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Presentation and implementation of a new approach of age from biological indicators (Bayesian inference procedure) Luc Buchet, Henri Caussinus, Daniel Courgeau (CNRS-Nice University, Toulouse University, Ined) 27th IUSSP International Population Conference, Busan, Korea. 26 – 31 August 2013

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Context and methods used in the past To estimate the age structure of a population of the past, paleodemographers usually only have the structure by stage given by biological indicators. To estimate the age structure of a population of the past, paleodemographers usually only have the structure by stage given by biological indicators. To be able to make such an estimation, To be able to make such an estimation, they also need a reference population, for which a reference matrix gives both ages and stages. With these two elements, different methods were used in the past to make the estimation With these two elements, different methods were used in the past to make the estimation

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A first method (ALK), used in paleodemography since the 70s, consists in reconstituting from the observed data a matrix showing the greatest closeness with each cell of the reference matrix. However, these methods rest on an unreliable assumption of resemblance between the reference and the target populations and do not take into account the invariance hypothesis whereby for any remains of a given age at death, the likelihood of being classified in a given stage only depends on that age, and that it is constant over time. To consider this invariance hypothesis, even if it may be disputable, other methods have been proposed to permit a more satisfactory estimation. It is, for example, on this hypothesis that IALK method or similar methods (PFP) are based. Then we can deduce, from the given stage structure, a populations age structure. Then we can deduce, from the given stage structure, a populations age structure.

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Nevertheless, the observed population is usually of small size, so that the results obtained with these methods are often inconsistent, giving for example an estimated null probability for some age groups. In order to avoid such inconsistencies it is possible to introduce a continuous rather than discrete age and use a parametric model, as the Rostock manifesto recommended at the beginning of the 2000s (Hoppa and Vaupel). In order to avoid such inconsistencies it is possible to introduce a continuous rather than discrete age and use a parametric model, as the Rostock manifesto recommended at the beginning of the 2000s (Hoppa and Vaupel). Another approach (IBPFP) was more recently proposed by Bocquet- Appel and Bacro (2008). To escape the constraints of a uniform distribution (reference matrix), the authors search a solution in a universe of mortality models. Another approach (IBPFP) was more recently proposed by Bocquet- Appel and Bacro (2008). To escape the constraints of a uniform distribution (reference matrix), the authors search a solution in a universe of mortality models. All these methods introduce some a priori considerations in their problems solution. However, they are still using a frequentist paradigm. We propose instead a typically Bayesian statistical method which results in substantial improvement.

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Proposed method Here we address the case where skeletal remains are classified into r stages and ages into c classes. Denote by the probability that an individual from the target population is at stage i and age j ; Here we address the case where skeletal remains are classified into r stages and ages into c classes. Denote by the probability that an individual from the target population is at stage i and age j ; the sum over i is, the sum over j is and the conditional probability of stage i when age j is known is They are related by the relationship: We will use here a full Bayesian statistical method in order to estimate from prior distributions of the different parameters, suitably chosen by the user, a posterior distribution of the probabilities that the individuals of the target population are of We will use here a full Bayesian statistical method in order to estimate from prior distributions of the different parameters, suitably chosen by the user, a posterior distribution of the probabilities that the individuals of the target population are of age j. The observed frequencies in the reference population are considered as a set of random variables multinomial; its the same for each column of the reference matrix.

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For the prior distribution of it is natural among statisticians to adopt a Dirichlet distribution. This distribution depends on c parameters; since we are dealing with a mortality distribution, we choose these parameters so that the average of the Dirichlet distribution is equal to mortality standard as defined for preindustrial societies. Other prior distributions were considered but they do not always give satisfactory results, especially if the site in question is atypical. For each j, the prior distribution of is a Dirichlet distribution which parameters are provided by the reference data using a Bayesian approach entirely conventional.

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From these prior distributions, and knowing the distribution stages in the target population, it is possible to calculate a posterior distribution of the age structure of the target population. From these prior distributions, and knowing the distribution stages in the target population, it is possible to calculate a posterior distribution of the age structure of the target population. The posterior means give point estimates of each but, beyond that, we can estimate the accuracy of the estimate by means of the posterior densities or the credibility intervals (Bayesian equivalents of confidence intervals of the frequentist estimation). The posterior means give point estimates of each but, beyond that, we can estimate the accuracy of the estimate by means of the posterior densities or the credibility intervals (Bayesian equivalents of confidence intervals of the frequentist estimation). From these prior distributions, and knowing the distribution stages in the target population, it is possible to calculate a posterior distribution of the age structure of the target population. From these prior distributions, and knowing the distribution stages in the target population, it is possible to calculate a posterior distribution of the age structure of the target population. The posterior means give point estimates of each but, beyond that, we can estimate the accuracy of the estimate by means of the posterior densities or the credibility intervals (Bayesian equivalents of confidence intervals of the frequentist estimation). The posterior means give point estimates of each but, beyond that, we can estimate the accuracy of the estimate by means of the posterior densities or the credibility intervals (Bayesian equivalents of confidence intervals of the frequentist estimation).

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Results To evaluate such a method, we have already applied it to different theoretical cases and to a number of past populations: Neolithic site of Loisy-en-Brie, nuns of Maubuisson (17 th and 18 th century), cemetery of Antibes (19 th century). To evaluate such a method, we have already applied it to different theoretical cases and to a number of past populations: Neolithic site of Loisy-en-Brie, nuns of Maubuisson (17 th and 18 th century), cemetery of Antibes (19 th century). We examine now the results obtained in Frénouville during the Gallo- Roman period (3 rd to 5 th century) and the Merovingian period (until late 7 th century). The samples were respectively of 69 and 200 remains. We examine now the results obtained in Frénouville during the Gallo- Roman period (3 rd to 5 th century) and the Merovingian period (until late 7 th century). The samples were respectively of 69 and 200 remains. Frénouville, 3 rd – 7 th c. A.D. Frénouville, 3 rd – 7 th c. A.D. Distribution by stages of synostosis. Distribution by stages of synostosis. Distribution by stages of synostosis. Distribution by stages of synostosis.

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Frénouville, survival functions for the Gallo-Roman (G) and Merovingian (M) periods and the pre-industrial standard (asterisks) in the left curve, and credibility intervals with 50% (red) and 90% (green) in the right curve. By this method, we see that it is highly likely that the two types of mortality are radically different, with a very high mortality of young individuals in the Merovingian period and a survival distribution close to the standard in the Gallo-Roman period.Indeed, the 90% credibility intervals are disjoint for the survival, at least up to the age of 40. By this method, we see that it is highly likely that the two types of mortality are radically different, with a very high mortality of young individuals in the Merovingian period and a survival distribution close to the standard in the Gallo-Roman period. Indeed, the 90% credibility intervals are disjoint for the survival, at least up to the age of 40. The differences noted in distribution of death by age from one period to the other can be explained either by a variation in the age mortality distribution or by differences in population structure (by sex and age) to which the same distribution is applied. By this method, we see that it is highly likely that the two types of mortality are radically different, with a very high mortality of young individuals in the Merovingian period and a survival distribution close to the standard in the Gallo-Roman period.Indeed, the 90% credibility intervals are disjoint for the survival, at least up to the age of 40. By this method, we see that it is highly likely that the two types of mortality are radically different, with a very high mortality of young individuals in the Merovingian period and a survival distribution close to the standard in the Gallo-Roman period. Indeed, the 90% credibility intervals are disjoint for the survival, at least up to the age of 40. The differences noted in distribution of death by age from one period to the other can be explained either by a variation in the age mortality distribution or by differences in population structure (by sex and age) to which the same distribution is applied.

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The comparison of mortality by sex, during the Merovingian period (figures), as well as for the Gallo-Roman period, seems to show a higher mortality for females than for males. However the credibility intervals are very wide (because of the small size of the samples) and superimposed on one another. Therefore data do not allow to conclude any real differences. The comparison of mortality by sex, during the Merovingian period (figures), as well as for the Gallo-Roman period, seems to show a higher mortality for females than for males. However the credibility intervals are very wide (because of the small size of the samples) and superimposed on one another. Therefore data do not allow to conclude any real differences. Frénouville, Merovingian period. Estimated survival functions for males (M), females (W) and both sexes (circles), plus pre-industrial standard for reference (asterisks). Frénouville, Merovingian period. Estimated survival functions for males (M), females (W) and both sexes (circles), plus pre-industrial standard for reference (asterisks). Frénouville, Merovingian period. Fifty percent (solid line) and 90% (dashed line) credible intervals for male and female survival functions (female ones are shifted to the right). Frénouville, Merovingian period. Fifty percent (solid line) and 90% (dashed line) credible intervals for male and female survival functions (female ones are shifted to the right).

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The statistical results obtained for the Frénouville cemetery confirm the historical and archaeological evidence: traces of ancient land surveying and grave goods. According to the archaeological data, the settlement of the new Roman colonists occurred with no mark of violence and the entire rural community lived relatively peacefully. These living conditions are echoed in the estimated survival curves for this period. According to the archaeological data, the settlement of the new Roman colonists occurred with no mark of violence and the entire rural community lived relatively peacefully. These living conditions are echoed in the estimated survival curves for this period. The establishment of Merovingian society after the fall of the Roman Empire was marked by major social changes. The new political and socioeconomic context explains the fact that the chances of survival of young adults were worse than during the previous period. The establishment of Merovingian society after the fall of the Roman Empire was marked by major social changes. The new political and socioeconomic context explains the fact that the chances of survival of young adults were worse than during the previous period. Conclusions

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Compared with the previously proposed methods, our statistical method is significantly better. This was clearly demonstrated by calculating the estimates from two sites where mortality by age was available from records (Maubuisson and Antibes, France), as well as by simulation studies. It has the advantage of properly taking into account the random nature of all the available data. The associated credible intervals are therefore reliable, which is far from being true for the confidence intervals associated with the methods used before. It has the advantage of properly taking into account the random nature of all the available data. The associated credible intervals are therefore reliable, which is far from being true for the confidence intervals associated with the methods used before. Finally it is important to emphasize that although the Bayesian character of the proposed method has many advantages, it does require the user to abandon previous habits. We always have to keep in mind that the prior distribution, and thus the "standard" mortality, has an origin" function in relation to which the data of the target site show the tendency of it to deviate more or less. The difference will be all the more sensitive since the target moves away from standard conditions, but also since the data are more numerous.Finally it is important to emphasize that although the Bayesian character of the proposed method has many advantages, it does require the user to abandon previous habits. We always have to keep in mind that the prior distribution, and thus the "standard" mortality, has an origin" function in relation to which the data of the target site show the tendency of it to deviate more or less. The difference will be all the more sensitive since the target moves away from standard conditions, but also since the data are more numerous.

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Subject to accept the invariance hypothesis, paleodemographic studies could adopt our method even if their archeological populations are far from the reference population, in time and space. Of course, our Bayesian method can be used with any biological age indicator, provided that the documentary reference samples are carefully validated. Thank you very much for your attention

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References Bocquet-Appel, J.-P. (dir), 2008, Recent advances in paleodemography. Data, techniques, patterns, Dordrecht/London, Springer Verlag. Caussinus, H., Courgeau, D., 2010, Estimation des âges au décès en paléodémographie, Population-F, 65 (1), pp. 117-145; English version, 2010, Estimating age without measuring it : A new method in paleodemography, Population-E, 65 (1), pp. 117-144. Hoppa, D., Vaupel, J.W. (eds.), 2002, Paleodemography. Age distributions from skeleton samples, Cambridge, Cambridge University Press. Konigsberg, L.W, Frankenberg, S.R., 1992, Estimation of age structure in anthropological demography, American Journal of Physical Anthropology, 117, pp. 297-309. Masset, C., 1973, La démographie des populations inhumées: essai de paléodémographie, LHomme, XIII (4), pp. 95-131. Séguy, I., Buchet, L., 2011, Manuel de paléodémographie, Paris, Ined. Séguy, I., Caussinus H., Courgeau D., Buchet, L., 2013, Estimating the Age Structure of a Buried Adult Population: A New Statistical Approach Applied to Archaeological Digs in France, American Journal of Physical Anthropology, 150, pp. 170-183.

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