AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 89935-256 (1992)

Estimation of Age Structure in Anthropological Demography LYLE W KONIGSBERG AND SUSAN R FRANKENBERG Department of Anthropology, Uncversrty of Tennessee, Knoxville, Tennessee 37996

Paleodemography, Iterated age-length key, ComKEY WORDS puter simulation ABSTRACT The past decade has produced considerable debate over the feasibility of paleodemographic research, with much attention focusing on the question of reliability of age estimates. We show here that in cases where age is estimated rather than known, the traditional method of assigning individuals to age classes will produce biased estimates of age structure. We demonstrate the effect of this bias both mathematically and by computer simulation, and show how a more appropriate method from the fisheries literature (the “iterated age length key”) can be used to estimate age structure. Because it is often the case that ages are also estimated for extant groups, we suggest that our results are relevant to the general field of anthropological demography, and that it is time for us to improve the statistical basis for age structure estimation. We further suggest that the oft noted paucity of older individuals in skeletal collections is a simple result of the use of inappropriate methods of age estimation, and that this problem can be rectified in the future by using maximum likelihood estimates of life table or hazard functions incorporating the uncertainty of age estimates. o 1992 Wiley-Liss, Inc

Over the past decade the field of paleodemography has undergone a resurgence of activity that has led to a number of contributions to method and theory (Meindl et al., 1983; Sattenspiel and Harpending, 1983; Buikstra et al., 1986; Johansson and Horowitz, 1986; Boldsen, 1988; Gage, 1988; Milner et al., 1989; Paine, 1989a,b; Siven, 1991; Whittington, 1991). Much of the recent interest in paleodemography has focused on analytical problems that plague this area of research. In the main, these analytical difficulties are not unique to paleodemography, but rather are symptomatic of anthropological demography in general. The stumblingblocks faced in many anthropological demography studies, and virtually all paleodemographic studies are that: 1)population growth rate may not be known, 2) samples may not be representative, and 3) ages are usually estimated rather than known. The first problem has been discussed by Sattenspiel and Harpending (19831, Johansson and Horowitz (1986), 0 1992 WILEY-LISS. INC

Buikstra et al. (1986), Gage (19881, Horowitz et al. (1988), Paine (1989a,b), and Milner et al. (19891, and some suggestions have been made concerning remedies to the problem of unknown growth rates. The second problem has been discussed by Weiss (1973:12-13) for living and skeletal populations and by Walker et al. (1988) in the paleodemographic setting. The third problem of unknown ages has seen heated debate in the paleodemographic literature (BocquetAppel and Masset, 1982, 1985; Van Gerven and Armelagos, 1983; Bocquet-Appel, 1986; Greene et al., 1986; Lanphear, 1989; Piontek and Weber, 1990), and serves as the basis for this paper. While the quality of age estimates is much better for extant populations with systems of vital registration, in the absence of written records anthropological demographic studies must rely on reported or estimated Received August 6,1991; accepted March 9.1992.

236

L.W. KONIGSBERG AND S.R. FRANKENRERG

ages that are often inexact (see e.g., Roberts, 1956:328; Chagnon, 1972:270; Nee1 and Weiss, 1975:27; Cavalli-Sforza, 1977:276). Indeed, Townsend and Hammel(1990) have recently suggested that the age structure of subadults might be best studied using dental age estimates (from numbers of erupted teeth) rather than reported ages. If age estimates in extant populations are determined on the basis of morphology or behavior, then the distinction against paleodemography begins to blur. As an example, consider the rase o f the Semai, where Fix (1977.37) argues that many subadults were incorrectly aged because the census-takers defined age classes on behavioral or social characteristics. Van de Walle’s (1968:13) comments on demography in Tropical Africa are also particularly telling. He noted that “all African demographic surveys share the problem of trying to record the ages of people who do not know their exact ages and are not fundamentally interested in knowing them.” Only recently have researchers emphasized that uncertainty in age estimates does (and should) cast some doubt on results from anthropological demography studies. For example, Gage et al. (1989:48) note “The high rate of aging in the paleodemographic tables and in the Yanomama is particularly intriguing. It may be a result of the error intrinsic in estimating ages in ethnographic populations and ages at death in paleopopulations, or it might represent some real difference in the dynamics of aging in these populations.” In a similar vein, Howell (1982) presented the following scenario: Two alternative interpretations arise from this consideration of the Libben life table. The more straightforward interpretation is to conclude that we now have empirical evidence in the form of this skeletal collection that life was far more difficult for prehistoric people in North America than we have observed it to be anywhere in the world during the 19th and 20th centuries. The alternative interpretation is to treat the results as a “reductio ad absurdum” shedding doubt an the literal accuracy of the life table itself. Perhaps the wisest course at this point i s to try to design research which tests hypotheses derived from the first alternative, while research on techniques of aging and sexing skeletons and studies on the preservation probabilities of human skeletal material continue.

TABLE 1. Example of reference sample age estimation information from McKeni and SterciartS (1957) third component of the pubic symphysis --____ Age a t death‘ 0 _____.-.__-.

Component stage . 1 .. 2 __-..___.._ 3 4 5 _____

16.50-24.49 24.50-30.49 30.50-39.49 39.50 +

29 14 0 0

202 0 0 0

~

2 29 5 0

3 16 1 2 0

0 2 2 6 5

0 0 2 2

‘Years,

It is this latter endeavor that WP p n r w e here. We depart, however, from the recent concern with identifying better “age markers” and methods of scoring them (see, e.g., Iscan, 1989a1, and instead focus strictly on the statistical basis for estimating age structure of populations. Starting from the position that much of anthropological demography, and all of paleodemography, is based on age data that are not known with certainty, we examine recent developments from wildlife management that will be useful in analyzing problems in human biology. In particular, we borrow heavily from the fisheries literature, where the problem of age structure estimation from morphology has been dealt with in an elegant manner (Kimura, 1977; Westrheim and Ricker, 1978; Clark, 1981; Bartoo and Parker, 1983; Fournier and Breen, 1983; Kimura and Chikuni, 1987). We begin with a discussion of the traditional methods for population age structure estimation applied in paleodemography, and then proceed to consider more appropriate methods that should have broad applicability in both paleodemography in specific, and anthropological demography in general. Following a discussion of a univariate method for discrete age indicators (due to Kimura and Chikuni, 1987), we present our extensions to the continuous indicator and multivariate cases. TRADITIONAL AGE ESTIMATION: A BAYESlAN APPROACH

As an example of a traditional means for estimating an age distribution in an anthropological sample, we present in Table 1 a collapsed version of NIcKern and Stewart’s

ESTIMATION OF AGE STRUCTURE

(1957) tabulation of component I11 pubic

symphyseal development against age for their sample of Korean War dead. We refer to the Korean War dead, or any other sample of known aged individuals as a “reference” sample, as this sample provides the reference information on age development which we wish to apply to a sample with unknown ages, The sample with unknown ages we will refer to as a “target” sample. The target sample could be a living anthropological population or a paleodemographic sample, as in lhe cuiieiii e~diiiple.:xi the case of a living population, the target sample may be aged on the basis of a morphological indicator (see e.g., Townsend and Hammel, 1990) or with reference to reported ages. The reference sample will have its oyn age distribution, which we symbolize as d,, where a represents the discontinuous age classes indexed from 1to ui. There are consequently w number of age classes, not necessarily of equal length. In the example from McKern and Stewart (see Table 11, there are four age classes, with da, the proportion of deaths in each age interval equal to 0.6762, 0.1748, 0.1289, and 0.0201. Using the information on distributions of pubic symphyseal component stages against ages, we wish to estimate the age distribution [or a target sample, which we symbolize as d,. Many different ways have previously been suggested for assigning ages on the basis of indicator states, but we will use a fairly simple method here. If in the target sample we observe an individual to be in a particular stage of the indicator state, say stage 2 of the third pubic symphyseal component, then we will probabilistically assign this individual to each age category on the basis of the reference sample. For this example, if we observe an individual to be in stage 2, then the reference sample tells us that the individual has 2/36 (or 0.056) chance of being age 16.50 to 24.49, 29/36 (or 0.806) chance of being age 24.50 to 30.49, 5/36 (or 0.139) chance of being age 30.50 to 39.49, and 0136 (or 0.000) chance of being 39.50 years old or older. The fact that an individual is fractionally assigned to age intervals should pose no problem, as there is no reason to require that the number of indi-

237

viduals in intervals be restricted to whole numbers. By summing the column for stage 2, and then dividing the number in each age group by the column total, all we have done is apply Bayes theorem. Specifically, we have found the probability of being in a particular age class conditional on our knowledge of the indicator state. We can formalize this procedure with the inclusion of just a few extra symbols. Lettingp,, be the probability from the reference sample of being age a conditions! on being in indicator state t, and conversely pla be the probability of being in indicator state i conditional on being in age interval a, Bayes theorem gives:

Returning to our original example, if a pubic symphysis is observed to be in stage 2 of the third symphyseal component, then we can find the probability that the individual is between 24.50 and 30.49 years of age as

(0.4754 x 0.1748)/(0.0085 x 0.6762 + 0.4754 x 0.1748 + 0.1111 x0.1289) = 0.81, identi-

cal with the previous calculation. Although the use of Bayes theorem above appears to be an annoying and tedious way t o write something that we can find on much simpler terms, use of the theorem allows us to examine mathematically the validity of various arguments put forth originally by Bocquet-Appel and Masset (1982) and by Van Gerven and Armelagos (1983). Additionally, use of the theorem demonstrates that this more or less traditional approach to age estimation is identical to the “age length key” method first published in the 1930s in the fisheries literature. The “age length key” method, which relies on the distribution of discretized measurements of fish lengths against known ages in a reference sample to estimate ages from lengths in a target sample, is identical to the Bayesian approach we have just described. More importantly, the method has been shown to produce biased estimates of the age distribution in most reasonable circumstances

238

L.W. KONIGSBERG AND S.R. FRANKENBERG

(Kimura, 1977; Westerheim and Ricker, 1978),and has consequently been revised by Kimura and Chikuni (1987). While our descriptions of the Bayesian method, or “age length key,” has so far centered on the estimation of ages for single individuals, the method can easily be applied to a complete target sample of individuals. If fl is the frequency of individuals in the target sample observed to be in age indicator state i, where i i_sindexed from 1 to n, then the estimates of d, in the target sample are:

An important point to note in equation 2 is that the estimates of the age distribution in the target sample appear t o be at least partially dependent on the age distribution of the reference sample. Clearly, this is not a desirable property of an estimator, as we would like to think that the reference sample age distribution and the estimate of the target sample age distribution are independent. In order to clarify the nature of the relationship between target and reference age distributions, we consider a number of special cases in the following sections. AN ABYSMAL AGE INDICATOR

Bocquet-Appel and Masset (1982)make the argument that if age prediction is very uncertain, then the estimated age distribution for a target sample will approximate the age distribution of the reference sample. Mensforth (1990)has referred to this as a problem of “age mimicry,” and attempts to show through an empirical example that his specific target distribution differs from his reference. We show here that the (estimated) target and reference sample age distributions will be equal if the age indicator is abysmal (i.e., unrelated to age). If a suggested “age” indicator bears no relationship whatsoever to age, thenp,, = lln for all a, where n is the number of indicator

states. In other words, for any age a we expect a uniform distribution of indicator states. Substitutingp, = lln in equation 2 yields:

(3)

so that the estimated target sample age distribution exactly equals the reference sample age distribution. This is the ultimate “age mimicry,” but of course we can safely assume that no “age”indicator is completely independent of age. Consequently, we consider an equally unlikely special case, where the age indicator is perfectly correlated with age.

A PERFECT AGE INDICATOR

If an age indicator assigns individuals to age classes with complete certainty, then for an age class a, there will be only one probability p l a equal to one, while all other p L afor the age class will equal zero. Furthermore, it must be the case that the number of indicator states is equal to the number of age classes. In this case, equation 2 can be rewritten as:

where 6, equals 1 when a = i, and 0 otherwise. The reference age distribution is nowhere to be found in equation 4, and in point of fact not only are the target and reference age distributions independent, but the target distribution is estimated with complete certainty. Unfortunately, the search for a perfect age indicator is probably futile, so this second case is no more realistic than the first.

ESTIMATION OF AGE STRUCTURE

BIAS IN THE TRADITIONAL METHOD OF AGE ESTIMATION

Both the cases of an abysmal age indicator and a perfect age indicator are clearly unrealistic. They represent opposite ends of a continuum, with any real age indicator lying somewhere in between. Age indicators are neither perfect, nor are they completely unrelated to age. As a consequence of this fact, when the Bayesian approach (or “age length key”) is applied to an anthropological or paleodemographic target sample, we will get an estimated age distribution which is neither a complete “mimic” of the reference sample nor completely independent of the reference. Given this fact, there is little reason to make empirical comparisons of reference and target age distributions (e.g., Van Gerven and Armelagos, 1983; Mensforth, 1990) or of estimated and real age distributions (Piontek and Weber, 1990; Lanphear, 1989). We know a priori (see Kimura, 1977; Westrheim and Ricker, 1978) that the estimated target age distribution will be biased in virtually all cases. Using the “age length key” (or Bayesian method), an unbiased estimate of the age structure will only be produced when the reference and target sample have the same age distribution, when the reference sample has a uniform age distribution, or when the age indicator is perfect. Bocquet-Appel (1986) suggested that if the reference sample has a uniform age distribution, then the target sample age distribution will be estimated independent of the reference. In Bayesian terms, Bocquet-Appel argues for selecting an “uninformative prior” in order to avoid the relationship between the target and reference age distributions. In this case, d, = l f w for all a, where w is the number of age classes. Returning to equation 2:

239

where, like in equation 4,the reference age distribution is not included. While this solution does consequently remove the problem of dependence between the target and reference age distributions, it is not in general a practical way to proceed. The chief problem with selecting a reference sample with a uniform age distribution is that this requires discarding data, which certainly cannot be an efficient way to proceed. Fortunately, Kimura and Chikuni (1987) have recently shown that an earlier simple method due to Chi!c,uni, known as thc “”icratcd agc !cn,“,E: key” solves the problems of the Bayesian estimator described above, while not requiring a uniform age distribution in the reference sample. In the following, we describe the “iterated age length key” within the framework of maximum likelihood estimation, and then extend the technique to cover multiple age indicators, as well as continuous indicators (such as counts of Haversian canals, or numbers of incremental bands in cementum). At each juncture we will demonstrate the use of maximum likelihood estimation in comparison to more traditional means of age estimation when applied to simulated data sets. MAXIMUM LIKELIHOOD ESTIMATION OF THE AGE DISTRIBUTION

In order to obtain maximum likelihood estimates (NILE) of the target sample age distribution, we need to use information from the reference sample in order to obtain the target sample age distribution most likely to have produced the observed distribution of target sample (age) indicator states. We can write the indicator state distribution for the target sample as F,, where now the capital F denotes counts within each indicator state i. The sum of the F, is equal to N , the total number of individuals, or in the case of paleodemography, skeletons. Again, we let a lower case n be the number of indicator states. From these parameters the probability of obtaining an individual in the target sample who is in a particular indicator state i (symbolized asp,) is:

240

L.W. KONIGSBERG AND S.R. FRANKENBERG

where the probability now depends on the unknown age distribution of the target sample (rather than on the age distribution of the reference sample) and the conditional probabilities pL0of indicator state given age in the reference sample. In other words, we make one of Howell’s (1976) uniformitarian assumptions (that the target and reference samples age in the same way) but do not impose aspects of the reference sample age distribution on the target sample. Using equation 6 and the multinomial theorem, the probability of obtaining the entire set of individuals (or skeletons) arrayed across indicator states in the target sample is:

that the “iterated age length key” is an EM algorithm that provides maximum likelihood estimates of d, has already been presented in Kimura and Chikuni. The interested reader is referred to their important work, as well as t o Dempster et al. (1977). Following Kimura and Chikuni (1987), we do repeat the “iterated age length key” method here in simple terms so that it can be used to obtain maximum likelihood estimates of d,. In the “iterated age length key” method xve begin with an initial estimate of d,, for which Kimura and Chikuni (1987) suggest a uniform distribution. In point of fact, any distribution that does not contain zeros and which spms to one is acceptable. From this initial d , distribution the estimated probability of being age a conditional on being in age indicator state i is calculated from Bayes theorem as:

This is the likelihood of the estimated target sample age structure conditional on the observed indicator state data, which we symbolize as L(d,lFJ. The maximum likelihood estimate of d, is then the vector of proportions in age classes for the target sample a t which L(d,lFJ is maximized. Because this maximum will also be identified if the likelihood is converted into a log scale, and because the factorial term in equation 7 is invariant t o changes in d,, the maximum likelihood estimate will also occur when the following function is maximized:

This is an expectation step (or “E” step), in that the expected values ofp,, are found conditional on the current estimates of d, for the target sample. This “age length key” from equation 9 can then be applied to the observed distribution of indicator states in the target sample to obtain a new estimate of the target sample age distribution:

The maximum of this equation (and the corresponding maximum likelihood estimates) can be found by numerical methods, but by far the simplest method is t o apply an iterative algorithm due to Kimura and Chikuni (1987). Kimura and Chikuni point out that an iterative application of the “age length key” method used in the fisheries literature constitutes an expectation-maximization (or “ E M ) algorithm. EM algorithms are twostep iterative methods that lead to maximum likelihood solutions in many practical applications (Dempster et al., 1977). Proof

We use the prime symbol to indicate that equation 10 gives a new estimate of the age distribution. This is a maximization step (or “ M step) in that we are finding the maximum likelihood estimate d, a t the current estimates of pa,. Now the current maximum likelihood estimate of the age distribution (from equation 10)can be inserted into equation 9 to obtain a new estimate of the probability of being age a conditional on being in age indicator state i. The result from equation 9 can be reapplied in equation 10, and the cycling continued until the estimated age distribution converges. Because the log-

241

ESTIMATION OF AGE STRUCTURE

TABLE 2. Probabilities of being in uarious stages of McKern and Stewart’s 119571 third pubic symphyseal component conditional on known age Comnonent stare ~

Ace class’

0

1

2

3

4

5

16.5-21.4 21.5-22.4 22.5-23.4 23.5-24.4 24.5-25.4 25.5-26.4 26.5-27.4 27.5-28.4 28.5-30.4 30.5-39.4 39.5-50.0

0.9882 0.7619 0.5172 0.1875 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0118 0.2381 0.4828 0.5000 0.3750 0.3846 0.3333 0.1538 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.1250 0.3750 0.5385 0.2500 0.5385 0.6000 0.1111 0.0000

0.0000 0.0000 0.0000 0.1875 0.2500 0.0769 0.4167 0.3077 0.2667 0.2667 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1333 0.5778 0.7143

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0000 0.0444 0.2857

‘Years.

likelihood can be calculated a t any step using equation 8, the simplest procedure is to define convergence as the point where the change in the log-likelihood between any two steps is less than some very small value. In the following section, we apply both the “iterated age length key” and the Bayesian age estimation method to simulated data in order to demonstrate the desirable properties of the former method. APPLICATION OF THE MAXIMUM LIKELIHOOD METHOD FOR ESTIMATING AGE STRUCTURE We can continue with the example of age

estimation begun previously with the third pubic symphyseal component in Table 1.Table 2 lists the probabilities of being in varrous component stages (McKern and Stewart originally numbered these 0 to 5, so we preserve their coding here) conditional on age. These probabilities were obtained from McKern and Stewart’s (1957) Table 25, and they represent the age information from the reference sample (p,,) necessary to apply the “iterated age length key” method. Although age ranges are not as greatly collapsed as in Table 1, we have grouped some ages together where the pubic stage cannot discriminate between age classes or where the data were sparse. In order to simulate a target sample whose age structure differed from the McKern and Stewart sample, but whose individuals aged in similar fashion, we undertook the following Monte Carlo procedure.

We first drew a sample of 500 deaths from a mortality profile (see Moore et al., 1975, for this procedure) which differed substantially from that of the McKern and Stewart war dead. We then stochastically applied the information in Table 2 for each individual to probabilistically assign a pubic stage conditional on their age. Following this, the assigned pubic stages were used to estimate the age-at-death structure for the simulated target sample. The d, distribution was estimated both by the Bayesian method and using the maximum likelihood “iterated age length key” method. We repeated this simulation 100 times. Figure 1 shows a comparison of the McKern and Stewart probability of death function, the generating probability of death function, and the average estimated probability of death function (across 100 replicates) using the Bayesian method. Comparison of these three curves clearly demonstrates that the Bayesian method recovers the reference sample age distribution rather than the target sample through much of the later years. This is an undesirable feature of a technique that we consider to most closely approximate that traditionally used by paleodemographers. In contrast, in Figure 2, where the age structure is obtained by maximum likelihood estimation, there is a much better fit between the estimated target sample age distribution and the true age distribution of the target. While the example given here is drawn from a paleodemographic setting, applications for living populations are obvious. In

L.W. KONIGSBERG AND S.R. FRANKENBERG

242 1.0

,

1=

0.9

4

a, u

0.8

3

- h

(c

0

2?

.a

0.7

._I

0.6

(d

a

2

0.5

Q

0.0 17

22

23

24

25

26

27

28

29

31

40

Age Fig. 1. Comparison of the probability of death functions from the Bayesian method of estimation from the pubic symphysis, the generating target sample, and the reference sample (McKern and Stewart, 1957). 1.o

- McKern and Stewart

J=

0.9-

- ---- -- -. Target

Q

0.8

3 a, 2?

.5

II I

MLE method

Y-

O

f

0.7 0.6 -

(d

a

g o

.-

0.5 0.4

’c

0

Q)

0.3-

Q

;

0.2

CD

a

0.1 00

~~

17

22

23

24

-

~

25

26

-

27

_ _ - 28

-

29

31

__

40

Age Fig 2 Comparison of the probability of death functions from the maximum likelihood method of estimation (MLE) from the pubic symphysis, the generating target sample, and the reference sample (McKern and Stewart, 1957)

cases where morphological or behavioral data have been used to determine ages, the researcher must know the conditional distribution of the indicator against known age in a reference sample. This information can then be used to find the maximum likelihood estimates of the age distribution for the tar-

get sample. In cases where the researcher is using reported ages to estimate the demographic structure, it may be possible to specify the conditional distribution of reported against real age for a portion of the population which is subject to vital registration. For example, Bairagi et al. (1982) and Cald-

ESTIMATION OF AGE STRUCTURE

well (1966) present such information for children’s ages in Bangladesh and Ghana, respectively. Additionally, Bhat (1990) has recently presented a method for estimating these conditional distributions which does not require a reference sample, but which does have stringent data requirements and carries a number of assumptions. MULTIVARIATE FORM OF THE “ITERATED AGE LENGTH KEY” The results from Figures 1 and 2 suggest that thc maximurr, likclihoad mcthor! pro-

duces unbiased age structure estimates, while the traditional method is biased. This demonstration is, however, rather simplistic because only a single age indicator is used. There have been a number of calls now for “multifactorial aging” methods (Acsadi and Nemeskeri, 1970; Meindl et al., 1983; Lovejoy et al., 1985; Iscan, 1989131, correctly noting that more than just one criterion should be applied when estimating ages from morphological observations. In this section we extend Kimura and Chikuni’s (1987) results to cover more than one age indicator, so we can examine the performance of “multifactorial aging” methods. In the Bayesian method, if we consider (for example) three age indicators represented as i, j , and k , then equation 2 generalizes to:

243

sified by age indicators i, j , and k. Similarly, the EM algorithm or “age length key” equations shown in equations 9 and 10 generalize to:

a=l

and

where f i f h represents the frequency of individuals in the target sample observed to be in the ith,jth, and kth states of the three age indicators (summing to one across all i, j , and k). AN EXAMPLE OF THE MAXIMUM LIKELIHOOD ESTIMATION OF AGE STRUCTURE FROM MULTIPLE DISCONTINUOUS CHARACTERS In order to apply equation 11 (the Baye-

sian method) or equations 13 and 14 (the maximum likelihood method) it is necessary to have reference sample information where multiple indicator states are cross-classified against known age. Because of the immense size of such a table, this kind of information is rarely if ever published in the literature. For example, the original tabulations for the McKern and Stewart war dead study i1957) present numerous classifications of single indicators against age, but never a crossclassification of even so few as two indicawhere n,, nJ, and nk signify the number of tors against age. Similarly, this kind of instates within age indicators i, j , and k, and formation is unavailable from most of the more extensive modern studies of age deterPqkrr is the probability from the reference sample of being in the ith,jth, and kth states mination methods (Suchey et al., 1984; of the three age indicators conditional on Webb and Suchey, 1985; Loth and Iscan, being in the uth age class. For the maximum 1989;Meindl and Lovejoy, 1989) in humans. likelihood estimator of d,, the log-likelihood As a result of the paucity of this kind of function shown in equation 8 can be general- information in humans, in the following example we utilize multiple morphological obized to: servations on a reference skeletal sample from the Cayo Santiago rhesus macaque colony. Falk et al. (1989) have recently reported on the progression of endocranial suture clowhere FUkindicates the counts of numbers of sure using a sample of 330 rhesus monkeys individuals in the target sample cross-clas- from the Cay0 Santiago skeletal collection.

L.W. KONIGSBERG AND S.R. FRANFXNBERG

244

TABLE 3. Cross-tabulation of number of m,oiikeys o f known age against endocranial suture closure states for Cay0 Santiago rhesus macauues‘

No. -

Age iyr -I

0 00-0 99

1

Age (yr! __ __ ___

No. __ 1 1

5 00-5 99

7 1 1 2 2 2

1

1

1.00-1.99

11 2

1 7 3 3 1 1

2.00-2.99

3.00-3.99

4.00-4.99

i0,1,0) (0,1,1) 10,2,01 (0,2,1) (1,LO) (1,1,1) (1,2,0) i2,1,1) (2,2,1) (0,1,0) (O,l,l) (0,2,0) (0,2,1) (0,2,2) (1,1,0) il,1,1) (1,2,0) (1,2,1) (2,1,1) (2,2,0) (2,2,1) (0,1,0) 1) i2.2,O) (2,2,1) (2,2,2) (3.2.1)

2 2 6

1

8.00-8.99

1 1 9 4

9.00-9.99

1 10 6 1 1 1

10.00-14.99

99

i y

2‘5 1 9 20 1 1 2

15.00 +

4

2 1 2 2 2 6 2 3

After Falk et al., 1989. of listed sutures is hasilar, sphenotemporal, and caudal squamosal. 0 = open, 1 = closing, 2

They scored a series of sutures as open (O), closing (l),closed ( 2 ) ,or obliterated (3) (see Falk et al., 1989, for further definitions of these states). For this example, we consider closure of three sutures: the basilar, sphenotemporal, and caudal squamosal sutures,

=

closed, 3 = ohliterated.

because their development spans much of macaque ontogeny. Table 3 contains a summary of the tabulation of suture closure states against age for 340 rhesus macaques. An additional 10 macaques over the original Falk et al. (1989) sample are included in this

ESTIMATION OF AGE STRUCTURE

245

10

.r,

0.9

CEl

Q)

~3

0.8

O,o-?--

~-r-

-

0

1

2

3

5

4

6

7

8

9

10

15

Age Fig 3 Comparison of the probability of death functions from the Bayeslan method of estimation from cranial suture closure in rhesus macaques, the generating target sample, and the reference sample (Falk et a1 , 1989)

table because of the less stringent requirements concerning missing data. Although with 12 age classes and three indicators each with four states there should be 768 cells within Table 3, 670 of the cells contained no observations, so they are not included here. In order to form target samples against which to apply the Bayesian and maximum likelihood methods, we formed bootstrap samples out of the original observations on the 340 macaques. So that the age structure of the target samples would systematically differ from the reference sample, we first simulated 340 deaths using the life table obtained from census data (Sade et al., 1976). The life table from census data and from the skeletal sample are by no means equivalent, because of the nonrandom factors that have affected formation of the skeletal sample and collection of the endocranial suture closure data. Specifically, newborns are substantially underrepresented in the skeletal sample used here, in large measure because endocasts (the source for the suture closure data, see Falk et al., 1989)cannot usually be made on these younger animals. After forming a target sample age-at-death distribution, we sampled with replacement from each age class in the reference sample, se-

lecting the number of individuals based on the simulated number of deaths within intervals. Because of the fact that we sampled with replacement, this is a bootstrap sample, or more correctly, a bootstrap sample which is stratified on age. From the tabulations of suture closure information in Table 3 and the simulated distributions formed by the stratified bootstrap, the target sample age distribution can be estimated using equation 11 ithe Bayesian method) or equations 13 and 14 (the maximum likelihood method). Figure 3 shows a comparison of the average probability of death function (for 100 replicates) estimated by the Bayesian method with the reference and generating target sample functions. As in the previous example from the McKern and Stewart war dead, the Bayesian method appears to recover a biased estimate of the probability of death function, largely mirroring the reference sample. In Figure 4, which shows the average of the maximum likelihood solutions, the estimates are clearly recovering the structure of the target sample, as we hoped. It is therefore readily apparent from this example, that the addition of multiple indicators to the problem of age estimation cannot “patch up” the Bayesian method.

L.W. KONIGSBERG AND S.R. FRANKENBERG

246

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Logit of True Target Mortality Fig. 5 . Plot of logits of estimated and observed mortality for Cay0 Santiago macaques against the logit of true target mortality.

To more clearly represent the relationship maximum likelihood methods), the true tarbetween the estimated target and reference get mortality, and the reference mortality, sample age distributions, Figure 5 shows a all plotted against the logits of the true tarplot of mortality in logit scale (see Brass get mortality. On the logit scale, a plot of the and Coale, 1968:127-129; Brass, 1969). estimated mortality profile (against the true Following Brass, the logit is defined as target) which precisely equalled the true %ln[(l - LJl,.., where 1, is survivorship to target profile would pass through the origin age x. Figure 5 is a plot of logits from the and have a slope of one (shown as a solid line estimated target mortality (by Bayesian and in Figure 5). Table 4 shows summary statis-

ESTIMATION OF AGE STRUCTURE

247

To apply the Bayesian method to a continuous age indicator we need to rewrite pa, represented by the parenthetical term in Source Coefficient S.E. Pl equation 2. The term pLain equation 2 now Bayesian method represents a univariate probability density constant --0,0548 0.0179 0.0157 0.1228 0.0049 function for the indicator i, conditional on target 0.5284 age. To make clear the fact that the indica0.0649 c _ 0.0001 0.5996 reference Maximum likelihood tor is now a continuous variable, we symbolmethod ize this conditional probability asp,,, where 0.0228 0.8867 constant -0.0033 0.1559 0.9060 x represents the observed value of the contarget 1.0190 0.0354 0.0824 0.6786 reference tinuous age indicator. If the indicator is nor‘Probability vdlues are for the hypotheses that the constant is zero, mally distributed within age classes, then the target’s coefficient IS one, and the reference’s coefficient is zero function N - 11age intervals Aajustea muitipie E* = v fi, ,liZ s ~ the~ prohahility ~ , density ~ ~ ~ forx ~ is conmethod regression and 0 997 for the MLE method regression ditional on the within age class variance and mean of the indicator from the reference sample. p,, is then given as: tics from the multiple regressions of logittransformed estimated mortality on the logit-transformed true target and reference mortality profiles. This table firmly establishes that the Bayesian method provides a significantly biased estimate of mortality, while the maximum likelihood method pro- where pa and CT’, are the expected value and vides an estimate which is not significantly variance ofx conditional on being in the ath different from the true target. Further, the age class. Up until this point, this method of size and sign of the coefficients indicates age estimation is identical with that given that the Bayesian method recovers a mortal- by Jackes (1985). ity profile which (like the comparison of the We modify Jackes’ (1985) method by takreference to the target) is too high, save for ing pa to be the predicted value of x from a the initial age interval. regression of the indicator on age in the reference sample, and use the midpoint of the ESTIMATION OF THE AGE DISTRIBUTION age interval to obtain the predicted value. FROM A CONTINUOUS INDICATOR We further assume homogeneity of the reAlthough we have to this point only con- gression residuals within age classes, so sidered age indicators that are scored dis- that C T ~is a constant across all age classes, continuously, there is nothing that pre- This is not a necessary assumption, but it is cludes application of the previous methods one that is traditionally used in anthropoto continuous indicators. A number of indi- logical age estimation. With this assump~ now the standard error of the esticators used in paleodemography, such as sub- tion, C T is adult long bone lengths (Sundick, 1978; mate of the regression of the indicator on Scheuer et al., 1980; Ubelaker, 1987), ce- age in the reference sample. Now pxa can be mental annulations (Charles et al., 19861, used (forp L J in equation 9 to extend the EM root dentine transparency (Drusini et al., algorithm to cover the case of a continuous 19911, and histomorphometry (Kerley, 1965; indicator. This usage is identical with the Thompson, 1981; Ericksen, 1991) are in fact finite mixture case (McLachlan and Basmore or less continuous in nature. Addition- ford, 1988; Titterington et al., 1985; Everitt ally, some ordinal categorical character ex- and Hand, 19811, where the estimates of da pressions have been treated as continuous represent mixing proportions and the pZa traits (Hanihara and Suzuki, 1978; Katz are normal densities that are not estimated. and Suchey, 1986). In this section we conse- In the next section we discuss a n example quently extend the Bayesian and the maxi- using a regression of cementa1 annulation mum likelihood methods to cover the case of counts on known age to estimate the age structure for a target sample. a single continuous age indicator. TABLE 4. Multiple regressions of logits of estimated mortality for Cay0 Santiago macaques on the target and reference sample lopits

973

L.W. KONIGSBERG AND S.R. F W K E N B E R G

248

AN EXAMPLE OF THE MAXIMUM LIKELIHOOD ESTIMATION OF AGE STRUCTURE FROM A CONTINUOUS CHARACTER For our example of a continuous character used in age estimation we consider counts of cementum annulations (Condon et al., 1986). Cementum annulations are annual rings that form in cementum, the material lining tooth roots. Although these rings are counted, and are therefore discrete, Condon et al. (1986) treat these variables by linear regression. L\ie consider the annuli as a continuous trait whose observation ultimately reduces the scale to a discontinuous count. As a consequence of this treatment, an annuli count of, for example, 20 rings is considered here to represent a trait value between 19.5 and 20.5. In a sample of 55 individuals of known age, Condon et al. (1986) obtained a correlation between annuli counts and age in years of 0.858. From the summary statistics provided in their article, the variance-covariance matrix for age and cementum annulation counts can be calculated as: =

p03.7095 1 5 5 . 4 0 2 ~ 155.4026 161.0389

(16)

where age is listed before annuli counts, and the mean age and cementum counts were 32.2 years and 32.95, respectively. These summary statistics describe the bivariate normal distribution which we take to represent the reference sample. In order to simulate target samples whose pattern of aging followed that of the reference sample, but whose age distribution differed we used the following procedure. Letting T be the Cholesky decomposition of V (i.e., T’T = V), p be a 2 x 1vector containing the mean age and mean cementum count (32.2 and 32.95, respectively), 2 be an N x 2 matrix of random standard normal deviates, and 1, be an N x 1vector of ones where n is the number of individuals, we can write the matrix R as:

R

= (T’Z’

+ pl’J‘

(17)

R is an N x 2 matrix whose first column contains simulated ages and whose second

column contains simulated cementum annulation counts. Further, the bivariate relationship from this simulated data will approximate that for the original Condon et al. (1986) sample. We simulated 500 R matrices, and then bootstrap sampled from each matrix 300 individuals with an age-at-death structure taken from Weiss’ (1973) model MT 15:30 (life expectancy at age 15 equal to 15 years and survivorship to age 15 equal to 0.30). Because the Condon et al. (1986) sample did not include any individuals under 10 years old we also eliminated any of these individuals. To estimate the age structure for these simulated target populations we applied the information from the Condon et al. (1986) reference sample. Specifically, from the summary statistics in Condon et al. (1986) the regression of cementa1 annulations on age would be x = 8.385779 + 0.762864Cy) where x is the cementum annulation count and y is the age in years. The standard error of the estimate from this regression is 6.579478 annulations. Consequently, for any observed annulation count the probability of that count conditional on age is given by the normal density in equation 15 with mean equal to 8.385779 + 0.762864(a), where a is the midpoint of the age interval, and standard deviation equal to 6.579478. We applied the EM algorithm t o this problem, and also took the more traditional route of estimating ages at death from simulated counts using the Condon ct nl. rcgrcssion o f y = 0.4 + 0.965~~3~). Figure 6 shows a comparison of the average (across 500) estimated age-at-death structure from this traditional approach against the generating Weiss model and the normally distributed reference age-at-death distribution. Clearly, the estimated age-at-death structure does not follow the probability of death function from which it was generated (the Weiss model), and in some respects appears to be followingthe structure of the reference sample instead. The comparable graph (Figure 7) shows the vastly superior performance of the maximum likelihood method, in which the generating age distribution is well estimated. A n interesting feature from Figure 6 is that the estimated age distribution of the

ESTIMATION OF AGE STRUCTURE

249

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target sample has drifted upwards towards more common name (due t o Galton, 1889)of the mean of the reference age distribution. regression (or reversion) to the mean. Under Masset (1989) has referred to this effect this phenomenon, when the correlation bewhen using the traditional method of age tween an indicator and age is less than one, estimation as an “attraction of the middle.” the estimated mean age for the target samWe refer to this phenomenon by its much ple will be biased towards the reference

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L.W. KONIGSRERG AND S.R PRANKENBERG

sample mean age. Specifically, if age is normally distributed in the reference and target samples, then by the Bayesian method the estimated mean age for the target will be:

where the subscripts t and r denote target and reference, p is the correlation between age and the indicator within the reference sample, u represents the standard deviation uf dge d i d uf the iiidieatui vi.itihiii the iefcience sample, a and x denote age and indicator, and the bar notation stands for average. As in equation 3, when the correlation between age and the indicator is zero, the estimated target mean age will precisely equal the reference mean age. Again, as in equation 4,if the correlation is one, then the estimated mean age for the target sample will be independent of the reference mean age, and will be an unbiased estimate. Because there are no indicators identified so far that are perfectly correlated with age, we strongly recommend that the maximum likelihood method (“iterated age length key”) be used in place of the incorrect traditional method of age structure estimation. ESTIMATION OF THE AGE DISTRIBUTION FROM MULTIVARIATE CONTINUOUS INDICATORS

The univariate continuous indicator method of age estimation can be extended to the multivariate case. For this extensionp,, is a multivariate probability density function for the vector of age indicators x, conditional on age in the reference sample. Assuming that the age indicators have a multivariate normal distribution within age classes, the density is:

(19) where p is the number of age indicators, p, is a vector of predicted indicator values from a multiple regression on age in the reference sample, and Z, is the residual variance-co-

variance matrix within age class a after regression on age. Again, we assume that these variance-covariance matrices are equal across age classes, and again we can use the density in equation 19 in the EM algorithm to obtain a maximum likelihood estimate of the age structure of a target sample. Unfortunately, because the information needed for equation 19 is rarely available in the literature, we cannot present a simulation of the multivariate continuous indicator approach to age structure estimation. SOME FUTURE DIRECTIONS

The previous discussion has clearly indicated how the more traditional means of age structure estimation from morphological characters can lead to biased estimates, whereas the maximum likelihood method does not generally suffer this shortcoming. E m u r a (1977:318) previously commented on the “age length key” that the method “will give biased results if applied to a population where the age composition differs from that of the population from which the age length key was drawn,” a point which has been reiterated in various forms by Bocquet-Appel and Masset in the human paleodemography literature (Bocquet-Appeland Masset, 1982; Bocquet-Appel, 1986; Masset, 1989). We have proven here that the previous methods of age estimation (the “age length key” or Bayesian method) applied in paleodemography will producc binscd estimates of age structure if: 1j the age indicators are imperfect, 2) the reference sample does not have a uniform age distribution, and 3 ) the target and reference samples differ in their age distribution. Unlike Bocquet-Appel and Masset, we suggest, following the fisheries literature, that appropriate methods for age structure estimation are currently available, and that it is vital that we switch to using the correct methods. We believe that our results concerning the effects of applying traditional methods of age assessment do make sense out of what has for many years been a puzzlement in paleodemography. A number of researchers have noted that means of age-at-death distributions are very low for the majority of prehistoric samples, possibly as a result of

ESTIMATION OF AGE STRUCTURE

paleodemographers underaging older adults (Weiss, 197359, Asch, 1976:41-43; Boddington 1987:188-191; Gage et al., 1989:48). We suggest that this problem is the result of using the traditional (Bayesian-like) approach to age estimation based on reference samples that have fairly young age distributions. Further, we suggest, as do BocquetAppel and Masset (1982), that many of the observed differences between paleodemographic life tables are due to the specific reference samples that were used to derive age estimates For Pxam-ple, in 8 stndy of26 life tables from prehistoric North America, Buikstra and Konigsberg (1985:329) summarized the probability of death functions by principal components analysis, and noted that: We believe that this first principal component is simply measuring the extent to which older individuals have been underaged or perhaps systematically excluded from the mortuary pattern. We note also that when we compare the various sites for their scores on the first principal component, there is no patterning with respect to ancient subsistence strategy, chronology, or geographic location.

In a recent publication, Buikstra (1991:178) went on to summarize this study, noting that “we . . . removed what we felt to be the effect of distinctive traditions of age estimation among our colleagues through principal components analysis.” We suggest here that the “distinctive traditions” are largely based on the particular reference samples emphasized in various analyses. Typically, researchers who emphasize reference samples with younger age distributions will recover target sample age distributions that are younger than what they would have obtained using an older aged reference sample. It is important to note, however, that the traditional approach only leads to bias in the age estimates, it does not lead to Mensforth‘s (1990) complete “age mimicry.” In any event, using the maximum likelihood aproach we have discussed here would remove this bias. We have not discussed the question of efficiency of estimates. We would obviously like to know not only what the “correct” estimates of life table parameters are, but what level of confidence we may have in their esti-

251

mation. Prior to the maximum likelihood methods described here, which include information on the imprecision of age estimation, all confidence intervals (as well as statistical tests) generated for life tables or survival analyses have been based on the assumption that ages were precisely known. Whittington (1991:174)incisively noted this shortcoming, when (in reference to his survival analysis for Copan) he wrote “these and all subsequent significance levels reflect only error due to finite sample size and do not take into account assi,=ment err(trc in sex, phase, or age, which were difficult, if not impossible, to quantify.” Clearly, the assumption that ages are known (when they are not) will lead to false power in tests and confidence intervals that are too small. The unpleasant consequence of this assumption is that in using traditional life table methods with individuals apportioned to intervals as if ages were known, we often may find “significant” results because we have not accounted for the fact that ages are estimated. The correct confidence intervals from the maximum likelihood method (that account for age estimation uncertainty) can, however, be obtained using standard likelihood methods. The standard errors of estimates are found as the square roots of the diagonal values from the inverse of the information matrix (see Edwards, 1984). The information matrix is the matrix of the negative of the second partial derivatives of the log-likelihood with respect to all possible pairs of parameters. This matrix can be found either by numerical methods or using the analytical derivatives given by Kimura and Chikuni (1987). A second area of interest we have not considered is the ability to compare different anthropological or paleodemographic samples to determine where important similarities or differences may lie. Although there has been considerable discussion of this problem (e.g., Lovejoy, 1971; Konigsberg, 1985; Gage, 1988; Paine, 1989a) all previous methods again start from the untenable assumption that ages are unambiguously known. In what is probably the commonest case in paleodemography, we wish to compare two death samples to determine whether or not they are significantly differ-

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L.W. KONIGSBERC AN D S.R. FRANKENBERG

ent. If we let d, be a vector representing the maximum likelihood estimate of the age-atdeath distribution for the first paleodemographic sample, d, be a similar vector forthe second paleodemographic sample, and d, a vector for the two samples pooled together, then the likelihood ratio test for equality of the two samples is:

native is to rewrite the log-likelihood equation (equation 8 or 12) parameterized with the hazards, and then numerically maximize this new likelihood. While the discussion from this paper presents a somewhat bewildering trail of suggestions for the future of paleodemography and anthropological demography, the truth of the matter is that we have been using -2[lnL(d,lF,) - lnL(dliFl) - lnL(cf,IF,)l (20) inappropriate methods for many years. We no longer need to use these incorrect methwhere the F vectors (or matrices) represent ods, and we certainly should not persist in infnrmatinn from age inrlicatnrs and InT, hlithely caltiilating and piihlishing life tnmeans log-likelihood. Under the null hy- bles based on uncertain data, presented unpothesis of equality, equation 20 is asymp- der the guise of certainty. As should be clear totically chi-square distributed with degrees from this paper, there is much work that of freedom equal to one less than the num- needs to be done in anthropological demogber of age intervals. One degree of freedom raphy, and the bulk of this work must focus is lost because of the linear constraint that on collecting and presenting in useable fashthe sum of the age distribution must equal ion age estimation data that can be widely one. For small samples, where the asymp- applied. While this may seem to be a droll totic approach to a chi-square distribution prospect for the future, we should be clear in may be unacceptable, the probability of ob- pointing out that the MLE methods distaining a likelihood ratio equal to or more cussed here all operate on finding the condiextreme than that in equation 20 can be ap- tional distributions of morphological or behavioral characteristics against known age. proximated by Monte Carlo methods. A future direction that we expect to see in This is the domain of human growth studies, anthropological demography and paleode- so the MLE method provides a strong impemography is the incorporation of uncer- tus for uniting human growth and demogratainty of age estimates into reduced param- phy within the common field of human bioleterizations of life table functions. For O g Y . It should also be clear from our work that example, hazards analysis, which reduces the mortality parameters to a small set, has much of the age assessment information recently been used in a number of anthropo- available for humans is not presented in a logical demography studies (Gage, 1988; form that is useful for paleodemography or Weiss, 1989; Whittington, 1991). There are anthropological demographic research. The two numerical methods for including our tables we present to summarize age indicacurrent results into hazards models or other tor information can serve as models for the models with reduced numbers of parame- kind of information that will be necessary to ters. The first, and most tedious method, is apply maximum likelihood methods to the to retain the EM algorithm. In this case, we problem of age structure estimation. Specifstart with some initial guess a t the hazards ically, in studies of reference samples (i.e., (or other) parameter; and estimate d,. Then samples with known ages) if the indicator is the distribution of d , is used in equation 9 a discontinuous character, then the tabula(the E step), followed by equation 19 (the M tion of the indicator states against known step) to produce a new estimate of d,. These age categories should be presented (see Tad, are then used to find the maximum likeli- ble 1). For multiple indicators, the multiway hood estimate of the hazards (or other) pa- cross-classification is necessary (see Table rameters, whish in turn are used to find new 3 ) .While the full multiway table would take estimates of d,. The process is then started considerable room to publish, in most cases anew. This may be a very slow procedure, many of the cells will be empty and consebecause the M step is computaticnally quently do not need to be shown. For a concostly, and many iterations may be neces- tinuous indicator the author should provide sary to reach convergence. A simpler alter- the requisite bivariate statistics (mean age,

ESTIMATION OF AGE STRUCTURE

mean indicator value, and the agehndicator variance-covariance matrix), while for multiple indicators the summary statistics for the complete multivariate distribution should be presented. In this latter case, the statistics consist of the vector of indicator means, the mean age, and the variance-covariance matrix between all indicators and age. In addition to the problem of how to present age indicator information, we must also deal with the problem of how to select age indicators that will be usefui ill a:itLi~pological demography. The problem of selection of age indicators ultimately relates to one of the “uniformitarian assumptions” (Howell, 1976), that all human populations age in similar fashion and at identical rates. If this assumption is unwarranted, then the reference sample probabilities used in equations 2 and 9 will lead to age estimation errors when applied to a target sample from a different population, and as a consequence the estimated age structure distribution will be biased. In some instances it may be possible to internally calibrate an age indicator in the target sample by comparison against other indicators in that sample (see Jantz and Owsley, 19921, but this does not completely remove the “uniformitarian assumption.” Much confusion on the “uniformitarian assumption” has arisen from the misconception that paleodemography and forensic anthropology t the source of many age indicator studies) have common goals in age estimation. A paleodemographer may well be satisfied with a method of age estimation which is unbiased when applied across different populations, but is not particularly sensitive to the details of age progression. From a statistical standpoint, the paleodemographer must emphasize lack of bias and the presence of high consistency in the selection of age indicators. Conversely, forensic anthropologists can often select age indicators that are population specific and thus emphasize indicators that give efficient estimates, provided that the anthropologist can identify the relevant population. As an example of these different goals in age estimation, it is instructive to look at a recent study of auricular surface aging applied in a forensic setting. Murray and Murray (1991:1162) note

253

that the auricular surface aging method is “equally applicable across race and sex,” which would of course make it a useful indicator in paleodemography. On the other hand, they further note that “the rate of degenerative change is too variable t o be used as a single criterion for the estimation of age; the range of estimation error is simply too large for forensic purposes” (Murray and Murray, 1991:1162). While this inefficient and inconsistent age estimator may not be useful forensically, it is precisely its stability across “race”and scx that malrcs it so attractive for paleodemography. The take home message is that for demographic purposes we must select indicators that are likely to fit Howell’s (1976) “uniformitarian assumption,” and then demonstrate that there is low between-group variation for these indicators. In closing, we call for a reexamination of the bases for paleodemography, and attempts to put the field on a more rational and scientific basis. Maples’ (1989:323) statement on age estimation procedures is particularly telling concerning the current status of data used in paleodemographic interpretations: “Age determination is ultimately an art, not a precise science. Many areas of scientific data must be evaluated, but the final best estimate results from a subjective weighting of the results of all of the techniques that were employed.” While we agree that the current status of age estimation in paleodemography is largely that of an “art,” we see no apparent reason why we should not strive to make it a science. ACKNOWLEDGMENTS

We thank Drs. John Blangero and Sarah Williams-Blangero for their comments on an earlier draft of this paper. The macaque cranial suture closure data used here were collected by Dr. R. Criss Helmkamp under funding from NIH (Public Health Service grant 7 R01 NS24904) to Drs. Dean Falk, James Cheverud, and Michael Vannier. We thank Dr. Falk for permission to use these data. LITERATURE CITED Acsadi G and Nemeskeri J (1970)History of Human Life Span and Mortality. Budapest: Akademiai Kiado. Asch D (1976) The Middle Woodland Population of the

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Lower Illinois Valley: A Study in Paleodemographic Methods. Northwestern University Archaeological Program, Scientific Papers, No. 1,Evanston, 111. Bairagi R, Aziz KMA, Chowdhury MK, and Edmonston B (1982)Age misstatement for young children in rural Bangladesh. Demography 19t447-458. Bartoo NW and Parker KR (1983) Stochastic age-frequency estimation using the von Bertalanffy growth equation. Fishery Bulletin 81:91-96. Bhat PNM (1990) Estimating transition probabilities of age misstatement. Demography 27t149-163. Bocquet-Appel J-P (1986) Once upon a time: Paleodemography. Mitteil. Berlin Gesell. Anthropol. Ethnol. Urges. 7r127-133. Bocquet-Appel