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Deconstructing death in paleodemography.

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Deconstructing Death in Paleodemography
Lyle W. Konigsberg* and Susan R. Frankenberg
Department of Anthropology, University of Tennessee, Knoxville, Tennessee 37996-0720
life tables; hazards analysis; human skeletal biology
In 1992 in this Journal (Konigsberg and
Frankenberg [1992] Am. J. Phys. Anthropol. 89:235–256),
we wrote about the use of maximum likelihood methods
for the “estimation of age structure in anthropological
demography.” More specifically, we presented a particular
method (the “iterated age-length key”) from the fisheries
literature and suggested that the method could be used in
human and primate demography and paleodemography as
well. In our paper (section titled “Some Future Directions”), we spelled out two broad areas that we expected to
see develop over the ensuing years. First, we felt that the
use of explicit likelihood methods would open up interest
in basic estimation issues, such as the calculation of stan-
dard errors for demographic estimates and the formulation of tests for whether samples differed in their demographic structure. Second, we felt that the time was ripe
for hazards analyses that would incorporate the uncertainty in estimation that follows from using age “indicators” rather than known ages. While some of these developments have occurred during the last decade, few have
been reported in the American Journal of Physical Anthropology. In this paper we resolve some issues from our
1992 paper, and attempt to redress this deficit in the
literature by reviewing some recent developments in paleodemography over the past decade. Am J Phys Anthropol 117:297–309, 2002. © 2002 Wiley-Liss, Inc.
In a 1992 article “Estimation of Age Structure in
Anthropological Demography,” published in this
Journal, we noted, “The stumbling-blocks faced in
many anthropological demography studies, and virtually all paleodemographic studies are that: 1) population growth rate may not be known, 2) samples
many not be representative, and 3) ages are usually
estimated rather than known” (Konigsberg and
Frankenberg, 1992, p. 235). Now, a decade later, it
seems reasonable to reevaluate these problems and
see what progress has been made in the last 10
years. Our primary focus in this paper will be on the
third problem of age estimation, though we also
touch tangentially on the second issue of how to
assess statistically whether “mortality” samples are
biased by nonbiological factors (for a brief discussion
of such “natural and cultural filters,” see Konigsberg
and Frankenberg, 1994). We will not deal with the
first problem of population growth, as others have
recently proposed promising solutions for the simultaneous estimation of growth rate and mortality
(described in Wood et al., 2002).
Our 1992 discussion of developments in the estimation of age structure in paleodemography was
roundly criticized by Bocquet-Appel and Masset
(1996) in an article entitled “Paleodemography: Expectancy and False Hope.” At the time this critique
appeared, we felt that the methods that we and
others were working on were sufficiently different
and advanced to overcome their objections. These
new developments in paleodemographic methods,
plus many others that we did not anticipate, were
recently published in a Cambridge University Press
volume coedited by Hoppa and Vaupel (2002). We
focus here on an erroneous application in our 1992
article that Bocquet-Appel and Masset (1996) did
not criticize, and that has been repeated by BocquetAppel and Bacro (1997) and by Jackes (2000). Specifically, we are culpable for introducing in this Journal some bad thinking in model estimation by
providing in our 1992 paper (our Table 2) an example application with negative 5 degrees of freedom.
More recently, Bocquet-Appel and Bacro (1997, their
Table 2) framed most of their analysis around model
estimation with negative one degree of freedom, and
Jackes (2000, her Fig. 15.7) presented an analysis
with negative 11 degrees of freedom. In part then,
this paper makes amends for some misapplications
from the past. This paper also argues that it is high
time to move past life table approaches in paleodemography.
We first briefly describe “contingency table” approaches to paleodemography, reanalyzing the data
presented in Bocquet-Appel and Bacro (1997) to motivate the discussion. These are essentially nonparametric methods in which one or more “age indica-
Grant sponsor: National Science Foundation; Grant number: SBR9727386.
*Correspondence to: Lyle W. Konigsberg, Department of Anthropology, University of Tennessee, 250 South Stadium Drive, Knoxville,
TN 37996-0720. E-mail:
Received 2 February 2001; accepted 11 October 2001.
DOI 10.1002/ajpa.10039
Published online in Wiley InterScience (www.interscience.wiley.
tors” are cross-tabulated against age categories in a
reference (known age-at-death) collection in order to
estimate the proportions of deaths in age categories
for a target (unknown age-at-death) sample. Contingency table models are limited by the fact that there
must be at least as many indicator states as the
number of age categories in order for the model to be
identified (i.e., estimable). Next, we will consider
hazard models as an alternative to contingency table
analyses. The hazard model is used with reference
sample information in order to predict the proportion of individuals that should be in each indicator
state in the target sample, and the hazard parameters are then adjusted until there is the best possible
fit between the expected proportions in indicator
states and the observed proportions. Because hazard
models have a small number of parameters, usually
considerably less than the number of age classes in
the contingency table approach, hazard models often
can be estimated where contingency table methods
would fail.
Within our discussion of hazard models, we will
consider a number of specific issues, including: 1)
estimation of confidence limits for hazard parameters when ages are estimated rather than known, 2)
calculation of mean age-at-death and its standard
error from hazard models, again incorporating the
uncertainty in ages, and 3) calculation of entropy
and its standard error for a target age-at-death distribution based on a hazard model. Then, again using hazard models, we will discuss the problem of
how to identify whether a target sample is a reasonable representation of deaths that occurred in a prehistoric population. Finally, we will discuss the overall problem of how age estimation should be done in
paleodemography. In addition to reviewing some
previous methods, we note the utility of fully parametric models where the target sample mortality is
fit using a hazard model and the development of the
“age indicator” is also fit with some form of parametric model.
In Konigsberg and Frankenberg (1992), we suggested using a technique originally developed in
fisheries management (Kimura and Chikuni, 1987)
referred to as the “iterated age length key” (IALK).
IALK was shown by Kimura and Chikuni (1987) to
be an expectation-maximization algorithm (Dempster et al., 1977) that leads to maximum likelihood
estimates for the age-at-death structure for a target
sample. Indeed, all of the methods we will consider
in this paper are based on the method of maximum
likelihood. IALK uses a contingency table of an age
indicator against known age classes in a reference
sample in order to infer the age-at-death structure
in a target sample. Bocquet-Appel and Masset
(1996) suggested that they had historical priority on
the development of the method, which they had developed out of an “iterative proportional fitting procedure” (IPFP) commonly used to fit log-linear mod-
els. Bocquet-Appel and Masset (1996) were trying to
fit an unobserved marginal distribution (age-atdeath), using the same type of information that we
were using in 1992.
The relationship between IPFP and IALK was
unclear in Bocquet-Appel and Masset (1996), because they obtained slightly different results using
the two techniques. A year later, Bocquet-Appel and
Bacro (1997, p. 571) wrote in this Journal, “One can
demonstrate analytically that both algorithms, although written differently, have the same step two
and lead in fact to the same solution.” Yet, when
they applied the two techniques to the same data,
they also obtained slightly different answers. We
programmed both procedures (see and, provided that they are
started from the same starting values for the estimated age-at-death distribution, the algorithms
take identical steps through the parameter space.
Though they are written differently, IPFP and the
IALK are the same, and following historical priority,
we will refer to the method as IPFP. IPFP does
differ, however, from an algorithm developed by
Hoenig and Heisey (1987) that accounts for finite
reference sample size.
Various authors (Konigsberg and Frankenberg,
1992; Bocquet-Appel and Bacro, 1997; Jackes, 2000)
could have avoided erroneous application of IPFP
resulting in overfitting (i.e., negative degrees of freedom) by reading more closely the following from
Clark (1981, p. 299):
If I ⬍ J (i.e., the number of length intervals is less than the
number of age-groups with distinct length distributions), there
will usually be a multiplicity of algebraic solutions and therefore
no useful estimate.
The recent analyses by Bocquet-Appel and Bacro
(1997) and Jackes (2000), which estimate the age
structure for a target sample as a single event,
should have detected this transgression in their
evaluation statistics. Using the goodness-of-fit test
given in Kimura and Chikuni (1987, p. 28) and in
Hoenig and Heisey (1987, p. 235), they would have
found that the test gives negative degrees of freedom
if there are more age classes than indicator stages.
Bocquet-Appel and Bacro (1997) and Jackes (2000)
also should have recovered excessively large calculated standard errors when they fit more age classes
than indicator states, or if they had looked at the
likelihood surface, they would have noted remarkably flat surfaces, a clear sign that the model is
overparameterized. (Goodness-of-fit and standard
errors were not calculated in the example in Konigsberg and Frankenberg (1992), because it was a simulation study.)
In her recent analysis, Jackes (2000) attempted to
estimate a model with 17 age classes (i.e., a model
with 16 parameters, as the sum of the proportions in
age classes must equal 1.0), using the SucheyBrooks six-phase system. Her results yielded a number of age classes with zero frequencies, a clear sign
TABLE 1. Distribution of classifiable adults by age and stage of
trabecular involution of the femoral head for the Coimbra
reference sample and Loisy-en-Brie target sample1
From Bocquet-Appel and Bacro (1997: their Table 2 and p. 573).
We use the values from within their Table 2, which sum to N ⫽
422, and not the N ⫽ 421 given in their footnote and text.
that the model was not identified. Bocquet-Appel
and Bacro (1997) presented a less extreme estimation problem in trying to fit a seven age-class model
using six stages. In the remainder of this paper we
will deal with their application, discuss some related
issues from Bocquet-Appel and Masset (1996), and
sketch some of the directions in which we see paleodemography now heading.
Bocquet-Appel and Bacro (1997) presented tables
of age and stage data for their reference sample and
target sample that we reproduce here in Table 1.
Their analyses for adults were based on stage data
for femoral head trabecular involution, using 422
known-age individuals from an anatomical collection (Coimbra) and 96 individuals from a “target”
archaeological sample (Loisy-en-Brie). Rather than
use IPFP to estimate the age structure of the target
sample here, we explicitly maximized the log-likelihood using the Nelder-Mead simplex algorithm
(Nelder and Mead, 1965) available in the “optim”
function of the statistical/graphics package “R.” “R”
is a major package distributed under the GNU Public License, and as a consequence it can be freely
downloaded (see for binaries and
source code, Ripley (2001) for an overview, and Cribari-Neto and Zarkos (1999) for a recent review).
Using “optim” to maximize the log-likelihood leads
to estimated proportions in each of the seven age
classes of 0.1645, 0.0269, 0.2781, 0.2061, 0.1197,
0.1935, and 0.0113.
“Optim” can also be used to find the standard
errors of these estimates by: 1) requesting calculation of the Hessian matrix (i.e., the matrix of second
partial derivatives of the log-likelihood with respect
to the parameters), 2) changing the sign of all elements of the matrix, 3) inverting the matrix, and
then 4) taking the square root of diagonal elements.
Rounding to the nearest whole number the standard
errors on these proportions are 576, 3,156, 9,496,
14,355, 25,786, 22,084, and 3,737, respectively. The
analytical standard errors for the estimated proportions are even larger. Clearly, these are not usable
results. As indicated above, the reason things went
awry here is that the model, with negative 1 degree
Fig. 1. Normed likelihood for proportion of deaths in the
40 –50-year age interval (curved line). Horizontal line gives asymptotic 95% confidence set at the line’s intersections with the
normed likelihood.
of freedom, is overparameterized. Another way to
see this is to plot the normed likelihood (Lindsey,
1996, p. 82) for one of the parameters, while fixing
all other parameters save one at their maximum
likelihood estimates.
In Figure 1, we plotted the normed likelihood for
the proportion of deaths in the 40 – 49-year age interval, allowing only the proportion of deaths in the
50 –59-year age interval to vary so that the age
distribution sums to 1. The asymptotic 95% confidence set for this parameter consists of all values of
the death proportion with a normed likelihood
greater than or equal to 0.1465 (see equation 7.20 in
Shao, 1999, p. 447), which we have plotted as a
horizontal line. From Figure 1 it is clear that the
normed likelihood never reaches this horizontal line
within the parameter space, so that the confidence
set is undefined. Further, because Figure 1 is drawn
as a profile likelihood, the normed likelihood appears even less flat than it actually should be. As
Lindsey (1996, p. 111) notes, “A profile likelihood
gives a narrower range of likely values of the parameter of interest than is necessarily appropriate” because it does not allow for uncertainty in the other
parameters (death proportions in the case used
One possible solution to this problem of overparameterization is to decrease the number of age
intervals by combining some adjacent age categories. For example, if we combine the last two age
categories (70 –79 and 80 – 89 years), then there are
six classes for the indicator state, and six age
classes. This leads to six equations with six unknowns, and can be solved using any of a number of
linear algebraic approaches (e.g., QR decomposi-
tion). In the more complicated situation where there
are more equations than unknowns, as discussed by
Clark (1981), some age classes could include negative estimated counts when fit by solving the system
of linear equations. When there are more unknowns
than equations, as in the original example in Bocquet-Appel and Bacro (1997), there is no unique
For the six age-class case (with the 70 –79-year
and 80 – 89-year age classes combined), we solved
the linear equations to estimate the proportions in
age classes as 0.1678, 0.0079, 0.3360, 0.1182,
0.2761, and 0.0939. Unfortunately, the associated
standard errors that we obtained from the Hessian
matrix both analytically and numerically are still
unacceptably large (0.2305, 0.7879, 1.0613, 0.7593,
0.5534, and 0.2237, respectively). In addition to the
fact that the 95% confidence intervals in this case
include negative values for the proportion of deaths
in every age class, the estimated age distribution is
very bumpy in nature. For example, about 17% of
deaths are estimated as occurring between 23–29
years, about 34% between 40 – 49 years, but less
than 1% of deaths occur between these two age
classes at 30 –39 years. This “bumpiness” occurs because in the contingency table approach to paleodemography, there is no parametric model describing
mortality. Next, we address this problem using hazards analysis.
A solution to the estimation problems we noted
above could have been developed from our 1992 article. There we suggested (Konigsberg and Frankenberg, 1992, p. 252), “A future direction that we expect to see in anthropological demography and
paleodemography is the incorporation of uncertainty
of age estimates into reduced parameterizations of
life table functions.” This was not a particularly
novel concept, as Gage (1988) had already shown the
utility of hazard models in paleodemography. We
further suggested that one could maximize the loglikelihood that accounted for age uncertainty in a
hazard model, an application inaccessible to Gage
(1988) working from life table tabulations that
lacked the osteological information used to estimate
ages. In the following, we show how hazard models
can be fit to the Loisy-en-Brie data while accounting
for the fact that ages are estimated rather than
The first step in this process is counterintuitive, in
that we will fit a model that perfectly recovers the
observed proportions of Loisy-en-Brie adult skeletons in the six femoral head stages, rather than
fitting a hazard model. In statistical parlance, such
a model is usually called a “saturated” model. This
model is easy to find, as we simply use the observed
proportions in the femoral stages for Loisy-en-Brie,
which are 0.02083, 0.08333, 0.32813, 0.42188,
0.12500, and 0.02083. The model has five parameters, because given the constraint that the propor-
tions must sum to 1.0, if we know any five parameters then we also know the sixth. The log-likelihood
for this saturated model is defined (up to an additive
constant) as the sum of the products of observed
counts with their natural log proportions. Consequently, we have for the log-likelihood of the saturated model: 2.0 ⫻ ln(0.02083) ⫹ 8.0 ⫻
ln(0.08333) ⫹ . . . ⫹ 2.0 ⫻ ln(0.02083), which equals
⫺130.3731. Lindsey (1996, p. 77–78) refers to this
quantity as the “empirical” log-likelihood.
Now we can fit a hazard model, which for simplicity’s sake will be a single-parameter exponential
model. We assume a survivorship of 1.0 at age 23
(i.e., life begins at age 23, so we subtract 23 years
from all ages). The exponential model gives survivorship to exact age t as:
S共t兲 ⫽ exp共⫺a 2 ⫻ t兲,
where a2 is the single parameter to be estimated.
(We use the subscript to differentiate this parameter
from later ones to be considered, following the numbering system in Gage, 1988.) If we assume a starting value for a2, say 0.1, then we can fit survivorship
at each of the age divisions given in the life table of
Bocquet-Appel and Bacro (1997), and then difference the series to get the expected proportion of
deaths in each age interval, which we place in a 7 ⫻
1 column vector d. We then define a row stochastic
matrix P ila from Table 1 that gives the probabilities
from the reference sample of being in the ith stage of
the indicator conditional on being in the ath age
interval. If ni is a column vector of counts of the
number of skeletons from the target sample in each
indicator state, then the log-likelihood is (up to an
additive constant) ln(d⬘Pila)ni, which with a2 equal
to 0.1 gives a log-likelihood of ⫺156.1552.
If we numerically maximize this model (again using “optim” to find the value of a2 at which the
log-likelihood is highest), then the maximum likelihood estimate for a2 is 0.02924. The log-likelihood at
this point is ⫺132.1952. Negative two times the
difference between the log-likelihood for this model
and the log-likelihood from the saturated model is a
number called the “deviance” or a “likelihood ratio”
(see Agresti, 1996, p. 96, or Kennedy, 1992, p. 59 –
62). The deviance is asymptotically distributed as a
chi-square random variable, with degrees of freedom
equal to the difference in the number of parameters
between the hazard model and the saturated model.
In this case, the deviance is equal to 3.6442 with 4
degrees of freedom, which gives an associated probability value of 0.4563. This relatively high probability value suggests that little is lost in “moving”
from the saturated model with five parameters
“down” to the more parsimonious exponential model
with only one parameter.
The one-parameter exponential model is generally
viewed as too simplistic to capture adult mortality,
even though it appears to do an adequate job of
recovering the observed proportions of individuals in
indicator states for Loisy-en-Brie. We can fit a
Fig. 2. Normed likelihood surface for the two hazard parameters found by using a Gompertz model of survivorship and the
likelihood of target skeletons in particular indicator states being within particular age intervals.
slightly less parsimonious model, which is the twoparameter Gompertz model. The Gompertz model
gives survivorship to exact age t as:
S共t兲 ⫽ exp
共1 ⫺ exp共b 3 ⫻ t兲兲 ,
again following the notation of Gage (1988), where a
and b are subscripted to indicate that they represent
the third component of mortality in a Siler model.
Numerically maximizing the log-likelihood in the
same way as for the exponential model, we find the
hazard parameters equal to 0.0129 and 0.0416, with
standard errors of 0.0065 and 0.0221, respectively.
Figure 2 contains a plot of the normed likelihood
surface for the two parameters, which shows that
the surface is quite convex. Again, the confidence set
in any one direction (i.e., for one of the parameters,
holding the other constant) is given where the
normed likelihood falls above 0.1465. The Gompertz
model gives a log-likelihood value of ⫺130.4562,
yielding a deviance of 0.1662 with 3 degrees of freedom, and an associated probability value of 0.9829.
This very high probability indicates that the Gompertz model does an excellent job of recovering the
observed proportions in indicator stages for Loisyen-Brie.
We also estimated a three-parameter Makeham
model, but the fit is trivially different from the
Gompertz model. The deviance for the Makeham
model is 0.0414 with 2 degrees of freedom, resulting
in a probability value of 0.9795 for the goodnessof-fit test. Because the Gompertz model is “nested”
within the Makeham, we can subtract the deviances
and degrees of freedom for one model from the other
to calculate an improvement ␹2. The resulting value
of 0.1248 with 1 degree of freedom gives a probabil-
Fig. 3. Kernel density plots of the two hazard parameters for
asymptotic normal distributions from the original analysis (solid
line), bootstrapped target (open circles), and bootstrapped target
and reference (filled circles) samples for 500 samplings.
ity value of 0.7239, indicating that there is little to
be gained by adding the additional parameter in the
Makeham model.
Simulation and confidence limits
Bocquet-Appel and Masset (1996, p. 574) raised
the question, “Besides the expectancy of an estimate, what is its variance of error?” They attempted
to answer this question using simulation studies.
We present a more modest simulation study here,
which takes the estimated Gompertz model for the
96 adults from Loisy-en-Brie and simulates in order
to establish the statistical properties of the estimates. This is largely an unnecessary exercise, as
we have presented a maximum likelihood estimator
whose statistical properties are fairly well understood. In order to simulate counts of the number of
individuals in each stage for the femoral head, we
drew 96 individuals from the multinomial age-atdeath vector d and then applied the appropriate row
from Pila to draw on the multinomial distribution of
indicator conditional on age. We repeated the simulation 500 times, and estimated the Gompertz parameters just as we did for the actual observations.
This simulation consequently is a parametric bootstrap in the sense that we take the estimated Gompertz parameters from Loisy-en-Brie to use as the
generating values in the simulation runs.
Figure 3 shows kernel density plots for the a3 and
b3 parameters from the 500 simulations, as well as
the asymptotic normal distributions from our original analysis. The normal distributions are drawn
using the estimated values as the means, and the
standard error of estimates from “optim.” While the
bootstraps show a bit more dispersion as well as
some asymmetry relative to the asymptotic results,
the results do not suggest that there is “no hope with
the expectation” (Bocquet-Appel and Masset, 1996,
p. 578). As Bocquet-Appel and Masset (1996) also
were concerned about the effects of a finite reference
sample, we bootstrapped the reference sample by
drawing from the two-dimensional contingency table in Table 1, and by following the previously described simulation procedure. As seen in Figure 3,
bootstrapping the reference has virtually no effect
on the simulation results.
Bocquet-Appel and Masset (1996, p. 576) argued,
“Bootstrapping the reference sample in order to generate an expectation for the estimates would be a
wrong approach, because the expectation would be
in the expectancy domain of the reference sample.”
If by this statement they meant that such a bootstrap inappropriately carries with it the “uniformitarian assumption” of equivalent aging rates in the
reference and target sample (Howell, 1976), then
this is an insurmountable stumbling block. It is not
possible to test the assumption of uniform rates of
aging in a univariate, or single-age indicator, setting. When there are multiple age indicators, in
contrast, it is possible to test for “inconsistency” in
information (Brown and Sundberg, 1987) in the target sample relative to the reference sample. Although one of us applied such an analysis to stature
estimation (Konigsberg et al., 1998), we are unaware of similar applications made for age estimation in paleodemography.
A statistical summary from hazard models:
mean age-at-death
Bocquet-Appel and Masset (1996, p. 578) felt on
the basis of their simulation work that “it is not
possible to estimate reasonably the shape of an age
distribution,” but that “one can nevertheless obtain
a fairly good estimation for the direction of the overall trend of an undefined distribution: its average.”
They consequently suggested calculating the mean
age-at-death for adults. While it is debatable
whether mean age-at-death and other summary
measures given in Bocquet-Appel and Masset (1996)
and Bocquet-Appel and Bacro (1997) are all that can
be calculated in paleodemographic studies, hazard
models do not preclude us from extracting such summary measures. In fact, any desired summary measures can be calculated from the hazard model parameters, although this is less immediate and direct
than working with the parameters themselves.
Assuming zero population growth, life expectancy
can be found from a hazard model by integrating the
survivorship, which we do in “R,” using the add-on
library “integrate.” For Loisy-en-Brie, the life expectancy at age 23 is estimated at 28.92, for a mean
age-at-death for those who die after age 23 of 51.92
years, which is close to estimates in Bocquet-Appel
and Bacro (1997) of 52.36 and 52.27. The standard
error of the mean that we calculated can be found
using the delta method (Shao, 1999, p. 45), for which
we will need numerical derivatives of the integral
with respect to the two parameters in the Gompertz.
We find these derivatives using “numericDeriv” in
the “R” add-on library “nls.” The asymptotic standard error is 3.18 years, similar to the figure of ⫾3
years that Bocquet-Appel and Bacro (1997) reported
from simulation work (see also Bocquet-Appel,
1994). The mean age-at-death we estimated from
the IPFP is 52.40 years, with a standard error also
from the delta method of 119,991 years, which is
much greater than the 3 years given in BocquetAppel and Bacro (1997).
Death and entropy in ancient France
Now we are faced with a dilemma. We have an
estimate of the mean age-at-death and its standard
error for Loisy-en-Brie, but we have no mortality
model. We could, for example, assume that everyone
at Loisy-en-Brie who survived past age 23 years had
the temerity to die at precisely 23.0 ⫹ 28.92 ⫽ 51.92
years of age, which would produce a rectangular
survival curve. Or we could assume that starting at
age 23 years, the instantaneous hazard is 1/28.92 for
the rest of life, which would produce a very different
negative exponential adult survival curve. By arguing that the shape of mortality cannot be estimated,
Bocquet-Appel and Masset (1996) deprive us of not
only the distribution, but also the dispersion of
deaths. Some idea of the dispersion can be recovered
by estimating the standard deviation for age-atdeath for those who died past age 23 years. From the
Gompertz model, we again need to integrate in order
to find the standard deviation for age-at-death. The
variance for age-at-death for those who die above
age 23 is:
t ⫻ S共t ⫺ 23兲 dt ⫺ e 23
where e23
is the square of life expectancy at age 23.
For Loisy-en-Brie, the estimated standard deviation
for age-at-death in those individuals surviving to
age 23 years is 16.41 years, with an asymptotic
standard error of 7.85 years.
But the standard deviation of age-at-death is not a
particularly useful demographic parameter, and it is
difficult to compare across populations or samples.
In place of the standard deviation, we suggest using
the entropy (Demetrius, 1978, 1979; Goldman and
Lord, 1986; Vaupel, 1986) of deaths at and over age
23, which is defined as:
S共t ⫺ 23兲 ⫻ ln共S共t ⫺ 23兲兲 dt
e 23
Entropy can range from a minimum of 0.0, in which
case all deaths occur at a single age, to a maximum
of 1.0, in which the hazard rate is a constant across
age. For Loisy-en-Brie the estimated entropy for
those who die past age 23 is 0.5218, with a standard
error of 0.1340. The 95% confidence interval for entropy easily includes most of the entropy values for
age-at-death for deaths past age 20 in the model life
tables of Coale and Demeny (1966).
One of the most niggling problems in paleodemography is the question of whether archaeological skeletal samples are truly representative of deaths that
occurred in a prehistoric population. We touched
very briefly on this issue before (Konigsberg and
Frankenberg, 1994), and Hoppa (1996) wrote his
dissertation on the topic. Bocquet-Appel and Masset
(1996, p. 574) ran their simulations of mortality
patterns “representing various models of age distributions which could possibly be found in cemeteries,” though their “bimodal” age distribution is not
well-fit by any reasonable model of human adult
mortality. If the point of their simulations was to
show that paleodemographic reconstruction falls
apart when samples are nonrepresentative, demographically unusual, or simply out of the range of
expectation, then it is difficult to agree with their
intent. There is a misperception in the field that the
analyst has no recourse (and no interest) when an
age structure estimated for a target sample is driven
by cultural and archaeological sampling. To the contrary, even though we will get back a Gompertz (or
any other) model when we fit a Gompertz (or any
other) hazard model, this does not mean that we
lack internal checks on reasonableness.
We have already seen that the Gompertz model
for Loisy-en-Brie yields a deviance value of 0.1662,
which with 3 degrees of freedom gives a probability
of about 0.9828. Clearly, the Gompertz does a quite
reasonable job of reproducing the observed distribution of the indicator state. As an example of the
failure of the Gompertz to reproduce an observed
distribution, we can take a made-up vector with
counts in the indicator states of 10, 20, 50, 100, 50,
and 10. Using the reference information in BocquetAppel and Bacro (1997), we would estimate a Gompertz model with a3 ⫽ 0.01855 and b3 ⫽ 0.0019.
While these parameters look reasonable, they give a
likelihood ratio chi-square of 13.86, which with 3
degrees of freedom produces a probability of about
0.003. The Gompertz consequently is not doing a
good job of reproducing the distribution of indicator
states in this fictitious example.
Loisy-en-Brie also contained the remains of individuals who died at ages less than 23 years, as well
as 12 individuals who died at ages greater than 23
years but for whom the femoral head was unobservable. Bocquet-Appel and Bacro (1997) chose not to
include these individuals in their analysis, presumably because of the problem of possible underenumeration of subadults. This approach is logically
TABLE 2. Loisy-en-Brie age distribution for those
not aged by the femoral head1
From Bocquet-Appel and Bacro (1997: their Table 1 and p. 573).
Fig. 5. Hazard rate for Loisy-en-Brie when four 0 –1-year-olds
are added to the sample, as compared with the hazard rate from
model West 3 of Coale and Demeny (1966) for females.
Fig. 4. Hazard rate for Loisy-en-Brie, with a 95% confidence
interval (solid line) as compared with hazard rate from model
West 3 of Coale and Demeny (1966) for females.
inconsistent, because it excludes the subadults out
of hand, without ever considering whether or not
these individuals are properly represented in the
assemblage. We can incorporate the subadults
(listed in Table 2) and the 12 adults in whom the age
indicator was unavailable, treating these individuals as having interval-censored ages-at-death. In the
case of the 12 adults, we treat their ages as being
equal to or greater than 23 years.
With this additional information, we can fit a fiveparameter Siler model (Gage, 1988; Wood et al.,
2002). The survivorship from this model is:
S共t兲 ⫽ exp ⫺
共1 ⫺ exp共⫺b 1 䡠 t兲兲
⫻ exp共⫺a 2 䡠 t兲exp
共1 ⫺ exp共b 3 䡠 t兲兲 .
The estimated parameters are 0.0220, 0.1144,
0.0054, 0.0026, and 0.0507, with standard errors of
0.0110, 0.1491, 0.0108, 0.0011, and 0.0098, respectively. Figure 4 shows the 95% confidence interval
around the hazard rate for Loisy-en-Brie, as well as
the hazard rate from model Coale and Demeny’s
(1966) West 3 for females. The only departure from
the model life table appears to be the lower hazard
rate for infants at Loisy-en-Brie. In Figure 5, we
show the hazard rate redrawn by adding four 0 –1year-olds to the Loisy-en-Brie sample, again compared to model West 3. The hazard rates are remarkably similar after the addition of these four
There have been a number of suggestions in this
Journal as to how age estimation should proceed in
paleodemography (Bedford et al., 1993; Jackes,
1985; Lovejoy et al., 1985; Meindl et al., 1985, 1990).
Any discussion of paleodemographic methods,
whether life table- or hazard analysis-based, cannot
ignore the central issue of determination of age-atdeath. Both the method proposed by Jackes (1985)
and that developed by Meindl et al. (1985) in this
Journal suffer from the problem that the reference
sample age structure will influence age-at-death estimation. Bocquet-Appel and Masset (1982) first discussed the magnitude of this problem in paleodemography. We illustrate this problem for the
methods of Jackes (1985) and Meindl et al. (1985,
1990), and suggest ways to conduct age-at-death
estimation that both avoid such “mimicry” (Mensforth,
1990) and allow incorporation of uncertainty of age
Jackes (1985) suggested that the mean and standard deviation for age-at-death within pubic symphyseal stages could be used to probabilistically assign ages-at-death in a target sample. To implement
this, she assigned individual probabilities to integer
ages, using the cumulative normal truncated to the
middle 95% of the distribution. This method is dif-
ing information on the reference sample age distribution. We can see this by writing the mean age and
variance of age within indicator state i as:
a៮ 兩i ⫽
a Pr共i兩a兲f共a兲 da
V a 兩i ⫽
共a ⫺ 共a៮ 兩i兲兲 2 Pr共i兩a兲f共a兲 da,
Fig. 6. Log-normal distributions of age for stages 0 –2 (“stage
0”) and 3–5 (“stage 1”) of third symphyseal component in the
Korean War dead of McKern and Stewart (1957) and Terry Anatomical Collection samples.
ficult to justify, as it is highly unlikely that age is
normally distributed within stages (discussed further below). We explore a slightly different approach
here, which is to use the full normal probability
density function for age within indicator states. As
an abbreviated example, we use a scoring of the
third component of McKern and Stewart (1957) of
the symphyseal surface, where we have simply dichotomized between stages 0 –2 and stages 3–5. Essentially, this divides pubic symphyses into those
with incomplete ventral symphyseal rims and those
with complete rims.
We have scores for this trait on casts of 351 male
pubic bones from the Korean War dead study of
McKern and Stewart (1957), and 413 male pubic
bones from the Terry Anatomical Collection. Rather
than using the distribution of raw age within these
two stages, we use the natural log of age. For the
Korean War dead, the mean and standard deviation
of log age are 3.07 and 0.156 within stage 0 (original
0 –2 of McKern and Stewart, 1957), and 3.44 and
0.173 for stage 1 (original 3–5 of McKern and Stewart, 1957). The comparable figures from the Terry
Collection are 3.47 and 0.361 for stage 0 and 3.90
and 0.324 for stage 1. Figure 6 shows a plot of these
log-normal distributions. Now we are presented
with quite a dilemma. If we plan to use this aging
information for a target sample, should we use the
log normal from the Korean War dead, or from the
Terry Collection? More importantly, why are these
distributions so different?
The reason that the information from these two
collections is so different is that conditioning on
indicator, instead of on age, has the effect of includ-
where Pr(i兩a) is the probability of being in stage i
conditional on being exact age a, and f(a) is the
probability density function for age-at-death. To
model Pr(i兩a) we use a probit regression of the indicator on log age in the Korean War dead and in the
Terry Collection. For the Korean War dead, the intercept is 18.88 and the slope is 5.61, whereas for the
Terry Collection, the intercept is 6.65 and the slope
is 1.99. Viewed as a transition analysis (Boldsen et
al., 2002), the mean age-to-transitions for the Korean War dead and Terry Collection samples are
exp(18.88/5.61) and exp(6.65/1.99), or about 28.9
and 28.3 years, respectively. The only essential difference between these two collections in the distribution of age-to-transition is in the dispersion, with
the log scale standard deviation being 1/5.61 and
1/1.99, or 0.178 and 0.502, for the Korean War dead
and Terry Collection samples, respectively.
These two samples have markedly different ageat-death structures, and it is this factor that contributes to the different distributions for age-at-death
within pubic stages. Fitting a Gompertz model to
mortality at and past age 16, we obtain a3 ⫽ 0.082
and b3 ⫽ 0.053 for the Korean War dead sample, and
a3 ⫽ 0.011 and b3 ⫽ 0.039 for the Terry Collection
sample. Applying Equation 6 with the appropriate
Pr(i兩a) for each sample, we can recover the mean and
variance of the age-at-death distribution within
stage for each collection. On the log scale, these
values for the Korean War dead are 3.06 and 0.186
within stage 0, and 3.43 and 0.175 within stage 1.
For the Terry Collection these values are 3.41 and
0.376 within stage 0, and 3.91 and 0.341 within
stage 1. All of these figures agree well with the direct
calculations of means and standard deviations given
above. As a consequence of the age determination
method of Jackes (1985, p. 294) depending on the
reference sample age distribution, we must disagree
with her statement that her method would “alleviate
the problems of adult age assessment.” In fact, her
method only serves to formalize the critique that
Bocquet-Appel and Masset (1982) leveled against
age estimation in paleodemography.
The suggestion (Lovejoy et al., 1985; Meindl et al.,
1985, 1990) to seriate target samples on separate
age indicators, apply point estimates to each skeleton, and then weight the estimates via a principal
components analysis, suffers from the same problem
that we have noted for the method of Jackes (1985).
We do not, in principle, disagree with seriating tar-
get sample skeletal material on the basis of morphological indicators, although later this may call into
question the usual statistical assumption of individual variate values being independently distributed.
However, a major problem arises when ages are
assigned across the seriated sample. These ages
must be assigned in regard to a reference sample,
and as such they again depend in part on the reference sample age distribution. For example, suppose
we seriated the bones (or casts) when we scored
Todd phases for the Korean War dead and Terry
Collection samples. As a consequence, within each
phase for each sample we would have a smear with,
e.g., phase III represented at one end by pubes that
look more phase two-ish and at the other end by
pubes that look more four-ish.
Now we have the nontrivial problem of deciding
how to establish breaks in the seriation so that we
know where one integer age ends and a new age
begins. Meindl et al. (1985, 1990) gloss over this part
of the analysis by referring to seriating “age,”
whereas in fact we have seriated the indicator. They
also provide no suggestion for mapping the morphological series into discrete integer ages. Should ages
be assigned uniformly within indicator class on the
basis of the reference information? Further, as we
have chosen to seriate, do indicator classes even
exist anymore? Finally, how should we treat reversals where older individuals are seriated morphologically into the younger segment? In desperation we
could use the mean and standard deviation of age
within indicator classes from the reference sample
to establish ages in the target sample, assigning
integer ages so that they are approximately normally distributed within the target indicator
classes. This brings us full circle to the original
problem with the method of Jackes (1985), which we
summarized in Equation 6.
It is becoming increasingly clear that the only
logical way to proceed in estimating age in paleodemography is via Bayes’ theorem (Milner et al., 2000;
Hoppa and Vaupel, 2002). Specifically, we have:
f共a兩i兲 ⫽
Pr共i兩a兲f共a兲 da
which includes the unknown age-at-death distribution f(a). If Pr(i兩a) is a proper probability function,
then the integral across age for Equation 7 shown in
the denominator gives P̂r(i) the estimated probability that an individual will be in the ith class of the
indicator. As a consequence, the sum of the products
of counts within indicator classes with log(P̂r(i)) is
the log-likelihood for the age-at-death distribution.
In this article, we assume that the age-at-death
distribution can be represented by a hazard model,
so that the task now is to maximize the log-likelihood across the hazard parameters. This is precisely
what we did in the example from Loisy-en-Brie, and
had we wished to estimate individual ages, we could
then apply Equation 7.
Although we make use of Bayes’ theorem, it is
important to understand that the age estimates we
obtain are maximum-likelihood estimates, not
Bayesian ones. This is true because Equation 7 uses
an estimated hazard, rather than a prior distribution. Figure 7 shows the full posterior densities for
age-at-death in Loisy-en-Brie conditional on being in
each of the six indicator states. The sawtooth appearance of these densities is due to the fact that we
used the multinomial distribution (following Bocquet-Appel and Bacro, 1997) for being in a particular
indicator state conditional on being in a particular
age class. Aside from producing a very ragged-looking distribution of age-at-death, such an assumption
does not make very good use of the reference sample
data, because we should have exact ages from that
sample. A number of authors (Boldsen et al., 2002;
Holman et al., 2002; Konigsberg and Herrmann,
2002; Konigsberg and Holman, 1999) are currently
using parametric models that relate the indicator to
exact age, while others (Aykroyd et al., 1999; Love
and Müeller et al., 2002) have used nonparametric
models for the same purpose.
Equation 7 also shows why the method of Jackes
(1985, 2000, shown as “95% CLP” in some of her
figures) for age estimation is difficult to justify. We
can simplify the equation a bit by removing the
normalization (division by the integral) and rewriting Pr(i兩a) as a likelihood L(a兩i) to get:
f 共a兩i兲 ⬀ L共a兩i兲f共a兲,
where f (a兩i) is the distribution for age within an
indicator stage (i.e., the age distribution posterior to
our knowledge about the indicator) and f (a) is the
prior distribution for age. The method of Jackes
(1985) relies on the posterior following a normal
distribution, but makes no stipulations about the
distributional forms for the likelihood or the prior.
We will assume the simplest case, where the likelihood follows a normal distribution. This is a reasonable assumption for all but the first and last stages,
provided that we have modeled the age indicator
using a cumulative probit model, and that the mean
ages-to-transition (see Boldsen et al., 2002) are not
too widely separated across stages. In this case, the
likelihood for the first stage follows a normal distribution function (a cumulative normal), while the
likelihood for the last stage is a normal survivor
function (the complement of a cumulative normal).
If the likelihood is normal, then the prior must also
be normal in order for the posterior to be normally
However, there is no justification for assuming
that mortality, which is what the prior represents, is
normally distributed. The argument we give here is
based on the Bayesian concept of “conjugacy” (Lee,
1989; Gelman et al., 1995), where once we have
established the form for the likelihood, we pick a
form for the prior such that the posterior has the
Fig. 7.
Full posterior densities for age-at-death in the Loisy-en-Brie sample conditional on indicator state.
same distributional form as the prior (hence the
prior is referred to as a “conjugate prior”). For a
normal likelihood, the conjugate prior is also a normal distribution. It is possible that we could, in
theory, combine some as yet unspecified likelihood
with some other prior to get a normal posterior
distribution. However, our practical experience from
examining distributions of known ages within osteological stages is that ages are not normally distributed.
There are two cautionary notes that should be
made in regard to age-at-death estimation. The first
concerns the nature of estimated individual ages-atdeath. Although Equation 7 shows how to provide
age estimates for individual skeletons once a hazard
model has been estimated for the target, it is difficult to think of examples where one would actually
want individual age estimates in paleodemography.
Unlike forensic anthropology, paleodemography has
no particular currency in the stock and trade of
individual age estimation. On the other hand, a researcher might want to look at the relationship between age-at-death and some other variable, or set
of variables, from the bioarchaeological context. If
this is the goal, then a simple substitution of a point
estimate for age-at-death for each skeleton is an
affront to the data. Substituting point estimates neglects the fact that ages are estimated rather than
known, and gives the analyst false power in tests.
Konigsberg and Holman (1999) previously made this
point for studies of prehistoric somatic growth, while
Milner et al. (2000) extensively considered the statistical basis for individual age estimates.
The second cautionary note concerns the fact that
our reevaluation of the analysis by Bocquet-Appel
and Bacro (1997) of Loisy-en-Brie is based on the
summary data they provided. Far more could be
learned from reevaluating the raw data. For example, models that replace the F matrix (which we
have written as Pila) of Bocquet-Appel and Masset
(1996) with smooth functions will usually require
access to the raw data from the reference sample. It
is for this reason that we make available on the Web
our raw reference sample data (
Bocquet-Appel (1994, p. 201) closed a chapter he
wrote on “Estimating the Average Age for an Unknown Age Distribution in Anthropology” by suggesting, “One can again take an interest in paleodemography.” We agree with the general tenor of this
statement, although it would appear from the voluminous response to the original critique by BocquetAppel and Masset (1982) that paleodemography is a
subject in which physical anthropologists have never
particularly lost interest. We would strengthen the
statement of Bocquet-Appel (1994) to, “We should
take an interest in paleodemography,” and we take
the remainder of this conclusion to “deconstruct”
this statement. First, the emphasis in paleodemography should be on “we,” not “me,” “I,” “one,” or “us”
(as vs. “them”). There have been too many fractious
statements made concerning paleodemography
(Bocquet-Appel and Masset, 1982, 1996; Petersen,
1975), and rarely do these comments provide suggestions on how to improve the field. In contrast, the
past few years have seen a number of collaborative
efforts in paleodemography (Hoppa and Vaupel,
2002; Paine, 1997), and these have had and should
continue to have a very positive effect in advancing
the field beyond the rote calculation of life tables.
Second, we should take an interest in paleodemography because of the inextricable link between demography and genetics in evolution, whether that
evolution is in humans, primates, or mammals (e.g.,
the treatment of the “evolution of the human life
cycle” in Bogin and Smith, 2000). Additionally, the
relationships between demography and ecology remain to be explored, and at least some of that exploration will likely include human prehistory as a
component (e.g., the comparative discussion of primate demography in Gage, 1998). There is a vast
amount to be learned about past demography, and
we are just beginning to enter a period when many
questions stand a reasonable chance of leading to
reasonable answers.
The Korean War dead and Terry Collection sample data were collected under funding from the Na-
tional Science Foundation (SBR-9727386). We are
especially grateful to Nick Herrmann and Danny
Wescott for helping the first author collect the data,
and to Dave Hunt at the Smithsonian for his tireless
assistance with the Terry Anatomical Collection. We
benefited from the useful and detailed comments
provided by Jim Wood, Emőke Szathmáry, Rob Hoppa,
Darryl Holman, and the anonymous reviewers.
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