AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 142:655–664 (2010)
Brief Communication: A Probabilistic Approach to Age Estimation From Infracranial Sequences of Maturation He´le`ne Coqueugniot,1,2 Timothy D. Weaver,2,3* and Francis Houe¨t1 1
UMR 5199—PACEA, Laboratoire d’Anthropologie des Populations du Passe´, Universite´ Bordeaux 1, avenue des Faculte´s, 33405 Talence cedex, France 2 Department of Human Evolution, Max Planck Institute for Evolutionary Anthropology, Deutscher Platz 6, D-04103 Leipzig, Germany 3 Department of Anthropology, University of California, Davis, CA 95616 KEY WORDS
Bayesian statistics; osteological collections; postcranium; skeletal maturation
ABSTRACT Infracranial sequences of maturation are commonly used to estimate the age at death of nonadult specimens found in archaeological, paleoanthropological, or forensic contexts. Typically, an age assessment is made by comparing the degree of long-bone epiphyseal fusion in the target specimen to the age ranges for different stages of fusion in a reference skeletal collection. While useful as a first approximation, this approach has a number of shortcomings, including the potential for ‘‘age mimicry,’’ being highly dependent on the sample size of the reference sample and outliers, not using the entire fusion distribution, and lacking a straightforward quantitative way of combining age estimates from multiple sites of fusion. Here we present an alternative probabilistic approach
based on data collected on 137 individuals, ranging in age from 7- to 29-years old, from a documented skeletal collection from Coimbra, Portugal. We then use cross validation to evaluate the accuracy of age estimation from epiphyseal fusion. While point estimates of age can, at least in some circumstances, be both accurate and precise based on the entire skeleton, or many sites of fusion, there will often be substantial error in these estimates when they derive from one or only a few sites. Because a probabilistic approach to age estimation from epiphyseal fusion is computationally intensive, we make available a series of spreadsheets or computer programs that implement the approach presented here. Am J Phys Anthropol 142:655– 664, 2010. V 2010 Wiley-Liss, Inc.
Age estimation is crucial in forensic contexts for identity determination, and it is required in archaeological or paleoanthropological contexts for further analysis of growth and development, pathology, or paleodemography. Infracranial sequences of maturation are commonly used to estimate the age at death of nonadult individuals found in archaeological, paleoanthropological, or forensic contexts, particularly when the dentition is not available or too few teeth are preserved for an accurate age assessment. Typically, an age assessment is made by comparing the degree of long-bone epiphyseal fusion in the target specimen to the age ranges for different stages of fusion in a reference skeletal collection (e.g., charts in Steele and Bramblett, 1988; Buikstra and Ubelaker, 1994; Bass, 1995; Mays, 1998; White and Folkens, 2000; Byers, 2002; Baker et al., 2005). While useful as a first approximation, this approach has a number of shortcomings. The age assessment may be subject to ‘‘age mimicry,’’ whereby the estimated age of the target specimen is inappropriately influenced by the composition of the reference sample (Bocquet-Appel and Masset, 1982; Love and Mu¨ller, 2002). For example, in the reference sample we use in this study there are no 13-yearold individuals, which means that the fusion range of an epiphyses can never begin or end at age 13. Additionally, the range is a statistic that is highly dependent on the sample size of the reference sample and outliers, and it conveys limited information about the fusion distribution of the reference sample. A method that considers the entire distribution would be expected to produce more precise age estimates. There is also no straightforward quantitative way of combining age estimates from multiple sites of fusion into a single composite age estimate. Range charts (e.g., Byers, 2002) can be helpful in this
regard, but it is still unclear how to weigh the different sites of fusion in the composite age estimate. A final, nonmethodological, issue is that commonly used standards for epiphyseal fusion are based on a limited number of documented skeletal collections (Coqueugniot and Weaver, 2007; Schaefer, 2008). This study has two primary objectives: first, to present a probabilistic approach to age estimation from epiphyseal fusion that overcomes the methodological shortcomings outlined above, and second, to evaluate the accuracy of age assessment from epiphyseal fusion. This article follows on our earlier paper documenting the ages of epiphyseal fusion in the documented skeletal collection from Coimbra, Portugal (Coqueugniot and Weaver, 2007).
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WILEY-LISS, INC.
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MATERIALS AND METHODS Sample and dataset Our analyses are based on data collected by HC and TDW on a documented skeletal collection from Coimbra, Portugal, that is housed in the Department of AnthropolFrancis Houe¨t: Deceased. *Correspondence to: Timothy D. Weaver, Department of Anthropology, University of California, One Shields Avenue, Davis, CA 95616. E-mail:
[email protected] Received 6 November 2009; accepted 24 February 2010 DOI 10.1002/ajpa.21312 Published online 28 April 2010 in Wiley InterScience (www.interscience.wiley.com).
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H. COQUEUGNIOT ET AL. TABLE 1. List of fusion sites and abbreviations used
Fig. 1. Age and sex composition of the skeletal sample used for this study.
ogy, Coimbra University (Coqueugniot and Weaver, 2007). This collection was assembled in the first half of the 20th century by Professor E. Tamagnini (BocquetAppel, 1984; Rocha, 1995; Coqueugniot and Weaver, 2007). The majority of the skeletons in the collection were excavated from the Cemiterio da Conchada, the largest cemetery in the city of Coimbra. The individuals were born between 1826 and 1922 and died between 1904 and 1938. The entire collection consists of 505 skeletons from both adults and subadults, with the ages at death ranging from 7- to 96-years old. Documentation for all the individuals is available in the Anthropology Department, Coimbra University, and it includes biographical information such as name, parents’ names, sex, marital status, occupation, birthplace, location and cause of death, age at death, and date of death. For this study, we used the 137 individuals from this collection aged between 7 and 29 years (69 females and 68 males, Fig. 1). None of these individuals showed any obvious signs of pathology that could have significantly disrupted growth and development, and based on the documentation, none of them were known to have died of tuberculosis. For each skeleton, the degree of fusion was recorded for the spheno-occipital synchondrosis and 63 infracranial sites of fusion on the sacrum and the left and right os coxae, scapulae, clavicles, humeri, radii, ulnae, femora, tibiae, and calcanae (Table 1). Fusion at each site was coded in three stages, ‘‘a,’’ ‘‘b,’’ and ‘‘c,’’ corresponding to open (no fusion), partial union, and complete union, respectively. More details on the scoring protocol, any difficulties in scoring, and interobserver error are given in Coqueugniot and Weaver (2007).
Age estimation approach The objective of our age estimation approach is to produce a distribution of probabilities that an unknown individual is a given age. These probabilities can then be used to produce either a single age estimate (point estimate) or a range of age estimates that span the ages with high probabilities (interval estimate). The first step is to use the reference dataset to estimate likelihoods, which are probabilities of observing a particular stage American Journal of Physical Anthropology
Fusion site
Abbreviation
Ilium pubis Upper ischium pubis Lower ischium pubis Ischium ilium Iliac crest Ischial tuberosity Anterior inferior iliac spine Medial sacral segments 1–2 Lateral sacral segments 1–2 Posterior sacral segments 1–2 Medial sacral segments 2–3 Lateral sacral segments 2–3 Posterior sacral segments 2–3 Medial sacral segments 3–4 Lateral sacral segments 3–4 Posterior sacral segments 3–4 Medial sacral segments 4–5 Lateral sacral segments 4–5 Posterior sacral segments 4–5 Coracoid Acromion Sternal end Humerus head Humerus medial epicondyle Humerus distal end Radius proximal end Radius distal end Ulna proximal end Ulna distal end Femur head Femur greater trochanter Femur lesser trochanter Femur distal end Tibia proximal end Tibia distal end Fibula proximal end Fibula distal end Calcaneus posterior end Spheno-occipital synchondrosis
Il_Pu U_Is_Pu L_Is_Pu Is_Il Ic It Aiis M_SS_1_2 L_SS_1_2 P_SS_1_2 M_SS_2_3 L_SS_2_3 P_SS_2_3 M_SS_3_4 L_SS_3_4 P_SS_3_4 M_SS_4_5 L_SS_4_5 P_SS_4_5 Crd Acm Ster Hum_Hd Hum_Me Hum_De Rad_Pe Rad_De Uln_Pe Uln_De Fem_Hd Fem_Gt Fem_Lt Fem_De Tib_Pe Tib_De Fib_Pe Fib_De Clc_Pe SOS
(‘‘a,’’ ‘‘b,’’ or ‘‘c’’) at a particular fusion site (e.g., iliac crest) if an individual is a particular age (e.g., 10-years old). Following Love and Mu¨ller (2002), we use kernel smoothing, a nonparametric regression procedure, to produce likelihood distributions for each site and age. Love and Mu¨ller (2002) call these likelihood distributions ‘‘weight functions,’’ because they can be thought of as functions describing the relationship between age and the probability of showing a particular stage of fusion. If we had an extremely large reference sample for which every age was represented by many individuals, then for each fusion site we could simply use the relative frequencies of the different fusion stages for individuals of a particular age as an estimate of likelihood distribution for that site and age. However, many reference samples, including the one we use here, are only on the order of 100 individuals, and the coverage is uneven across ages, making this approach problematic. For example, because there are no 13-year-old individuals in this study’s reference sample, simply using the relative frequencies as likelihoods results in an estimated likelihood of zero of showing any stage at any site for a 13-year-old individual. The weight functions alleviate this problem by expanding the data used to calculate the likelihood for a given age to adjacent age classes. Specifically, for each site, the likelihood distribution for a particular age is
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Fig. 2. Example of an asymmetrical posterior probability distribution.
estimated from the frequencies of each of the stages for the age, as well as the frequencies for adjacent ages. Here we use information from a 2-year window in each direction of an age (including 2 years younger and 2 years older), so for example, the likelihoods for age 13 would be based on information from ages 11 to 15. In the kernel estimation equations this window size corresponds to a bandwidth of 3 years, which is 12% of the range of ages in the reference sample. This percentage is close to the 10% recommended by Love and Mu¨ller (2002). To produce a distribution of probabilities that an unknown individual is a given age (posterior probabilities) we need to know not only the likelihoods but also the prior probabilities for each age (priors). Bayes’ Rule can then be used to calculate the posterior probabilities from the priors and likelihoods (Berry, 1996). The priors are based on contextual information (e.g., the unknown individual was a soldier) that makes certain ages more probable than others in advance of collecting data on the skeleton. When there is no such information, equal prior probabilities are assigned to all ages (uniform or uninformative priors). We discuss priors in more detail below in the context of evaluating the accuracy of age estimation from epiphyseal fusion. The last step is to combine information from multiple sites of fusion. To do this, we assume that an unknown skeleton’s stages of fusion at different sites are independent of each other given age (conditional independence). For instance, individuals showing Stage ‘‘a’’ at site no. 1 may be more likely to show Stage ‘‘a’’ at site no. 2, but if the conditional independence assumption is true, then the tendency for cooccurrence of Stage ‘‘a’’ at the two sites is only because the degree of fusion at both sites is correlated with age rather than because of specific genetic or developmental links between the two sites. By assuming conditional independence, we can calculate the posterior probabilities for a set of fusion sites by sequentially applying Bayes’ rule, so the posterior probabilities based on the 1st site become the prior probabilities for the 2nd site, the posterior probabilities based on the 2nd site become the prior probabilities for the 3rd site, and so on. The final posterior probability is the same regardless of the order of the sites in the sequence.
Evaluating accuracy Ideally, we would have a large reference sample to use for estimating likelihoods and multiple large target
samples to use for evaluating the accuracy of age estimation, but we only have data from a single sample. We could divide the sample in half and use one half as the reference sample and the other half as the target sample, but this procedure will tend to bias the results in the direction of poor performance, because the likelihoods are estimated from a very small sample. Additionally, the test of accuracy is based on a single target sample, which by chance may happen to be classified more or less accurately than a typical target sample. Instead, we use cross validation, which allows us to mimic using both a large reference sample and multiple target samples. The basic idea is to hold out a small subset of the sample (target sample), classify these individuals based on likelihoods estimated from the remaining portion of the sample (reference sample), and then repeat the procedure many times holding out a different target sample each time (Krzanowski, 2000; Roff, 2006). Importantly, for each iteration, the individuals in the target sample are different from those in the reference sample, so the results are not biased in the direction of good performance. There is some question as to the optimal number of individuals to hold out each time (Roff, 2006), so we performed the cross validation test twice: either leaving out 1 or 20 individuals. The results are extremely similar, so we present only the hold-out-20 results. To evaluate accuracy, we focus on point estimation rather than interval estimation, because otherwise it is difficult to separate accuracy from precision. Essentially, the larger the interval is, the lower the precision and the higher the accuracy, with the extreme case of an interval from 7- to 29-years old resulting in 100% accuracy. For each individual, we define the point estimate as the age with the highest posterior probability (a posterior mode point estimate). Sometimes, particularly for estimates based on a single fusion site, there is a block of ages with the same posterior probability. In this case, we define the point estimate as the average of the oldest and youngest ages with this probability. We could have defined the point estimate as the weighted average of all the ages where the weights are the posterior probabilities for each age (a posterior mean point estimate), but this approach biases the age estimates to be too old for very young individuals and too young for very old individuals. This bias results from the posterior probability distribution being asymmetrical when the mode is close to 7- or 29-years old, because the distribution is truncated below 7- and above 29-years old. For example, American Journal of Physical Anthropology
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Fig. 3. (See legend page 659.)
Figure 2 shows an asymmetrical posterior probability distribution for which the weighted average point estimate is 9-years old and the mode point estimate is American Journal of Physical Anthropology
7-years old. The mode point estimate seems more reasonable based on the overall shape of the posterior probability distribution.
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Fig. 3. Example of os coxae posterior probabilities for different stages of fusion without kernel smoothing. The distribution is based on uniform initial priors. Stages ‘‘a,’’ ‘‘b,’’ and ‘‘c’’ correspond to open, partial union, and complete union, respectively. The abbreviations used for the fusion sites (Il_Pu, U_Is_Pu, L_Is_Pu, Is_Il, Ic, It, and Aiis) are explained in Table 1. In Panel A, all seven os coxae fusion sites are open and the posterior probabilities for an age at death of 7-, 9-, 10-, and 12-years old are 0.383, 0.287, 0.255, and 0.076, respectively. The other panels give results for other combinations of maturation stages.
The initial prior probabilities are the final pieces of information that we need to calculate the posterior probabilities. We chose two different sets of prior probabilities to derive the point estimates for accuracy evaluation. First, we used the relative frequencies of different ages in the Coimbra sample as the initial priors (Coimbra priors), because these relative frequencies are representative of the distribution of ages of the ‘‘unknown’’ individuals in the target sample. However, because prior knowledge is never this informative (i.e., Coimbra priors are only representative of the distribution of ages in the Coimbra sample), point estimates derived this way will tend to overestimate the accuracy of age estimation in real applications. So, second, we used uniform priors (on the interval from 7- to 29-years old). Using uniform priors evaluates the accuracy of estimating the age of an isolated individual when the demographic structure of the unknown population from which the individual derives is unknown (Bocquet-Appel and Masset, 1996).
RESULTS The panels of Figure 3 show posterior probability distributions for different states of os coxae fusion when the likelihoods are estimated without kernel smoothing. Figure 4 gives distributions for the same fusion states with kernel smoothing. When the states of fusion are Il_Pu 5 a, U_Is_Pu 5 a, L_Is_Pu 5 a, Is_Il 5 b, Ic 5 a, It 5 a, and Aiis 5 a, the posterior probabilities are undefined without kernel smoothing (see Fig. 3B), because for
every age at least one of the site-stage combinations has an estimated likelihood of zero. In contrast, with kernel smoothing, ages 9- though 14-years old have nonzero posterior probabilities, with 11-years old being the most probable age (see Fig. 4B). The most striking contrast between Figures 3 and 4 is for the ‘‘B’’ panels (posterior probabilities are undefined without kernel smoothing), but the other panels illustrate other, albeit less extreme, problems with the posterior probability distributions without kernel smoothing. As we discuss in the introduction, with extremely large samples from all ages, kernel smoothing is unnecessary, but with smaller or unevenly distributed samples it leads to more reasonable posterior probability distributions that are less subject to ‘‘age mimicry.’’ Tables 2 and 3 summarize the results of the cross validation test of accuracy with the two different choices of priors. In these tables, we present results for individual sites and combinations of sites. Many other combinations are possible, and in any particular application, the exact combination will depend on which skeletal elements are present and how they are preserved. Rather than being comprehensive Tables 2 and 3 are meant to be representative of the types of site combinations that would occur in practice. We categorized age estimates into those that are within 15% and within 30% (double the error) of the actual age. We chose 15% as the first cut-off because this percentage corresponds to an absolute error of 1 year in either direction for our youngest age category (7-years American Journal of Physical Anthropology
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Fig. 4. (See legend page 661.)
old), and confusing a 7-year old for a 6- or 8-year old seems to be an acceptable margin of error for most purposes. The absolute error for the oldest individuals, American Journal of Physical Anthropology
who are 29-years old, is !4 years in either direction. These error ranges are comparable to, if not better than, those typically given for age estimates based on
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Fig. 4. Example of os coxae probabilities for different stages of fusion with kernel smoothing. The distribution is based on uniform initial priors. Stages ‘‘a,’’ ‘‘b,’’ and ‘‘c’’ correspond to open, partial union, and complete union, respectively. The ordering of the panels is the same as in Figure 3.
dental development. For example, Ubelaker’s (1978) classic chart of tooth formation and eruption stages gives error ranges of 2 years in either direction for 7year-old individuals and 3 years in either direction 15year-old individuals, which would correspond to 29% and 20 errors, respectively. Alternatively, the percentage errors can be thought of as levels of precision, which for 15% errors corresponds to levels of precision ranging from 3- to 9-year intervals for the youngest to oldest age classes, respectively. In practice, the errors for 7-year-old individuals will always be in the direction of being too old, and the errors for 29-year-old individuals will always be in the direction of being too young, because point estimates will never be younger than 7-years old or older than 29-years old (see Materials and Methods). Tables 2 and 3 show that while point estimates of age can be quite accurate based on the entire skeleton (82% within 15% error for both sets of priors) or when many sites of fusion are used (e.g., 79 and 77% within 15% error for the entire pelvis with Coimbra and uniform priors, respectively), there can be substantial errors in these estimates when they derive from one or only a few sites, particularly when good prior information is not available (ranging from 41 to 67% within 15% error for single sites with uniform priors). The age estimates tend to be more accurate with the Coimbra priors, but there can still be substantial errors for estimates from single sites (ranging from 49 to 78% within 15% error). The frequent poor performance of single sites is not unexpected given that with our coding scheme individuals can only group into three age cate-
gories, based on whether the epiphysis is open, partially fused, or completely fused. The percentage of individuals who are correctly classified approaches or surpasses 90% for many individual bones for both Coimbra and uniform priors when the percent error is increased to 30%. However, with this amount of error the precision ranges from 5 to 17 years for the youngest to oldest age classes respectively, which for many applications would be an unacceptably low. Age estimates for females tended to be biased in the direction of being too old for most loci, whereas for males they were often in the direction of being too young. This pattern is the result of pooling of the sexes in the reference sample, because bone maturation tends to be faster in females than in males in the Coimbra sample (Coqueugniot and Weaver, 2007).
DISCUSSION AND CONCLUSIONS A significant shortcoming of the reference sample that we use here is the absence of individuals \7-years old. To overcome this problem and other sample deficiencies, in the future, we would like to expand the reference sample to include numerous individuals from all subadult and young adult ages. Given that there appears to be interpopulational variability in epiphyseal fusion (compare, for example, Stevenson, 1924; McKern and Stewart, 1957; Veschi and Facchini, 2002; Coqueugniot and Weaver, 2007; Cardoso, 2008a,b; Schaefer, 2008) that may be related to ancestry or environmental factors such as socioeconomic status (Meijerman et al., 2007; Schaefer, 2008), these additional samples would ideally be geographically, American Journal of Physical Anthropology
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H. COQUEUGNIOT ET AL. TABLE 2. Accuracy of age estimation by fusion site and groups of sites with priors based on the Coimbra sample Percentage error
Site
Site group
615 (%)
630 (%)
615 (%)
630 (%)
615 (%)
630 (%)
615 (%)
630 (%)
Il_Pu U_Is_Pu L_Is_Pu Is_Il Ic It Aiis M_SS_1_2 L_SS_1_2 P_SS_1_2 M_SS_2_3 L_SS_2_3 P_SS_2_3 M_SS_3_4 L_SS_3_4 P_SS_3_4 M_SS_4_5 L_SS_4_5 P_SS_4_5 Crd Acm Ster Hum_Hd Hum_Me Hum_De Rad_Pe Rad_De Uln_Pe Uln_De Fem_Hd Fem_Gt Fem_Lt Fem_De Tib_Pe Tib_De Fib_Pe Fib_De Clc_Pe SOS
Os Coxae
62 58 49 59 69 78 59 52 73 62 68 76 57 74 70 59 73 55 59 58 66 67 68 56 56 60 68 56 63 73 64 63 64 73 63 66 72 58 67
80 79 70 80 79 91 81 71 89 82 89 91 80 90 88 81 91 74 79 80 85 85 83 79 78 80 82 79 79 89 84 83 84 87 85 87 88 80 88
84
93
79
91
82
95
74
91
74
92
82
93
67 79
85 95
85
96
78
93
75
93
75
91
80
93
76
91
76
91
58 67
80 88
67
88
Sacrum
Scapula Clavicle Humerus Radius Ulna Femur
Tibia Fibula Calcaneus Skull
Columns 3–10 give the percentage of age estimates that were within 15 or 30% of the actual age of the individual. The percentages in columns 5–10 correspond to age estimates based on the group of sites from the row of percentage to the row above the next percentage.
temporally, and nutritionally similar to the Coimbra sample. However, it would also be worthwhile to build-up a more diverse test sample for evaluating the accuracy, precision, and bias of age estimates when the unknown individuals come from a different population than the individuals in the reference sample. Given that interobserver error for epiphyseal fusion should be much lower than age indicators with more complicated patterns of changes, such as the pubic symphysis, it may be possible to pool together data collected by multiple researchers. In principle, data sharing along these lines could lead to a central repository of epiphyseal fusion data, which could be made generally available, leading to more accurate and consistent age estimates. Our results indicate that infracranial sequences of maturation, at least in some circumstances, can be used as accurate age indicators for nonadults. Not surprisingly, the general pattern is the more sites of fusion that are used for an age estimate, the more accurate the estimate (holding precision constant). However, this increase in accuracy with number of sites is not linear; some twobone combinations are about as accurate as age estimates based on all 11 infracranial bones and the sphenooccipital synchondrosis (Tables 2 and 3). A nonlinear American Journal of Physical Anthropology
increase in accuracy with number of sites suggests that the assumption of conditional independence of site fusion (see Materials and Methods) is not strictly correct, so after a few sites, each additional site adds very little new information about age. This assumption could be tested with very large sample sizes. Skeletal elements with more fusion sites, such as the pelvis (os coxae and sacrum) and humerus, can provide accurate age estimates across a wide range of ages, while elements with only a single site, such as the clavicle or calcaneus, will often be quite inaccurate (Tables 2 and 3). Nevertheless, single sites that fuse either particularly early or late, such as the clavicle (Szilva´ssy, 1980; Webb and Suchey, 1985; Black and Scheuer, 1996; Coqueugniot and Weaver, 2007), are not useless, because they can indicate than an unknown individual is older or younger than a particular age cut-off. This example about the clavicle illustrates the variety of uses of age estimation and the need to evaluate accuracy within the context of a particular application. Consequently, the test of accuracy we present here, depending on the objective, may or may not be applicable. Comparing the Coimbra priors results with those for uniform priors indicates that
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AGE ESTIMATION FROM INFRACRANIAL MATURATION TABLE 3. Accuracy of age estimation by fusion site and groups of sites with uniform priors Percentage error Site
Site group
615 (%)
630 (%)
615 (%)
630 (%)
615 (%)
630 (%)
615 (%)
630 (%)
Il_Pu U_Is_Pu L_Is_Pu Is_Il Ic It Aiis M_SS_1_2 L_SS_1_2 P_SS_1_2 M_SS_2_3 L_SS_2_3 P_SS_2_3 M_SS_3_4 L_SS_3_4 P_SS_3_4 M_SS_4_5 L_SS_4_5 P_SS_4_5 Crd Acm Ster Hum_Hd Hum_Me Hum_De Rad_Pe Rad_De Uln_Pe Uln_De Fem_Hd Fem_Gt Fem_Lt Fem_De Tib_Pe Tib_De Fib_Pe Fib_De Clc_Pe SOS
Os Coxae
41 52 47 56 60 67 56 50 55 43 55 67 52 70 56 54 60 51 45 49 63 52 65 54 57 54 64 54 59 62 62 55 61 65 59 62 67 52 62
69 80 73 86 76 90 83 73 83 68 78 84 80 91 83 79 86 67 65 74 81 64 78 81 88 81 80 81 75 85 86 81 80 85 87 82 84 80 78
70
94
77
93
82
97
73
89
62
88
75
93
52 72
64 91
76
93
67
91
65
91
62
86
64
89
66
89
69
88
52 62
80 78
62
78
Sacrum
Scapula Clavicle Humerus Radius Ulna Femur
Tibia Fibula Calcaneus Skull
Columns and rows are the same as in Table 2.
informative priors should be used when appropriate information is available, but it is important to keep in mind that prior information will never be as informative as in our test of accuracy with the Coimbra priors. Finally, age estimates based on sex-specific reference samples would be expected to be more accurate than those based on pooled-sex samples, but in practice, it is often impossible to determine the sex of nonadult individuals. In summary, although a probabilistic approach to age estimation from epiphyseal fusion is more computationally intensive than the typical approach based on age ranges, it has a number of advantages. Among these, it uses more information from the reference sample, allows data from multiple loci to be combined in a straightforward way into a single, composite age estimate, and generates an entire probability distribution that a researcher can then use to produce various point or interval age estimates, depending on the purpose of the age estimate. For these reasons, we recommend that researchers use a probabilistic approach to estimate the ages of nonadult skeletons from infracranial sequences of maturation. A series of Excel (Microsoft, Redmond, WA) spreadsheets or Matlab (Mathworks, Natick, MA) computer programs that implement
that approach presented here are available at either at http://www.pacea.u-bordeaux1.fr or http://anthropology.ucdavis.edu. We will provide further guidance about how to use the spreadsheets and programs upon request.
ACKNOWLEDGMENTS The authors thank Professors E. Cunha and N. Porto for access and permissions; Professors O. Dutour and C. Roseman for fruitful discussions; and Professor C. Ruff and two anonymous referees for helpful comments.
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