## Abstract

Physiological senescence is characterized by the increasing limitation of capabilities of an organism resulting from the progressive accumulation of molecular damage, which at group (cohort) level translates into, among other things, an increase in mortality risks with age. Physiological senescence is generally thought to begin at birth, if not earlier, but models of demographic aging (i.e., an increase in mortality risks) normally start at considerably later ages. This apparent inconsistency can be solved by assuming the existence of two mortality regimes: “latent” and “manifest” aging. Up to a certain age, there is only latent aging: physiological senescence occurs, but its low level does not trigger any measurable increase in mortality. Past a certain level (and age), molecular damage is such that mortality risks start to increase. We first discuss why this transition from latent to manifest aging should exist at all, and then we turn to the empirical estimation of the corresponding threshold age by applying Bai’s approach to the estimation of breakpoints in time series. Our analysis, which covers several cohorts born between 1850 and 1938 in 14 of the countries included in the Human Mortality Database, indicates that an age at the onset of manifest aging can be identified. However, it has not remained constant: it has declined from about 43 and 47 years, respectively, for males and females at the beginning of the period (cohorts born in 1850–1869) to about 31 for both males and females toward its end (cohorts born in 1920–1938). A discussion of why this may have happened ensues.

## Introduction

The literature on demographic aging1 frequently refers to an alleged starting age for mortality acceleration, the age when mortality risks begin to increase significantly and irreversibly. However, only a few studies have specifically addressed this topic.

Mortality acceleration was originally believed to begin at sexual maturity, between 12 and 20 years. Gompertz (1825) and Makeham (1860), for instance, applied their models to the ages from 20 onward. Brownlee, in 1919 (reported in Olshansky and Carnes 1997), conjectured that 12 years could mark the beginning of demographic aging. The identification of sexual maturity as the beginning of mortality acceleration is consistent with the evolutionary theorizations of aging (Kirkwood and Rose 1991; Medawar 1952; Williams 1957): after sexual maturity is reached, natural selection loses importance and momentum (Hamilton 1966), and senescence-related mortality progressively emerges. Later empirical findings, however, led scholars to delay the likely inception of mortality acceleration to an age between 30 and 50 (Carnes and Olshansky 1997; Carnes and Witten 2014; Carnes et al. 2006; Gurven and Kaplan 2007; Vaupel 2003, 2010), which implicitly challenges the evolutionary theorizations of aging (Baudisch and Vaupel 2012; Jones et al. 2014).

Although very few empirical studies have addressed this issue, human mortality is generally thought to reach a minimum between 9 and 15 years (Burger et al. 2012; Finch et al. 2014) and to remain roughly constant thereafter for several years. Initially believed to lie somewhere around 35 years (Burger et al. 2012; Vaupel 2003), the beginning of mortality acceleration has recently been raised to about 40 in hunter-gatherer populations (Gurven and Kaplan 2007) and to age 50 and older in European historical populations (Missov 2013).

In the literature on physiological aging, consensus exists that senescence results from the continuous accumulation of damages in the macromolecules of cells, which are not fixed (or not completely, or not timely enough) by the various “repair and maintenance” systems of the organism (Holliday 1998, 2006, 2010; Rattan 2006, 2008; Yin and Chen 2005). The accumulation of cellular damage likely starts very early: at birth; at conception; or even before, in the ova of the female fetuses that will generate the new generation (Abrams 1991; Carnes et al. 2008; Milne 2006). However, its effects on vital functions become manifest only much later. Bafitis and Sargent (1977), Shock (1981), and Weale (2004), for instance, found no relevant decrease in human vital functions until at least age 30. Similar thresholds have also been identified for other living species. Jones et al. (2008) found that in the majority of the 20 terrestrial vertebrate species they examined, individual fitness increases, instead of decreasing, after the age of sexual maturity; Péron et al. (2010) reached basically the same conclusions in their research on 72 species of birds and mammals.

Why does demographic aging occur so late, if senescence starts at birth or even before? A possible explanation is that molecular damage accumulation need not translate immediately into a marked increase of mortality: its effects may become manifest only beyond a certain threshold. Kirkwood and Austad (2000) explicitly considered this hypothesis,2 which appears consistent with the idea of a biological warranty period proposed by Carnes et al. (2003). It may therefore make sense to distinguish between latent aging, during which damage accumulates but remains low and with very limited impact on survival (and reproductive) chances, and manifest aging, when the accumulation of damage is such that the resistance of the organism is affected and mortality increases.

Detecting the onset of individual senescence (the accumulation of molecular damage) using survival data is extremely difficult (Levitis and Martinez 2013). The identification of the transition from latent to manifest collective aging, however, is easier: at the aggregate (cohort) level, there should be an age when mortality changes from stationary (β ≈ 0 in Gompertz’s model) to increasing, perhaps with constant acceleration (e.g., β ≈ 0.1).

Our purpose is verify whether in heterogeneous cohorts, a threshold for the start of mortality acceleration can be detected and, if so, whether it is constant over time and across countries. We do so by applying Bai’s (2010) method for the identification of structural breaks in panel data. The method yields reliable results if it is applied simultaneously to several time series with a common breakpoint (see the following section). We apply Bai’s technique to selected cohort life tables of the Human Mortality Database (n.d.): female cohorts born in 14 countries in predefined time intervals between 1850 and 1938 (the rationale for this selection is discussed in the upcoming Data section). The most relevant finding of our study is that the age at the onset of mortality acceleration may have markedly declined over time, from about age 48 at the beginning of the period to about age 32 at its end (see the upcoming section discussing the onset of mortality acceleration). The possible causes and implications of this change are discussed in the final section of this article.

## Applying Bai’s Method to Ages and Life Tables

The ages investigated in this article are those between 21 and 70 years. At younger ages—say, from age 0 to 12—mortality is mainly characterized by a downward trend, determined by a multiplicity of causes and connected with the progressive adaptation of the organism to the environment (Levitis 2010; Siler 1979). Then, up to approximately 20 years, mortality increases rapidly, mainly because of accidents and other external causes (Carnes et al. 2006; Gessert et al. 2002, 2003; Goldstein 2011; Remund 2012). At (very) old ages instead—say, 80 and older—mortality decelerates because of selection (Gavrilov and Gavrilova 1991, 2006, 2011; Vaupel et. al. 1979). The age range retained in this application should be, at most, only marginally affected by these disturbing processes, which permits us to better focus on the object of this article: the quest for the age when mortality starts to accelerate.

Let t = 21, 22, . . ., 70 be the age of an individual, x = t – 20 be its translated equivalent, and μi,x be the force of mortality for individual i at age x:
$μi,x=1−Di,xLi+Di,xSi,x+εi,x,$
(1)
where εi,x stands for the error term; Li is a constant representing the strength of mortality during the period of latent aging; and Si,x is a strictly increasing function describing how demographic aging evolves with age. Di,x (=1 if x > ki,0; zero otherwise), acting like a “switch,” is an indicator variable that triggers dysfunction: it is 0 as long as the accumulated damage remains below a given threshold (and therefore before age ki,0) and becomes 1 past this age, when mortality acceleration begins. Because ki,0 is an individual parameter, individuals may pass the threshold at different ages (heterogeneity in the beginning of mortality acceleration).
At the aggregate (cohort) level, Eq. (1) transforms into Eq. (2):
$μ¯x=1−DxL¯+DxS¯x+εx,$
(2)
where $μ¯x$ represents the aggregate force of mortality, and $L¯$and $S¯x$are the mean mortality levels during the phase of latent and manifest aging. Dx is an aggregate “dimmer,” which does not abruptly switch but gradually increases from 0 to 1 in the vicinity of the mean age, $k¯0$, at which the individuals of the cohort make their transition to the phase of manifest aging.3
Let us now assume that in Eq. (2), $L¯$and $S¯x$can be approximated by two Gompertz models:
$L¯=α1eβLx;S¯x=α2eβSx,$
(3)
where βL(=0) and βS are the rates of aging during the phase of latent (L) and manifest aging (S), respectively. Now let $δ¯x$, the log-difference of $μ¯x$, be our measure of the increase in the force of mortality:
$δ¯x=lnμ¯x−lnμ¯x+1=1−DxβL+DxβS+εi,x,$
(4)
where Dx is the aggregate “dimmer” introduced in Eq. (2).

The $δ¯x$series (at the aggregate level) can be calculated empirically from ordinary life tables, aptly called “life table aging rates” (LAR) by Horiuchi and Coale (1990). The $δ¯x$series are heteroskedastic because they depend on the number of deaths (dx) and, ultimately, on age. The variance of each $lnμ¯x$ is approximately 1 / dx (Brillinger 1986; Horiuchi and Wilmoth 1998). Assuming independence, the variance of $δ¯x$is close to $1dx+1dx−1$, which is roughly twice as high as the variance of each $lnμ¯x$.4

Bai’s methodology, when applied to several log-differentiated hazard functions $δ¯j,x$pooled into an assumedly homogeneous group (for instance, the Swedish cohorts born between 1850 and 1870), will yield the single break point that grants the best overall fit for the entire group. We will use this methodology to find the mean age at which individuals make their transition to manifest aging.

Bai’s least squares approach to the identification of break points proceeds by exhaustion of cases. For each candidate breakpoint b, b ∈ [1, 49], and for each cohort j in a group, one needs first to calculate the two subgroup means Mj,1 and Mj,2:
$Mj,1=1b+1∑x=1bδ¯j,xandMj,2=150−b+1∑x=b+150δ¯j,x.$
(5)
Then, for each series $δ¯j,x$, the sum Sj(b) of the squared residuals with respect to Mj,1 and Mj,2:
$Sjb=∑x=1bδ¯j,x−Mj,12+∑x=b+150δ¯j,x−Mj,22,$
(6)
while the sum of the squared residuals (SSR) for all the N series is
$SSRb=∑j=1NSjb.$
(7)
The least squares estimator for the break point emerges as
$b^=minbSSRb.$
(8)

Bai (2010) proved that this estimator is consistent, gave a simple procedure for the computation of confidence intervals, and indicated how to extend the methodology to the case of multiple breaks and regimes.5 The accuracy of the estimate of the age at the onset of mortality acceleration improves when the number of the series (N) is large; when the series are long; when the variability of each series is small; and when the “magnitude of the break” is large (that is, when the difference between the means calculated before and after the breakpoint is ample).

Bai’s estimation does not require any assumption about the distribution of the error terms of Eq. (3),6 but it requires that errors be stationary with respect to age. When this condition is not met, as in our case, the estimate of the breakpoint remains correct, but the estimate of its confidence interval may be biased (Bai 2010:79).

Finally, Bai’s technique assumes that errors in Eq. (3) are cross-sectionally independent. For practical purposes, however, this assumption may be relaxed as long as dependence is mild (Bai 2010:79). In our case, a limited cross-sectional dependence may be induced by international mortality crises, but they are rare, and only a few cohorts of our data set undergo abnormally high (or low) mortality levels in the same year and at the same ages (see the following section).

## The Data, Its Problems, and How to Circumvent Them

For our analysis, we use selected cohort life tables, taken from the Human Mortality Database (HMD), by gender. Selection is partly imposed by the data: not all countries have cohort life tables, which we prefer over period life tables. However, we select even further in order to limit the impact of the factors that may interfere with the “natural” course of mortality and therefore introduce biases in the estimation of the onset of mortality acceleration. These factors belong to four main typologies: mortality decline, “mortality hump,” mortality crises, and national idiosyncrasies.

### Mortality Decline

Most of the data in the HMD cover an epoch of protracted mortality decline in which both period and cohort effects can be detected but not always clearly distinguished (for recent reviews, see Barbi and Vaupel 2005; Doblhammer 2004; Yang 2008). Cohort effects are those induced by improvements in survival at young ages—for example, through a reduction in diseases (Beltrán-Sánchez et al. 2012; Finch and Crimmins 2004) or malnutrition (Ben-Shlomo and Kuh 2002)—that trigger a decline in the mortality of that cohort at all subsequent ages. Conversely, a sudden variation in mortality brought about by, for instance, a medical breakthrough, which benefits everybody from that moment on irrespective of their age or year of birth, is regarded here as a period effect.7

In the study of the evolution of mortality, and more specifically in the quest for breakpoints à la Bai, both effects constitute a problem. In period life tables, for instance, mortality may increase with age not only because of demographic aging but also because older individuals belong to previous cohorts that do not yet benefit from (cohort) mortality reduction. This phenomenon may introduce a bias in the identification of the age at which mortality acceleration begins. Cohort data, on the other hand, suffer from period effects: mortality reduction over time may attenuate the apparent magnitude of the break from latent to manifest aging, making its identification more uncertain.

We pursued both types of analyses, but in this article, we present our results only for cohorts for two reasons. First, cohorts are more homogeneous. Second, despite the presence of period effects, we do find breakpoints, which we take as an indication that they exist and are stronger than their confounders.

### Mortality Hump

The “mortality hump” is a recurrent, although probably not universal, increase in mortality that takes place at young ages (20 years or so). This phenomenon is due to a rise in extrinsic mortality (Carnes et al. 2006), which apparently affects women less than men (Remund 2012) or not at all (Goldstein 2011). The male hump may be provoked by a peak in male hormone production, which currently occurs at age 17–18, possibly down from about 21 years in the mid-nineteenth century (Goldstein 2011). Conversely, the mortality hump for women in the past was mainly due to maternal mortality (Remund 2012), has gradually declined over time, and virtually disappeared after World War II (Loudon 1988).

To minimize the effect of the mortality hump, our analysis deals mostly with women (aged 21 and over).

### Mortality Crises

Five major catastrophes appear to perturb the cohort trajectories in the HMD after 1850, when data on cohort mortality are generally believed to be more reliable: two major cholera epidemics in 1863–1875 and 1883–1887, the Spanish flu pandemic of 1918–1919, and two world wars. The effect of the two world wars is arguably less strong for women, the results for whom thus appear to be more robust. To cope with the three remaining factors without (completely) losing the historical depth of our analysis, we selected three “waves” of cohorts born in different periods, with correspondingly different levels of reliability.

The first wave is that of the cohorts born between 1850 and 1870: they passed through all three epidemics, but the strongest of them—the Spanish flu—hit these cohorts in their 50s, an age group that appears to have been comparatively less affected (Almond and Mazumder 2005). However, maternal mortality was still high in this period: indeed, this is the “wave” that provides the least reliable results for both genders.

Next, we consider the cohorts born between 1902 and 1919. These cohorts came into observation in 1923 and escaped the Spanish flu pandemic, although the exposure of women to maternal mortality is still not negligible.

Finally, the third wave is that of the cohorts born between 1920 and 1938, who suffer from none of the previously identified mortality crises. We consider this wave as the most reliable.

### National Idiosyncrasies

The cross-country variability of the $δ¯x$series is high for several reasons: different population sizes, specific historical events, different mortality regimes, different ways of collecting data on part of national statistical offices, different procedures of harmonization and imputation followed by the HMD, and so on. Because an excessive variability of the series may hamper the identification of the transition from latent to manifest aging, it is important to check whether some countries are “outliers” in this sense. To this end, Table 1 reports a small set of summary statistics for each of the countries covered by our analysis (women only).8

Let CA,w be the set of cohorts born in country A who belong to wave w, and let M ′ and M ′ ′ represent the mean $δ¯x$(increase in log mortality) experienced by these cohorts in two 25-year age intervals, 21–45 and 46–70, meant to broadly represent the phases with, respectively, latent and manifest aging. Their standard deviation is
$σA,w′=∑k∈CA,w∑x=125δ¯k,x−M′2n;σA,w′′=∑k∈CA,w∑x=2650δ¯k,x−M′′2n,$
(9)
where n is the product between the cardinality of CA,w (number of cohorts) and 25 (the number of ages considered in each of the two formulas).

Because the variability of our series depends, as a first approximation, on the size of the cohorts and on their mortality, Table 1 also reports the median cohort size and median mortality rate at age 21. Some countries are characterized by unusually high (H) or low (L) levels of variability. Denmark is, by far, the country with the highest variability in $δ¯x$ series for the third wave (1920–1938), with σ ′ = 0.3, three times the expected magnitude of the break from latent to manifest aging.9 Next comes the second wave (1902–1919) of French and Spanish women, who also have very high values of σ ′—roughly twice the expected magnitude of the break. By contrast, some other countries have $δ¯x$ series with very low variability (e.g., Australia in the second and third wave, and Denmark in the first wave), which is perhaps at least partly attributable to excessive data smoothing.

More can be learned by modeling the σA,w as a power law function of the typical number of deaths (Brillinger 1986; Horiuchi and Wilmoth 1998), as in Fig. 1, which also shows the 95 % confidence interval. What emerges is that the high variability of $δ¯x$in Denmark may simply depend on its comparatively small number of deaths (owing to a small population combined with a favorable mortality regime). Conversely, the second wave of women in Spain and France displays an “excessive” variability, possibly because of specific historical events (e.g., the Spanish Civil War). Other countries show an abnormally low variability of $δ¯x$, outside the confidence interval of our prediction: the data for these countries were perhaps smoothed at some point.

Table 1 may also be used to shed light on some other features of our data set. As expected, mortality is lower in more recent cohorts: the mean value of the rate of mortality at 21 years starts from 5.8 per thousand in the first wave (born in 1850–1870), declines to 3.5 per thousand in the second wave (born in 1902–1919), and drops to a mere 0.9 per thousand in the third wave (born in 1920–1938).

Within each wave, the typical mortality at age 21 does not differ significantly by country. Overall, our data set presents high variability between waves (owing to mortality decline) but only a modest variability within waves.

The variability of our series in the age interval 21–45 (σA,w) increases as we pass from older to more recent cohorts, while the opposite happens to the variability of the subsequent age interval 46–70 (σA,w′ ′). This (probably counterintuitive) outcome derives from our earlier observations that at each age x, the variability of our series depends roughly on the number of deaths (Brillinger 1986), and that because deaths tend to occur at older ages, σA,w increases over time as σA,w′ ′ decreases.

## The Onset of Mortality Acceleration

Before examining each wave and each country in detail, we will provide a general overview of our findings. In all the countries and in all the waves considered here (see Table 2 and Figs. 24), a sudden shift from low to high rates of aging can be detected somewhere between 30 and 50 years. This result corroborates our hypothesis about the existence of two distinct mortality regimes of latent and manifest aging. These results are also roughly in line with the estimates produced using the Gompertz model and judging its goodness of fit (Gurven and Kaplan 2007; Missov 2013; Vaupel 2003).

However, the age at which this transition takes place varies considerably: it typically appears late in the oldest cohorts of women (at about 48 years) and early (at about 32 years) in the youngest ones. A relatively strong variability also characterizes the mean rate of aging during the phase of latent aging. The mean rate of aging is not exactly zero (as expected); it is slightly negative for the most recent cohorts (mortality declines with age, up to the breakpoint) and slightly positive for the earliest ones.

Let us now consider the three waves in detail.

### First Wave (1850–1870)

The estimated age at the onset of mortality acceleration for the first wave of these cohorts is late: at about 48 years (Fig. 2, first wave). The inception of demographic aging in each country does not vary significantly, ranging between 54 years in Norway and 47 years in France (Fig. 3, first wave). The average mortality risk during the phase of latent aging lies between 7 and 9 per thousand. No correlation emerges between the age at the onset of mortality acceleration and the variability of the $δ¯x$series during the phase of latent aging, which suggests that the variability of the mortality rates does not obscure the transition to manifest aging.

The mean rate of aging experienced by the cohorts during the phase of latent aging, which was expected to be close to 0, is slightly but significantly positive in all the countries ($β¯L$= +0.012). This finding signals the existence of a weak aging process during this phase. As explained in footnote 4, with individual heterogeneity in the onset of manifest aging, this finding is consistent with the model presented in the earlier section discussing the application of Bai’s method to ages and life tables.

Similar reasoning can be applied to the rate of aging observed after the breakpoint, which has been estimated at approximately 0.071. This is less than the 0.1 posited by Baudisch and Vaupel (2010), but the difference can depend on the fact that a certain fraction of the cohorts start aging sometime after the average breakpoint identified here.

Although the quality of the data that refer to the first wave of our study may be questioned, the onset of mortality acceleration in a mortality regime similar to that in which our ancestors lived (Burger et al. 2012) appears to have occurred approximately at the end of the reproductive age span.

### Second Wave (1902–1919)

The cohorts of the second wave show a mean age at the onset of mortality acceleration of about 44 years (Fig. 2, second wave). The typical reduction with respect to the first wave is of about four years. Norway and Sweden present the largest shifts, of more than 10 years (Fig. 3).

This earlier appearance of aging accompanies a reduction in the death risks just before the break (from 7.5 % in first wave to 3.2 % in the second) and an increase in the variability of the series (from 0.07 in the first wave to 0.15 in the second) at young ages during the phase of latent aging. Several cohorts of the second wave experience a moderate but significant negative rate of aging during the phase of latent aging ($β¯L$= –0.009, on average), which is positive in two countries (Australia and the Netherlands) and not significantly different from 0 in five others.

The rate of aging measured during the phase of manifest aging ($β¯S$= 0.074) is slightly higher here than in the first wave (0.071), but it is still well below 0.1.

### Third Wave (1920–1938)

The cohorts of the third wave are characterized by the earliest onset of mortality acceleration—at about 32 years, on average (Fig. 2, third wave). Apart from Denmark (with very early onset of mortality acceleration but small sample sizes and high variability of the series), the onset of mortality acceleration in the various countries shows a limited variability: from a maximum of 33 years in Finland and Norway to a minimum of 26 in France (Fig. 3). In 9 of 14 countries, the range is much smaller: only 30 and 32.

In the third wave, the phase of latent aging is characterized by the lowest average risk of death (down to 1.3 %) and the highest variability (standard deviation = 0.192, which is almost twice as large as the break itself). The estimated rate of aging during the phase of latent aging (21–32 years) is even more negative than before ($β¯L$= –0.033), but the rate of aging in the phase of manifest aging remains roughly constant (0.071).

Figures 24 show the transition from latent to manifest aging at different levels of aggregation. Figure 2 refers to the average cohort in each of the three waves (the highest level of aggregation); Fig. 3 focuses on the average cohorts of selected countries, within each wave (that is, about 20 birth cohorts; an intermediate level of aggregation); and Fig. 4 considers only single cohorts (the lowest level of aggregation). At all levels of aggregation, all the series are characterized by a low (approximately constant) rate of aging before the breakpoint, followed by a high (approximately constant) rate after the breakpoint. Thus, the transition from latent to manifest aging appears to be a generalized phenomenon in our cohort life tables.

Conversely, some preliminary attempts at determining the onset of mortality acceleration on period life tables from 1950 onward did not give the same results (not reported here). When applied to this type of data, Bai’s technique tends to identify the first point of the series (age 21 in our analysis) as the single most important break point, while the search for multiple breaks leads typically to the identification of very many of them. These results are all in all consistent with our findings for cohort data. Because period data encompass several cohorts and because cohorts have different threshold ages for the beginning of mortality acceleration, period data appear to be characterized by multiple onsets of aging—possibly even one for each cohort.

We applied our analysis to different age ranges, too. Considering the age range 13–70 in which the lower limit coincides, more or less, with the age of minimum mortality, the results (not reported here) resemble those of Table 2, with the only exception of the break point estimated for the United States (third wave), which declines to 21.

As an additional check of the robustness of our results, we applied Bai’s methodology also to males. As explained in the earlier data section, the data for men are probably more affected by the several disturbing processes of the periods examined. Table 3 presents the estimated break points for the three waves covered by our analysis: the onset of mortality acceleration seems to occur slightly earlier in men than in women. Men, however, follow the same historical trend as women, with a marked decline in the age at the onset of mortality acceleration.

Finally, we also tested the robustness of our findings by estimating the onset of mortality acceleration in Sweden from 1820 to 1920 (birth cohorts). The idea here was to check whether the onset of mortality acceleration had changed before mid-nineteenth century. To do so, we grouped the cohorts as follows: 1820–1839, 1821–1840, . . . , 1919–1938. For each of these groups of 20 birth cohorts, we estimated the onset of mortality acceleration, and we attributed this value to the central year of the period considered (as in a moving average).

As Fig. 5 shows, the onset of mortality acceleration was probably close to 50 years in earlier birth cohorts, and a declining trend appeared only for the cohorts born in the final quarter of the nineteenth century for both men and women. Readers may note that Bai’s technique could not detect any breakpoint for female cohorts born between 1881 and 1888 (actually between 1871–1990 and 1878–1997). We conjecture that this may be related to the specific historical context: a significant part of these cohorts groups enter our observation window (at age 21) slightly before the 1919 Spanish flu pandemic, which particularly affected young adults (Almond and Mazumder 2005).

## Discussion

In short, the more recent cohorts are, the lower their starting mortality level (hardly a surprise) and the earlier the onset of their demographic aging, which was probably unexpected. But why should cohorts with higher life expectancy suffer from an earlier onset of mortality acceleration?

First, it is important to recognize that this does not seem to be a statistical artifact. For instance, the age-specific death risks of selected (and grouped) Swedish female cohorts (Fig. 6) strongly suggest that the inception of mortality acceleration in the third wave takes place significantly earlier (at 32 years) than in the second (40) or the first (53). Nevertheless, the qx of the third wave are always below those of the first and second wave—indeed, largely below, at all ages.

The literature offers a few indirect elements in support of these findings. Missov (2013), for instance, reported that when applied to ages 30 and over, Gamma-Gompertz models produce more accurate estimates of life expectancy in recent than in remote periods, which is consistent with a younger onset of mortality acceleration. Li et al. (2013) also found evidence of this in male (although not in female) Swedish cohorts born in 1885 and 1905, which they tentatively attributed to an increase in smoking rates.

In a different conceptual framework, Luder (1993) and Dolejs (1997) presented results that bear similarities with ours. The analysis of general mortality estimates the inception of mortality acceleration at about 30–35 years, but when the focus shifts to intrinsic mortality, the onset of mortality acceleration appears to take place before—at about 20 years.10 Luder ventured that extrinsic mortality may obscure the rise in intrinsic mortality. What he probably meant is that the variability of extrinsic mortality, which is dominant over intrinsic mortality up to about 30–35 years, may obscure the rise of intrinsic mortality. Our results, however, seem to contradict this hypothesis: the higher the variability of $δ¯x$, the earlier the breakpoint (from 48 in the most remote cohorts to about 32 in the most recent ones). Moreover, within each wave, different countries show remarkably different values of variability but similar (estimated) ages at the onset of mortality acceleration. In short, the decline observed in the onset of mortality acceleration cannot be easily explained as a consequence of the different variability of our series during the phase of latent aging.

This may be checked by regressing the estimated age at onset of mortality acceleration in a given country against its level and variability of mortality during the phase of latent aging (see Table 2). In Table 4, Model 1 indicates that the onset of mortality acceleration depends on the level of mortality but not on the variability of the $δ¯x$series. Model 2 shows that the (estimated) magnitude of the break, another possible source of bias, does not significantly affect the onset of mortality acceleration.

A different explanation for the decline in the onset of mortality acceleration may come from the hidden heterogeneity theory (Vaupel et al. 1979). In a typical population of the ancient regime, survival at age 20 was about 50 %; today, this proportion reaches 98 % to 99 %. Thus, frailer individuals, once destined to die early, now reach age 21 (and older) and enter the analysis. If frailer individuals are characterized by an earlier onset of mortality acceleration, the aggregate mean age at the onset of manifest aging will be lower in more favorable mortality regimes. In short, the hidden heterogeneity theory can explain not only mortality deceleration at older ages (its standard use) but also the earlier onset of mortality acceleration. This hypothesis would be consistent with the results presented by Luder (1993): by focusing on intrinsic mortality and on specific intrinsic causes of death, Luder’s analysis may have selected subpopulations characterized by higher frailty levels and earlier onset of manifest aging.

The hidden heterogeneity theory may contribute to explain the downshift of the inception of mortality acceleration also in a different way: Wrigley-Field (2014) recently showed that in a Gompertz model, even a small level of selection can cause an apparent deceleration of mortality. Similarly, the onset of mortality acceleration can be delayed by a selection process taking place at very young ages. Under this hypothesis, the best estimate of the onset of manifest aging would be that identified in the third wave, not only because the data are more reliable, but also because mortality is very low and the possible effects of selection correspondingly weaker.

The slightly negative values of the rate of aging that we observe during the phase of latent aging (i.e., at young ages) may be due to period effects because in times of intense mortality decline, the cohort hazard functions may (appear to) decelerate. Although no such deceleration is detectable at older ages during the phase of manifest aging (the mean rate of aging remains basically constant in all the three waves examined here), improvements in survival conditions have not been the same at all ages. Diseases that predominantly hit in early adulthood (e.g., maternal mortality, tuberculosis, and syphilis) have gradually disappeared, and better nutrition and better general health conditions may have improved the capacity of the immune system to resist infections (Finch and Crimmis 2004; Fogel 2004a, b).

Whatever the correct explanation, two main conclusions emerge from our analysis: First, from age 21, the human hazard function appears to be formed by (basically) two distinct mortality regimes characterized by significantly different rates of aging (almost flat initially, increasing later). Second, the passage from the former regime to the latter has occurred at younger and younger ages in more recent cohorts.

## Acknowledgment

The research work has been financed by the P.O.R. Sardinia F.S.E. 2007–2013 in the context of research project 13/D3-2 developed at the University of Sassari.

## Notes

1

In this text, demographic aging denotes the phase when individual mortality risks accelerate irreversibly. Instead, senescence or physiological aging indicates the process characterized by the increasing limitation of capabilities of an organism due to the progressive accumulation of molecular damage.

2

“Variation in initial cell number and damage rate will in turn affect the time taken before a threshold for dysfunction is crossed,” according to Kirkwood and Austad (2000:237; emphasis added).

3

Because $k¯0$is the mean age at which manifest (individual) aging begins, some individuals will start aging before or after $k¯0$. Thus, the aggregate (cohort) rate of aging will begin to increase some time before $k¯0$and will continue to do so for a while after that, gradually passing from its initial level (supposedly, 0) to its final value: for instance, β = 0.1 in Vaupel’s (2010) hypothesis (see also Baudisch and Vaupel 2010). This is consistent with the empirical estimates of β made by, among others, Horiuchi (1997), Horiuchi et al. (2003), and Li et al. (2013). In short, $k¯0$is not the age when mortality starts increasing following exactly a Gompertz pattern, with a possibly constant rate of β = 0.1.

4

Noisy series should therefore be treated with special caution.

5

Bai’s methodology works separately on the log-differentiated hazard functions of each cohort: mortality selection can be thought to operate within each cohort, but not at the group level. Note that $b^$ is calculated on the mean evolution of the rate of aging in each cohort and is therefore not affected by the size of cohorts, who have all the same weight in the procedure.

6

The distribution of the error terms, however, can be proved to be approximately normal.

7

Period effects can also be negative: wars and epidemics are the most relevant examples. In this context, we consider only positive period effects, such as medical breakthroughs, because we focus on mortality reduction.

8

In the following, with the exception of Table 3 and Fig. 5, we focus only on women to avoid multiple presentations of parameters, tables, and figures and to minimize the effects of WWI and WWII.

9

The expected magnitude of the break is βS − βL = 0.1 – 0 = 0.1.

10

This pioneering analysis follows Finch et al.’s (1990) and Finch’s (1994) suggestion of focusing on intrinsic mortality. It may, however, suffer from a few biases because (1) it is based on period data; (2) it does not account for heterogeneity; and (3) mortality acceleration is supposed to begin when the first individuals of the (fictitious) cohort start aging.

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