often entails estimates of the sizes of fossil populations. Such estimates cannot be accurately obtained for most fossil assemblages
because they are time-averaged. But even
where time-averaging does not distort the
relative sizes of populations, efforts to count
biological individuals have been hampered
by the problem of dealing with multiple body
parts. The two general approaches to the
problem that have been attempted previously
define only the extremes in the range of possible ways of counting individuals. The work
undertaken here places the MNI and XNI approaches into the context of a general mathematical theory of counting multi-part individuals. The theory shows that the
relationship between number of individuals
and number of body parts is nonlinear, with
the shape of the curve relating the two variables being a function of yet a third variable:
the size of the sampling domain. The good
news is that the expected number of unique
individuals can be rigorously obtained for a
specified size of sampling domain. The bad
news is that the size of the sampling domain
can never be rigorously specified. It is for
precisely this reason that the best that has
been done thus far is to estimate numbers of
individuals using the XNI and MNI approaches. The XNI approach effectively assumes that the sampling domain is infinitely
large, or at least that it is much larger than
the number of body parts in the fossil assemblage. And the MNI approach effectively assumes that the sampling domain is the smallest possible, for the assemblage under
258
NORMAN L. GILINSKY A N D J BRET BENNINGTON
consideration, for only at the smallest sampling domain does the MNI method give estimates that even approach the correct value.
So the theory developed here functions to
clarify the nature of the problem of counting
unique individuals, and to pinpoint why we
have not been able to progress much further
than the MNI and XNI methods.
The counting theory does not provide a
recipe for counting unique individuals, but
it does give some hints regarding more accurate counts. For instance, if the actual sample of valves (or plates, etc.) is much smaller
than the total number of individuals that were
preserved in the sampling area (and one can
often tell simply by collecting from the rock),
then the theory prescribes counting each valve
as a unique individual (the sample falls in the
near-linear part of one of the curves in fig.
4). Conversely, if the size of the sample is
large relative to the size of the sampling domain, e.g., if sampling is exhaustive, then the
theory tells us that counting each valve as an
individual is not appropriate; the sample
probably falls out on the flattened part of the
curve.
One of the roles of taphonomy is to define
the limits of resolution of the paleontological
record, for instance, to define just how closely
we might be able to come to obtaining that
coveted ecological snapshot. Counting unique
individuals and thereby obtaining that "class
portrait" for a fossil community might seem
to be one of the minimal prerequisites for
obtaining such a snapshot. But the theory developed here demonstrates that the problem
of the size of the sampling domain places real
taphonomic limits on our ability to resolve
ancient communities. These limits might be
breached, as discussed above, by making certain realistic assumptions regarding the sampling domain. But this number, by its very
nature, can never be rigorously obtained.
Acknowledgments
We thank R. K. Bambach for discussing the
topic of this paper with us. We also thank H.
Cummins, A. Miller, C. Peterson, and an
anonymous reviewer (twice) for their constructive criticisms.
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