Presentation and Presentation and implementation of a new implementation of a new

approach of age from biological approach of age from biological indicatorsindicators

(Bayesian inference procedure)(Bayesian inference procedure)

Luc Buchet, Henri Caussinus, Daniel CourgeauLuc Buchet, Henri Caussinus, Daniel Courgeau(CNRS-Nice University, Toulouse (CNRS-Nice University, Toulouse UniversityUniversity, Ined), Ined)

27th IUSSP International Population Conference, Busan, Korea. 27th IUSSP International Population Conference, Busan, Korea. 26 – 31 August 201326 – 31 August 201327th IUSSP International Population Conference, Busan, Korea. 27th IUSSP International Population Conference, Busan, Korea. 26 – 31 August 201326 – 31 August 2013

Context and methods used in the Context and methods used in the pastpast

• To estimate the age structure of a population of the past, To estimate the age structure of a population of the past, paleodemographers usually only have the paleodemographers usually only have the structure by stage structure by stage given given by biological indicatorsby biological indicators..

• To be able to make such an estimation, To be able to make such an estimation,

they also need a they also need a reference populationreference population, ,

for which a for which a reference matrix reference matrix gives both gives both

ages and stages.ages and stages.

• With these two elements, With these two elements, different methods different methods were used in the were used in the past to make the estimationpast to make the estimation

A first method A first method (ALK)(ALK), used in paleodemography since the 70’s, , used in paleodemography since the 70’s, consists in reconstituting from the observed data consists in reconstituting from the observed data a matrix a matrix showing the greatest closeness with each cell of the reference showing the greatest closeness with each cell of the reference matrix. matrix.

However, these methods However, these methods rest on an unreliable assumption of rest on an unreliable assumption of resemblance between the reference and the target populations resemblance between the reference and the target populations and and do not take into account the do not take into account the “invariance hypothesis” “invariance hypothesis” whereby whereby for any remains of a given age at death, the likelihood of being 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 classified in a given stage only depends on that age, and that it is constant over time. constant over time.

To consider this invariance hypothesis, even if it may be To consider this invariance hypothesis, even if it may be disputable, other methods have been proposed to permit a more disputable, other methods have been proposed to permit a more satisfactory estimation. satisfactory estimation. It is, for example, on this hypothesis that It is, for example, on this hypothesis that IALKIALK method or similar methods method or similar methods (PFP) (PFP) are based.are based.

Then we can deduce, from the Then we can deduce, from the given given stage structurestage structure, a , a

population’s population’s age structureage structure..

Then we can deduce, from the Then we can deduce, from the given given stage structurestage structure, a , a

population’s population’s age structureage structure..

18-29 30-39 40-49 50-59 60-69 70-79 80+stage A 0,35 0,19 0,17 0,13 0,09 0,06 0,03stage B 0,10 0,16 0,17 0,20 0,13 0,18 0,06stage C 0,05 0,08 0,13 0,17 0,24 0,22 0,11stage D 0,03 0,05 0,07 0,15 0,30 0,26 0,14stage E 0,01 0,05 0,05 0,12 0,35 0,23 0,19

18-29 30-39 40-49 50-59 60-69 70-79 80+stage A : 48 16,78 8,93 7,97 6,09 4,32 2,68 1,23stage B : 24 2,36 3,76 4,20 4,76 3,09 4,38 1,45stage C : 14 0,72 1,05 1,83 2,44 3,31 3,14 1,52stage D : 16 0,47 0,72 1,19 2,45 4,80 4,20 2,16stage E : 23 0,25 1,17 1,17 2,65 8,01 5,36 4,38

% 11,76 8,93 9,34 10,51 13,45 11,30 6,14

Archaeological Matrix (Rouen, France)

Reference Matrix M-F

Nevertheless, the observed population is usually of small size, so Nevertheless, the observed population is usually of small size, so that the results obtained with these methods are often that the results obtained with these methods are often inconsistent, giving for example an estimated null probability for inconsistent, giving for example an estimated null probability for some age groups .some age groups .

• In order to avoid such inconsistencies it is possible to introduce a In order to avoid such inconsistencies it is possible to introduce a continuous rather than discrete age and use a continuous rather than discrete age and use a parametric modelparametric model, as , as the Rostock manifesto recommended at the beginning of the 2000’s the Rostock manifesto recommended at the beginning of the 2000’s ((Hoppa and Vaupel)Hoppa and Vaupel)..

• Another approach Another approach (IBPFP)(IBPFP) was more recently proposed by Bocquet- was more recently proposed by Bocquet-Appel and Bacro (2008). To escape the constraints of a uniform Appel and Bacro (2008). To escape the constraints of a uniform distribution (reference matrix), the authors search a solution in a distribution (reference matrix), the authors search a solution in a universe of mortality models.universe of mortality models.

All these methods introduce some All these methods introduce some a priori a priori considerations in their considerations in their problem’s solution. However, they are still using a problem’s solution. However, they are still using a frequentist frequentist paradigmparadigm. We propose instead a typically . We propose instead a typically Bayesian statistical method Bayesian statistical method which results in substantial improvement.which results in substantial improvement.

Proposed Proposed methodmethod

• Here we address the case where skeletal remains are classified into Here we address the case where skeletal remains are classified into rr

stages stages and ages into and ages into cc classes classes. Denote by the probability that an . Denote by the probability that an

individual from the target population is at individual from the target population is at stage stage ii and and age age j j ; ;

the sum over the sum over ii is is ,, the sum over the sum over jj is is and the conditional and the conditional

probability of stage probability of stage ii when age when age jj is known is is known is

They are related by the relationship:They are related by the relationship:

• We will use here a We will use here a full Bayesian statistical method full Bayesian statistical method in order to in order to estimate from estimate from priorprior distributions of the different parameters, suitably distributions of the different parameters, suitably chosen by the user, a chosen by the user, a posteriorposterior distribution of the probabilities that distribution of the probabilities that the individuals of the target population are of the individuals of the target population are of

age age jj..

The observed frequencies in the reference population are considered The observed frequencies in the reference population are considered as a set of random variables as a set of random variables multinomial; it’s multinomial; it’s the same for each the same for each column of the reference matrix.column of the reference matrix.

ijp

jp ijip

ijij

j pp

For the For the priorprior distribution distribution of it is natural of it is natural among statisticians among statisticians to to adopt a adopt a Dirichlet distributionDirichlet distribution. This distribution depends on . This distribution depends on cc parameters; since we are dealing with a mortality distribution, we parameters; since we are dealing with a mortality distribution, we choose these parameters so that the average of the Dirichlet choose these parameters so that the average of the Dirichlet distribution is equal to distribution is equal to mortality “standard”mortality “standard” asas defined for defined for preindustrial societies. preindustrial societies.

Other Other prior prior distributions were considered but they do not always distributions were considered but they do not always give satisfactory results, especially if the site in question is give satisfactory results, especially if the site in question is atypical.atypical.

For each For each j,j, the the priorprior distribution of is a Dirichlet distribution distribution of is a Dirichlet distribution which parameters are provided by the reference data using a which parameters are provided by the reference data using a Bayesian approach entirely conventional.Bayesian approach entirely conventional.

jp

jip

Age groups 18–19 20–24 25–29 30–34 35–39 40–44 45–49 50–54 55–59 60–64 65–69 70–74 75–79 80 +Mean 0.019 0.047 0.052 0.052 0.057 0.060 0.066 0.075 0.086 0.099 0.107 0.107 0.086 0.086 Standard deviation 0.035 0.055 0.057 0.057 0.060 0.061 0.064 0.068 0.073 0.077 0.080 0.080 0.073 0.073

Prior distribution (standard)

• From these From these priorprior distributions, and knowing the distribution stages distributions, and knowing the distribution stages in the target population, it in the target population, it is possible to calculate a is possible to calculate a posteriorposterior distribution distribution of the age structure of the target population.of the age structure of the target population.

• The The posteriorposterior means means give give point estimates of each but, beyond point estimates of each but, beyond that, we can estimate the accuracy of the estimate by means of that, we can estimate the accuracy of the estimate by means of the posterior densities or the credibility intervalsthe posterior densities or the credibility intervals ( (Bayesian Bayesian equivalents of confidence intervals of the frequentist estimation).equivalents of confidence intervals of the frequentist estimation).

• From these From these priorprior distributions, and knowing the distribution stages distributions, and knowing the distribution stages in the target population, it in the target population, it is possible to calculate a is possible to calculate a posteriorposterior distribution distribution of the age structure of the target population.of the age structure of the target population.

• The The posteriorposterior means means give give point estimates of each but, beyond point estimates of each but, beyond that, we can estimate the accuracy of the estimate by means of that, we can estimate the accuracy of the estimate by means of the posterior densities or the credibility intervalsthe posterior densities or the credibility intervals ( (Bayesian Bayesian equivalents of confidence intervals of the frequentist estimation).equivalents of confidence intervals of the frequentist estimation).

jp

ResultsResults• To evaluate To evaluate such a method, such a method, wwe have e have already applied it already applied it to different theoretical cases and to a to different theoretical cases and to a

number of past populations: Neolithic site of number of past populations: Neolithic site of Loisy-en-BrieLoisy-en-Brie, nuns of , nuns of MaubuissonMaubuisson (17 (17thth and 18 and 18thth century), cemetery of century), cemetery of AntibesAntibes (19 (19thth century). century).

• We examine now We examine now the results obtained in the results obtained in FrénouvilleFrénouville during the during the Gallo-RomanGallo-Roman period period (3(3rdrd to 5 to 5thth century) and the century) and the Merovingian period Merovingian period (until late 7(until late 7thth century). The samples were respectively of 69 century). The samples were respectively of 69 and 200 remains.and 200 remains.

Frénouville, 3Frénouville, 3rdrd – 7 – 7thth c. A.D. c. A.D.

Distribution by stages of synostosis. Distribution by stages of synostosisDistribution by stages of synostosis. Distribution by stages of synostosis . .

Stage M-F F MA 92 33 30B 29 6 9C 22 6 11D 27 3 15E 30 13 9

200 61 74

N/stageMerovingian period

Stage M-F F MA 20 12 5B 10 0 8C 14 3 10D 13 5 3E 12 6 6

69 26 32

Gallo-Roman periodN/stage

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 By this method, we see that it is highly likely that that it is highly likely that the two types of the two types of mortality are radically differentmortality are radically different, with a very high mortality of , with a very high mortality of young individuals in the Merovingian period and a survival young individuals in the Merovingian period and a survival distribution close to the “standard” in the distribution close to the “standard” in the Gallo-Roman period.Gallo-Roman period. Indeed, the Indeed, the 90% 90% credibility intervalscredibility intervals are disjoint for the survival, at are disjoint for the survival, at least up to the age of 40. least up to the age of 40.

The differences noted in distribution of death by age from The differences noted in distribution of death by age from one period to the other can be explained either by a variation in one period to the other can be explained either by a variation in the age mortality distribution or by differences in population the age mortality distribution or by differences in population structure (by sex and age) to which structure (by sex and age) to which the same distribution is the same distribution is applied.applied.

By this method, we see By this method, we see that it is highly likely that that it is highly likely that the two types of the two types of mortality are radically differentmortality are radically different, with a very high mortality of , with a very high mortality of young individuals in the Merovingian period and a survival young individuals in the Merovingian period and a survival distribution close to the “standard” in the distribution close to the “standard” in the Gallo-Roman period.Gallo-Roman period. Indeed, the Indeed, the 90% 90% credibility intervalscredibility intervals are disjoint for the survival, at are disjoint for the survival, at least up to the age of 40. least up to the age of 40.

The differences noted in distribution of death by age from The differences noted in distribution of death by age from one period to the other can be explained either by a variation in one period to the other can be explained either by a variation in the age mortality distribution or by differences in population the age mortality distribution or by differences in population structure (by sex and age) to which structure (by sex and age) to which the same distribution is the same distribution is applied.applied.

• The The comparison of mortality by sex comparison of mortality by sex , during the Merovingian period , during the Merovingian period (figures), as well as for the Gallo-Roman period, seems to show a higher (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 mortality for females than for males. However the credibility intervals are very wide (because of the small size of the samples) and superimposed on very wide (because of the small size of the samples) and superimposed on one anotherone another. Therefore data . Therefore data do not allow to conclude any real differences. do not allow to conclude any real differences.

Frénouville, Merovingian period. Estimated Frénouville, Merovingian period. Estimated survivalsurvival

functions for males (M), females (W) and both functions for males (M), females (W) and both sexes (circles),sexes (circles),

plus pre-industrial standard for reference plus pre-industrial standard for reference (asterisks).(asterisks).

Frénouville, Merovingian period. Estimated Frénouville, Merovingian period. Estimated survivalsurvival

functions for males (M), females (W) and both functions for males (M), females (W) and both sexes (circles),sexes (circles),

plus pre-industrial standard for reference plus pre-industrial standard for reference (asterisks).(asterisks).

Frénouville, Merovingian period. Fifty percent (solidFrénouville, Merovingian period. Fifty percent (solidline) and 90% (dashed line) credible intervals for line) and 90% (dashed line) credible intervals for

male andmale andfemale survival functions (female ones are shifted female survival functions (female ones are shifted

to the right).to the right).

Frénouville, Merovingian period. Fifty percent (solidFrénouville, Merovingian period. Fifty percent (solidline) and 90% (dashed line) credible intervals for line) and 90% (dashed line) credible intervals for

male andmale andfemale survival functions (female ones are shifted female survival functions (female ones are shifted

to the right).to the right).

The statistical results obtained for the Frénouville cemetery confirm The statistical results obtained for the Frénouville cemetery confirm the historical and archaeological evidence: traces of ancient land the historical and archaeological evidence: traces of ancient land surveying and grave goods.surveying and grave goods.

• According to the archaeological data, According to the archaeological data, the settlement of the new the settlement of the new Roman colonists occurred with no mark of violence and the entire Roman colonists occurred with no mark of violence and the entire rural community lived relatively peacefully. These living conditions rural community lived relatively peacefully. These living conditions are echoed in the estimated survival curves for this period.are echoed in the estimated survival curves for this period.

• The establishment of Merovingian society after the fall of the Roman The establishment of Merovingian society after the fall of the Roman Empire was marked by major social changes. The new political and Empire was marked by major social changes. The new political and socioeconomic context explains the fact that the chances of survival socioeconomic context explains the fact that the chances of survival of young adults were worse than during the previous period.of young adults were worse than during the previous period.

ConclusioConclusionsns

Compared with the previously proposed methods, our Compared with the previously proposed methods, our statistical method is significantly better. This was clearly statistical method is significantly better. This was clearly demonstrated by calculating the estimates from two sites demonstrated by calculating the estimates from two sites where mortality by age was available from records where mortality by age was available from records (Maubuisson and Antibes, France), as well as by simulation (Maubuisson and Antibes, France), as well as by simulation studies. studies.

• It has the advantage of properly taking into account the random nature of all It has the advantage of properly taking into account the random nature of all the available data. The associated credible intervals are therefore reliable, the available data. The associated credible intervals are therefore reliable, which is far from being true for the confidence intervals associated with the which is far from being true for the confidence intervals associated with the methods used before.methods used before.

• Finally it is important to emphasize that although the Bayesian character of the 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 proposed method has many advantages, it does require the user to abandon previous habits. We always have to keep in mind that the previous habits. We always have to keep in mind that the prior prior distribution, and distribution, and thus the "standard" mortality, has an “origin" function in relation to which the 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 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 difference will be all the more sensitive since the target moves away from standard conditions, but also since the data are more numerous.standard conditions, but also since the data are more numerous.

Subject to accept the invariance hypothesis, paleodemographic Subject to accept the invariance hypothesis, paleodemographic studies could adopt our method even if their archeological studies could adopt our method even if their archeological populations are far from the reference population, in time and populations are far from the reference population, in time and space.space.

Of course, our Bayesian method can be used with any biological Of course, our Bayesian method can be used with any biological age indicator, provided that the documentary reference samples age indicator, provided that the documentary reference samples are carefully validated. are carefully validated.

Thank you very much for your attentionThank you very much for your attention

ReferencesReferences

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