CAIRN-INT.INFO : International Edition

1The use of family reconstitution for studying the demography of the Ancien Régime has, up to the present, generally been limited to fertility and nuptiality. Though this approach has proved to be the most appropriate for these two phenomena, it has raised numerous problems for the estimation of adult mortality [1]. It is obviously not possible to study mortality by using only the data on women’s deaths in a village, as this would lead to an overestimate, since migration results in a bias in favour of women who died at young ages.

2It has been well nigh impossible to obtain an accurate estimate of local adult mortality because of these problems [2] or to make comparisons between different villages or regions (except from the point of view of fluctuations : mortality crises, etc.).

3We shall begin by studying the indices which have so far been proposed, and go on to present a certain number of estimates and analyze their stability.

I – Proposed methods of estimation

4Two authors have proposed methods for estimating local mortality. L. Henry [3] has described the construction of two extreme tables. The first is obtained by assuming that all individuals whose date of death is unknown died at the oldest age at which they were observed alive (i.e., at the birth of a child, in nominative lists, etc.). The second table is obtained by assuming that these same individuals died at the lowest age at which they were mentioned as deceased (death of a child, marriage of a child, etc.). While these tables have at least the advantage of being unaffected by any particular error or bias, they only make use of a very small proportion of data and do not, therefore, reflect the overall mortality in a given village correctly. They can, furthermore, lack precision, if the rate of mobility is too high, and this can render their use difficult [4].

The method proposed by J. Dupâquier

5It was with the aim of mitigating these disadvantages that J. Dupâquier suggested a method of estimating adult mortality, which consists of delimiting the observation periods for a woman or a man by the birth of his or her first child and the last death observed of an individual’s child aged less than fifteen years ; this age was chosen as it was assumed that a child aged fifteen had not yet left its parental home. This method defined the dates of the end of the observation period independently of the death of the person under consideration and, consequently, all types of family cards could be used. J. Dupâquier had noted, as did J.P. Bardet at a later date, that mortality had been greatly underestimated, particularly for the younger age groups.

6These estimates are in fact biased, not by any possible interactions between mortality and fertility, but by the fact that the observation period thus defined is strongly related to the level of mortality or, more precisely, the probability of being observed is not independent of the age at death. In effect, the more children an individual has, the greater is the likelihood that at least one will die and, thus, the greater the probability of his or her being observed. Since individuals who live longer tend to have more children, a selection bias has been introduced favouring those who live longest and mortality observed at the reproductive ages is underestimated.

7A more formal expression of this reasoning is given in Annex 1. We show how only a fraction of the women observed are representative of the population and that it is not possible to correct the bias by weighting different groups of women. The extent of the bias, under given conditions of nuptiality, fertility and mortality, can, however, be estimated by simulation.

Study of the indices

8We used Network 100 of Ledermann’s life tables with various life expectancies (between 25 and 45 years) as the mortality pattern [5]. In Figure 1 below, we show the relation between the estimates obtained from the original method and the actual five-year probabilities of dying between the ages of 20 and 45 years (q1x\qx for men and women, with the notation of Annex 1). There is, indeed, a considerable degree of underestimation, which increases with life expectancy and decreases with age. While this underestimation is low when life expectancy at birth is 25 years (approximately 0.8 for 5q20), the corresponding value for a life expectancy of 45 years is 0.4 for females and as low as 0.3 for males.

9These simulations, therefore, show that the original method is impracticable in its present form. To conclude this approach, we can check the table calculated by J.P. Bardet in his study of Rouen [6] as shown in Table 1.

10The underestimate is no doubt higher at younger ages, for both males and females, and decreases with age. (We have calculated the ratio between the probabilities for Rouen and those for the whole of France between 1750 and 1759 [7]. Of course, this ratio does not give a true value for the underestimate, but its development over time can reveal actual trends). On studying Figure 1, it may be seen that the estimated probabilities appear to correspond to those of Ledermann’s life table with a life expectancy at birth slightly in excess of 30 years (this result should, however, be regarded with caution since it depends on the hypotheses used in the simulations of nuptiality and fertility).

Table 1

Probability of dying (per 1,000) of married persons in Rouen (18th century) and in the whole of France (1750-1759)

Table 1
Probability Men Women Rouen (1) Whole of France (2) Ratio (1)/(2) Rouen (1) Whole of France (2) Ratio (1)/(2) 5q25 24 59 0.41 39 60 0.65 5q30 31 64 0.48 44 69 0.64 5q35 32 72 0.44 62 75 0.83 5q40 44 90 0.49 88 80 1.10 5q45 49 111 0.44 104 88 1.19 5q50 108 124 0.87 100

Probability of dying (per 1,000) of married persons in Rouen (18th century) and in the whole of France (1750-1759)

The probabilities for the whole of France are taken from Y. Blayo, op. cit. in fn. 7.

II – New methods of estimation

11Given the deficiencies of the methods described in section I, we must look for alternatives. We shall deal with the mortality of women married locally [8] (M cards) [9].

The localities used in the study

12To illustrate the proposed estimates, we use the villages of Champigny (located in the North-East region of the INED sample) and Saint-Aignan (North-West region). In Table 2, we show the number of women by 50-year cohorts in the locality [10], classified by whether they are alive or dead, married or unmarried, and by the number of locally-born children [11]. We note that a majority of the women who died outside the locality, but who were married within it, were childless. It is likely that most of them were women who married in their locality of origin, but who left immediately to join their husbands in their place of residence.

Figure 1

Ratio of estimated quinquennial probabilities of dying to actual probabilities by age and for various levels of life expectancy at birth

Figure 1

Ratio of estimated quinquennial probabilities of dying to actual probabilities by age and for various levels of life expectancy at birth

Table 2

Number of women in the sample(1),(2)

Table 2
Champigny Saint-Aignan Number of children cohorts cohorts 1650-1699 1700-1749 1650-1699 1700-1749 (1) (2) (1) (2) (1) (2) (1) (2) 0 36 68 51 42 45 57 77 71 1 28 15 33 10 63 10 48 9 2 29 14 23 5 47 6 45 5 3 26 9 31 6 49 5 51 4 4 27 7 28 6 46 5 43 2 5 38 7 37 0 44 0 55 2 6 39 3 39 2 28 0 37 0 7 33 4 34 0 27 0 27 2 8 23 3 32 3 24 0 20 2 9 15 3 20 1 16 0 17 0 10 or more 22 1 32 0 30 1 18 1 Total 316 134 360 75 419 84 438 98

Number of women in the sample(1),(2)

The calculations were carried out after imputation of missing ages (cf. Annex 2).
(1) Women who died or remarried or whose husbands predeceased them in the village, and who were married in the village (FC cards and some MO cards) (see fn. 9 for notations).
(2) Women who married in the village but who did not die there (majority of the MO cards).

A – Mortality of women married in the village

13These women are shown on the MF and MO family cards. Missing ages have been imputed using the method described in Annex 2. It is important to use as many data cards as possible and, thus, to be able to assign an age to every woman. Each woman was observed from the time of her marriage in the locality studied. If the date of migration were known, it would be possible to estimate the mortality of these women accurately. A woman would be traced from her marriage to her death or departure. This did not prove to be possible but, for each woman, various dates are available which make it possible to assess whether or not she was present in the village. Women who died in the village were considered to be at risk from the date of their marriage to that of their death [12]. For other women, we know a last event which proved their presence in the village. We shall consider only births or widowhoods as such events [13]. It is this information which we propose to use for the study of mortality.

A maximum estimate of mortality

14To begin with, we may assume that a woman whose death has not been observed emigrated immediately after her last recorded birth, or marriage if no birth had been recorded [14]. Such women are considered to have been at risk between the date of their marriage and that of their last recorded birth. This assumption is completely justifiable for childless women (who, in the main, are those who came to be married in the parish but returned to their husband’s parish immediately after their marriage) [15] ; it is less so for the others. The women who died were exposed to risk between the dates of their marriage and their deaths. The women who remarried were at risk between the dates of their marriage and remarriage for first marriages, and between remarriage and leaving observation (death, or birth of last child, or end of period of observation) for the others. Entries into or exits from observation are summarized in Table 3.

15Mortality can then be estimated as follows :

17where Vx are the women present at age x, Ex exits from observation as defined above, Dx deaths observed at such an age, and Ix entries into observation. Naturally, this index will overestimate the mortality of women married in the locality since the number of women present at age x is underestimated.

18To be certain of obtaining a maximum estimate of mortality, temporary absences should also be taken into account. In effect, a woman is not necessarily present in the locality during the period between two births, and it is impossible to be certain that a woman, who had married and died in the locality, was truly at risk between her marriage and her death. If the proportion of women temporarily absent at each age were known, the number present would be overestimated in the same proportion and, consequently, the probability of dying underestimated in proportion. However, this proportion is not known, but could be estimated by using the proportion of "lost" births [16]. This proportion can be put equal to the proportion of women temporarily absent and the probabilities of dying adjusted accordingly.

Table 3

Women married in the locality who leave observation — Maximum estimate of mortality

Table 3
Data card nomenclature Description Exit from observation Women whose death is observed I) Married in the locality deceased in the locality death II) Married in the locality, remarried in the locality remarriage Women whose death is not observed III) Married in the locality, husband died in the locality, woman died later outside the locality widowhood IV) Married in the locality, no child born in the locality marriage V) Married in the locality, n children born in the locality birth of the nth child

Women married in the locality who leave observation — Maximum estimate of mortality

Similar simplified methods

19The hypothesis that women migrate after the last observed event is basic to methods which consist of reallocating deaths to migrant women by iteration, based on their estimated mortality at the previous stage. This is demonstrated and explained in Annex 3. It is possible to conceive of more indirect methods than have been described. In particular, it would be feasible to determine the age distribution at death of women who died locally, depending on the number of their children : from these, the distribution of ages at death of women with 0 or more children, 1 or more children, etc., could be inferred. It would then be possible to apply the distribution of ages at death of women who had had n or more children and who had died locally to women with the same number of children who died outside the locality. It is therefore possible to re-estimate the previous distributions based on the experience of all women, and to apply them again to women who died outside the locality. This procedure is then repeated until convergence. This approach, which has the advantage of simplicity (in particular, it is not necessary to impute missing ages in advance), is more or less equivalent to what has just been described. In effect, the strong link between birth order and mother’s age is equivalent to using mortality according to the age of the last observed child.

Empirical results

20In Table 4, we show life tables obtained for Champigny and Saint-Aignan. In one set, only data relating to women deceased or remarried in the village have been used ; in the other, we have added information about women married in the village but who were still living. Both the methods described previously were applied [17]. A comparison between the two sets of tables shows that the final bias resulting from migration was very slight. Adjustment for temporary absences yields column II’. The slight difference between columns II and II’ does not therefore cast any doubts on the value of this approach [18].

Table 4

Women married in the village : life table with maximum mortality(I),(II),(II’)

Table 4
Age (years) Champigny Saint-Aignan cohorts cohorts 1650-1699 1700-1749 1650-1699 1700-1749 (I) (II) (II’) (I) (II) (II’) (I) (II) (II’) (I) (II) (II’) 20 32.2 33.1 32.6 36.2 36.8 36.1 24.1 24.9 24.7 28.3 29.7 29.3 25 28.6 29.2 28.7 33.4 33.8 33.1 22.0 22.7 22.6 25.4 26.7 26.3 30 25.2 25.6 25.2 30.3 30.6 30.0 20.3 20.9 20.7 24.4 25.4 25.1 35 23.1 23.3 22.9 27.5 27.7 27.1 19.3 19.7 19.6 22.6 23.4 23.1 40 20.3 20.4 20.0 24.8 25.0 24.5 18.3 18.7 18.5 20.8 21.4 21.1 45 17.2 17.2 16.8 21.0 21.1 20.6 16.1 16.4 16.3 17.8 18.3 18.0 50 15.0 15.0 14.7 17.5 17.6 17.2 13.3 13.5 13.4 14.6 15.1 14.8 55 12.8 12.8 12.6 14.2 14.3 13.9 11.1 11.3 11.2 12.7 13.0 12.8 60 9.7 9.7 9.4 11.4 11.5 11.1 9.1 9.3 9.2 10.0 10.2 10.0 65 7.2 7.2 7.0 8.9 9.0 8.7 8.0 8.2 8.1 7.7 7.8 7.7 70 6.1 6.1 5.9 6.5 6.6 6.4 6.8 7.0 6.9 5.6 5.6 5.5 75 5.6 5.6 5.4 4.8 4.9 4.7 5.6 5.8 5.7 4.9 4.9 4.7

Women married in the village : life table with maximum mortality(I),(II),(II’)

(I) Life expectancy obtained from data relating only to women who died in the village.
(II) Life expectancy obtained from data relating to women married in the village, on the assumption that those who emigrated did so immediately after the last event observed.
(II’) II corrected for temporary absences.
The tables were originally calculated for single years of age.

A minimum estimate of mortality

21The women in the village are observed at the birth of each child. In the demographic regime of 18th-century France, women under the age of 45 had frequent births (we recall that the average interval was around 20 to 30 months, depending on age). Women who had left the locality, and whose last birth occurred before their 45th birthday, will, in the majority of cases, have migrated before having a further birth, and the age distribution of women at this birth can be discovered.

Simplified description

22For the sake of clarity, we begin by assuming that intervals between births are independent of mother’s age and birth order, and are identical for all women. We shall also at first disregard cases of definitive sterility before the age of 45 and those of women widowed before that age. A married woman will thus have a child at regular intervals (say, every two years), between the date of her marriage and her death, or her 45th birthday provided she survived to that age.

23Consider a woman who had her last child locally at age a and for whom no trace can be found at a later date. This woman must therefore have left the locality before dying, between the ages of a and a’ = a + 2. This provides a first possible method of estimating a life table. It is sufficient to suppose that the woman left at age a + 2, or just before next giving birth. This is the analogue of the previous method.

24At this stage of the analysis, we have two extreme estimated life tables, based on data relating to married women in the village.

Complete description and application

25In real life, births do not occur at regular intervals and no information is available about births which occur after women have left the locality. Furthermore, a proportion of women become definitively sterile after each birth. Finally, the reproductive life span of some women is interrupted by widowhood.

26However, by using information about women who die or remarry locally, and who have reached the age of 45, it is possible to estimate the interval between each birth, the interval between marriage and first birth and definitive sterility. If it is then assumed that the fertility pattern of women who left the locality is the same as that of those who remained, it is possible to estimate the proportion who had a later birth among those who left, as well as the age at which such births occurred.

27More precisely, suppose that the interval between last birth and sterility depends only on age of the woman [19]. We can then pair an emigrant woman with a woman who had died in the village after her 45th birthday and who had a birth at the same age at which the woman who emigrated had her last observed birth, or if she had no children born in the locality, married at the same age (in practice, within a short interval, for example +/– 2.5 years). This woman will be chosen at random from the whole group, as in the imputation of ages (cf. Annex 2) [20]. The exits from observation are then determined as in Table 5. The "unobserved" events are those which occur to the paired woman, after the event which is common to both women.

Table 5

Exits from observation of women married in the village

Table 5
Data card nomenclature Description Exit from observation Women whose death is observed I) Married in the locality, deceased in the locality death II) Married in the locality, remarried in the locality remarriage Women whose death is not observed III) Married in the locality, husband died in the locality, died later outside the locality before the "unobserved" death but after widowhood IV) Married in the locality, no children born in the locality before the 1st child or the "unobserved" death V) Married in the locality, n children born in the locality before the (n + 1)th child or the "unobserved" death*

Exits from observation of women married in the village

* For these women, when no "unobserved" child is born, we impute a death : it is then assumed that the exit from observation so "imputed" occurs immediately before this death.

28Consider, for example, a woman married locally, who has had n children in the locality ; her nth child is born when she is aged a and neither her death nor her widowhood is observed. We therefore associate with this woman another woman who has reached her 45th birthday in the village, had a child of order p at an age close to a ; if this woman then has a child of order p + 1 at age a’, the former woman’s exit from observation is assumed to occur at age a’.

29Consider, for example, a woman married locally, who has had n children in the locality ; her nth child is born when she is aged a and neither her death nor her widowhood is observed. We therefore associate with this woman another woman who has reached her 45th birthday in the village, had a child of order p at an age close to a ; if this woman then has a child of order p + 1 at age a’, the former woman’s exit from observation is assumed to occur at age a’.

30To be rigorous, the procedure described should be repeated when the fate imputed is a death. The associated age at death is, in effect, drawn from the ages at death of women who died locally ; this procedure introduces a bias which, however, is not significant. Furthermore, we have not taken temporary absences into account. The mortality levels obtained are, therefore, even further underestimated. However, they could be incorporated into our calculations as was done previously, but the correction would not be very significant.

Empirical results

31The "minimum" life table thus obtained is compared with the previous table in Table 6.

Table 6

Women married in the village : maximum and minimum life tables

Table 6
Champigny Saint-Aignan Age cohorts cohorts (years) 1650-1699 1700-1749 1650-1699 1700-1749 (II’) (III) (II’) (III) (II’) (III) (II’) (III) 20 32.6 34.1 36.1 37.6 24.7 25.6 29.3 31.0 25 28.7 30.3 33.1 34.5 22.6 23.5 26.3 27.9 30 25.2 26.6 30.0 31.3 20.7 21.7 25.1 26.7 35 22.9 24.3 27.1 28.4 19.6 20.6 23.1 24.7 40 20.0 21.4 24.5 25.6 18.5 19.6 21.1 22.7 45 16.8 18.2 20.6 21.7 16.3 17.3 18.0 19.6 50 14.7 15.9 17.2 18.3 13.4 14.4 14.8 16.4 55 12.6 18.6 13.9 14.9 11.2 12.2 12.8 14.3 60 9.4 10.4 11.1 12.0 9.2 10.2 10.0 11.4 65 7.0 7.9 8.7 9.5 8.1 8.9 7.7 8.9 70 5.9 6.6 6.4 7.0 6.9 7.6 5.5 6.5 75 5.4 6.0 4.7 5.2 5.7 6.3 4.7 5.5

Women married in the village : maximum and minimum life tables

(II’) Life expectancy obtained using information about women married in the village, assuming that the women who did not die in the village left immediately after the last observed birth (reminder of Table 4).
(III) Life expectancy obtained using information about women married in the village, assuming that the women who did not die in the village left immediately before the first "birth following their departure".

32We have incorporated the life expectancies at various ages in Figure 2 (hereafter) for the 1700-1749 cohorts and for the two estimates. The differences between the extreme tables, for the same village, are not significant, and we can, therefore, obtain a good estimate of mortality. It should, however, be noted that it would be possible, after having checked that the results are sufficiently stable, to construct for more detailed analysis a single table obtained by assuming that the women who left observation at a given age left in the middle of the interval between the last observed event and the first subsequent event (or preferably, uniformly over the interval).

Figure 2

Life expectancy in Champigny and Saint-Aignan – minimum and maximum mortality (women married locally – cohorts 1700-1749)

Figure 2

Life expectancy in Champigny and Saint-Aignan – minimum and maximum mortality (women married locally – cohorts 1700-1749)

Possible improvements and age-specific migration

33An alternative approach would permit this type of estimate to be improved further, particularly when differences between the two tables thus obtained are large, and an age distribution of those who left was available. Assume that there is no relation between emigration and the birth of a child, but that emigration depends on age only.

34If we assume, as previously, that women emigrate immediately after their last observed birth, we can construct a preliminary profile of age-specific emigration, defined by a function of presence, in the absence of mortality, P1(x). In the same way, a survival function can be estimated l1(x). The probability that a woman who has a last birth at age a and is absent at age a’ ("birth of the next child") leaves at age x can be written :

36In effect, the probability of being present and surviving at age a is l1 (a) ∙ P1 (a). The probability of emigrating between a and x + 1 is thus l1 (a) ∙ P1 (a) – l1 (x + 1) ∙ P1 (x + 1). We may then deduce P1 (x/a, a’), which is the corresponding conditional probability. The procedure can be repeated, assuming that each woman who emigrated between a and a’ did so at age x with the probability calculated during the previous stage.

B – Direct estimates and extension of these approaches to a larger number of women

37It would now be feasible for us to estimate mortality by using only the age-specific distributions of marriages, remarriages and deaths. In theory, this type of approach would lead to the supposition that the age distributions of those who left the locality and of new settlers are identical and that their mortality patterns, too, were the same.

38Such a method would make it unnecessary to reconstitute the families, provided that marital status at death was registered satisfactorily. More specifically, the number of deaths on the E (see fn. 9 for explanation of symbols E and M) data cards is increased so that it equals that of the M or E data cards where the woman had not died (with the age distribution remaining the same).

39However, this type of estimate can only be applied at the regional level, when a group of villages is considered as a whole. In effect, we cannot be certain of measuring local mortality by proceeding in this way : the greater the difference between mortality in the locality of origin and the locality of arrival, the older will the immigrants be. We must, therefore, suppose that in-migrants and out-migrants are the same, an hypothesis which can rarely be correct at a local level.

40Furthermore, we have not taken into account in the previous part of this article these E cards (immigrants). The methods we have suggested above could be extended to cover such women, but this would only complicate matters (for example, it is risky to impute ages at arrival to the women who die locally, but who neither had any children in the locality nor married there). The attempts which we made in this direction showed that any alterations resulting from such an extension would only be very slight.


41An accurate estimate of local mortality may be obtained by using the various methods presented in this article. The fact that the life expectancies obtained by using different methods are very similar suggests that the estimates are reliable.

42It would, therefore, be interesting to construct several tables in this way with and without using the E cards. In practice, it would be possible to use only information on those women who had married locally (M cards), since the use of the E cards would only result in complications, without making any difference to the analysis. It would be necessary to explain any significant variations between these different tables, but, where the results are sufficiently alike, it would be sufficient to describe mortality by using the tables obtained in this way. Local variations or variations over time could thus be brought out. Other methods, based on a simple count of ages at marriage, remarriage and death could be investigated more closely by an example at a regional level.

43In the present article, the methods we have suggested have been applied only to the mortality of women. They can also be used to study the mortality of men, but since the relation between age and fertility is not so clear-cut, the results are likely to be less specific.

Annex 1

Dupaquier’s suggested method of estimation

44In this Annex, we attempt to explain more formally and generally, the reason why the method suggested by J. Dupâquier leads to erroneous results, and to use our formal analysis for the purposes of correction.

1 – Representativeness of the women

45Consider the set of women aged x who have already had a child. The suggested method sub-divides this category into three groups :

  • mothers who have, at this age, at least one surviving child aged under 15 but who will die before the 15th birthday.
  • those who do not belong to the first group above but who, after their xth birthday, will have at least one child who will die before the age of 15.
  • those who belong to neither of these two groups.

46The third group consists of women who either did not have a child who died before the age of 15, or who are no longer under observation at age x because those of their children who have not yet reached their 15th birthday died before the mothers’ xth birthday. In the method proposed by J. Dupâquier, women in Groups 1 and 2 are under observation at age x, while women in Group 3 are not.

47Consider the case where the mothers’ mortality is independent of their fertility and of the mortality of their children. On this assumption, women in Group 1 are representative of all women aged x as regards mortality. This is not the case for women in Group 2, since all the women in this group will die after the birth of another child. Since it is not possible to use Group 3 to study the mothers’ mortality, only Group 1 can be used for this purpose without leading to bias ; using Groups 1 and 2 together could only lead to an underestimate. However, in practice, Group 1 is likely to be small and it is not really possible to base an estimate on these women only.

2 – Formal analysis

48We consider a cohort of N0 individuals who, we shall assume, came under observation at a given age, x0. We shall designate the probability, π (u, x, θ), conditional on dying at age x, of being under observation at age u and of having certain characteristics θ at the time of death. We shall designate qx as the true probability of dying and write equation im11 for our estimates.

49The probability estimated from the observed deaths (deaths of women still under observation) and women present (under observation) can be written as follows :

51designating by ηu,x = Σθ π (x,u,θ)/Σθ π (x,x,θ), d (x, θ) the number of deaths observed at age x, for women with characteristics θ and N (x, θ) the number of women observed at age x with the same characteristics. The factor ηu,x thus introduces a bias.

53if we write Pr (u/v) for the conditional probability of u given v.

54Let us assume then that Pr (u/(θ, x)) is independent of x and can be estimated ; we shall call it f (u, θ) and we can then rewrite the expression (A-I) as follows :

56with Pr (θ/x) thus being the probability of having the characteristic θ at time of death, given that death occurred at age x.

57If our observations are then weighted (deaths and observed population) by the coefficient 1/f (x, θ), where x is the age at death of the woman under observation, an unbiased estimate of qx can be obtained.

58Let us now apply these results to our problem.

59We begin by assuming that a woman remains under observation from the moment she comes under observation up to her death. Let p (x, n) be the probability that a woman will have had n children at the time of her death, given that she dies at age x, s be the probability of surviving to age 15, and lx the survival function.

60The probability π (x, u, n) can be expressed in the following way :

62(f here is independent of u). An estimate of the probability therefore gives :

64with :

66The direction of the bias can be found if we note that ηn,x increases with age, and that, therefore, the underestimate of qx1 is greater, the smaller ηn,x that is, at the younger ages.

67However, as we have already shown above, if each individual is weighted by 1/(1 – lsn) where n is the completed fertility, a second estimate q2x is obtained. Returning to the deaths, we obtain :

69So far, we have assumed that the women under observation had been specified as in the method described, but were observed until their death. This is not true in practice. We shall now use conditions which are closer to reality : a woman comes under observation at the birth of her first child ; the date of the end of the observation period is either the date of death, if she has a child who dies before the age of 15 but after her own death, or else the last death of a child aged under 15.

70Consider the set of parameters consisting of the ages of the mother at the births of her various children, (x1,…,xn). The factor f (x, u, x1, …, xn) can then be expressed as :

72on condition that xxi ∈ [0, 15] ; if xxi < 0, we replace it by 0 and if xxi > 15 we replace it by 15. It is obvious that it is impossible to put this system of weighting into practice. It is, therefore, impossible to correct the bias.

Annex 2

Imputation of missing ages

73In order to impute the missing ages to the women on MO and MF cards, we have ordered these cards by date of marriage.

74To every woman of unknown age, married at date t, who is the mother of n children, who died in the locality, we have paired a woman of known age, who is the mother of n children, who died in the locality, and who married at the date closest to t.

75In the case of women who did not die in the village, we proceeded in the same way, but we paired them with all the women who had had at least the number of children the first woman had had.

Annex 3

Indirect estimate of mortality

76The object of this Annex is to demonstrate that an iterative estimate of mortality undertaken by imputing to emigrants the mortality estimate obtained in a previous stage of the analysis is equivalent to assuming that these emigrants left immediately after the last event that indicated their presence (in the present case, the last birth of a child in the village). More specifically, suppose that, to begin with, we estimate mortality for only those women who die in the village ; we, therefore, obtain a survival function lox and probabilities qox We then repeat the procedure, assuming that, at the nth iteration, the emigrants are subjected to mortality ln–1x calculated during the last step (n- 1th). We thus estimate :

78in which Vx is the number of individuals under observation at age x, Eu the number leaving observation at age u (i.e., the number of emigrant women who had their last registered child at age u). The equilibrium solution of this equation then becomes :

80In effect, this solution is given by :

82Or :

84We may deduce :


  • [*]
    Translated by Bobbie Le-Texier.
  • [**]
    R. Price, Observations on reversionary payments, London, 1776.
  • [***]
    J.A. Mourgue, Essais de statistique, Paris, Maradan, year IX, 76 p.
  • [****]
    We should like to thank L. Henry and J. Houdaille whose advice and suggestions enabled us to undertake this study.
  • [*****]
  • [1]
    Cf. in particular, L. Henry, Techniques d’analyse en démographie historique, Paris, INED, 1980 ; J. Dupâquier, "Réflexion sur la mortalité du passé : mesure de la mortalité des adultes d’après les fiches de familles", Annales de démographie historique, 1978, pp. 31-48.
  • [2]
    Infant mortality can, however, be ascertained. Cf. J. Houdaille, "La mortalité des enfants en France rurale de 1670 à 1779", Population, 1, 1984, pp. 77-106.
  • [3]
    Henry, op. cit., in fn. 1.
  • [4]
    For instance, J.P. Bardet did not consider this method suitable for his study of Rouen. J.P. Bardet, Rouen au xviie et xviiie siècles, Paris, SEDES, 1983.
  • [5]
    The simulation programs used are part of the programs described by H. Le Bras ; cohorts of males and females aged from 15 upwards were simulated. (H. Le Bras, "L’évolution des liens de la famille au cours de l’existence", In Les âges de la vie, Paris, INED/PUF, 1981. (Travaux e Documents, Cahier 96)).
  • [6]
    Bardet, op. cit., in fn. 4.
  • [7]
    Y. Blayo, "La mortalité en France de 1740 à 1829", Population, 1975, no spécial, pp. 123-142.
  • [8]
    In the remainder of this article, we shall use the term "locally" to designate events registered in the locality studied (births, marriages, deaths).
  • [9]
    The methods which we suggest will be applied to women only. We shall return to this point in our conclusion. The notations we shall use for family cards are the following : M cards correspond to families for which marriage has been recorded in the village, E cards to other families. On E cards, therefore, the first recorded event is a birth of a child. M and E cards can be "open" (MO and EO cards) when it is impossible to time precisely the end of a marital union, or "closed" (MF and EF cards) when the date of this end is known (e.g., by the recorded death of one of the relatives).
  • [10]
    This division by cohorts will be used in the remainder of the present article. It is quite arbitrary, and is not in any way related to actual population trends. But our aim is not so much a study of actual mortality, but of the indices which make it possible to understand mortality trends.
  • [11]
    We have considered as having been born in the locality the so-called "black" children and "red" children, provided the births of the latter are preceded and followed by those of "black" children. A "black" child is a child whose birth has been noted in the village, while a "red" child is one whose birth is discovered at a later date, from data relating to a marriage or a death (we remind the reader that these terms originate from the rules used in transcribing the cards).
  • [12]
    In practice, for a woman who married and later remarried locally, two family data cards were available : the first opened on marriage and closed with the first husband’s death, the second opened on remarriage. A woman who remarried was considered at risk between her marriage and her remarriage ; remarriages were, therefore, treated in the same way as marriages.
  • [13]
    Other documents which mentioned their presence could also have been taken into account. However, we preferred homogeneity to various selection biases which might have been introduced by using other documents.
  • [14]
    "Red" births, which were those of the last child born to the women, have been excluded because of the limitations mentioned in the preceding note. In the majority of cases in the MO data cards, they were children born outside the village, who arrived at a later date.
  • [15]
    Cf. Henry, « Mobilité et fécondité d’après les fiches de famille », Annales de démographie historique, 1976, pp. 280-302.
  • [16]
    "Lost" births are those which :
    • took place outside the locality ;
    • did, in fact, take place within the locality, but were not registered, or were entered in registers or on sheets which were later lost or destroyed.
    If Mr, is the number of marriages of children born locally, belonging to a data card MF, Mp the number of marriages of children born outside (or not registered) but married locally and belonging to a data card MF, the proportion of lost births will be the ratio Mr,/Mp + Mr.
  • [17]
    The estimate was firstly obtained by using single-year probabilities.
  • [18]
    The estimated proportions of "lost" births were as follows : for the 1650-1699 cohorts : 3.5 % in Champigny and 1.1 % in Saint-Aignan and, for the 1700-1749 cohorts, 4.9 % and 2.7 % respectively.
  • [19]
    We know that age at marriage and fertility are related, as are birth order and birth interval. We shall not take these relationships into account.
  • [20]
    The only difference is that, here, we are imputing not only an age, but also an outcome (the woman will have a child at a later date, the woman will die without having any further children, etc.).

Because of migration and underregistration of deaths, it is impossible to discover accurately the level of mortality in a given place during the 17th and 18th centuries. This is all the more regrettable, since it would appear that mortality did, indeed, vary considerably from village to village, as has been highlighted by contemporary authors, such as R. Price[**] or J.A. Mourgue[***].
Thus, to obtain an estimate of local mortality, either limiting hypotheses (constant underregistration, or restriction to children, independence of migration and mortality) have to be made, or else a single age group chosen, for which it becomes possible to make more accurate calculations. In the present paper [****], Alain Blum[*****] has opted for the second alternative and has obtained very accurate results for the mortality of adults, by taking into account any indication denoting their presence before they disappear either through death or migration.

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