1The modal age at death is a specific indicator of the “typical” length of life, corresponding to the age at which deaths are most numerous. In contemporary low-mortality populations, the majority of deaths occur at advanced ages. In populations of the past, however, deaths were most numerous in the first year of life and were much less concentrated at adult ages than they are today. After explaining how the adult modal age at death sheds light on mortality conditions at advanced ages, Nadine Ouellette, Jean-Marie Robine, Robert Bourbeau and Bertrand Desjardins draw upon a very comprehensive database, the Registre de la population du Québec ancien (population register of historical Quebec), to estimate changes in this modal age over the second half of the eighteenth century. They show that it rose by almost three years for both men and women, an unprecedented increase in such early populations, and propose an explanation based on the growth in the share of non-urban dwellers over the period.
2In demography, the term longevity may refer either to the individual capacity to survive to an advanced age, or to the survival of the population as a whole. At the population level, longevity is generally measured by life expectancy at birth, i.e. the mean length of life estimated from life table data. However, the modal length of life, an additional measure of the central tendency of the age-distribution of deaths, is especially useful for studying adult longevity.  Indeed, unlike life expectancy at birth, the adult modal age at death is not influenced by mortality conditions at young ages, so is much more sensitive to changes within the population of older adults (Kannisto, 2001; Horiuchi, 2003).
3The difference between life expectancy at birth and adult modal age at death may be considerable, especially when infant mortality is high. For example, the difference was 30 years (both sexes combined) in Sweden over the period 1860-1869 (Figure 1). It is especially important to distinguish the mode of the distribution at age 0 – the age at which the highest numbers of deaths occur, all ages included – from the mode at around 75 years used here, which is the equivalent for adult ages and is the focus of this article. Figure 1 also shows that the median age at death (or median length of life), another central tendency indicator, while less influenced than life expectancy at birth by the high infant mortality of that period, is still affected by mortality at young ages. Life expectancy at age 20, which summarizes mortality conditions beyond that age, does not give an accurate picture of mortality at advanced ages relative to the period in question, unlike modal age at death. Even life expectancy at any advanced age – 60 or 65 years for example – which only captures the mortality conditions at this age and beyond, has limitations for the study of adult longevity, the main one being that age is both arbitrary and fixed. In a context where mortality conditions at advanced ages are improving (or deteriorating), we would prefer this age to rise (or fall) accordingly, since when it remains fixed the increase (or decrease) in adult longevity is underestimated. In other words, the modal age at death, unlike life expectancy at a given age does not underestimate the increase or decrease in longevity. The mathematical proof is provided by Horiuchi, Ouellette, Cheung and Robine (2012). The modal age at death is not only a valuable summary indicator for studying longevity, but is also useful for understanding and applying various mathematical models describing the age trajectory of mortality, such as the Gompertz and Weibull models (Robine et al., 2006; Thatcher et al., 2010). 
Figure 1Modal (M) and median (Md) ages at death, and life expectancy at birth (e0) based on the age distribution of deaths estimated from the complete life table for Sweden, both sexes, 1860-1869
4As early as the late nineteenth century, the concept of “normal life durations” that emerged from the pioneering work of Lexis (1877, 1878) identified the modal age at death as the most “central” and “normal” characteristic of human longevity. The contribution of Lexis, received favourably at the time by Bertillon (1878) and several other statisticians and economists (Véron and Rohrbasser, 2003), went almost unnoticed from more than a century. It was not until the turn of the twenty-first century that Kannisto (2000, 2001) took a fresh look at Lexis’ work and popularized the use of modal age at death in studies of human longevity. Since then, modal length of life has been a regular focus of researchers’ attention (Kannisto, 2000, 2001, 2007; Robine, 2001, 2011; Cheung et al., 2005; Cheung et al., 2008; Cheung et al., 2009; Cheung and Robine, 2007; Canudas-Romo, 2008, 2010; Thatcher et al., 2010; Ouellette and Bourbeau, 2011; Brown et al., 2012; Ouellette et al., 2012).
5Thanks to these studies, we now have a detailed knowledge of changes in the most frequent adult length of life since the mid-nineteenth century in several low-mortality countries. In the second half of the nineteenth century, it rarely exceeded 80 years. It then increased sharply in the twentieth century, especially after 1950, and has recently reached 90 years for women in Japan and France, and 85 years for men in several developed countries.
6Our knowledge of the adult modal age at death in past population remains limited, however, because detailed and reliable historical data on mortality before the mid-nineteenth century are scarce. For example, we do not know whether the most frequent adult age at death increased, fell, fluctuated or remained generally stable over a very long period in the past. The main empirical observation currently at our disposal concerns Sweden, the only developed country to possess national statistical series on mortality covering a large part of the eighteenth century.  According to these Swedish data, over the period of more than a century between 1751 and 1875, the modal age at death appears to have fluctuated around 72 years for women and 69 years for men (Robine et al., 2006; Robine and Cheung, 2008; Robine, 2011). But these observations are difficult to generalize as they cannot be compared with those for other historical populations. Note also that the Swedish data available for the 1751-1860 period are of lesser quality than those for subsequent years (Glei et al., 2012) since data on deaths and population size are given by five-year age group only. The data by single year of age needed to construct the complete life tables used by Robine and Cheung (2008) and Robine (2011), had to be estimated beforehand (Wilmoth et al., 2007, pp. 9-15 and Appendix B). A further problem is that of age heaping in the data on deaths and population size,  which has an even stronger adverse effect on the quality of the historical Swedish data. Age heaping is more pronounced before 1800, but does not disappear completely until 1860.  Estimates of the most frequent adult length of life in Sweden before 1860, in terms of both level and trends, are thus liable to be influenced by these data imperfections. Indeed, they may perhaps explain the apparent stagnation in modal age at death in Sweden between 1751 and 1875, by favouring values that remain close to 70 years, for example.
7However, from a strictly analytical perspective, Canudas-Romo (2010, Appendix B) has shown that if mortality gains occur solely at ages below the modal age at death, then this measure remains unchanged. In other words, the modal age at death can only be raised by mortality gains above the modal age, i.e. among older adults. The fact that the most frequent length of life in Sweden between 1751 and 1875 fluctuated around 72 years for women and 69 years for men could be explained by a stagnation of mortality beyond these ages over the period. Given that the unexpected and unprecedented mortality decline at very advanced ages (beyond 80 years) did not really begin until 1950 in most developed countries (Kannisto et al., 1994; Vaupel and Lundström, 1996), this is a credible hypothesis. On this basis, we might even be tempted to presume that the most frequent adult length of life barely changed before the mid- or even the late nineteenth century in most historical populations.
8The Registre de la population du Québec ancien (RPQA) (Desjardins, 1998), is a population register of historical Quebec that provides a unique source of historical data, reputed for its reliability. It is an ideal starting point for an additional empirical exploration of this question, enabling us to observe the population of historical Quebec over a long period. In this article we will use these data to determine levels and trends in the modal age at death of French-Canadians during the eighteenth century and to compare these findings with those described earlier for Sweden. Beforehand, in the manner of Robine and Cheung (2008) and of Robine (2011), we will first track the changes in the age distribution of adult deaths, focusing on the modal age at death, on the basis of existing life tables for various European populations of the seventeenth, eighteenth and nineteenth centuries. The results of this initial research will provide an introduction to the study of the age distribution of deaths among French-Canadians in the eighteenth century, and form the first section of this article. The three subsequent sections will shed new light on adult longevity in the eighteenth century through analysis of data from the RPQA. We will first present data and methods, followed by the results obtained in the particular context of the French-Canadian population of that period.
I – From the absence to the emergence of a modal length of life
9Halley (1693) was the first to construct a life table based on empirical data. Thirty years earlier, Graunt (1662a) had come up with the brilliant idea of grouping several variables – age, number of deaths and numbers of survivors – in a single table. However, the mortality records of the city of London available to Graunt did not include the decedents’ age at death. To construct his life table, Graunt made several questionable assumptions and relied on an “arbitrarily defined arithmetic function” (Dupâquier, 1996, p. 74) to describe changes in mortality by age. Figure 2A shows an excerpt of the age distribution of deaths from the life table (both sexes combined) established by Halley for the city of Breslau (Lower Silesia, now Wroclaw in Poland).  It shows that a similar number of deaths occur at each age between 40 and 75 years. Slight peaks are observed around ages 50 and 70, but it is impossible to discern a modal age at death corresponding to the most frequent adult length of life at that time. This observation is consistent with that of Robine and Cheung (2008), who point out the impossibility of “theorizing the existence of a usual length of life for adults” (Robine and Cheung, 2008, p. 949) from the works of Halley.
10This research inspired a number of scholars to construct their own life tables, although the tables generally concerned closed, selected populations – more specifically those of annuitants – which were not representative of the population as a whole. It was not until 1760 that Deparcieux published national life tables calculated from deaths in the years 1754, 1755 and 1756 in Sweden (Deparcieux, 1760).  Figure 2B shows an excerpt of the age distribution of deaths by sex drawn from these Swedish life tables. This time, the curves obtained from the abridged tables show that the most frequent adult length of life in Sweden was between 70 and 75 years in the mid eighteenth century, for men and women alike, and the complete life tables give an age of 72 years for both sexes.
Figure 2Age distribution of deaths (from age 20), Breslau (A), 1687-1691, and Sweden (B), 1754-1756
Note: In the Deparcieux life tables (1760), the data for Sweden at age 37 are illegible and those between ages 38 and 43 are missing.
11The distributions of deaths shown in Figure 2B can today be compared with the most recent data for Sweden and other countries, thanks notably to the Human Mortality Database (2012). However, to our knowledge, no complete life table established on the basis of statistical series of deaths and population by single year of age and by sex is available for the eighteenth century.  For the nineteenth century, at least three other countries in addition to Sweden, namely France, the Netherlands and Switzerland, possess such tables (Figure 3). On these four illustrations, the bulge in the curve around the modal age is clear for both sexes, and the age at which most deaths occur is always, without exception, between 70 and 75 years (Table 1), as was the case 100 to 115 years earlier in Sweden (Figure 2B). While it is hazardous to deduce that the adult modal length of life for both sexes barely changed over more than a hundred years, when set against the data for France, the Netherlands and Switzerland,  the Swedish data suggest a stagnation in the modal length of life.
Age at which the most deaths occurred for each sex, France, Netherlands, Switzerland and Sweden, nineteenth century
Age at which the most deaths occurred for each sex, France, Netherlands, Switzerland and Sweden, nineteenth century
Figure 3Distribution of deaths by age (from 20 years) and sex for various periods of the nineteenth century, France, Netherlands, Switzerland and Sweden
12The RPQA provides an ideal data source for pursuing our analysis of adult longevity in historical populations. Recognized for its reliability, the register enables us to track changes in the most frequent adult length of life within a single population over more than half of the eighteenth century. Until now, this was only possible for Sweden, where data are available from 1751 but are of lower quality up to 1860. The Quebec register thus provides an opportunity to verify whether modal age at death also tended to remain stable over a long period in the past, as suggested by the Swedish data. The abundance of information contained in the register enables accurate description of modal age at death among the French-Canadians since deaths and the population counts are detailed by single year of age, sex and calendar year for the entire study period. We will determine whether the most frequent length of life in this population also stood at 70-75 years, as indicated by eighteenth and nineteenth century data for Sweden, and by nineteenth century data for several other European countries.
II – Data and methods
The data in the Registre de la population du Québec ancien
13The RPQA is a database of information taken from baptism, marriage and burial certificates of the seventeenth and eighteenth centuries, and from burial records of the period 1800-1850 concerning persons born before 1750 (Légaré, 1981; Desjardins, 1998). The major task of collating these data in the form of a computer database was accomplished by the Programme de recherche en démographie historique, (research programme in historical demography, PRDH) of Université de Montréal (Légaré, 1981). The register concerns all individuals who settled in the Saint-Lawrence valley at the time of French colonization, and their descendants. It covers the entire territory occupied by the French-Canadian population, thereby limiting problems of selection and bias that might otherwise have resulted due to inter-regional and inter-parish migration, for example. Moreover, the quality of the register has been confirmed on numerous occasions, so the data are based on very reliable foundations. The database thus provides an exceptional historical resource, that can be used to establish a cross-sectional portrait of the Quebec population from the arrival of the first settlers in 1620 through to the year 1799. The ultimate objective of the PRDH is to achieve similar coverage for the nineteenth century.
14Because single individuals are difficult to observe,  only married persons, who represent the vast majority of adults at that time, were used in our study. Here, the term “married” covers all individuals who had married at least once, whatever the outcome of their union and their resulting marital status – marriage, widowhood possibly followed by a second union, etc. While it is very difficult to determine the frequency of singlehood by sex in this population, we know that permanent celibacy was extremely rare among women (excluding religious celibacy), and almost exclusively concerned women with serious health problems. Among men, who were more mobile and sometimes engaged in informal unions with Amerindian women, a higher proportion of single men would be more apparent than real. In short, marriage was the norm in Quebec at that time, and celibacy was religious or confined to very specific situations.
15Our initial population comprised 126,825 married individuals in historical Quebec before 1799. Immigrants, Amerindians and Blacks were first excluded for the sake of population homogeneity and because their date of birth was generally unknown (most were born outside the territory covered by the register). They represent a total of 15,460 individuals (12.2 %). The remaining 111,365 individuals (Table 2) comprise 110,750 non-emigrants (99.4 %) and 615 emigrants (0.6 %). We include emigrants because they are exposed to the risk of dying so long as they remain in the territory of historical Quebec. Omitting them would introduce bias into our analysis since any prospective emigrants who died before actually departing would be counted, unlike those who emigrated before death, and this would lead to an overestimation of mortality. Next, for both non-emigrants and emigrants, the year of birth is unknown for a minority of individuals (4,492 non-emigrants, i.e. 4.0%; 38 emigrants, i.e. 6.2%) due to random register losses, so these individuals must also be excluded. In addition, among non-emigrants born before 1750, those whose year of death is missing must be removed likewise because all burial certificates dated between 1800 and 1850 concerning individuals born before 1750 have been recorded by the PRDH. Random loss or lack of death records are the main reason why the exact destiny of these individuals is unknown (Charbonneau et al., 1996). Here, their proportion is estimated at 9.2% (note in Table 2), a very exceptional level in a historical context of this kind. We were nonetheless able to estimate the year of death of 2,720 (52.1%) of these 5,221 non-emigrants with missing records born before 1750 via the civil records of other family members. So our final sample contains 104,334 individuals, i.e. 103,757 non-emigrants (101,037 + 2,720) and 577 emigrants.
Descriptive statistics of data on married persons taken from the RPQA
Descriptive statistics of data on married persons taken from the RPQANote: Death certificates were not found for 5,221 persons (2,720 + 2,501). They represent 9.2% of the 56,491 non-emigrants born before 1750. Despite the absence of death records, the year of death could be estimated for 2,720 of them.
16Here we give some additional details on the way in which migration is handled in the RPQA database. In general, “emigrant” status is given to all persons who died outside the territory. In cases where the death certificate was not found, couples where the husband was an immigrant and for which we have no trace in the registers after the British Conquest of 1759, are also given emigrant status (both husband and wife) because, in all likelihood, they moved back to France. Additional cases are also identified via indirect sources, notably notarial acts. In all other cases where death records are missing, the individuals are not given emigrant status, even though they might have emigrated. However, given the information available to us, and the care that was taken in identifying departures, we can confidently state that the great majority of missing deaths are due to the loss of some registers. In any case, given that we have information on 90% of deaths, the few missing departures will in no way adversely affect the results.
17Figure 4 illustrates the strong increase in the population of married individuals born in historical Quebec between 1640 and 1799, reflecting overall population growth. Around the mid-seventeenth century, inhabitants of the St Lawrence valley totalled a thousand at most, but following a new wave of colonization over the period 1663-1673  the total population rose to almost 20,000 individuals by the early eighteenth century and around 200,000 at the turn of the nineteenth century (Henripin and Péron, 1973). The married population included in our study totalled around 100, 3,000 and 60,000 individuals, respectively, at these three same dates, both sexes combined. The apparent imbalance between married men and women on this figure was expected, since we only included Canadians in our study. Practically no women are excluded since female immigration stopped after 1673, while male immigration continued.
Figure 4Married population by sex (in selected sample), historical Quebec, 1640-1799
18Figure 5 shows the variations in annual numbers of deaths in the population for each sex. The sharp increase in deaths observed throughout the period is explained mainly by the regular arrival of new settlers in the territory. The annual variations, typical of the old mortality regime, reflect deaths due to exceptional circumstances such as epidemics, severe winters, poor harvests and wars.
Figure 5Annual deaths of married persons by sex, historical Quebec, 1640-1799
Complete life tables for married French-Canadians in the eighteenth century
19Starting from death and population counts by single year of age, sex and calendar year, we can construct complete life tables for married persons. To attenuate the effect of annual variations linked to period conditions that were typical of mortality at that time (Figure 5), the best option was to construct tables for 15-year periods. These life tables beginning at the exact age of 20 years will shed light on changes in the age and sex distribution over time of deaths of married adults in historical Quebec. Because of the small numbers observed up to the early eighteenth century (Figures 4 and 5), the first life table concerns the period 1740-1754, and the last table is based on data for the period 1785-1799. Eight successive life tables covering more than half of the eighteenth century are thus calculated, four for each sex. Table 3 summarizes the data used in our analyses, i.e. the total number of deaths and exposure to risk (of dying) by period for each sex.
Population exposed to the risk of dying, and deaths observed from age 20 by period and sex, historical Quebec, 1740-1754 to 1785-1799
Population exposed to the risk of dying, and deaths observed from age 20 by period and sex, historical Quebec, 1740-1754 to 1785-1799
20The death rates by age, sex and period are first obtained by dividing the number of deaths occurring over the period by the sum of exposure to risk estimated for each year of the period. The death rate at age x (in completed years) for sex s and period [T, T + 14] equals
22where and represent, respectively, the deaths observed and the exposure to risk at age x for the sex s during year i. Exposure to risk is estimated by the number of married persons of age x and sex s on 1 July of the year i.
23The death rates by age, sex and period obtained from equation (1) are then transformed into probabilities of dying using the linear method (also called actuarial method), i.e. assuming a uniform distribution of deaths within each one-year age interval
25Among adults, this assumption is often satisfied, except at advanced ages where more sophisticated methods are generally used (Thatcher et al., 1998). Given that the sole purpose of these life tables is to provide an initial idea of the age distribution of deaths of married persons by sex, the linear method suffices for our demonstration. Last, the survivors and the deaths of the life table are obtained directly from the probabilities of dying defined above.
Studying mortality with the P-spline smoothing method
26To obtain precise estimates of the modal age at death by sex and period, we use the P-spline smoothing method described by Ouellette and Bourbeau (2011), based on the work of Eilers and Marx (1996), Currie et al. (2004) and Camarda (2008). Given the typically erratic age distribution of deaths around the mode in the life table, other methods could have been applied. For example, we could have used a quadratic model (Pearson, 1902; Kannisto, 2001, 2007; Canudas-Romo, 2008, 2010; Cheung et al., 2008; Thatcher et al., 2010; Brown et al., 2012) or a model based on a normal distribution (Cheung and Robine, 2007; Cheung et al., 2009) to obtain a smooth distribution of deaths from the adult life table. We preferred to obtain smoothed death rates by age for each sex and period using P-splines for greater flexibility (Ouellette and Bourbeau, 2011, pp. 597-601). In short, the P-spline method yields a non-parametric estimate of the density function which describes the age distribution of deaths. The age at which this density function reaches its maximum thus corresponds to our estimate of modal age at death. Some additional details of this smoothing method, to be applied separately by sex and period, are given below. For the sake of clarity, the indices relative to sex and period are omitted here. First, assuming a constant force of mortality µx  within each age interval , the deaths observed by age Dx can be seen as realizations of a Poisson distribution with mean , expressed as
28where Ex refers to the population exposed to the risk of dying at age x.
29To estimate the force of mortality underlying the data on observed deaths, we can use a Poisson regression model. The parameters of this model are estimated with a nonparametric approach combining the concepts of B-splines (de Boor, 1978), and penalized likelihood function, hence the name P-splines. These two concepts are complementary since the B-splines, comprising polynomial segments connected at points on the x-axis called “knots”, provide considerable flexibility in the modelling process. On the other hand, the penalized likelihood ensures that the estimated force of mortality will be smooth. The idea behind the P-splines is therefore to use a relatively high number of B-splines, resulting from several knots evenly distributed over the modelling domain, while limiting (i.e. penalizing) variations in the parameters associated with the adjacent B-splines. While it is important to use a sufficient number of knots,  the role played by the penalized likelihood function is such that the exact number of knots used has little or no impact on the final smoothing result. For our analysis we used cubic B-splines, made of pieces of third-degree polynomials, and a knot every 5 years on the age domain x, such that x = 20.
30The vectors of death and risk exposure are denoted D and E, respectively. We thus obtain, based on equation (2)
32where B is a matrix which represents the B-splines basis evaluated at the different ages x, and the vector corresponds to the estimated parameters associated with each of the B-splines included in the basis B. The vector of parameters a is estimated using the maximum likelihood method, and the penalized log-likelihood function to be maximized is
34The first term of this equation corresponds to the standard log-likelihood of a generalized linear model. The second is a penalty term intended to achieve a certain regularity in the vector of the estimated parameters by restricting the values of the estimated parameters associated with adjacent B-splines to change abruptly. The compromise between smoothness and precision (fit to the observed data) is controlled in the model by a smoothing parameter included in the penality matrix P.  The higher the smoothing parameter, the greater the priority attributed to smoothness over precision, and vice versa. In our analyses, we rely on the Bayesian Information Criterion (BIC) to select the smoothing parameter. The resulting estimated force of mortality is smooth. For a more detailed description of this penalized log-likelihood function, see Currie, Durban and Eilers (2004, pp. 282-284) or Camarda (2008, chapter 2).
35From equations (3) and (4) we make use of P-splines to arrive at the following expression for the smoothed force of mortality by age
37Based on the relationships between the force of mortality µ(x), the survival function S(x) and the density function f(x) (Klein and Moeschberger, 1997, chapter 2), we can write
39In other words, from the smoothed force of mortality drawn from equation (5), we compute the corresponding smoothed density function using standard numerical integration methods. As describes the age distribution of deaths, we estimate the modal age at death as follows
III – Results
41The age distribution of married adult deaths drawn from the complete life tables by sex established for the periods 1740-1754 to 1785-1799 is illustrated in Figure 6. Given the erratic variations in the four deaths series shown on each of these graphs, it is difficult to determine a single modal age at death in these cases. However, at first sight, it would appear that the modal age at death over all these periods always remained between 70 and 80 years. We also see that over time, the age distribution of deaths for both sexes shifts to the right, i.e. towards older ages. We still need to confirm, by means of P-spline smoothing, whether these changes in the age distribution of deaths for each sex are reflected by the most frequent length of life in eighteenth century historical Quebec.
Figure 6Age distribution of deaths of married adults by sex based on life tables for the periods 1740-1754 to 1785-1799, historical Quebec
42As an example, the density function describing the age distribution of deaths estimated with P-spline smoothing among married men for the period 1740-1754 is shown in Figure 7. We have adjusted this smoothed density function to the scale of 100,000 survivors at age 20 so that it can be compared with the series of deaths in the complete life table of married men for the period 1740-1754 also shown on the figure. While it is difficult to determine the most frequent length of life from the distribution of deaths based on the complete life table, it can be accurately identified using the smoothed density function. As this function reaches its maximum at 70.4 years, we consider that this is the most frequent length of life of married men in historical Quebec over the period 1740-1754.
Figure 7Age distributions of deaths based on the life table and on P-spline smoothing, married men, historical Quebec, 1740-1754
43The modal age at death in historical Quebec estimated by sex and period between 1740-1754 and 1785-1799 is shown in Figure 8. We see immediately that the most frequent length of life did not remain stable over these periods. In fact, the modal age at death increased steadily for women, rising from around 73 years in 1740-1754 to almost 76 years in 1785-1799 (Table 4). Among men, the most frequent length of life also increased regularly, from 70.4 years in 1740-1754 to 74 years in 1785-1799. The most frequent age at death thus differs by sex in historical Quebec and is systematically higher for women.
Figure 8Trends in modal age at death of married adults by sex, estimated by P-spline smoothing, historical Quebec, 1740-1754 to 1785-1799
Modal age at death of married adults by sex and period, estimated by P-spline smoothing, historical Quebec, 1740-1754 to 1785-1799
Modal age at death of married adults by sex and period, estimated by P-spline smoothing, historical Quebec, 1740-1754 to 1785-1799
IV – Discussion
44In the last ten years, the concept of modal length of life has received increasing attention in studies of human longevity. In line with this trend, the main aim of the present article was to reduce the uncertainty regarding the level and trend over time of modal age at death in historical populations. Our knowledge of this topic remains limited, mainly due to the scarcity of reliable and detailed historical data before the early-to-mid nineteenth century. Thanks to the RPQA, an exceptional historical data source, recognized for its broad coverage of the population of historical Quebec and for its reliability, we were able to track changes in the modal length of life of French-Canadians in the second half of the eighteenth century. Previously, most of the information available to us came from Sweden, where national data are available from 1751, but are of lower quality up to 1860. Free of such data imperfections, the register data, generally exploited longitudinally, were ideal for our cross-sectional study of the most frequent adult length of life.
45Our results reveal that the most frequent adult length of life among French-Canadians increased steadily between 1740-1754 and 1785-1799. The increase is notable for each sex, with a gain of almost three years for women and three and a half years for men. At first sight, these findings are surprising for two reasons. First, no increase in modal age at death has been detected in Swedish data for the years between 1751 and 1875 (Robine, 2011; Robine and Cheung, 2008). Rather, it tends to oscillate around 72 years for women and 69 years for men throughout the period, without any substantial increase before 1875. Second, it has been mathematically proven that the modal age at death cannot increase in the absence of mortality gains beyond this age (Canudas-Romo, 2010). On this basis, we tended to assume that the modal age at death would vary little in historical populations. Despite the flaws in the Swedish data for the years 1751-1860, notably the problems of age heaping liable to influence levels and trends in modal age at death, the stagnation observed over more than a century appears to be highly plausible. So how can the considerable increase in historical Quebec be explained?
Robustness of our modal age at death estimates
46At this first stage of reflection, it is essential to measure the variability of our modal age at death estimates among married French-Canadians to attest the accuracy of the estimated values and of the associated time trends. For this purpose, we repeated all the previous analyses on the population of historical Quebec, this time considering periods of 10 years (1740-1749 to 1790-1799) rather than 15 years. The estimates obtained for the modal age at death under the new configuration are shown in Figure 9, along with those for the 15-year periods to facilitate comparison. The two series of results for each sex are very similar everywhere, with the exception, for men, of the period 1750-1759 preceding the British Conquest (1759) which was punctuated by various events with a negative impact on individual survival. The adverse impact of the conditions particular to that time on our modal length of life estimates is less visible for the 15-year periods as two periods are concerned (1740-1754 and 1755-1769) rather than one, each of which is also longer. In short, our modal age estimates appear to be reliable, and the question of the considerable increase in the most frequent length of life remains unanswered.
Figure 9Modal age at death estimates by sex and period, 10-year periods (1740-1749 to 1790-1799) and 15-year periods (1740-1754 to 1785-1799), married adults, historical Quebec
Specific living conditions of French-Canadians in the eighteenth century
47Our results must be viewed in the specific context of the French-Canadian population of that time. The sustained increase in modal age at death among French-Canadian adults in the second half of the eighteenth century indicates that the mortality conditions at ages above the modal age, i.e. among older adults, improved in historical Quebec over that period. In fact, recent research on French-Canadians shows a decrease in mortality at advanced ages in successive generations of the late seventeenth century and the first quarter of the eighteenth century (Gagnon and Mazan, 2009; Lacroix, 2009). Our findings are consistent with these conclusions, since it was the individuals of these cohorts who formed the elderly population in the second half of the eighteenth century.
48The years of conflict in historical Quebec that culminated in the British Conquest in 1759 were particularly unfavourable to individual survival due to large-scale losses in the militia and, above all, the brutality of the British army. Moreover, following the French defeat, the military and merchant elite, comprising administrators and their support personnel, traders and artisans, returned to France. As these people were essentially city-dwellers, the proportion of rural French-Canadians doubtless increased, since they stayed behind in Quebec. Not surprisingly, mortality was higher in cities than in rural areas at that time due to overcrowding and poor hygiene. Quebec was a landing point for migrants arriving by ship from France, some of whom carried contagious diseases which spread rapidly through the city. Nevertheless, we know that mortality was lower in the colony than in mainland France, since Quebec had several advantages, including low population density, abundant clean water, a harsh but healthy climate, plentiful game and fish. The lower rural mortality and the simultaneous increase in the proportion of rural inhabitants may thus explain the increase in the most frequent length of life of French-Canadians between 1740-1754 and 1785-1799.  The subsequent massive urbanization of the colony, from the early nineteenth century, contributed to a deterioration of living conditions and may have adversely affected individual survival. This suggests that the pace of increase in modal age at death among French-Canadians in the second half of the eighteenth century may have slowed down after 1800.
49The PRDH team is pursuing its efforts to reconstitute the French-Canadian population and it should eventually be possible to extend our analysis of changes in the modal length of life up to the end of the nineteenth century. Through the combination of these data with those of the vital event registration that begins in Quebec in 1926, a continuous portrait of the longevity of Quebecois adults will be achievable. According to a recent study based partly on data from Quebec’s vital event records, the modal age at death in Quebec was around 77.5 years for women and slightly above 76 years for men in the early 1930s (Ouellette et al., 2012). This also suggests that the pace of increase in modal age at death in the second half of the eighteenth century in historical Quebec was not sustained throughout the nineteenth century. For the time being, however, it is impossible to determine whether the modal age continued to increase, but more slowly, during the nineteenth century, or whether it levelled off or decreased.
50While the RPQA is an exemplary source of historical data, it would be useful to make comparisons with other sources that can be used to follow historical populations over several years. The living conditions of the population of historical Quebec in the second half of the eighteenth century were quite atypical, so it is difficult to determine whether our results on the modal length of life of French-Canadians can be generalized. In the first part of this article, we showed that the data in the Human Mortality Database (2012) were of little help in this respect. Among the 37 countries (or regions) currently included in this large international database, Sweden is the only one with statistics covering the eighteenth century (in part). Ten other countries have data on population and death counts dating back to the nineteenth century, but only those of France, the Netherlands and Switzerland give details by single year of age for that period. Among the other data sources worth exploring is the genealogical reconstitution of English families by the Cambridge Group for the History of Population and Social Structure, based on information from Anglican parish registers (Wrigley et al., 1997). Last, alongside the analysis of historical populations in countries which now have the lowest mortality levels, the study of modal length of life of contemporary populations at different stages of economic development would also be an interesting avenue of research.
51 Acknowledgements: We wish to thank the three anonymous reviewers for their comments, as well as the Social Sciences and Humanities Research Council of Canada and the Fonds québécois de la recherche sur la société et la culture for their financial support.
In the present article, the expressions “modal length of life”, “modal age at death”, “the most frequent adult length of life” and “most frequent adult age at death” are used interchangeably. All refer to the age at which the largest number of deaths occur in the age distribution of deaths (excluding infant and early childhood deaths) in a life table.
For more information on the development of mathematical expressions for modal age based on these various parametric mortality models, see: Gumbel (1937), Pollard (1991, 1998), Pollard and Valkovics (1992), Robine et al. (2006), Canudas-Romo (2008), Thatcher et al. (2010).
The national statistical system in Sweden dates back to 1749.
The number of deaths and the population size for the five-year age groups beginning with a multiple of 10 (for example, 40-44; 50-54; …; 70-74) are systematically higher than those for other age groups.
Glei, Lundström and Wilmoth (2012) recommend using abridged life tables (by five-year or ten-year age groups) rather than complete tables for Sweden for these years, especially before 1800.
As we are focusing on the adult modal length of life, deaths in the first 20 years of life have been omitted in the remainder of this article.
The abridged life tables by sex published in his book were communicated to him by Wargentin. Deparcieux smoothed them to obtain complete tables.
The configuration of the observed number of deaths and of population size by year of age is very valuable here as the modal age at death can be documented more accurately; the data do not need to be estimated beforehand from broader age groups.
The apparent stagnation of the mode may simply reflect the fact that adult mortality in France, the Netherlands and Switzerland in the mid- or late nineteenth century had barely reached the level observed in the mid eighteenth century in Sweden.
Measuring mortality among never-married adults is more complex than among married adults as their deaths were more frequently under-reported. Never-married individuals were generally more mobile than married ones and therefore more likely to be lost to observation, especially when they travelled over large distances (Charbonneau, 1975).
This period marks the arrival of the “King’s Daughters” (Filles du roi), women, often orphaned, who were recruited specifically to marry and form a family in the colony where women were in short supply.
The force of mortality, also called “instantaneous mortality rate” is such that (Thatcher et al., 1998, Appendix A).
An insufficient number of knots would describe a set of B-splines (called B-splines basis) which would not capture the variations within the observed data.
More precisely, , where ? is the smoothing parameter to be selected and Dk corresponds to the matrix representation of a difference operator of order k acting on the parameters associated with the B-splines. Here we use a quadratic difference operator such that
To our knowledge, the mortality differential between urban and rural regions of Quebec in the eighteenth century has never been measured. However, the cohort life tables available for the French-Canadian population born before 1750 reveal large differences in life expectancy (e20 at age 20, e65 at age 65) between rural and urban areas, to the advantage of the former (e20: 39.9 years versus 35.0 years for women, and 43.7 years versus 36.7 years for men; e65: 11.8 years versus 10.5 years for women, and 11.2 years versus 9.9 years for men) (Lacroix, 2009, Appendix Tables 3 and 7).