1How does an individual’s risk of death increase over time? The Gompertz model postulates that it increases exponentially with age, i.e. at a constant relative rate. The rate of senescence, ?, which measures this relative increase, is estimated to be around 10% per year of age. However, this hypothesis is difficult to verify as researchers only dispose of aggregate data that mix the experiences of individuals with very different biological profiles and life histories. In this article, Giambattista Salinari and Gustavo De Santis propose new estimates based on data from the Human Mortality Database for cohorts born in the late nineteenth and early twentieth centuries in five European countries. These are the first cohorts for which sufficiently reliable data series are available. Using a Gamma-Gompertz model refined to take account of period effects and selection effects due to differences in individual frailty, they estimate the value of b by cohort, sex and country. They take advantage of this very accurate and well-documented database to demonstrate both the robustness and the limits of the Gompertz model. Senescence rates vary little across cohorts or countries, and are systematically higher for women than for men.

2At adult ages x, the force of mortality [1] ? increases more or less exponentially with age (Gompertz, 1825), and the parameter associated with age, ?, can be used to gauge the rate of ageing (or senescence, as we will call it in this article) of the members of a cohort j (Finch, 1994):

3

4Vaupel (2010) has recently advanced the hypothesis that, with rare exceptions (e.g. individuals with “accelerated-ageing disorder”), the rate of senescence at the individual level ? may be constant through space and time, and not far from 0.1, so that mortality doubles approximately every seven years (Baudish and Vaupel, 2010). If equation (1) holds, and if the rate of senescence ? is indeed a constant for all generations, the decline in adult mortality of the past two to three centuries must be captured by a decrease of parameter a_j in equation (1), that is a downward shift of generational mortality, so that “the process of senescence is being delayed rather than being decelerated” (Vaupel, 2010, p. 539).

5Several studies have tried to estimate the rate of senescence and to test Vaupel’s hypothesis. A possible solution is to simply interpolate the observed (aggregate) force of mortality

6 with the Gompertz model of equation (1). For instance, Finch et al. (1990) and Finch (1994) estimate an average doubling time for human mortality of about eight years, corresponding to a rate of senescence of about 0.085. More recently, Bronikowski et al. (2011) have produced estimates for ? in the contemporary United States that range between 0.086 for males and 0.096 for females (doubling times of, respectively, 8.07 and 7.20 years), with a significant difference between the sexes. Shortly before, Gurven and Kaplan (2007) had estimated the doubling time of mortality at adult ages among hunters-gatherers to be somewhere between six and nine years.

7The shortcoming of this approach is that it ignores heterogeneity (frailty [2]): if only the most resilient individuals reach older ages, the hazard function of an entire cohort increases more slowly than that of individuals (Vaupel et al., 1979; Figure 1). Unfortunately, it is impossible to pass from observed (aggregate) mortality risks to the individual notion of “force of mortality” without a few assumptions. Probably the most popular assumptions are those of the so-called Gamma-Gompertz model, a proportional hazard model in which the baseline hazard is supposed to evolve exponentially with age, and frailty is assumed to be Gamma-distributed. Recent findings on the evolution of mortality at very advanced ages seem to support these assumptions. Gampe’s (2010) estimates of mortality at very advanced ages (110+), for instance, reveal the existence of a sort of “mortality plateau” where the probabilities of dying are basically constant. It can be proved that this plateau is consistent with the hypothesis that individual mortality follows a proportional hazard model (as opposed to accelerated time failure models; Finkelstein and Esaulova, 2006) with Gamma-distributed frailty (Missov and Finkelstein, 2011). Furthermore, Missov (2013) shows that, at ages 50+, Gamma-Gompertz can yield very good predictions of the remaining life expectancy of a cohort.

Figure 1

The effect of heterogeneity (frailty) and selection on the aggregate dynamics of mortality

The effect of heterogeneity (frailty) and selection on the aggregate dynamics of mortality

Note: The dashed lines indicate the assumed individual hazard functions, which stop when the person dies. The solid line indicates the resulting aggregate hazard function.

8Missov (2013) uses the Gamma-Gompertz model to estimate the average rate of senescence of Swedish women over the period 1891-2010, and his estimates range between 0.10 and 0.14. Similar results have also been obtained by Barbi (2003) for the Italian cohorts born between 1873 and 1895 (

9 = 0.13 for females and 0.11 for males), and by Barbi et al. (2003) for the French cohorts born between 1820 and 1879, for whom

10 ranges between 0.10 and 0.12.

11The hypothesis of a constant rate of senescence can also be tested by comparing populations subject to different mortality regimes, in a quasi-natural experimental framework. Zarulli (2012, 2013), for instance, compares the rate of senescence of a few populations subject to a sudden shock in mortality (treated) with that of populations who escape that shock (controls), and finds only small differences in the observed rate of senescence.

12There is also the non-trivial question of when senescence begins or, at least, becomes statistically significant. The first conjecture was that senescence may begin when individuals attain sexual maturity, i.e. at about 12 years (Olshansky and Carnes, 1997), but it is now generally agreed that senescence starts much later, possibly at the end of the reproductive period. In the case of hunter-gatherers, for instance, the beginning of senescence has been identified at about 40 years (Gurven and Kaplan, 2007).

13In short, there are theoretical (biological) reasons to think that the rate of senescence in human populations may be constant, and empirical estimates do not patently contradict this conjecture (although neither do they strongly corroborate it). They give values of ? that oscillate around 0.1, although it should be kept in mind that with exponential functions even small differences in the parameters may result in significantly diverging paths of death risks with age.

14This article contributes to this discussion in two ways: first, it proposes a (relatively) simple method to estimate the rate of senescence ? when frailty operates and, secondly, it offers a few empirical estimates for ?, for selected cohorts from the life tables of the Human Mortality Database (individuals born in five European countries between 1878-1912, and for whom original – that is, not smoothed – data are available). What emerges from the analysis is that the average rate of senescence is actually close to the predicted value of 0.1, but there are also signs of statistically significant differences between groups, times and ages, although these differences are very small in absolute terms, especially in the case of women.

15Section I of this article considers the effects of selection in a Gamma-Gompertz framework. In Section II the problem of how to deal with period effects is tackled. Data reliability is scrutinized in Section III, and the data are analysed in Section IV.

I – Effects of heterogeneity

16This article sets out to estimate the rate of individual senescence starting from the aggregate information of cohort life tables. The statistical units of analysis are cohorts j, whose (aggregate) force of mortality

17 is repeatedly observed at various ages x (x = 65, 66, …, 98 years). The technique employed here is a standard longitudinal panel data analysis (Cheng, 2003; Frees, 2004) with fixed effects and weighted least squares estimation. In this article, each cohort j is univocally identified by three characteristics: sex, year of birth and country of residence (e.g. Swedish women, born in 1900).

18The problem arises from the fact that if individuals are characterized by an inherently different resistance to the pressure of the environment (frailty), only the fittest survive to older ages, and this selection process originates a compositional change in each cohort. As a consequence, the hazard function for the entire cohort increases more slowly than any individual hazard function (Vaupel et al., 1979; Figure 1). Let us assume that, starting from a properly chosen initial age a (the choice of which will be discussed below), the force of mortality evolves exponentially (Gompertz) and frailty is distributed as a Gamma. With these assumptions, the observed log force of mortality

19 for a given cohort j is (Vaupel 1979, Equation 11):

20

21where x = age – a, ?_j is the logarithm of the initial [3] mortality of cohort j,?²_j represents the initial variance of its frailty, is the value of the survival function at age x, and ?_j,x is a normally distributed random variable with zero mean. [4] Equation 2 – and all the models presented in this and the following section – can be estimated with a weighted least squares approach (WLS), where weights are given by the number of deaths of each cohort at each age. WLS, as opposed to ordinary least squares (OLS), reduces the risk of bias due to heteroscedasticity (Horiuchi and Wilmoth, 1998).

22Estimating the parameters of Equation (2) proves problematic, however, not so much because of their high number, but because the presence of several cohort-specific parameters (?_j and ?²_j) may bias the estimate of the unique common parameter ?₁ (incidental parameter problem; see Neyman and Scott, 1948; Lancaster, 2000).

23This difficulty is frequently circumvented by assuming that all cohorts share the same initial variability of frailty, so that the ?²_j parameters collapse into a unique parameter ?². But this simplification does not seem to be warranted here, where “initial” means “at age a = 65”, and cohorts differ profoundly (by sex, country and epoch). The alternative that we suggest is to assume that the cohorts’ initial variability of frailty ?²_j is correlated with the overall mortality experienced by the cohorts up to age a, so that

24

25where is the value of the observed log survival function at age a, and where the ?₃ coefficient measures by how much the different initial variability of frailty observed between cohorts can be explained by the proportion of survivors at age a. [5]

26Substituting (3) into (2) gives

27

28Equation (4) can be better interpreted if one considers its three parts separately. The first, ?_j, describes a cohort effect: the logarithm of the initial force of mortality for cohort j. The second, ?₁x, describes the evolution of the individual force of mortality with age. The third, , captures the effect of heterogeneity on the aggregate dynamic of mortality. The advantage of this equation is that it halves the number of cohort-specific parameters by estimating ?₂ and ?₃ instead of ?²_j and therefore reduces the possible bias due to the “incidental parameter problem”.

29Unfortunately, the parameters of Equation (4) are difficult to interpret because the ?₂ effect of the focal variable is mediated by the ?₃ effect of the moderating variable and cannot be considered only in itself (Braumollerm, 2004; Brambor et al., 2006). In these cases, Ozer-Balli and Sorensen (2010) suggest replacing the original moderating variables with their deviations from the mean: their interpretation is the same, but the advantage is that at (or, more loosely, around) their mean, their effect is zero, and the original meaning and interpretation of the ?₂ parameter is restored. Therefore, the estimates of this paper are based on Equation (5), which replaces Equation (4).

30

31where and are deviations from the mean of the corresponding original variables, and , respectively.

32Section IV of this article provides (and compares) empirical estimates of the rate of senescence under two different assumptions:

• that the bias due to the incidental parameter problem is negligible (Equation 2);
• that the starting level of frailty is affected by the observed proportion of survivors at age a (Equation 5).

II – Period and group effects

33Equations (2) and (5) neglect period effects, but these may have a strong impact on the estimates of the rate of senescence. Imagine a significant technological innovation that triggers a sudden and enduring reduction of death risks: for instance, the discovery of penicillin. From that moment on, all individual hazard functions will be shifted downward as in Figure 2, or more irregularly, and this will bias (in this case, will depress) the estimated value of ?₁. [6]

Figure 2

Impact of period effects on the estimation of senescence

Impact of period effects on the estimation of senescence

Note: The dashed lines indicate the assumed individual hazard functions. The solid line indicates the aggregate hazard function. The two panels show two cohorts characterized by the absence (A), or the presence (B), of period effects. When these are present, a simple linear interpolation of the curve (panel B) biases the estimated slope downwards.

34An easy way to take period effects into account is to include in Equations (2) and (5) a set of dummies indicating P disjoint (five-year) periods. [7] The inclusion of these dummies transforms Equations (2) and (5) into, respectively, (6) and (7):

35

36which can be used to estimate the rate of senescence, net of the disturbances due to period effects and under different assumptions on the initial distribution of frailty.

37The purpose of this article, however, is not only to estimate the rate of senescence, but also to compare it between cohorts. This can be done in several ways. One possibility is to form two groups (males and females, for instance), estimate their rates of senescence (?_m and ?_f), and see if they differ significantly. With several groups, however, it is more efficient to proceed in a different way. Let {g_k}^K_k=1 be a set of dummy variables whose value is 1 for the cohort of interest (k, out of K groups of cohorts) and 0 in all other cases. Equations (6) and (7) can then be rewritten as follows:

38

39where ?_1,k represents the rate of senescence within each of the K (groups of) cohorts under consideration.

40Once again, it is probably worthwhile to consider that Equations (8) and (9) can be broken down into four different components:

1. a cohort-specific initial level of mortality ?_j for each cohort j (identified by country, sex and year of birth);
2. a period effect,
   , which basically modifies the intercept of the regression and represents a variation in the level of mortality – due to technological improvements – and not accounted for by the Gamma-Gompertz model;
3. heterogeneity: the parameters ?₂ and ?₃ (only in Equation 9) take into account (under different assumptions) the fact that individual mortality increases faster than aggregate mortality;
4. a group effect, , representing the rate of senescence (?₁) observed in cohorts belonging to different groups (for instance males and females).

41The advantage of this approach is that it becomes relatively easy to focus on point (4), which is what interests us here, and to test whether different groups have significantly different rates of senescence (?₁). Vaupel’s hypothesis, for instance, can be tested with an F-test on the variance explained by models with and without group effects (Equations 8-9 vs. Equations 6-7). If the more complicated models (8) and (9), with group effects, turn out to explain a significantly greater variance than their simpler counterparts, (6) and (7), the hypothesis of a unique rate of senescence is rejected, and the alternative hypothesis of significant differences between groups is corroborated.

42The focus of this article is on point (4), and on four possible sources of variability in the rates of senescence: sex, country, year of birth, and age.

III – The Human Mortality Database

43The data used in this paper come from the Human Mortality Database (HMD), covering several countries over a long time span, in some cases dating back to the mid-eighteenth century. But the choice of the countries that fit our requirements is constrained in several ways.

44In the first place, the HMD contains very many period life tables, which will not be used in this application because of their intrinsic heterogeneity: cohort data will be preferred instead. The HMD has cohort life tables for 11 countries (see Appendix Table A.1), in some cases with details on sub-populations. In France, or England and Wales, for instance, military personnel can be separated from the rest of the population (as is the case here, incidentally). For New Zealand there are two series: one is on the entire population while the other excludes the Maoris; only the latter has been considered here because it is longer.

45The data collected by the HMD has generally been produced by national institutes of statistics with different procedures, which, on top of this, have sometimes changed over time. The HMD has harmonized the original series in order to make them (more) comparable, redressing primarily five shortcomings:

• The original series did not always report all the information, and year of birth, age, or year of death were occasionally missing. Depending on the information collected, therefore, the death rates may refer to triangles, squares or parallelograms in a Lexis diagram. In order to make the data comparable, the HMD has estimated the number of deaths by triangle, with an imputation procedure that may have somewhat smoothed the data;
• The original data occasionally used multi-year age classes. In this case, too, the HMD has imputed data by single year of age. This procedure has also probably smoothed the original series, at least to some extent;
• The last age-class of the table is frequently an open one, for instance 90+, or 100+. In this case, too, the number of deaths in the last open class has been imputed for each single year, up to age 109;
• The most recent cohorts, born between 1900 and 1919, are not extinct yet. The HMD has thus extrapolated these series up to the year 2029, imputing the same mortality rates observed for adjacent cohorts;
• The population exposed to the risk of dying has also been “adjusted” in various ways before appearing in the HMD. This is the case, for instance, when the age structure of a population is known only from censuses (and is thus missing in inter-censal periods), or by age classes instead of single years of age.

46All these operations are reasonable, accurate and well-documented. But all of them introduce a possible bias in the relationship between age and mortality. In this article, therefore, the analysis refers exclusively to the countries and epochs for which the original series can be represented in either triangles or parallelograms on a Lexis diagram. Furthermore, only the series where deaths were originally classified by single year of age have been used (except for the last, open class). For these reasons, England and Wales, Island, Italy and New Zealand were excluded from the analysis (see Appendix Table A.1). Finally, France and the Netherland were also dropped from the dataset because their last open age class starts too early.

47The five selected countries (Table 1) have series that span over significantly different periods. The longest are those of Norway, which fit our criteria since 1846, while the shortest are those of Finland, which are acceptable (by our standards) only from 1917 on. Since a balanced panel (with series of the same length for all countries) is to be preferred, this paper uses only cohorts born after 1878. The age interval chosen for the analysis is 65-98 years (see below): our cohorts are therefore observed between 1943 (when they turn 65) and 2008.

48The observation ends at age 98 because after then the data are of dubious quality and have been processed in various ways (especially because of the final, open age class). The choice of the age a from which the process of senescence is assumed to begin should not be too important: if the increase in the log force of mortality were indeed linear past a certain age a, starting the analysis at age a or later would not affect the results, but it would be risky to begin earlier. Therefore, it appears safer to start at a conveniently high age. After a few attempts, it was decided to start at age a = 65, which has the additional advantage of confining the window of observation to, basically, the post-war period (1943-2008).

49In short, this analysis is carried out on 5 countries (Denmark, Finland, Norway, Sweden, and Switzerland), 2 sexes, 35 cohorts (born between 1878 and 1912), and 34 years of age (65 to 98), in the calendar years 1943 to 2008 (Table 1). The logarithms of the force of mortality and the evolution of cohort life expectancy at 65 years are shown in Figures 3 and 4, respectively.

Table 1

The cohort life tables selected from the HMD dataset

Country Sex First cohort Last cohort Nbr of cohorts Lexis triangles Parallelograms Max. age Annual statistics Switzerland F 1876 1918 43 1876 1876 98 1876 M 1876 1918 43 1876 1876 98 1876 Denmark F 1835 1918 84 1921 1916 99 1906 M 1835 1918 84 1921 1916 99 1906 Finland F 1878 1918 41 1917 1917 100 1941 M 1878 1918 41 1917 1917 100 1941 Norway F 1846 1918 73 1994 1846 max 1846 M 1846 1918 73 1994 1846 max 1846 Sweden F 1751 1919 169 1895 1895 max 1970 M 1751 1919 169 1895 1895 max 1970

The cohort life tables selected from the HMD dataset

Note: Columns “First cohort” and “Last cohort” identify, respectively, the year of birth of the first and the last cohort for which data are available. “Lexis Triangles” and “Parallelograms” specify the year when data were first collected for annual Lexis triangles or parallelograms (without subsequent interruption). “Max. Age” specifies the age at which the last open class starts (with reference to the year since when the available data can be represented as parallelograms or triangles). When different ages are used to define the last open class, “Max. age” is the lowest among them. The modality “max” indicates that there is no open-ending class. “Annual stats” refers to the year since when annual uninterrupted population statistics are available.

Figure 3

The force of mortality in the countries analysed, cohorts 1878 to 1912

The force of mortality in the countries analysed, cohorts 1878 to 1912

Note: The grey region in the panels shows the ages (both sexes) on which this analysis focuses. The five countries are Denmark, Finland, Norway, Sweden, and Switzerland.

Figure 4

Evolution of life expectancy at age 65 (e₆₅) in the five countries analysed (cohorts of 1878 to 1912)

Evolution of life expectancy at age 65 (e₆₅) in the five countries analysed (cohorts of 1878 to 1912)

IV – Analysis

50The aim of this analysis is to test if human groups who lived in different epochs and regions experienced the same senescence processes, as measured by the ?₁ parameter of the various equations of this article. Let us first consider the issues of heterogeneity and period effects (Tables 2, 3 and 4).

51Table 2 reports the estimated mean rate of senescence obtained with three different methods: the simple Gompertz model of Equation (1) ignores heterogeneity and, as expected, its estimated rate of senescence (0.094) is significantly lower than that of all other models, where heterogeneity is accounted for.

Table 2

Rate of senescence estimated with different assumptions on heterogeneity

Equation 1 Equation 2 Equation 5 All cohorts All cohorts All cohorts Rate of senescence [?ˆ1] 0.094 0.099 0.100 Controls for heterogeneity ln(s¯x) [?ˆ2] [0.059] 0.072 ln(s¯a)?ln(s¯x)?[?ˆ3] – 0.293 Parameters 351 701 353 Adj. R2 0.9853 0.9939 0.9903 BIC – 17,230 – 24,468 – 22,037 Note: In all the models, the dependent variable is the logarithm of the force of mortality. The data covers 5 countries, 2 sexes, 35 cohorts and 34 years of age (11,900 observations in total). All the coefficients shown in the table are statistically significant (p-value < 0.001). The intercepts (?ˆj) are not shown. In model 2 the estimates of the initial variability of frailty for each cohort (?ˆj2) have been omitted: the mean value of the estimated ?j2 coefficients is reported instead [in brackets]. The number of each model corresponds to that of its equation in the text. BIC = Bayesian information criterion, which balances the total variance explained with the number of parameters employed. Models with a lower BIC should be preferred.

Rate of senescence estimated with different assumptions on heterogeneity

52In models (2) and (5) the rate of senescence is estimated with the Gamma-Gompertz model, with different assumptions on the initial variability of frailty. In model (2), each cohort has its own specific initial variability of frailty (estimates not reported in the table). This approach is defensible, but the high number of parameters in the model may bias the estimate of ?₁ (incidental parameter problem). In model (5) the rate of senescence is estimated under the assumption that the initial variability of frailty depends on the proportion of survivors up to age a, which halves the number of parameters to be estimated, and correspondingly reduces the potential bias of the incidental parameter problem.

53Models (2) and (5) produce very similar estimates of the rate of senescence, both very close to the predicted value of 0.1. This suggests that the incidental parameter problem may not be such a problem, after all – at least not in this case. Note that, as expected (cf. footnote 3), the initial variability of frailty is negatively correlated with survival up to age a.

54Table 3 gives the same estimates separately for females and males. The two models (2 and 5) still produce very similar estimates of the rate of senescence ?₁, but a significant difference now appears between women (whose senescence is higher: 0.104) and men (whose senescence is lower: 0.090 or 0.091). Women also appear to be more heterogeneous than men: the estimates of initial variance of frailty range between 0.054 and 0.056 for women, and between 0.043 (model 2m) and 0.039 (model 5m) for men. And, of course, women’s mortality is lower: their mean force of mortality at 65 years is 0.018 as against 0.030 for men (not shown in the table).

Table 3

The rate of senescence by sex, with different assumptions on heterogeneity

Equation 2f Equation 2m Equation 5f Equation 5m Females Males Females Males Rate of senescence [?ˆ1] 0.104 0.091 0.104 0.090 Controls for heterogeneity ln(s¯x) [?ˆ2] [0.054] [0.043] 0.056 0.039 ln(s¯a)?ln(s¯x)?[?ˆ3] – 0.379 – 0.142 Parameters 351 351 178 178 Adj. R2 0.9956 0.9928 0.9946 0.9904 BIC – 13,901 – 12,412 – 14,071 – 12,095 Note: In all the models the dependent variable is the logarithm of the force of mortality. The data covers 5 countries, 1 sex, 35 cohorts and 34 years of age, for a total of 5,950 observations. All the coefficients shown in the table are statistically significant (p-value < 0.001). The intercepts (?ˆ j) are not shown. In models 2f and 2m the estimates of the initial variability of frailty for each cohort (?ˆj 2) are not shown: the mean value of the estimated ? j 2 coefficients is reported instead [in brackets]. The number of each model corresponds to that of its equation in the text. BIC = Bayesian information criterion.

The rate of senescence by sex, with different assumptions on heterogeneity

55The difference in the rate of senescence between men and women might depend, a least in part, on period effects, that is on the different impact that technological and medical innovation may have had on these two sub-populations. This conjecture is tested in Table 4, where controls for period effects are introduced.

56The introduction of period effects entails a significant improvement in the overall fit of the models, as shown by the BIC statistics – which are always lower in Table 4 than in Table 3 – and by the highly significant values of the F-statistics. This notwithstanding, the effect of period variations on our estimates of the rate of senescence appears rather limited and does not affect the comparison between men and women: the rate of senescence remains significantly higher among the latter.

Table 4

Estimates of the rate of senescence by sex, with different assumptions on heterogeneity and including period effects

Equation 6f Equation 6m Equation 7f Equation 7m Females Males Females Males Rate of senescence [?ˆ1] 0.103 0.091 0.105 0.087 Controls for heterogeneity ln(s¯x) [?ˆ2] [0.055] [0.053] 0.067 0.020 ln(s¯a)?ln(s¯x)?[?ˆ3] – 0.272 – 0.197 Parameters 364 364 191 191 Adj. R2 0.9963 0.9941 0.9958 0.9935 BIC – 14,807 – 13,500 – 15,390 – 1,188 ANOVA Models 2-6(f) Models 2-6(m) Models 5-7(f) Models 5-7(m) F-statistics 80.3 96.1 120.9 199.8 p-value < 2.2e-16 < 2.2e-16 < 2.2e-16 < 2.2e-16 Note: In all the models the dependent variable is the logarithm of the force of mortality. The data covers 5 countries, 1 sex, 35 cohorts and 34 years of age, for a total of 5,950 observations. All the coefficients shown in the table are statistically significant (p-value < 0.001). Intercepts (?ˆ j) and period effects (?ˆ 4,p) are not shown. In models 6f and 6m the estimates of the initial variability of frailty for each cohort (?ˆj 2) are not shown: the mean value of the estimated ?j 2 coefficients is reported instead [in brackets]. The number of each model corresponds to that of its equation in the text. BIC = Bayesian information criterion. In the ANOVA section, the label indicates what models have been compared: e.g. 2-6(f) means that the overall variances explained by Models 2f and 6f have been compared. All the values of the F-statistics are highly significant which means that the overall variance explained by the more detailed model is considerably higher.

Estimates of the rate of senescence by sex, with different assumptions on heterogeneity and including period effects

57Figure 5 shows the estimated values of period effects in models 7m and 7f (that is, net of the other effects included in the Gamma-Gompertz model). Between 1947 and 1972, male mortality appears to have increased before subsequently declining, [8] while for females no general trend can be detected. The fact that period effects do not reveal a steadily declining trend suggests that mortality decline is mainly explained by a reduction of the ? parameter (initial mortality) in the Gamma-Gompertz model.

Figure 5

Analysis of period effects

Analysis of period effects

58Models 6f and 6m of Table 4 corroborate one of our conjectures: the (almost) linear correlation existing between the initial variability of frailty and (log)survival – see also Equation 3, footnote 4, and Figure 6, where we plot, separately for males and females, the estimated variability of frailty of each cohort (calculated with models 6m and 6f) against log survival at age 65. The (negative) relation between

59 and at the initial age a is close to linear, especially for women, which corroborates our proposal to introduce Equation (3) and what derives from it (models 5, 7 and 9). [9]

Figure 6

Estimated relation between initial variability of frailty and survival

Estimated relation between initial variability of frailty and survival

60Table 5 shows the estimates of the rate of senescence for each of the five countries of this analysis. Men and women appear once again to age differently: in all the countries considered, independently of the model adopted, men undergo a slower senescence process than women (their values of ? are smaller).

Table 5

Estimates of the rate of senescence by country including period effects

Equation 8f Equation 8m Equation 9f Equation 9m Females Males Females Males Rate of senescence [?ˆ1] Switzerland 0.101 0.082 0.105 0.087 Denmark 0.099 0.091 0.102 0.090 Finland 0.102 0.071 0.104 0.075 Norway 0.111 0.098 0.107 0.094 Sweden 0.104 0.096 0.105 0.094 Controls for heterogeneity ln(s¯x) [?ˆ2] [0.056] [0.026] 0.068 0.031 ln(s¯a)?ln(s¯x)?[?ˆ3] – 0.251 0.036 Parameters 368 368 195 195 Adj. R2 0.9965 0.9954 0.9962 0.9947 BIC – 15,158 – 14,817 – 15,774 – 15,363 ANOVA Models 6-8(f) Models 6-8(m) Models 7-9(f) Models 7-9(m) F-statistics 93.4 356.6 104.9 325.2 p-value < 2.2e-16 < 2.2e-16 < 2.2e-16 < 2.2e-16 Note: In all the models the dependent variable is the logarithm of the force of mortality. The data covers 5 countries, 1 sex, 35 cohorts and 34 years of age, for a total of 5,950 observations. All the coefficients shown in the table are statistically significant (p-value < 0.001). Intercepts (?ˆj) and period effects (?ˆ4,p) are not shown. In models 9f and 9m the estimates of the initial variability of frailty for each cohort (?ˆj2) are not shown: the mean value of the estimated ?j2 coefficients is reported instead [in brackets]. The number of each model corresponds to that of its equation in the text. BIC = Bayesian information criterion. In the ANOVA section, the label indicates what models have been compared: e.g. 6-8(f) means that the overall variances explained by Models 8f and 6f have been compared. All the values of the F-statistics are highly significant which means that the overall variance explained by the more detailed model (where the rate of senescence has been estimated separately by country) is considerably higher. In other words, the rate of senescence appears to differ by country.

Estimates of the rate of senescence by country including period effects

61In model 8, the variability of the rate of senescence between countries appears to be relatively small among women (from a minimum of 0.102 in Denmark to a maximum of 0.107 in Norway), and larger among men (from a low of 0.075 in Finland to a high of 0.094 in Norway and Sweden). In both cases, however, the highly significant F-tests suggest that this variability is not produced by chance only, and that something more systematic is at work.

62In order to analyse the rate of senescence by year of birth (Table 6; see also Appendix Table A.2 for detailed statistics on model 9f), the cohorts born in 1878-1912 were broken down into eight different sub-groups, normally of five years, except for the first (1878-1880) and the last (1911-1912) which span only three and two years, respectively. For each of these sub-groups, the rate of senescence was estimated with Equations (8) and (9). A slight but continuous downward trend in the rate of senescence emerges – both without controls for period effects (not shown here) and with these controls (as in Table 6) – with the exception of model 9f. Once again, the rate of senescence appears to be more variable among men than among women, and declines less (if at all – model 9f) among the latter.

Table 6

Analysis of the rate of senescence by year of birth including period effects

Equation 8f Equation 8m Equation 9f Equation 9m Females Males Females Males Rate of senescence [?ˆ1] Birth cohorts 1878-1880 0.108 0.104 0.105 0.100 1881-1885 0.105 0.101 0.104 0.097 1886-1890 0.104 0.095 0.105 0.095 1891-1895 0.105 0.094 0.105 0.093 1896-1900 0.103 0.089 0.105 0.090 1901-1905 0.104 0.087 0.105 0.088 1906-1910 0.102 0.086 0.105 0.086 1911-1912 0.099 0.083 0.104 0.084 Controls for heterogeneity ln(s¯x) [?ˆ2] [0.059] [0.065] 0.069 0.058 ln(s¯a)?ln(s¯x)?[?ˆ3] – 0.267 – 0.233 Parameters 371 371 198 198 Adj. R2 0.9963 0.9943 0.9958 0.9937 BIC – 14,780 – 13,559 – 15,347 – 14,319 ANOVA Models 6-8(f) Models 6-8(m) Models 7-9(f) Models 7-9(m) F-statistics 4.5 16.1 2.3 26.8 p-value 4.98e-05 < 2.2e-16 0.025 < 2.2e-16

Analysis of the rate of senescence by year of birth including period effects

63Finally, Table 7 shows how the estimates of the rate of senescence vary by age, using six five-year age classes (65-69, 70-74, …, 90-94) and one four-year age class (95-98). The rate of senescence systematically increases with age, both for women (from 0.103 to about 0.106) and men (from about 0.088 to 0.095).

Table 7

Estimates of the rate of senescence by age class, including period effects

Equation 8f Equation 8m Equation 9f Equation 9m Females Males Females Males Rate of senescence [?ˆ1] Birth cohorts 65-69 ans 0.101 0.092 0.103 0.088 70-74 ans 0.101 0.092 0.102 0.088 75-79 ans 0.102 0.093 0.104 0.089 80-84 ans 0.103 0.094 0.105 0.090 85-89 ans 0.104 0.095 0.106 0.092 90-94 ans 0.104 0.097 0.106 0.094 95-98 ans 0.104 0.098 0.106 0.095 Controls for heterogeneity ln(s¯x) [?ˆ2] [0.070] [0.102] 0.088 0.076 ln(s¯a)?ln(s¯x)?[?ˆ3] – 0.267 – 0.202 Parameters 370 370 197 197 Adj. R2 0.9963 0.9942 0.9958 0.9936 BIC – 14 875 – 13 544 – 15 439 – 14 250 ANOVA Models 6-8(f) Models 6-8(m) Models 7-9(f) Models 7-9(m) F-statistics 18.9 15.1 16.2 18.6 p-value < 2.2e-16 < 2.2e-16 < 2.2e-16 < 2.2e-16 Note: In all the models the dependent variable is the logarithm of the force of mortality. The data covers 5 countries, 1 sex, 35 cohorts and 34 years of age, for a total of 5,950 observations. All the coefficients shown in the table are statistically significant (p-value < 0.001). Intercepts (?ˆ j) and period effects (?ˆ 4,p) are not shown. In models 9f and 9m the estimates of the initial variability of frailty for each cohort () are not shown: the mean value of the estimated ?j 2 coefficients is reported instead [in brackets]. The number of each model corresponds to that of its equation in the text. BIC = Bayesian information criterion. In the ANOVA section, the label indicates what models have been compared: e.g. 6-8(f) means that the overall variance explained by Models 8f and 6f has been compared. All the values of the F-statistics are highly significant which means that the overall variance explained by the more detailed model (where the rate of senescence has been estimated separately by age cohort) is considerably higher. In other words, the rate of senescence appears to differ by age cohort.

Estimates of the rate of senescence by age class, including period effects

64These results do not change if the analysis is carried out with controls for period effects or separately by country (not shown here). Swedish males are the only exception: senescence appears to remain basically unchanged as they age.

V – Discussion and conclusion

65Assessing the implications of our findings is not an easy task. From a purely statistical point of view, all the tests reject the hypothesis of a constant rate of senescence ?₁, but at the same time the variability of senescence among the different cohorts examined is very limited. We cannot rule out the possibility that minor departures from the model assumptions (e.g. frailty may not be exactly Gamma-distributed, errors may not be perfectly normal, or the incidental parameter problem may somehow persist) lead to rejection of the null hypothesis when the null hypothesis is in fact true. Caution is needed in this case.

66The most important and systematic difference in the rate of senescence is that between men and women, whose senescence is faster (see also Table 8, for a summary of our findings), a result which is in line with previous estimates (e.g. Bronikowsky et al., 2011; Barbi, 2003).

Table 8

Summary of findings (estimates of ?₁)

Females Males Minimum Maximum Minimum Maximum Overall 0.105 0.105 0.087 0.087 By country 0.102 0.107 0.075 0.094 By year of birth 0.104 0.105 0.084 0.100 By age 0.103 0.106 0.088 0.095

Summary of findings (estimates of ?₁)

Note: The table shows the minimum and maximum values of the rate of senescence estimated for both sexes using models 7 and 9 in Tables 4-7.

67Kirkwood and Holliday’s (1979) biological theory of the “disposable soma” can be used, at least in part, to explain this difference between males and females. This theory suggests that in each species there may be a trade-off between the energy allocated to reproduction and that necessary for somatic maintenance and repair (“antagonistic pleiotropy”). A few studies (Doblhammer and Oeppen, 2003; Gagnon et al., 2009) have indeed identified a significant positive correlation between fertility and post-reproductive mortality. Among females, in particular, late-life mortality may largely depend on the biological costs of repeated pregnancies and births, while among males the biological costs of producing and raising offspring are small and thus the effects on late-life mortality negligible.

68In this theoretical framework, the greater costs of reproduction for women may be responsible for their faster ageing.

69While a considerable variability in the rate of male senescence can be observed across countries and cohorts, women’s senescence seems to remain relatively constant. Besides, our assumption of a linear relation between the initial variability of frailty and log survival works better for females than for males (Figure 6), and women are less affected by period effects (Figure 5). In short, female mortality conforms better to the expectations that derive from the constant senescence hypothesis advanced by Vaupel.

70Last, senescence appears to increase with age, for both sexes. This phenomenon had indeed already been observed in literature. Horiuchi (1983, 1997), for instance, noticed it for women aged 36-80 in Italy, Japan, the Netherlands and Spain. Later on, a slightly accelerated senescence process was also identified by Yashin and Iachine in their study of Danish twins (Yashin and Iachine, 1997; Vaupel et al., 1998), and by Doblhammer and Oeppen (2003) in their study of British peerage. Horiuchi (2003) suggests that if the mechanisms for somatic maintenance and repair deteriorate over time too, then mortality risks should increase faster than exponentially, which is in line with our findings.