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Estimation of the most likely number of individuals from commingled human skeletal remains.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 125:138 –151 (2004)
Estimation of the Most Likely Number of Individuals
From Commingled Human Skeletal Remains
Bradley J. Adams1 and Lyle W. Konigsberg2*
1
2
Central Identification Laboratory, Hickam Air Force Base, Oahu, Hawaii 96853
Department of Anthropology, University of Tennessee, Knoxville, Tennessee 37996-0720
KEY WORDS
Lincoln index; MNI; forensic anthropology; skeletal biology; pair-matching;
recovery probability
ABSTRACT
This study examines quantification
techniques applicable to human skeletal remains, and
in particular the Lincoln index (LI), the minimum number of individuals (MNI), and what we refer to as the
most likely number of individuals (MLNI), which is a
modification of the LI by Chapman ([1951] Univ. Calif.
Publ. Stat. 1:131–159). As part of the study, a test of
pair-matching between commingled homologous elements, e.g., right and left femora, was performed based
on gross morphology. The results show that pair-matching can be accurately performed, and that the MLNI is
a useful technique for dealing with well-preserved commingled remains recovered from archaeological excavations and/or forensic investigations. Our results show
that it is potentially misleading to draw population
A wide variety of quantification techniques are
available to the zooarchaeologist. However, physical
anthropologists tend to focus almost exclusively on
calculation of the minimum number of individuals,
or MNI, as the best way of quantifying the number
of individuals represented by commingled human
remains (e.g., Ubelaker, 1974; Willey, 1990). The
Lincoln index (LI) has been effectively used to quantify zooarchaeological samples and should be directly applicable to the analysis of commingled human remains. The key difference between these two
quantification techniques is that the LI estimates
the original number of individuals represented by
the osteological assemblage, while the MNI only estimates the number of individuals represented by
the recovered assemblage. In cases involving significant taphonomic loss, the MNI may provide misleading number estimates. The LI, on the other
hand, will provide a more accurate estimate of the
original population size useful for paleodemographic
or forensic purposes. In the following, we discuss the
MNI, variants of the LI, and methods that can be
applied to multiple paired elements to estimate the
most likely number of individuals (MLNI) to have
generated a commingled human skeletal assemblage.
©
2004 WILEY-LISS, INC.
conclusions based on the MNI, except in instances
where recovery is near 100%. The MLNI was found to be
the best method to compensate for the potential underestimates of the MNI and potential bias in the original
LI estimates resulting from small sample sizes. We
demonstrate the use of MLNI in estimating the number
of individuals from Lodge 21 at the Larson site, a late
protohistoric structure at which the inhabitants were
massacred and subsequently had their skeletal elements commingled by further taphonomic processes. We
also show how to calculate estimates and standard errors for the recovery probabilities of skeletal elements.
Am J Phys Anthropol 125:138 –151, 2004.
©
2004 Wiley-Liss, Inc.
METHODS FOR QUANTIFYING
NUMBERS OF INDIVIDUALS
Minimum number of individuals
The MNI is arguably the most popular method of
quantification in any type of commingled osteological analysis. Many researchers credit White (1953)
with its initial use for abundance studies in archaeology. For interpreting population size from a skeletal assemblage, the MNI (as the name suggests)
presents the minimum estimate for the number of
individuals that contributed to the sample. In order
to deal with fragmentary remains, specific segments
of an element (e.g., distal femur) can be used for the
Grant sponsor: National Science Foundation; Grant numbers:
GS837, GS1635; Grant sponsor: National Geographic Society; Grant
number: 699,912.
*Correspondence to: Lyle Konigsberg, Department of Anthropology,
University of Tennessee, 252 South Stadium Hall, Knoxville, TN
37996-0720. E-mail: lylek@utk.edu
Received 8 November 2002; accepted 4 July 2003.
DOI 10.1002/ajpa.10381
Published online 12 January 2004 in Wiley InterScience (www.
interscience.wiley.com).
QUANTIFICATION OF HUMAN REMAINS
calculation of the MNI. Every fragment must share
a unique landmark to ensure that fragments of the
same element of the same individual are not counted
as two distinct individuals. The basic principle of an
MNI estimate is to avoid counting the same individual twice.
There has been considerable debate in the zooarchaeological literature regarding the MNI and other
quantification techniques (e.g., Grayson, 1973, 1978,
1979, 1981, 1984). Many criticisms of the MNI for
the analysis of zooarchaeological remains (and the
recommended alternatives) are not germane to the
study of human remains (e.g., differential transport
and butchery of body parts). In most instances, human remains would have been initially deposited at
a site in their entirety, rather than in parts.
Regardless of the target population being studied,
an important shortcoming of the MNI is that in
many instances it will not provide an accurate estimate of the original number of individuals. In their
discussion of faunal remains, Fieller and Turner
(1982, p. 56) wrote that “the very presence of unmatched bones indicates that the MNI estimate is
necessarily an underestimate of the number comprising the death assemblage.” The MNI simply
states how many individuals would have been necessary to provide the recovered skeletal elements,
but says nothing about the original death population. As we will show, the MNI varies depending on
the recovery probability (i.e., the percentage of recovered elements), and for this reason it may be of
limited value for the quantification of osteological
assemblages.
Calculation of MNI
There are three main variations for the calculation of the MNI, where L signifies the number of left
bones, R signifies the number of right bones, and P
signifies the number of pairs. The MNI estimators
are:
Max 共L, R兲
(1)
共L ⫹ R兲/2
(2)
L ⫹ R ⫺ P
(3)
The most common variant of the MNI used for the
analysis of human remains is the Max(L, R) method.
This value is obtained by sorting elements into lefts
and rights, and then taking the greatest number as
the estimate. It is equivalent to assuming that all of
the less frequently observed bones are paired with
the more frequently observed bones, and subsequently provides a rather unlikely estimate. The
variant (L ⫹ R)/2 will usually provide the most minimal counts. It is an average for a paired element,
and unless the lefts and rights are equal, the value
will always be less than the maximum side. The L ⫹
R ⫺ P method usually provides a higher estimate
than the other two options, because unpaired bones
from different sides are assumed to come from dif-
139
ferent individuals. Horton (1984) referred to Equation (3) as the Grand minimum total.
We will not deal extensively with MNI calculations in this paper. Under a simple model where
each bone has a probability r of being recovered
(which we refer to as the recovery probability) and
there are N individuals, we would expect to recover
Nr lefts, Nr rights, and Nr2 paired bones. Equations
(1) and (2) then underestimate the actual N by (r ⫺
1) ⫻ 100%, while Equation (3) underestimates the
true N by (r(2 ⫺ r) ⫺ 1) ⫻ 100%. For example, when
the recovery probability equals 0.7, Equations (1)
and (2) underestimate the true N by 30%, while
Equation (3) underestimates it by 9%. Equation (3)
would appear preferable to Equations (1) and (2),
but as we will see below, there are better methods
than the MNI for estimating the actual number of
individuals when considering paired elements. Further, because the recovery probability is unknown in
most osteological assemblages, we have no real way
of determining to what extent MNI methods underestimate the true N. This general trend in the MNI’s
behavior indicates the need for caution when using
MNI to quantify an abundance of human remains.
Lincoln/Petersen index (LI)
The Lincoln index (LI) has been almost exclusively applied to zooarchaeological remains and population studies of living animals (Allen and Guy,
1984; Borchers et al., 2002; Chapman, 1951; Chase
and Hagaman, 1987; Fieller and Turner, 1982; Horton, 1984; Klein and Cruz-Uribe, 1984; Plug, 1984;
Ringrose, 1993; Seber, 1973; Turner, 1980, 1983,
1984; Turner and Fieller, 1985; Wolter, 1990; and
many others). The LI was first developed for population studies of living animals based on capturerecapture techniques, and was later adapted for application to zooarchaeological faunal assemblages.
Fishery biologists refer to it as “Petersen’s method,”
which was used by C.G.J. Petersen in 1889, while
ornithologists and mammalogists call it the “Lincoln
index” due to its use by F.C. Lincoln in 1930 (LeCren,
1965). Although the distinction may be trivial, it is
important to be aware that both names refer to the
same technique. The same logic as used in the LI is
also applied to human census data in order to estimate the number of missed individuals from a large
census. In this setting, the method is referred to as
“dual-system estimation,” and is used when there is
both a census and a “post-enumeration survey”
(Freedman, 1991).
The key reason to use the LI for skeletal remains
is that accurate estimates of the original population
can be derived from samples in which taphonomic
biasing has occurred. The theoretical basis of the
formula is that one is studying populations in which
all of the animals (living or dead) need not or cannot
be observed. This is of particular utility with archaeological or paleoanthropological samples in which
site sampling takes place, or in forensic situations
140
B.J. ADAMS AND L.W. KONIGSBERG
when remains may have been exposed to taphonomic processes.
The LI, as an inherent feature of its derivation,
accounts for some degree of data loss, but it is important that the data loss occurs randomly. Ringrose
(1993) stated that if both sides of an element were
initially deposited on a site or transported to it, and
neither was subsequently transported away, then
the LI is an effective tool. Ringrose (1993, p. 129)
concluded that this technique “is perhaps the best
way we have of trying to get back to the Death
Assemblage.” A similar conclusion was drawn by
Fieller and Turner (1982, p. 59), who wrote that:
“if matches can be made and if bias in selection has not operated,
then the method is capable of reconstructing the original death
assemblage size even though a number of selection processes may
have operated. This is of particular importance to the archaeologist dealing with material which has been subjected to selection
during initial deposition, preservation and eventual recovery.”
Of course, estimates will be more precise with
high levels of recovery, but it is extremely important
to realize that this technique does account for random data loss.
Calculation of the LI and MLNI
from single elements
When working with skeletal elements, the calculation of the Lincoln index requires a determination
of element pair-matching. Pair-matching involves
the comparison of right and left elements to decide if
they are from a single individual. In theory, any
paired element in the body could be used for the
calculation of the LI. In practice, though, particular
parts of the skeleton will be more useful than others.
For example, teeth may not be appropriate for this
technique due to the fact that asymmetric wear and
antemortem loss resulting from decay or trauma
could confound the process. Important criteria to
consider when choosing appropriate skeletal elements are element size, the presence of distinct morphological traits, the potential for age and sex determination, and the likelihood of survival.
Numerous researchers (e.g., Brain, 1976; Lyman,
1993, 1994; Waldron, 1987; Willey et al., 1997) found
that the larger and denser bone elements have a
better chance of survival than the delicate elements.
With respect to human remains, some of the best
bones to include would be the femur, tibia, humerus,
and os coxa due to their morphology, survivability,
and/or potential for providing significant biological
information.
The bones from one side of the skeleton, e.g., left
(L), are analogous to the initial capture stage in a
capture-recapture study. The bones from the other
side of the skeleton, e.g., right (R), are analogous to
the second stage of the capture-recapture study. It is
irrelevant which side of the skeleton is treated as
the initial “catch.” The number of elements which
can be matched from the right and left sides (P),
indicating that they originated from the same indi-
TABLE 1. Contingency table representations for sampling of
paired elements1
a. Sampling layout assuming known population size
Sampling
of lefts
⫹
⫺
Total
Sampling of rights
⫹
⫺
Total
P
R⫺P
R
L⫺P
N⫺L⫺R⫹P
N⫺R
L
N⫺L
N
b. Example discussed in text
Sampling of rights
Sampling
of lefts
⫹
⫺
Total
⫹
⫺
Total
12
8
20
6
4
10
18
12
30
c. MNI approach based on observed numbers of lefts, rights,
and pairs
Sampling
of lefts
⫹
⫺
Total
Sampling of rights
⫹
⫺
Total
P
R⫺P
R
L⫺P
ⱖ0
ⱖL ⫺ P
L
ⱖR ⫺ P
ⱖL ⫹ R ⫺ P
d. MNI approach based on observed lefts and rights (with L ⱖ
R), ignoring the existence of pairs
Sampling of rights
Sampling
of lefts
⫹
⫺
Total
⫹
⫺
Total
R
0
R
L⫺R
ⱖ0
ⱖL ⫺ R
L
ⱖ0
ⱖL
1
(L,number of lefts; R, number of rights; P, number of pairs; N,
population size. ⫹ and ⫺, observed (sampled) and unobserved
(not sampled), respectively.
vidual, is analogous to the recapture of initially
tagged animals. An estimate of the original death
assemblage N̂ represented by the skeletal elements
is:
N̂ ⫽
L ⫻ R
,
P
(4)
In the event that no pairs are discovered, Fieller and
Turner (1982) and Turner (1983, 1984) suggested
that the formula should be modified to:
N̂ ⫽ 共L ⫹ 1兲共R ⫹ 1兲.
(5)
While the typical presentation of the LI is in terms
of capture-recapture studies, we will use a contingency table presentation (Cormack, 1989) focused on
the specific example of estimating the most likely
number of individuals from paired elements. In a
hypothetical death assemblage of N individuals, a
number of right (R) and a number of left (L) elements of a particular bone will be recovered. Some of
the elements can be matched and suggested to have
derived from single individuals, where the number
of such left with right pairings is P. Note that
max(L, R) ⱖ P. If we know the original size of the
population, a 2 ⫻ 2 contingency table can be constructed, as shown in Table 1a. As an example, if the
recovered sample contained 18 left and 20 right
QUANTIFICATION OF HUMAN REMAINS
141
bones, of which 12 are paired and the original population consisted of 30 individuals, the contingency
table in Table 1b results. In Table 1b, the marginal
totals present the numbers of left and right bones
that were sampled or observed, and the numbers
that were not sampled or observed. The fourth cell in
the table gives the number of individuals from whom
neither a right nor a left was sampled, and who were
thus unobserved. If we assume independence for
sampling of rights and lefts, then the expected number of paired bones from Table 1a is:
E共P兲 ⫽
冉 冊冉 冊
R
N
L
N.
N
(6)
Replacing the number of expected pairs with the
observed number and solving for N gives the Lincoln
index shown in Equation (4). Chapman (1951) referred to this estimator as N0, and noted that as it
must take an integer value, the correct estimate is
given by the integer that satisfies the following inequality:
L ⫻ R
L ⫻ R
⫺ 1 ⬍ N0 ⱕ
,
P
P
(7)
where N0 is the integer value used in place of N̂. In
an alternative scheme, where we only wish to find
the minimum number of individuals allowing for
pairs, we assume that N ⫺ L ⫺ R ⫹ P is at least zero,
as shown in Table 1c. The total (L ⫹ R ⫺ P) is then
the MNI based on observed left, right, and paired
bones (as in Equation (3)). Curiously, if we ignore
the existence of pairs, then this is equivalent to
saying that all of the bones from the less frequently
occurring side must be paired with all of the bones
from the more frequently occurring side. In this
case, P ⫽ min(L, R), so that L ⫹ R ⫺ P becomes
max(L, R) ⫹ min(L, R) ⫺ min(L, R) ⫽ max(L, R) (as
in Equation (1)). The case where there are more lefts
than rights is shown in Table 1d for this final setting, where the MNI is simply L.
The LI was shown to be the best asymptotically
normal estimator for N, given L, R, and P (Chapman, 1951). However, under conditions of small
sample size, the index is well-known to produce biased estimates (Bailey, 1951; Chapman, 1951; Robson and Regier, 1964). Chapman (1951) and Bailey
(1951) both proposed modifications to the LI, but as
Robson and Regier (1964) emphasized, these modified estimators have substantial negative biases
when the number of recovered pairs is low. The
variant by Chapman (1951) of the Lincoln index is:
N* ⫽
共L ⫹ 1兲共R ⫹ 1兲
⫺ 1 ,
P ⫹ 1
(8)
where the symbol   means “floor” (i.e., truncation to
the integer value). The estimator of Chapman (1951)
is the value we will use to estimate the “most likely
number of individuals” (MLNI).
In order to demonstrate the relationship between
the original LI (Equation (4)) and the estimator by
Fig. 1. Comparison of percent bias for N̂ (original Lincoln
index) and N* (modification of LI by Chapman, 1951) against
recovery probability when true N is equal to 50.
Chapman (1951) (Equation (8)), Figure 1 shows a
plot of the expected percent bias for each estimator.
The plot shows the percent bias against recovery
probability for a case where the true N is 50, and
under the simplifying assumption that L ⫽ R. The
graph was constructed assuming that the probability function for P conditional on L (and R, which is
equal to L) is specified by the hypergeometric distribution. Figure 1 shows that at very low recovery
probabilities, the original LI (N̂) first underestimates the true N, and then overestimates the true
N. In contrast, the estimator of Chapman (1951)
(N*) initially underestimates the true N, but then
rapidly becomes an unbiased estimator. Figure 1
demonstrates that the LI estimator of Chapman
(1951) (Equation (8)) outperforms the traditional index, and given the simplicity of its calculation, it is
preferable to both the traditional LI and any MNIbased approach. However, the estimator by Chapman (1951) is not immune to bias. Figure 2 shows
the expectation for the estimator of Chapman (1951)
across different population sizes (N ⫽ 15, 30, and 50)
for various probabilities of bone recovery and for
various expected numbers of pairs. The salient point
in these graphs is that this estimator is unbiased,
provided that the recovery probability is 0.5 or
greater and/or there are five or more expected pairs.
More correctly, the estimator is unbiased if L ⫹ R ⱖ
N. Robson and Regier (1964) pointed out that the
estimator of Chapman (1951) was unbiased if L ⫹
R ⱖ N, but as neither N nor the recovery probability
is known in applications of the LI, it is necessary to
rely on some other criterion. Robson and Regier
(1964, p. 217) suggested that if the number of observed pairs is 7 or more, then “with 95 percent
confidence the bias is negligible.” This is born out in
Figure 2, where there is no appreciable bias when
there are 5 or more pairs (the lower number of 5 as
opposed to 7 is again due to our assumption that L ⫽
R). The results of Robson and Regier (1964) indicate
142
B.J. ADAMS AND L.W. KONIGSBERG
for N, which is used to specify exact intervals on
estimates, and which allows us to extend the estimator of Chapman (1951) to multiple elements. To
find the probability function for N, we must write
the probability of N conditional on L, R, and P. This
probability can be found by combining the probability of getting P number of pairs conditional on L, R,
and N with an uninformative prior for the recovery
probability r. The probability of getting P number of
pairs conditional on L, R, and N is found using the
hypergeometric distribution (Burford, 1967; Evans
et al., 1993; Feller, 1950; Freund, 1992; Harris and
Stocker, 1998; Hays, 1988; Hogg and Tanis, 1983;
Larson, 1969; Lee, 1997; Mendenhall et al., 1990;
Mosteller et al., 1970; Neter et al., 1982; Nolan and
Speed, 2000; Sokal and Rohlf, 1981; Wadsworth and
Bryan, 1974; Weiss, 1961; Zehna, 1969), while the
uninformative prior for the recovery probability is a
uniform distribution between zero and one, specified
by a beta(1,1) distribution (Evans et al., 1993;
Freund, 1992; Harris and Stocker, 1998; Hogg and
Tanis, 1983; Larson, 1969; Lee, 1997; Mendenhall et
al., 1990; Mosteller and Tukey, 1977; Nolan and
Speed, 2000; Wadsworth and Bryan, 1974). Using
results from Roberts (1967, Equation 2.11; see also
Lee, 1997, p. 200 –202), we write the probability of N
conditional on L, R, and P as:
Pr共N兩L,R,P兲
⫽
Fig. 2. Comparison of percent bias for modification by Chapman (1951) of LI graphed against recovery probability and expected number of pairs with N ⫽ 15, 30, and 50.
that, provided there are at least 7 pairs present in
an assemblage, the variant by Chapman (1951) of
the LI (Equation (8), above) will have only a trivial
negative sampling bias.
In addition to estimates of the number of individuals, it would be useful to form confidence intervals
around these estimates. Seber (1973) discussed the
calculation of the variance of N, which can be used to
form approximate confidence intervals. However,
the distribution for estimates of N is generally asymmetric, and it is also a discrete distribution, making
the confidence intervals based on variance estimates
misleading. The main advantage of the original LI N̂
and of the modified form by Chapman (1951) of the
LI (N*) is that they can be easily and quickly calculated. If correct calculation of confidence intervals is
at issue, then it is far better to use a probabilistic
model, as we show below.
Estimation of the MLNI from the complete
probability function
Here, we derive the estimator of Chapman (1951)
to use as the MLNI. The derivation is necessary in
order to generate the complete probability function
L!R!共N ⫺ R兲!共N ⫺ L兲!
共L ⫺ P兲!共R ⫺ P兲!共N ⫺ L ⫺ R ⫹ P兲!
共N ⫹ 1兲!共P ⫺ 1兲!
⫽ Pr共P兩L,R,N兲 ⫻
(9)
P
,
共N ⫹ 1兲
where Pr(P兩L,R,N) is the probability from the hypergeometric distribution of getting P pairs, upon drawing L lefts and R rights from N individuals. Equation (9) is easy to implement because many
statistical packages as well as some spreadsheets
include the hypergeometric probability function. We
implemented Equation (9) in an Excel™ spreadsheet (available from http://konig.la.utk.edu/
MLNI.html).
To find the maximum likelihood estimator from
(9), we write the ratio of adjacent probabilities as:
共N ⫺ R兲共N ⫺ L兲
Pr共N兩L,R,P兲
⫽
.
Pr共N ⫺ 1兩L,R,P兲
共N ⫺ L ⫺ R ⫹ P兲共N ⫹ 1兲
(10)
For values of N less than the maximum likelihood
estimate, this ratio will be greater than one, while
for values of N greater than the maximum likelihood
estimate, this ratio will be less than one. We can
solve the inequality:
共N ⫺ R兲共N ⫺ L兲 ⱖ 共N ⫺ L ⫺ R ⫹ P兲共N ⫹ 1兲
(11)
143
QUANTIFICATION OF HUMAN REMAINS
TABLE 2. Probability values from Equation (9) and cumulative
probabilities, for L ⫽ 21, R ⫽ 15, and P ⫽ 11, tabled
at various N values
N
P
Cumulative
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
0.0457
0.0930
0.1196
0.1251
0.1167
0.1017
0.0847
0.0686
0.0545
0.0427
0.0332
0.0257
0.0198
0.0153
0.0118
0.0091
0.0071
0.0055
0.0043
0.0033
0.0026
0.0020
0.0016
0.0013
0.0010
0.0008
0.0457
0.1386
0.2582
0.3833
0.5000
0.6017
0.6864
0.7550
0.8094
0.8521
0.8853
0.9111
0.9309
0.9462
0.9580
0.9671
0.9742
0.9797
0.9839
0.9872
0.9898
0.9919
0.9935
0.9948
0.9958
0.9966
to find that the maximum likelihood estimate occurs
at:
共L ⫹ 1兲共R ⫹ 1兲
⫺ 1 ,
P ⫹ 1
identical with Equation (8). We refer to Equation (8)
as the MLNI because it gives the value of N with the
highest posterior probability after starting from a
uniform prior for N and a uniform prior for the
recovery probability.
Table 2 shows the probability values across N
from Equation (9) for the case where L ⫽ 21, R ⫽ 15,
and P ⫽ 11. These probability values were calculated in the Microsoft Excel™ spreadsheet at http://
konig.la.utk.edu/MLNI.html, and are shown from
N ⫽ 25 (the MNI) up to N ⫽ 50. Table 2 also contains
the cumulative sum of the probability function for N,
from which we can establish intervals around the
maximum likelihood estimate of N. Because N follows a discrete distribution, it is not usually possible
to give customary confidence intervals, such as 95%
intervals. We can, however, determine the “highestdensity region” (HDR) (see Lee 1997, p. 49), which
we prefer to classic confidence intervals. The concept
of an HDR is best treated graphically, which we do
in Figure 3. At the top of Figure 3 we draw the
frequency histogram for N from the probabilities in
Table 2. We then move a horizontal line (dashed line
in Fig. 3) down through the histogram until it comes
to rest on a block of our choosing, and we hatch the
histogram columns that terminate above this line.
By doing so, we find all values of N at which the
probability is larger than the “baseline” value established by our horizontal line (hence, the “highest
density region”). This region (Fig. 3, top) includes
70.9% of the distribution and runs from N ⫽ 26 to
N ⫽ 32, inclusive. The percentage in the HDR is
found as F(32) ⫺ F(25), where F() is the cumulative
density up to and including the number in parentheses. A 70.9% HDR is rather small, so we drop the
horizontal line even further, until we obtain Figure
3 (bottom). This second HDR runs from N ⫽ 25 (the
MNI) to N ⫽ 38, inclusive, and contains 94.6% of the
distribution (equal to F(38) ⫺ F(24), where F(24) ⫽
0). From Figure 3, or the values presented in Table
2, we can see that the maximum likelihood estimate
is at 28 individuals, since this is the highest part of
the histogram. This is also the value we would obtain using Equations (7) or (8).
As an alternative to direct calculation of the distribution for N (Equation (9)), George and Robert
(1992) and Manly (1997) discussed Markov chain
Monte Carlo (MCMC) estimators in mark-recapture
settings. The edited volume by Gilks et al. (1996)
gives a general account of MCMC methods. In addition to the spreadsheet calculations described above,
we used a Metropolis algorithm (Gelman et al.,
1995, p. 323–324) to sample from the distribution for
N. We used Equation (10) to calculate the “acceptance probability” for proposed “jumps” to N ⫺ 1 or
N ⫹ 1 in the Monte Carlo chain. This MCMC
progresses rather slowly, so we cannot recommend it
over the exact analytical calculations from Equation
(9).
Estimating the recovery probability
In addition to estimating N, it may be desirable to
estimate the “recovery probability,” or “r.” The recovery probability (r) is defined here as the probability that a bone will make its way into the sample
being analyzed. The recovery probability can be affected by myriad factors, including mortuary practice, taphonomy, and archaeological recovery methods. We use the term here simply to represent the
final probability of recovery in a sample. In the
mark-recapture setting, the recovery probability is
usually referred to as the “probability of capture” or
“catchability.” Assuming that the probability of recovering a left bone is equal to that of recovering a
right, the maximum likelihood estimate of r is:
r̂ ⫽
2P
,
L ⫹ R
(12)
and the asymptotic standard error of the estimate is:
冋
册
共r̂ ⫺ 1兲2 共r̂ ⫺ 2兲2 r̂2
s.e.共r̂兲 ⫽ 2
r̂ 共L ⫹ R兲共3 ⫺ 2r̂兲 ⫹ 2P共2
⫺ 6r̂ ⫹ 3r̂2)
1
2
(13)
As an example, with L ⫽ 21, R ⫽ 15, and P ⫽ 11,
Equation (12) gives an estimated recovery probability of 0.6111, and Equation (13) gives a standard
error of 0.0957. The calculations for the analytical
estimate and standard error of the recovery proba-
144
B.J. ADAMS AND L.W. KONIGSBERG
Fig. 3. Graphical representation of calculation of “highest-density regions” (HDR) for N when L ⫽ 21, R ⫽ 15, and P ⫽ 11. Top:
70.9% HDR. Bottom: 94.6% HDR.
bility are available in the spreadsheet we provide at
http://konig.la.utk.edu/MLNI.html, and the data in
the spreadsheet can be edited to solve different problems.
Extension of the MLNI to multiple elements
The methods we described above can be easily
extended to cover multiple paired elements. In the
mark-recapture literature, such an application is
most analogous to what is known as a mark-resight
study. In mark-resight studies, animals are first
marked (generally with a radio transmitter), and
then at later times, animals are tallied in a survey
as to whether or not they have a radio transmitter
(White and Garrott, 1990). The multiple resightings
are analogous to having multiple elements available
for estimating N. In the wildlife literature, N is
estimated using a joint hypergeometric estimator
(JHE) (Bartmann et al., 1987; Neal et al., 1993;
White, 1993, White 1996; White and Garrott, 1990).
In our setting, this means that we use the probability distribution for N (Equation (9)) across each element, multiply the probabilities across elements
for each value of N, and then normalize the probabilities so that they sum to 1.0 (we give a specific
example from the Larson site, below). The assumption in making this calculation is that the recovery
of, say, femora is independent of the recovery of, say,
tibiae. White (1996) wrote and made available the
program NOREMARK (see http://www.cnr.colostate.
edu/⬃gwhite/software.html) which fits the JHE, and
his program can also be used for the single-element
estimation problem described above. His likelihood
calculation is from the product rule applied to inde-
QUANTIFICATION OF HUMAN REMAINS
pendent hypergeometric distributions, and is consequently a generalization of the original Lincoln index. Our calculations for multiple elements are
available in a spreadsheet from http://konig.la.utk.edu/MLNI.html, and serve as a generalization of
the estimator of Chapman (1951) to the multiple
element case.
The use of multiple elements to estimate N introduces additional complications in the analysis, particularly because the assumption of independence in
recovery cannot be tested. It is possible to use a
likelihood ratio chi-square test to test for consistency of the estimator (see http://konig.la.utk.edu/
MLNI.html), but this test assumes independence in
recovery of elements. Provided the likelihood ratio
chi-square test is not significant, the multiple element estimator stands a reasonable chance of being
unbiased. However, if the assumption of independence is violated, then the HDR calculation will
provide regions that are too narrow.
APPLICATIONS OF MLNI AND RECOVERY
PROBABILITY ESTIMATION
Test for the accuracy of pair-matching
As accurate pair-matching is critical in the estimation of the LI, MLNI, and recovery probabilities,
it was necessary to carefully examine this facet of
the technique. In order to examine the reliability of
pair-matching bones of the same element type (e.g.,
right and left femora), an experimental test was
performed using an archaeologically recovered assemblage of well-preserved human skeletal remains.
The observation of gross morphological characteristics was the only criterion used in the pair-matching
process. It should be noted that only pair-matching
of the same element was examined. Morphological
sorting between element types (e.g., associating a
femur with a humerus) may be feasible when only a
small number of individuals are commingled and
there is significant size variation between individuals, but in more complicated situations this technique can be highly subjective and problematic. It
should be noted that morphological sorting between
different element types is not necessary for the calculation of the MLNI or LI.
In order to perform a verifiable test of gross pairmatching abilities, a random sample was drawn
from an archaeological skeletal collection housed at
the University of Tennessee, Knoxville. Larson
Cemetery (39WW2) is an Arikara site associated
with the Postcontact Variant of the Coalescent Tradition (Owsley, 1994; Owsley et al., 1977). The
Arikara were a Caddoan-speaking tribe who constructed dome-shaped, earth-covered lodges (Bass
and Rucker, 1976; Catlin, 1989; Denig, 1961). Initial
occupancy at Larson is suspected to have begun
about A.D. 1680 and to have ended by A.D. 1700
(Johnson, 1994). Larson Cemetery was excavated
during the 1966 –1968 field seasons under the direction of William M. Bass through the University of
145
Kansas. In total, 621 individuals were recovered,
most as primary flexed interments (Owsley et al.,
1977). It is believed that this sample represents
approximately 90% of the cemetery population (Bass
and Rucker, 1976; Owsley and Bass, 1979). Based on
the demographic analysis of the remains, the cemetery appears to have been the sole repository for the
village dead (Bass and Rucker, 1976).
Two randomly generated lists simulated a 60%
recovery of tibiae, femora, and humeri from both
adult and subadult primary burials, one derived
from an original sample of 15 skeletons, and one
from 30 skeletons. The samples drawn for the test
consisted of both complete and fragmentary elements. The randomly selected elements were commingled on a table without any assistance by the
observer (B.J.A.), and any identifying labels (e.g.,
catalog numbers) were covered. These procedures
removed any chance of bias by the observer during
the test. The observer also did not know the recovery
probability or the original number of individuals
represented by the remains.
After an inventory of the test sample was completed, fragments from the same element were conjoined, and then pair-matching was performed. This
task involved placing all specimens of a particular
element (e.g., complete and fragmentary tibiae) on a
table. The bones were then sorted by side, age, and
robusticity, to assist in the conjoining of fragmentary remains and eventual pairing of elements.
A pair-match can be determined based on general
morphology and taphonomic indicators. Morphological indicators include robusticity, muscle markings,
epiphyseal shape, bilateral expression of a periosteal reaction, and general symmetry between elements. Furthermore, biological information (e.g.,
age and sex) may also be useful for pair-matching.
Taphonomic variables may include the state of preservation (e.g., degree of weathering and color), presence of burning, cut marks, and animal damage.
Because of the variation that is possible from differential preservation, taphonomic indicators should
not be weighted as heavily as gross morphological
features for pair-matching.
Results of the 15-individual group drawn from
Larson Cemetery (Table 3) show that all femora, all
tibiae, and all but one pair of humeri were matched
correctly. The error with the humeri was due to the
fact that one of the specimens exhibited significant
remodeling resulting from a probable dislocation.
Table 3 also contains the realized recovery probabilities (L ⫹ R divided by two times the known N) and
their standard errors from the binomial distribution. Table 3 also lists the estimated recovery probabilities and their standard errors from the observed
counts of pairs and from the true counts. There is
good agreement of the estimated recovery probabilities with the realized probabilities. The standard
errors of the estimated recovery probabilities are
slightly higher than for the realized recovery probabilities, but this is to be expected because of the
146
B.J. ADAMS AND L.W. KONIGSBERG
TABLE 3. Results of 15-individual test for pair-matching
TABLE 4. Results of 30-individual test for pair-matching
a. Bone counts, realized recovery probabilities, and their
standard errors
a. Bone counts, realized recovery probabilities ,and their
standard errors
Element
Lefts
Rights
Pairs
Realized
recovery
probability
Femur
Tibia
Humerus
9
7
7
9
7
9
6
4
3 (4)1
0.600
0.467
0.533
Realized
standard
error
Element
Lefts
Rights
0.089
0.091
0.091
Femur
Tibia
Humerus
15
19
20
21
18
18
b. Estimates of recovery probabilities (r) and their standard
errors, based on observed pair-matches and actual (correct) pair
match for humerus
Pairs
Realized
recovery
probability
Realized
standard
error
11
10 (11)1
11 (12)1
0.600
0.617
0.633
0.063
0.063
0.062
b. Estimates of recovery probabilities (r) and their standard
errors, based on observed pair-matches and actual (correct) pair
match for tibia and humerus
Element
r
s.e.
Element
r
s.e.
Femur
Tibia
Humerus
Humerus (correct)
0.667
0.571
0.375
0.500
0.128
0.158
0.154
0.153
Femur
Tibia
Tibia (correct)
Humerus
Humerus (correct)
0.611
0.541
0.595
0.579
0.632
0.096
0.099
0.096
0.095
0.092
1
Error was made, and number in parentheses is correct answer.
Fig. 4. Probability function for N in 15-individual test case,
where solid line is from observed number of pairs, and dashed line
is from actual (true) number of pairs.
additional uncertainty that arises from not knowing
the value of N. Figure 4 shows the probability histogram for N for both the observed pair counts and
the true pair counts. The maximum likelihood estimate (MLNI) for the number of individuals is 14 for
the observed pair counts and 13 for the true pair
counts (using the combined probability across bones
from Equation (9)). In both cases, the true value of
15 individuals is within the (approximate) 95%
HDR. For the observed pairs, the 95.7% HDR is from
13–19 individuals, while for the true pair count, the
96.3% HDR is from 12–17 individuals.
Results of the blind pair-matching test involving
30 individuals (Table 4) revealed that one pair of
humeri was missed, one pair of tibiae was missed,
and all femora were correctly pair-matched. Again,
there is good agreement of the estimated recovery
probabilities with the realized probabilities, and
again the standard errors of the estimated recovery
1
Error was made, and number in parentheses is correct answer.
Fig. 5. Probability function for N in 30-individual test case,
where solid line is from observed number of pairs, and dashed line
is from actual (true) number of pairs.
probabilities are slightly higher than for the realized
recovery probabilities. The maximum likelihood estimate (MLNI) for the number of individuals is 31
(with a 94.8% HDR of 28 –36 individuals) for the
observed pair counts, and 29 (with a 93.5% HDR of
27–33 individuals) for the true pair counts (see Fig.
5).
Results of the pair-matching test derived from the
Larson Cemetery sample indicate that pair-matching can be accurately performed for all of the elements examined. Furthermore, if errors in pairmatching are committed, they are more likely to
occur from overlooking true pairs, as opposed to the
pairing of unrelated elements. It is possible that this
tendency may change if the sample size is substantially increased, since variation between individuals
may not be as obvious. The main reason for difficulty
in pair-matching is that fragmentation or weather-
147
QUANTIFICATION OF HUMAN REMAINS
TABLE 5. Results of analysis of Larson Lodge 21
a. Element counts and estimates of N
Element
Lefts
Rights
Pairs
Max(L, R)
L⫹R⫺P
Tibia
Os Coxa
Humerus
Femur
Overall
30
37
31
43
34
39
37
36
20
29
22
31
34
39
37
43
43
44
47
46
48
48
MLNI1
45,
47,
47,
48,
48,
50,
49,
51,
49,
50,
62
55
62
54
52
%
HDR
94.8%
95.2%
94.6%
94.5%
95.2%
b. Estimates of recovery probabilities (r) and their standard errors
Element
r
s.e.
Tibia
Os coxa
Humerus
Femur
0.625
0.763
0.647
0.785
0.071
0.054
0.067
0.051
1
Central number in triplet is MLNI, while first and last numbers give highest-density region (HDR). Column to immediate right gives
percentage in HDR.
ing, even minimal, can obliterate key areas used for
identifying a match. Overall, the Larson Cemetery
pair-matching test provided encouraging results. If
pair-matching and conjoining are performed by an
experienced osteologist, accurate results can be derived, especially when the original number of individuals is relatively small.
Estimation of MNLI and recovery probabilities
for Larson Village Lodge 21
A sample of commingled skeletal remains from
Larson Village (39WW2) were utilized as a test for
the validity of the quantification methods described
above. Larson Village is an Arikara habitation site
that is associated with Larson Cemetery (described
above, in Test for the Accuracy of Pair-Matching).
Larson Village was excavated during the summers
of 1963 and 1964 as part of an archaeological salvage program of the Smithsonian Institution River
Basin Surveys. Although 29 circular lodge depressions were visible on the ground surface at the village site, only three (Lodges 1, 21, and 23) were
completely excavated (Owsley et al., 1977).
Discovery of scattered human remains on the
lodge floors initially led to suspicion that villagers
had been struck down by an epidemic disease
(smallpox; see Bass and Rucker, 1976), but subsequent analysis revealed extensive perimortem
trauma suggesting warfare (Owsley et al., 1977).
The state of the skeletons encountered within the
lodges implied that intentional burial did not occur.
Evidence suggested that the individuals were either
massacred and placed within the lodges, or they
were left to decompose inside the lodges where they
had been killed. The massacre at Larson Village is
associated with the terminal occupation period of
the site (A.D. 1700). Collapse of the lodge roofs buried the remains, but considerable commingling had
occurred prior to this sealing of deposits. Owsley et
al. (1977, p. 121) stated, “Due to disturbance and
mixture of the skeletons . . . it was necessary to treat
the bones as if they came from an ossuary.” The
majority of the remains were discovered on the floor
of Lodge 21, which had an MNI calculated at 44
individuals (Owsley et al., 1977).
In order to estimate the MLNI and recovery probabilities for Lodge 21, four paired elements were
selected for study. Sorting and pair-matching procedures were performed on the humeri, os coxae, femora, and tibiae. Counts of left, right, and paired
elements are given in Table 5. The MNI results from
the current study (Table 5) are reasonably consistent with MNI totals obtained by Owsley et al.
(1977, p. 121), which were derived by “counting the
major bones, and dividing them into rights and
lefts.” This suggests that the Max(L, R) variant of
the MNI was employed. Sorting of the femora as
part of the current study revealed an MNI of 43. It is
suspected that this difference may be due to the fact
that a complete inventory of the entire skeleton was
not performed for the present research; Owsley et al.
(1977) did not indicate which element gave them
their MNI of 44. Table 5 also lists the MNI values
(from L ⫹ R ⫺ P) and the MLNI values for each
element. The largest MNI value, 48 individuals, was
derived from the femur. The MLNI values are 50
from the tibiae, 49 from the os coxae, 51 from the
humerii, and 49 from the femora. Figure 6 graphs
the probability distributions across N for each bone
(from Equation (9)), as well as the combined probability across all four elements. Figure 6 shows the
MLNI across all elements at 50 individuals, and the
95.2% HDR is from 48 –52 individuals (see spreadsheet at http://konig.la.utk.edu/MLNI.html for the
calculations). The likelihood ratio chi-square comparing the MLNI estimates from each element (50,
49, 51, and 49 from the tibia, os coxa, humerus, and
femur, respectively) to the overall estimate of 50 is
0.3634, which with three degrees of freedom (number of elements minus one) yields a probability value
equal to 0.9477. Each element therefore provides a
consistent estimate of the number of individuals.
Table 5 also contains the estimated recovery probabilities and their standard errors. The estimated
148
B.J. ADAMS AND L.W. KONIGSBERG
Fig. 6. Graph of probability function for N from tibiae, os
coxae, humerii, and femora from Larson Lodge 21. Graph also
shows combined probability function for all bones.
recovery probabilities are 0.625, 0.763, 0.647, and
0.785 for the tibia, os coxa, humerus, and femur,
respectively, which represent fairly high recovery
probabilities. Indeed, from these probabilities the
probability of obtaining no tibiae, os coxae, humerii,
or femora from an individual would be about 4.55 ⫻
10⫺5. Figure 7 shows a comparison of the distribution function for the recovery probability (the cumulative probability), assuming a normal distribution
against 1,000 nonparametric bootstrap samples
(Borchers et al., 2002, p. 113) for each of the elements. The agreement between our parametric approach (Equations (12) and (13)) and the nonparametric bootstrap is quite good, indicating that
parametric estimation can be used in place of the
more tedious bootstrap.
Results from the Larson Village analysis suggest
that in situations of good skeletal preservation, near
complete recovery, and deposition resulting from a
single event, the MNI estimates will accurately reflect the original population numbers. Similarly, in
this type of situation, MLNI estimates should also
be accurate. Agreement between the two techniques
can be used to suggest that a high degree of recovery
was achieved. However, it is considered advisable to
compute the MLNI whenever feasible, since it will
be much more accurate than the MNI when recovery
probabilities drop.
DISCUSSION
Data loss from skeletal assemblages can occur as
a result of many different taphonomic forces, including disarticulation, dispersal, bone weathering, and
fragmentation (Lyman, 1987, 1994; Haglund and
Sorg, 1997, 2002). Furthermore, recovery procedures, curational factors, and analytical issues need
to be considered as possible taphonomic forces that
can potentially bias both ancient archaeological
sites, as well as more recent forensic contexts. Any
study that attempts to draw conclusions related to
numbers of individuals based on an analysis of commingled human remains needs to implement the
most appropriate quantification techniques available.
This study compared two different quantification
techniques, the MNI and the LI, as well as a generalization of the LI that we refer to as the “most likely
number of individuals,” or MLNI. With respect to
the analysis of human skeletal remains, typically
the only method considered for estimating the original population size has been the MNI. The results
described here show that the LI and MLNI are better estimators of the original death assemblage. The
strength of these techniques lies in the fact that they
can account for taphonomic biases common to both
archaeological samples and forensic contexts, with
negligible effects upon the estimate.
Rogers (2000) described another maximum likelihood estimator (“analysis of bone counts by maximum likelihood,” or “abcml”) that can be used to
estimate the MLNI and taphonomic parameters of
commingled zooarchaeological skeletal remains. His
method could presumably be applied to samples of
human remains. However, “abcml” first requires external estimates of recovery probabilities in order to
generate probabilities of what Rogers (2000) referred to as “configurations.” In fact, we could have
applied “abcml” using our internally estimated recovery probabilities, but this would be a completely
circular analysis. In the future, if recovery probabilities are shown to have some generality for human
skeletal remains, then they could be used as fixed
parameters in applications of the “abcml” method to
commingled human remains. Further, the “abcml”
method has the distinct advantage over other quantification methods in that it is calculated directly
from bone counts, and does not require the timeintensive and often subjective demands of element
pair-matching.
We have not yet discussed variation in recovery
probability across individuals, but it may be the case
that there is variation in this parameter across an
archaeological site. For example, the skeletal material may have been recovered from an ossuary pit
where part of the pit was filled with a heavy, poorly
drained clay, while the other part was filled with a
well-drained loess. We would expect poor preservation within the clay with a concomitant low recovery
probability, and good preservation within the loess
with a concomitant high recovery probability. The
effect that this variation has on the LI depends on
how these different regimes correlate with individuals. For example, if there is complete correlation,
such that all individuals were exposed to only one
depositional environment, then the LI should be
calculated for each context and summed. On the
other hand, if the depositional environment is ran-
QUANTIFICATION OF HUMAN REMAINS
149
Fig. 7. Distribution function for recovery probabilities of tibiae, os coxae, humerii, and femora from Larson Lodge 21. Smooth lines
are from cumulative normal distributions with means and standard deviations from Equations (12) and (13), while step functions are
empirical cumulative distributions from nonparametric bootstraps with 1,000 samples drawn for each graph.
dom with respect to individuals, then the LI should
be calculated for the entire site. If, for example, the
fact that the left femur from an individual ended up
in the clay does not affect whether the right femur
ended up in the clay, then ignoring the heterogeneity in recovery/preservation has no effect on the LI
calculated for the entire site. It is only when there is
an intermediate correlation that estimation of the LI
is subject to unknowable error.
Due primarily to precedence and ease of calculation, the MNI will likely continue to serve as the
quantifier of choice with commingled human remains. However, as demonstrated here, superior
methods of quantification are available. The LI
and MLNI provide a more realistic reconstruction
of population counts from commingled samples,
and supply researchers with more accurate estimates from which to present demographic results.
It is suggested that the MNI should be reported
alongside the LI or MLNI. Similar results between
techniques substantiate the reliability of estimates and indicate a high recovery probability.
Discrepancies between methods demonstrate the
MNI’s correlation with the recovery probability
and its inability to estimate the original population when data loss has occurred. The LI’s and
MLNI’s behavior is not as susceptible to variable
recovery probabilities, and their estimates provide
a more accurate reconstruction of the original
numbers of dead.
150
B.J. ADAMS AND L.W. KONIGSBERG
Overall, positive results were found for the reliability of pair-matching from selected skeletal elements. It should be stressed that the pair-matching
of homologous elements (e.g., right and left femora)
was tested, and not the association of different element types with each other. If fragmentation is extensive or preservation is extremely poor, so that
accurate pair-matches are impossible to determine,
the LI and MLNI are prone to gross miscalculations
due to the multiplicative nature of the procedures.
In this type of situation, no method (save for “abcml,” with its own stringent data requirements) can
provide accurate estimates of original populations.
In situations of reasonable recovery and good
preservation, the LI or MLNI provides an accurate
assessment of the actual population that contributed to the recovered osteological assemblage. The
total population can be divided into subgroups (e.g.,
age cohorts), using modifications to the LI or MLNI
for more detailed demographic reconstructions. It is
also possible to apply HDRs to these estimates, a
feature not available with some other techniques. As
recovery probabilities surpass 50%, the HDRs can
become quite small. Overall, the LI and MLNI provide results that are of much more value for interpretation than any other quantification technique
currently employed.
ACKNOWLEDGMENTS
We thank William M. Bass, Richard L. Jantz, and
Walter E. Klippel for their input on B.J.A.’s M.A.
thesis, which provided the genesis for this paper. We
also thank Gary C. White for providing L.W.K. with
the source code for the program NOREMARK. Finally, we thank the anonymous reviews for their
helpful comments and Susan R. Frankenberg for her
comments on the final draft. The Larson Cemetery
was excavated under grants from the National Science Foundation (GS837 and GS1635) and a grant
from the National Geographic Society (699,912), all
to William M. Bass.
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