Isabelle SÉGUY 1,2 Luc BUCHET 2,1 Henri CAUSSINUS 3 , Daniel COURGEAU 1

1. Indirect age estimation: probabilistic models and statistical approach Isabelle SÉGUY1,2 Luc BUCHET2,1 Henri CAUSSINUS3, Daniel COURGEAU1 1 Institut National d’Études Démographiques (seguy@ined.fr ; courgeau@ined.fr) 2 CEPAM-UMR 7264, CNRS-Université de Nice-Sophia-Antipolis (luc.buchet@cepam.cnrs.fr) 3 Université Paul Sabatier, Toulouse(henri.caussinus@math.univ-toulouse.fr) Indirect age estimation requires a model linking the object to be estimated (calendar age) to the available data (biological indicator of age). We cannot hope to find adeterministic relation: the model is thus probabilistic and the estimation relies on statistics. Formalisation of the problem The joint probability distribution of a unit’s biological age (discretised as a stage class i, i=1 …r) and chronological age (defined as an age class j, j=1 … c), is described in table 1 where: iis the (marginal)probability of biological stageiand pjthe probability of chronological age j pj/iis the probability of chronological age j given biological stagei pi/jis the probability of biological stageigiven chronological age j pijis the probability of biological stageiand chronological age j Table 1. Probabilitylaw of the "age – biological indicator" pair. Theseprobabilitiessatisfy the formulae(1) and (2) An ‘‘invariance hypothesis’’ assuming that the conditional distribution of stage indicators at a given age is constant over time allows the estimation of probabilities pi/jby frequencies nij/n.jdrawn from a reference data sample (table 2) Table 2. Observed distribution of the "age – biologicalindicator" pair in a referencedata sample. Twokinds of questions arise How do we estimate the age structure of a population, i.e. the probabilitiespj (j=1,…c) ? Credibility intervals are a convenient way to summarise this precision and compare several target sites by means of any function of the pj, e.g. the survival function (figure 1). A Bayesian method has been proposed: it starts from suitable prior probability distributions for the pi/jand pj . As well as a point estimate (e.g. the posterior expectation) the posterior distribution of each pjprovides useful information about the precision of this estimate. The iare estimated by means of anthropological observations on the target site which yield the numbers (m1,…, mi, …, mr ) of skeletons in the different biological stages in an m-sample. The formula (1), together with this information and that provided by the reference data, allows us to estimate the pj . 20-24 30-34 40-44 50-54 60-64 70-74 80 + Groupes d’âges Figure 1. Frénouville, 4th century (left) and 6th-7th centuries AD (shifted to the right); 90% (green) and 50% (red) credibility intervals for the survival function. (Caussinus, Courgeau, 2010; Séguy, Caussinus, Courgeau, Buchet, 2013; session paper 292). How do we estimate the chronological age of an isolated individual whose biological stage i is known, i.e. the probabilities pj/i for a given i, and j =1,…c ? Formula (2) provides a point estimate of pj/i,by replacing the probabilities in the second member by point estimates deduced from the previous study. But it is important to evaluate the reliability of this estimate. With the probability distribution of the pi/j provided bythe reference dataand the (posterior) distribution of thepjobtained above, we draw random samples for these parameters and formula (2) provides a random sample for the pj/i which simulates their distribution. Applications in funeral archaeology and paleopathology Two men were buried, not in the necropolis of Frénouville (France), but not far from it, near a ford (at Bellengreville).Their robust morphology distinguishes them from Gallo-Roman inhabitants of Frénouvilleand the observed degenerative osteoarthritis suggests that they are rather old individuals, whereas their biological stage (3) would classify them as mature. Were they over 50 years old as suggested by their bone status; or severely disabled young adults? We can estimate the chronological age of these men, given their biological stage i = 3, if we assume that they are drawn for the same population as that of the Gallo-Roman necropolis, for which the age structure has been estimated using our method. (cf. session paper 292) The estimated probability that the chronological age is above 50 years is 0.701. The estimated probability that the chronological age is below 30 years is 0.075. The precision of these estimates is shown graphically by the densities in figure 2 and some numerical values of the quantiles are given in table 3. - The first probability (age >50 years) is estimated with rather low precision (between .60 and .75 with a 50% probability); however, there is only a 5% chance that it is below .495 (about ½) and a 25% chance that it is below .606. - The second probability is clearly low, estimated with rather high precision around the median .08; there is a 5% chance that it is above .212 and a 25% chance that it is above .131. Figure 2. Distribution (density) and  5%, 50% , 95% quantiles (redlines) of the estimated probability that the chronological age is above 50 years (left) or below 30 years, for an individualwhose biological stage i = 3 Age over 50 yearold, knowing the biological stage: 3 Table 3. Somequantiles corresponding to the densiities in figure 2 XXVII IUSSP International Population Conference, Busan, South Korea, August 25-31, 2013 Thus, the statistical approach confirms, with a quantifiable degree of confidence, that these men have reached an advanced age in spite of their very active and perhaps dangerous way of life, as evidenced by paleopathological observations. Age under 30 yearold, knowing the biological stage: 3 Applications in Forensic and Social Sciences • As part of a multidisciplinary workgroup " Measures of age without civil registers ", supported by Fondation du Campus Condorcet, we examined the application of these methods to problems arising in different domains but with a similar formal presentation. They can be applied easily, at the cost of reconsidering the prior probability distribution of the pj. For example: • in forensicmedecine, when an expert has to estimate a civil age j,on the basis of anatomical observations i, on a dead or living subject whose age is unknown (Chariot, 2010), • in demography when it is sometimes necessary to reattribute a civil age to persons whose declared age is subject to opportunistic variation. Références : CHARIOT Patrick. Quand les médecins se font juges: la détermination de l'âge des adolescents migrants. Chimères, 2010, no 3, p. 103-111. CAUSSINUS Henri, and COURGEAU Daniel. Estimating age without measuring it: A new method in paleodemography. Population (English edition), 2010, vol. 65, no 1, pp. 117-144. SÉGUY Isabelle, CAUSSINUS Henri, COURGEAU Daniel, BUCHET Luc. 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, 2013, p. 170–18 SÉGUY Isabelle, BUCHET Luc. With the contributions of Henri Caussinus and Daniel Courgeau.Handbook of Paleodemography. Springer Series: INED Population Studies, Vol. 2. (Original French edition published by INED, Paris, 2011). 2014, V, 220 p.