## Abstract

Demographic studies of mortality often emphasize the two ends of the lifespan, focusing on the declining hazard after birth or the increasing risk of death at older ages. We call attention to the intervening phase, when humans are least vulnerable to the force of mortality, and consider its features in both evolutionary and historical perspectives. We define this quiescent phase (Q-phase) formally, estimate its bounds using life tables for Swedish cohorts born between 1800 and 1920, and describe changes in the morphology of the Q-phase. We show that for cohorts aging during Sweden’s demographic and epidemiological transitions, the Q-phase became longer and more pronounced, reflecting the retreat of infections and maternal mortality as key causes of death. These changes revealed an underlying hazard trajectory that remains relatively low and constant during the prime ages for reproduction and investment in both personal capital and relationships with others. Our characterization of the Q-phase highlights it as a unique, dynamic, and historically contingent cohort feature, whose increased visibility was made possible by the rapid pace of survival improvements in the nineteenth and twentieth centuries. This visibility may be reduced or sustained under subsequent demographic regimes.

## Introduction

The demographic literature has traditionally emphasized the age pattern of mortality at the two ends of the life span. Mortality rates in middle and older ages have been modeled by mathematical demographers for nearly two centuries (Gompertz 1825; Thiele and Sprague 1871; Wachter 2003), and reducing infant and child mortality has long been a key public health concern (Bengtsson 2004; Black et al. 2010; Livi-Bacci 2012). However, the shape of the mortality trajectory between childhood and midlife is seldom the focus of analysis. Nonetheless, an increasingly long and well-defined phase of low mortality can be observed between the two ends of the lifespan (see Fig. 1; see the online version of the article to view all the figures in color). This quiescent phase (Q-phase, in our terminology) spans the ages when humans are least vulnerable to the force of mortality.

This interim phase is frequently conceptualized as a period of latency for the development of chronic diseases that manifest in mortality trajectories at older ages (Barker 2004; Beltrán-Sánchez et al. 2012; Crimmins and Finch 2006). The same age range also encompasses the time when individuals invest both in themselves (e.g., via physical, cognitive, and emotional maturation, as well as via the accumulation of human capital through education and work) and in their relationships with others (e.g., via forming families and participating in communities). Given the significance of the Q-phase as a foundation for subsequent life course events, a characterization of its span and features will both enhance our understanding of the human mortality trajectory and provide potential insight into the interacting biological and social factors that shape patterns of population health.

Here, we formally define the Q-phase in human mortality via life table functions and the Siler model, and empirically document its existence using complete mortality data for Swedish cohorts born in 1800–1920. We trace trends in the Q-phase’s beginning and end under our definition and describe consistencies and changes in its characteristics over the course of the dramatic demographic and epidemiological changes that took place as these cohorts lived and died. Although the overlap between the Q-phase and the ages of prime reproductive potential suggests a strong evolutionary rationale for a phase of low, stable mortality at these ages, we show that the Q-phase does not appear in period data and in fact emerged in relatively recent history as a unique feature of human cohorts living through unprecedentedly rapid survival improvements.

## The Age Pattern of Human Mortality

Hazard curves summarize the relationship of mortality with age at the population level. All human populations display both an exponential decrease in mortality with age immediately after birth and an exponential increase in mortality with age in adulthood. In high-longevity populations, widespread improvements in survival have shifted the hazard curve downward over time. However, the basic “U” or “bathtub” shape of the hazard trajectory (see Fig. 1) has remained relatively consistent even as mortality rates declined. This mortality pattern is seen across geographic and cultural boundaries and in a variety of nonhuman species, including fruit flies, beetles, mice, primates, and other mammals (Caughley 1966; Gage 1998; Gurven and Kaplan 2007; Vaupel et al. 1998). The consistency has inspired scholars across the biological and social sciences to model the age patterns of mortality and explain its relationship to reproduction, development, and senescence on a population level (Finch 2012; Wachter 2003; Wachter et al. 2014).

The most widely used model of mortality is the Gompertz (1825) exponential function. The Gompertz model captures the trajectory of increasing mortality in adulthood but not the pattern of mortality in early life or at the oldest ages. William Makeham (1889) expanded the Gompertz model by including a non-age-dependent parameter that captures varying levels of mortality over time, but he retained the focus on adult ages. Advances in modeling mortality at younger ages came primarily from researchers in the fields of biology and ecology. Pearl and Miner (1935) provided a classification of hazard and survival patterns across various nonhuman species, and Siler (1979) later drew on that work to extend the Gompertz-Makeham model via components that describe the exponentially declining mortality hazard during childhood and a flat background component, reflecting the overall level of mortality. This five-parameter model captures the U-shaped pattern of mortality across the lifespan more fully than the Gompertz-Makeham model and its extensions (Engelman et al. 2014), and it does so in a relatively parsimonious fashion. Other models have deemphasized the relative flatness of the hazard trajectory before the onset of increasing mortality in adulthood and instead highlighted the young-adulthood mortality humps (Heligman and Pollard 1980) that occur in roughly the same age range as the Q-phase and disrupt the otherwise smooth trajectory.

More recent scholarship in biodemography has, however, identified an intervening age range of low mortality consistent with our observation of a Q-phase. Finch (2012:11) described the age range from 10 to 40 as a phase of “basal” mortality, when the risk of death is lowest relative to the rest of the lifespan. This basal phase is preceded by a phase of decreasing mortality from birth to puberty (which Finch positioned at age 9) and is followed by a phase (at ages 40–80) of exponentially accelerating mortality that is frequently used to define senescence in population biology and reliability theory. The fourth and final phase is one of plateauing (or decelerating) mortality. Comparisons with other species suggest that the basal or Q-phase of mortality is uniquely long in humans compared with other mammals, whose trajectory of senescence begins at younger ages (Finch 2012; Gurven and Kaplan 2007). In Finch’s description, the cut points between different mortality phases appear relatively static, suggesting that they reflect intrinsic, biologically, and evolutionarily determined characteristics of the mortality hazard. Here, we further examine these points of inflection and investigate the extent to which they are indeed a static or dynamic feature of the human mortality trajectory.

## Mortality Hazards in Evolutionary and Historical Perspectives

The classic evolutionary account of the age pattern of mortality emphasizes the importance of surviving to and through the ages of prime reproductive fitness. This biologically rooted account casts declining health and the increased risk of death characteristic of older ages as the product of the reduced force of natural selection at ages when fertility is unlikely or impossible (Charlesworth 1994; Hamilton 1966; Medawar 1952, 1957; Vaupel 2003).

Lee (2003) influentially argued that a more complete evolutionary explanation of human mortality patterns should account not only for the biology of fertility but also for the long-term economic and social investments required to raise offspring into maturity. These intergenerational transfers of money, time, and energy help explain both why mortality declines with age from birth until some threshold of maturity (Levitis 2011) and why humans survive beyond their reproductive years but with a steadily rising risk of mortality in later life. Thus, continued investments in the development of younger kin yields lower mortality among growing children while also promoting postreproductive survival for parents, grandparents, and other adult relatives. Together, these processes yield an optimal population mortality pattern that is U-shaped (Chu et al. 2008; Robson and Kaplan 2003).

What remains to be elucidated, however, is the exact morphology of that U shape. How might we characterize the age when mortality stops declining in early life or starts to rise steeply in midlife? How might we characterize the duration through which mortality remains relatively flat? From an evolutionary perspective, it seems intuitive that the low mortality risk characteristic of the basal or Q-phase would coincide with the ages of maximum reproductive potential. The low levels of mortality right before, during, and after puberty allow increased exposure to mating and facilitate higher potential fertility. Roughly the same age range is also the prime time for investment in one’s own human capital (e.g., via the accumulation of skills and work experience) and in social relationships with partners, family members, children, and friends that can promote both personal and familial well-being.

In contrast to the emphasis on a smooth U-shaped trajectory, some researchers have also offered evolutionary explanations for the persistence of a mortality hump—the increase in mortality in adolescence and young adulthood that is particularly apparent in data for men (Goldstein 2011) and is most explicitly depicted by the Heligman and Pollard (1980) model. The pronounced hump in male mortality trajectories is traditionally attributed to injuries related to risk-taking behaviors, which are in turn thought to be biologically triggered via a testosterone surge in puberty (an attribute common to multiple species). This hormonal change allows males to build muscle mass, but it is also hypothesized to render them more likely to engage in fighting and other potentially violent activities designed to establish their status among males and increase their attractiveness to females (Bribiescas 2001; Gage 1998).

However, here, too, an emphasis on biology alone would overlook the key influence of social and historical factors on observed demographic phenomena. In particular, social norms of masculinity are also likely to be a powerful influence on young men’s risk-taking behaviors, and social conventions have long tended to concentrate on men—particularly young men—in activities that carry relatively high risks of injury and death, including both civilian occupations (e.g., mining, logging, construction, agriculture, and other jobs requiring the operation of heavy machinery) and enlistment in the military or police or membership in other organized groups with regular exposure to combative situations. A consideration of female mortality patterns over time similarly highlights the impact of historical changes in public health knowledge and medical practices surrounding childbirth on the notable reductions in maternal deaths (Loudon 1992).

In contrast to the expansive horizon of evolutionary biology, consideration of the social determinants of mortality and longevity points to key contextual influence on human mortality patterns that have changed dramatically in relatively recent history. Over the past two centuries, improvements in sanitary infrastructures, increases in the availability and accessibility of nutritious food, advances in medical (especially obstetric) practices, rising educational attainment, the growing prevalence of effective health behaviors, economic development on local and national scales, and the implementation of effective environmental safety regulations have all contributed to survival improvements at all ages (Livi-Bacci 2012; Riley 2001). These secular changes in the physical and social environment have been hypothesized to prompt earlier sexual maturation for males and females over a rather short period (Goldstein 2011) and may have similarly influenced the onset of senescence. Taken together, these socially induced yet biologically manifested changes may have changed the specific parameterization of the U-shaped mortality trajectory. Despite a likely strong evolutionarily influenced and biologically determined pattern of human mortality, changes in environmental, social, and behavioral contexts may thus have an important role in determining the characteristics of the mortality hazard’s distinct phases.

## Data and Methods

### Cohort Versus Period Life Tables

The Human Mortality Database (HMD 2015) contains detailed time series of mortality data and life tables for populations with virtually complete vital registration and census data. For parsimony, we focus our analysis on male and female life tables from Sweden. However, our supplementary analyses found similar patterns of change in at least 14 other high-longevity countries with data compiled in the HMD (results available upon request). Although the insights gained from analyzing the slow and steady process of mortality improvement in Sweden during the nineteenth and twentieth centuries may not be generalizable to populations that have not enjoyed a similar pattern of rising life expectancy in the context of economic growth and an expanding welfare regime, Sweden’s high-quality and exceptionally long-spanning data provide an important and useful case study and a baseline for further comparisons.

The age-specific mortality probabilities compiled in life tables are well suited for exploring the existence and morphology of a Q-phase. However, the results of such an analysis depend considerably on whether the data summarize the mortality experiences of real cohorts or synthetic ones. Cohort life tables reflect the mortality experiences of individuals born in the same year who move together through successive periods. The longitudinal data they provide is highly valuable for addressing questions regarding mortality risks that unfold along the actual life course of a cohort. However, compiling a cohort life table requires approximately a century—the time until the large majority of birth cohort members have died. Period life tables, on the other hand, comprise cross-sectional data that are more easily collected and thus more commonly analyzed. Period life tables document the mortality experiences of a synthetic cohort—a mixture of individuals of different ages observed at a single point in time. Although convenient and useful for summarizing mortality conditions at a particular time (Wilmoth 1989), period data do not represent true longitudinal survival experiences and may yield inaccurate impressions of underlying mortality changes (Guillot 2011). With some assumptions, translation between period and cohort data is possible (Goldstein and Wachter 2006; Jones 1961; Ryder 1964). But do the two types of life tables offer equivalent depictions of the age trajectory of mortality, including the Q-phase?

Figure 2 illustrates a sharp difference between the two data sources: the hazard trajectory constructed from the 1915 period life table suggests a linear increase of the mortality hazard from early adolescence into adulthood, and a prominent hump, consistent with the hazard pattern described by Medawar (1952, 1955), Heligman and Pollard (1980), and Austad and Partridge (1997). However, this pattern does not match the trajectories of mortality among the cohorts (born in 1875–1905 and aged 10–40 in 1915) that make up the relevant age range in the period data. Each of these cohorts instead displays the relatively flat Q-phase (with some quasi-quadratic bumps) that has become increasingly noticeable for cohorts with successively better survival conditions.

The linear increase in the period mortality hazard is mathematically reconcilable with a Q-phase in the cohort curves if the constant cohort hazard rates are decreasing (Jones 1961), as they were during the dramatic and relatively rapid demographic and epidemiological transitions of the past century. If we assume, following Finch (2012), that the cohort mortality hazards are constant between ages 10 and 40 and that this constant hazard decreases for successive cohorts, then the corresponding period mortality hazard will encompass the experience of four distinct 10-year cohorts. In this case, the period curve shows a linearly increasing trend, despite the fact that each of the cohort curves are in fact quiescent, or relatively flat between the ages of 10 and 40.

This relationship between period and cohort data suggests that the Q-phase is a unique feature of the longitudinal experience of real cohorts that is, however, not replicated in the cross-sectional period life tables (Bongaarts and Feeney 2002; Jones 1961). Although the cohort data reflect successive cohorts’ improved survival, the mixing of cohorts in period data creates a different and potentially misleading impression of the age pattern of mortality under conditions of improving survival. Given these findings, as well as prior research suggesting that cohort measures are more useful for understanding the dynamics of mortality change (Guillot 2003, 2011), our analyses rely on data for cohorts born between 1800 and 1920 to best reflect the underlying age pattern of human mortality during a time of rising longevity.

The HMD presents cohort life tables for extinct cohorts—those whose members are deceased by the end of the observation period—and some almost-extinct cohorts, where death rates for the very high ages not yet observed are based on the average experience of previous cohorts. The most recent complete cohort life tables available for Sweden—the country with the longest and most complete set of vital statistic records—are from 1920.

### The Quiescent Phase: A Formal Definition

In life table notation, the hazard or force of mortality at age x is denoted as μ(x), and the number of individuals or proportion of the birth cohort surviving to age x is given by l(x). The two functions completely determine each other, given that the mortality hazard may be calculated as the relative derivative of the life table survival function:
$μx=−dlnlxdx.$
1
Assuming a constant μ(x) over one-year age intervals, we can use Eq. (1) along with d(x), the number of life table deaths at age x, to obtain the following key relationship between μ(x) and q(x), which is the conditional probability of dying in the age interval [x, x + 1) having survived up to age x:
$qx=lx−lx+1lx=dxlx=1−exp−μx.$
2
Having defined the fundamental entities of a life table, we now turn to the quiescent (Q) phase. Theoretically, the Q-phase comprises the age range when humans are least vulnerable to the force of mortality. It typically encompasses the age interval between puberty and the onset of senescence. Mathematically, we define the Q-phase as the smallest interval of age, [a, b], such that for any age x < a, the mortality hazard has a negative slope (i.e., it decreases monotonically with age) and for any age x > b, the mortality hazard has a positive slope (i.e., it increases monotonically with age). In other words, the Q-phase is the age interval that lies between the monotonic decline of the early childhood mortality hazard and the monotonic increase of the mortality hazard in the senescent ages. Formally, we write this as follows:
$Qε=smallest intervala≤x≤b:gμx≤−ε1∀x∈0aandgμx≥ε2∀x∈b∞,$
3
where a and b are inflection points marking the endpoints of the Q-phase, and $gμx=ddxln[μx]$ is the derivative of the log of the hazard function. ε1 and ε2 are positive real numbers that are tolerances in the magnitude of the derivative of the log hazard, chosen to be small relative to the magnitude of the slopes of the childhood and senescence phases of mortality hazard. Our definition intends the derivative of the log hazard to be within (−ε12), although the observed derivative may be outside these intrinsic bounds because of fluctuations or humps due to extrinsic causes of death.

This definition is deliberately broad and flexible, given that it neither presumes the existence of a Q-phase nor imposes assumptions about the characteristics of the mortality hazard within the Q-phase age range. Instead, it describes the behavior of the mortality hazard outside the Q-phase. This definition allows the hazard to take a number of possible shapes during the Q-phase age range, including a flat hazard (see, e.g., Finch 2012), quasi-quadratic bumps (see, e.g., Goldstein 2011; Heligman and Pollard 1980), or other types of irregular fluctuations (e.g., spikes due to epidemics or wars, but not necessarily a failure of internal homeostasis). Equation (3) also allows the possibility that the Q-phase age range is an empty set, as would be the case if senescent mortality immediately followed childhood mortality as hypothesized by Medawar (1952) and Austad and Partridge (1997). This flexible definition allows us to explore the existence of a distinct phase of mortality between childhood and senescence without making assumptions about the form it may take.

### The Siler Model

The general aforementioned definition of the Q-phase can be operationalized in a number of ways, including parametric models, an examination of the first derivative of the empirical hazard function for inflection points, or segmented regressions with two data-driven break points. The latter two methods require either a visual (“eyeball”) identification of the Q-phase boundaries (for the derivative-based approach) or numerous additional constraints to maintain consistency with the definition of the Q-phase (for the segmented approach). The Siler (1979) model offers a compelling and easily reproducible alternative: its parameters can be used to precisely calculate the beginning and endpoints of the Q-phase based directly on the definition in Eq. (3).

We chose the Siler model, rather than the more commonly used Gompertz (1825) model, because the latter includes information only on adult mortality risks, whereas the Siler model includes three competing but noninteracting components that characterize mortality across the full age spectrum. Other models cover the full lifespan (Heligman and Pollard 1980; Thiele and Sprague 1871), but the Siler model is simpler and less encumbered by mathematical identifiability problems (Booth and Tickle 2008). Compared with other models, the Siler model has been favored by researchers interested in characterizing mortality across all ages and has been shown to provide an equally good or better fit to mortality data for humans and other primates (Engelman et al. 2014; Finch 2012; Gage and Dyke 1986; Gage and Mode 1993; Gurven and Kaplan 2007).

Siler argued that living beings are exposed to three types of hazards throughout the lifespan: (1) a hazard that decreases from birth onward as the animal adjusts to its environment, likely as a result of maturation; (2) a hazard that increases with age, reflecting the growing risk of death as a result of senescence; and (3) a constant hazard, reflecting a set of risks present in the “background” and to which susceptibility does not vary with age. Combining the three components, Siler’s additive model produces a hazard trajectory that decreases in early life, increases monotonically at older ages, and remains flat in the interim:
$μx=f1x+f2x+f3x=eα1−β1x+eα2+β2x+eα3,$
4
where the α intercepts describe the hazard levels at birth, and the β slopes represent fixed rates of change in the mortality hazard over age. The first term on the right side of the Siler model represents the exponentially declining mortality hazard during childhood, with the α1 parameter representing the magnitude of “immature” mortality at the moment of birth and β1 representing the rate at which this immature mortality decreases during the childhood developmental phase. The first term describes the increased ability of a child’s immune and physiological systems to adjust to environmental challenges with age. The second Siler term (which is the conventional Gompertz model) represents the exponentially increasing mortality hazard in older adulthood or the senescent phase, with α2 representing the initial magnitude of “senescent” mortality at birth and β2 representing the rate of increase in senescent mortality with age. Finally, the third term is the background mortality component, with α3 reflecting the background (non-age-dependent) level of mortality. The three components additively create a U-shaped hazard function for mortality across the lifespan, and all parameters are measured in units of mortality hazards.

The Siler model does not capture the hump in early adult mortality frequently attributed to accidents (particularly for males) and maternal causes (for females). Notably, the magnitude of these conditions has declined substantially as improved sanitation, better-informed obstetric practices, higher standards of living, and well-enforced safety regulations have become widespread in high-longevity societies (Livi-Bacci 2012; Riley 2001). Thus, while the absence of the hump may be viewed as a potential shortcoming of the Siler model in fitting some empirical data, it may also be an advantageous feature in terms of providing a theoretically informed focus on mortality from causes that have not thus far succumbed to social or medical intervention.

The Siler model parameter estimates can be used to calculate the two end points of the Q-phase as the inflection points of the smooth Siler hazard function. However, because of the Siler model’s functional form, its derivative never equals 0 to indicate a constant hazard. Nonetheless, using the aforementioned definition for the Q-phase, we can approximate the endpoints by setting the magnitude of the derivatives of the developmental and senescent phase hazard functions to be much smaller than the background hazard. We operationalize this as follows:
$df1x=adx=k1f3x=a$
5
$df2x=bdx=k2f3x=b,$
6
where k1 and k2 are small, positive scale factors.
The left endpoint point of the Q-phase, a, represents the age marking the end of infant mortality. Based on Eqs. (5) and (4), we calculate this age as
$a=α̂1−α̂3+logβ̂1−logk1/β̂1.$
7
The right end point of the Q-phase, b, represents the age marking the beginning of senescence, as defined by a sharp increase in the mortality hazard slope. It can be calculated from Eqs. (6) and (4) as follows:
$b=α̂3−α̂2−logβ̂2+logk2/β̂2.$
8

We conducted sensitivity analyses to examine how the Q-phase endpoints [a, b] would change with different scale factors. While the specific endpoints and the length of the Q-phase change somewhat under different specifications, the qualitative characteristics of the Q-phase across cohorts are consistent. In defining the Q-phase in Eq. (3), we choose ε1 = ε2 = 0.01. This corresponds approximately to k1 = 1 / 100 and k2 = 1 / 50, in Eqs. (5) and (6), which define a and b, respectively. Here, k1< k2 because the slope of the mortality hazard is greater in magnitude during the childhood phase than during the senescent phase.

### Parameter Estimation

Life tables are unique in containing complete information (in terms of person-years of exposure and number of events) on the event history for the population of interest, rendering the direct computation and estimation of continuous hazard functions more tractable relative to other survival data.

We obtain the Siler model parameters in two ways: (1) directly fitting the Siler hazard to the empirically observed hazards from life tables via nonlinear least squares (NLS) estimation, and (2) maximizing the likelihood function for the probability of death—that is, via maximum likelihood estimation (MLE). The two approaches for parameter estimation are asymptotically equivalent, but empirically the parameter estimates and their statistical properties (e.g., standard errors) might differ slightly because of the finite sample size (especially for the oldest ages) and to the discretization of continuous age into one-year intervals.

We used the optimx package (Nash and Varadhan 2011) in R to obtain both the NLS and MLE estimates. We found that the NLS estimation approach performed slightly better than the MLE approach in terms of (1) convergence of iterative algorithms for parameter estimation and (2) closeness of the fitted values to the empirically observed values. Thus, the results we report rely on the NLS estimates. (See the appendix for more information on estimation with optimx and a comparison of the NLS and MLE results.) We also obtained robust standard errors for the parameter estimates using a sandwich variance estimator (White 1980; Zeileis 2006). These robust estimators provide a consistent estimate of the standard errors for parameters even when the underlying distribution of the errors is not Gaussian.

We first investigate the Q-phase using smoothed mortality data for female and male Swedish cohorts born in 1800–1920, with no parametric model assumptions. We then turn to the Siler model and calculate the beginning and endpoints of the Q-phase based directly on the aforementioned definitions.

## Results

Our analyses reveal the emergence of a distinct quiescent phase in the mortality trajectories of men and women. First, we look solely at smoothed mortality data, using as many cohorts as possible from the Swedish data and imposing no parametric assumptions. Figure 3 presents separate plots of empirical hazard trajectories for Swedish women and men, with each trajectory representing an average across 15 birth cohorts. Thus, for example, the 1860 trajectory averages the hazards of all birth cohorts from 1860 to 1874. By borrowing information across multiple cohorts, these trajectories smooth out the random variation in individual cohort hazards, revealing a clearer systematic pattern of change over age and time.

For the earliest documented male cohorts, born in the eighteenth century, the risk of mortality reaches its lowest point sometime in early adolescence and immediately begins a gradual increase. The hump appears and becomes increasingly prominent for cohorts born during the nineteenth century. For these cohorts, mortality declines in childhood, rises sharply in adolescence, and then levels off somewhat before rising sharply again after age 50. For the most recent cohorts, those born in the latter part of the nineteenth and early twentieth centuries, the young adulthood hump appears more like a divergence from the phase of lowest mortality. Mortality for these cohorts declines markedly in childhood, rises in adolescence, but declines again between ages 20 and 40, almost matching the lowest level of pre-hump mortality before beginning its exponential increase in the fifth decade of life.

For female cohorts, the emergence of the hump is gradual. The earliest cohorts display a small but steady increase in mortality after the low point of childhood mortality (sometime in early adolescence), followed by a more steep increase in mortality afterward. For the most recent set of cohorts, however, the curve begins to flatten between early adolescence and roughly age 40. An increasingly symmetric hump emerges, with mortality increasing in adolescence and subsequently declining between ages 20 and 40. This small hump, like its more pronounced male counterpart, looks increasingly like a divergence from an underlying flat Q-phase.

For both Swedish men and women, then, a postpuberty Q-phase emerges in the hazard trajectories of cohorts born during the latter half of the nineteenth century and becomes more pronounced thereafter. The hump near age 20 is more apparent for later cohorts of men as well as women because the overall risk of mortality between ages 20 and 40 is lower than in previous cohorts.

A similarly emergent Q-phase can be discerned in cohort trajectories for men and women in 14 other high-longevity countries with cohort data compiled in the HMD (figures available upon request). For parsimony, subsequent analyses focus on data for cohorts born after 1800 to more closely examine the pattern of mortality change during the demographic transition. Trajectories fitted to single-year cohort data using the Siler model (Fig. 4) agree with the prior descriptive analysis and likewise indicate the presence of a relatively flat and increasingly long Q-phase for both women and men. (As noted earlier, the Siler model does not allow for the hump, which is both an advantage and limitation.)

Figure 5 shows the pattern of change in the inflection points defining the Q-phase for women and men in successive cohorts. The figure includes both the empirical calculation of the inflection points and a loess-smoothed trend line. The starting age of the Q-phase (bottom) has fluctuated somewhat over time for both women and men, showing a general decline from approximately age 15 for cohorts born after 1800. This starting point rose again for cohorts both in the 1830s and 1840s, stabilized for a while, and decreased again for cohorts born in the latter part of the nineteenth century.

At the other end of the Q-phase, the age marking the beginning of the exponential or senescent mortality phase increased across cohorts experiencing improved survival. Women saw a gradual increase of the age marking the onset of senescence from the mid-20s for cohorts born around 1800 to the late 30s for cohorts born a century later. Men experienced a dramatic increase of this age from the late teens to the early 30s between 1800 and 1840. This inflection point stabilized for several decades thereafter. For the most recent cohorts (born in 1900–1920) of women and men alike, we detect a small increase at the age of Q-phase onset and a notable decline in the age of senescence onset.

As a result of these dynamics, the length of the Q-phase has not been stable for successive cohorts. Figure 6 tracks the changing length of the Q-phase (in years) for Swedish male and female cohorts, highlighting the dramatic increases during the first part of the nineteenth century for women (12.36 years for those born in 1800 to 23.12 years for those born in 1837) and men (5.03 years for those born in 1800 to 22.31 years for those born in 1839). This was followed by a decline for cohorts born in the 1840s and 1850s, and then gradual increase (men) or stability followed by a more rapid increase (women) for cohorts born in the latter part of the nineteenth century. The length of the Q-phase then declined again for men and women born between 1900 and 1920. Women have a consistently longer Q-phase over time, but men have experienced a more dramatic increase in the Q-phase length since the beginning of the nineteenth century. The gap between women and men narrowed during the first half of the nineteenth century but began and continued to widen through the end of the nineteenth and beginning of the twentieth century.

## Discussion

We introduced the term quiescent phase for the age range when humans are least vulnerable to the force of mortality and developed a framework for characterizing its features. We also illustrated the utility of this approach by examining the changing morphology of the Q-phase using cohort data from Sweden.

The cohorts we follow in this study came of age during the nineteenth and twentieth centuries and witnessed varied stages of Sweden’s dramatic economic and social transformation. The earlier cohorts lived through times of agricultural restructuring and early industrialization, whereas later cohorts saw the rise of urbanization and rapid economic growth coupled with the emergence of Sweden’s modern education, health, and welfare programs (Schön and Schubert 2010). These historic changes coincided with similarly dramatic declines in child and infant mortality and, subsequently, gradual gains in adult survival. Although the absolute magnitudes of the mortality hazards in early life and adulthood have declined, their trajectories have remained notably consistent across successive cohorts. In contrast, although an intervening Q-phase of steady, low mortality appears to be an important characteristic of real Swedish cohorts, its morphology has shifted as mortality levels at all ages have decreased. Over time, the Q-phase’s starting point has occurred at younger ages, and the onset of senescence has shifted to older ages. As a result, the length of the Q-phase has increased substantially for successive Swedish cohorts.

However, the trajectory of survival improvements was neither constant nor smooth. Substantial declines in Q-phase length were observed for Swedish cohorts born in the 1840s and 1850s as well as for cohorts born in the first decades of the twentieth century. The first decline in the length of the Q-phase was driven by an increase in the age marking the end of childhood mortality. Although child mortality began to drop in the late eighteenth century, the mid-nineteenth century saw a temporary setback with rising mortality from airborne infectious diseases, especially scarlet fever (Fridlizius 1989; Kermack et al. 2001). A more sustained decline in child mortality and infectious disease mortality throughout the lifespan began in the mid-nineteenth century as standards of living for broad segments of Sweden’s rural and urban populations saw marked improvements (Bengtsson and Lindström 2000).

The second decline in the Swedish Q-phase length was the result of an epidemiological reversal at the other end of the Q-phase, the age marking the onset of adult mortality. The cohorts born in the early twentieth century benefited from the rapid declines in early life mortality but came up against the mortality plateau observed from the 1960s up through 1986 during their adult years (Ouellette et al. 2014) as progress against cancer and cardiovascular diseases stalled. The high prevalence of tobacco smoking for cohorts born in the early twentieth century is one commonly noted explanation for this divergence from the trend of improved survival (Fridlizius 1989). Researchers have also found that the same Swedish cohorts were the first to experience substantial socioeconomic disparities in adult mortality (Bengtsson and Dribe 2011). Reasons for the emergence of the social gradient within these cohorts are not fully understood, but changes in work environments and employees’ reduced control over work processes in occupational sectors such as manufacturing and farming may be implicated in the unequal burdens of chronic disease morbidity and mortality (Bengtsson and Dribe 2011; Marmot et al. 1997; Schön and Schubert 2010).

The Siler model offers one intuitive and consistent way of modeling mortality across the life course, and it provides us with plausible estimates of the Q-phase’s beginning and endpoints. But the identification of a Q-phase is not contingent on the validity of the parametric Siler model for modeling mortality hazards. Indeed, the smoothed, nonparametric estimates of mortality hazards in Fig. 3 provide important evidence in support of the emergence of a distinctive Q-phase. By modeling the mortality hazards of cohorts over the developmental, quiescent, and senescent phases of the life course with the Siler model, however, we gain additional insights into how varying secular forces interact with more stable features of human biology.

In particular, the comparison of hazard trajectories across genders highlights one aspect of the complex interaction among genetic, physiological, behavioral, and social causes of mortality (Kruger and Nesse 2006). The force of mortality appears relatively quiescent for several decades following puberty. Men have a more pronounced hump in mortality after puberty relative to women but also a flatter presenescent hazard curve. For women, the historical pattern of gradually increasing mortality between adolescence and age 40 (likely due to mortality associated with childbearing) has given way to declining mortality rates during the third and fourth decades of life. These changes, along with the symmetrical and transitory nature of the hump in more recent cohorts, are consistent with an interpretation of the young adulthood hump as representing deaths due to conditions (such as accidents, wars, pestilence, and poor obstetric practices) more influenced by external, period-related factors than by biological characteristics. Thus, as risks associated with injuries (for men) and childbirth (for women) were reduced, the hazard trajectories seem to have approached a flatter Q-phase that may more closely reflect an optimal potential of human organisms (Carnes and Olshansky 1997; Chu et al. 2008), regardless of whether that potential is realized.

From an evolutionary perspective, the relative flatness of the mortality hazard during the Q-phase for women as well as men may minimize the endogenous threats to survival during a cohort’s prime years of reproduction, caregiving, and investment in physical, financial, and social capital. Nonetheless, the dramatic changes over the course of the demographic and epidemiological transitions have served to make the Q-phase more visible and pronounced than it was prior to these social and environmental changes. The pattern we observe in the morphology of the Q-phase clearly reflects the historical retreat of specific causes of death—those from external causes that do not seem linked to the underlying biology of senescence. Deviations from the trend toward a longer and flatter Q-phase are likewise consistent with periods when historical survival improvements have stalled, thus underscoring the interplay of biology and historical changes in the physical and social environment in determining patterns of population health.

The difficulty of separating endogenous from external influences on mortality has been acknowledged for as long as explanations for population mortality patterns have been offered. In popularizing the term “senescence” to denote mortality increases due to declining physiological function with age, Medawar (1955:12) described it as “a change which is of innate origin, which is built into the developmental structure of an animal, and which could take place even under the most ‘favourable’ conditions it is possible to envisage. The force of mortality does indeed measure this inborn decline, but the difficulty is that it must also measure another sort of decline, of purely contingent origin, as well.” Increased risk of mortality, however, is only one measure of senescence. Other measures drawing on physical, cognitive, and social domains of function are necessary for a fuller characterization of aging—for individuals as well as populations.

The recognition that social, psychological, and economic factors operate in conjunction with biological ones is also increasingly integrated into evolutionary explanations of mortality and reproduction (Lee 2003). In trying to interpret the Q-phase and the implications that its changing boundaries have for our understanding of human development, maturation, and senescence, we must keep in mind the degree to which the biological and social factors that influence mortality are deeply intertwined—not only because they are difficult to disentangle empirically but also because social processes influence the development and manifestation of biological factors, while biological factors likewise affect social behaviors and structures over time.

Our analyses rely on cohort data to best reflect the complete age pattern of human mortality. Improved survival at any specific age clearly points to the progress we have made in health and living standards over the past few centuries. It does not, however, imply that our intrinsic biological fitness has improved over the relatively short span of two centuries. Notably, neither period nor cohort life tables directly support a biologically or more broadly evolutionarily based interpretation of age-related changes in individual mortality risks. Both types of aggregate data are subject to the impact of mortality selection and compositional changes that, over time, may yield a population trajectory that is not representative of the longitudinal experiences of individuals (Vaupel and Yashin 1985). Although period life tables are widely used for mortality analyses that highlight the imprint of historical changes on multiple age groups at once (Ouellette et al. 2014; Wilmoth 1989), they comprise a heterogeneous mixture of cohorts at a point in time and thus may yield unreliable answers to questions about how the risk of mortality unfolds with age (Guillot 2011). Although cohort life tables comprise a similarly heterogeneous mixture of periods, these periods constitute the actual historical contexts through which the study cohorts have aged.

As shown in this article, during a time of rapid survival improvements, cohort life tables reflect the emergence of a Q-phase, but period life tables do not capture this feature of the mortality trajectory (see also Jones 1961). The dynamic cohort trends underscore the extent to which the Q-phase features are historically contingent rather than static and primarily determined by the biology of human development. Cohort analysis allows us to explore true aggregate mortality experiences and surmise about underlying optimal trajectories, but it is the historic demographic and epidemiological transitions of the early twentieth century that have made it possible to observe the increased flattening of the Q-phase. In the absence of such dramatic changes compressed into a relatively short time, the cohort trajectories might well have looked less flat and more similar to the period trajectories.

We postulate that advances in public health; medicine; and, more broadly, standards of living have ameliorated many of the external causes of mortality in early- and midlife, offering us a glimpse into the optimal trajectory envisioned by Medawar (1955)—that which characterizes cohort mortality hazards under beneficial environmental, social, and behavioral conditions. This optimal cohort mortality hazard appears to have a Siler-like morphology, with cohort members developing toward and attaining their maximum physical potential at puberty, operating at their prime during the Q-phase, and then experiencing increased debilitation during the senescent phase. The Q-phase’s pattern of increased length for successive cohorts—a result of lower childhood mortality and delayed adult mortality—seemed to reverse for the most recent cohorts in our analysis, raising questions about likely trends for cohorts born later in the twentieth century.

Multiple scenarios seem plausible. For one, continued reductions in mortality from external causes could further flatten the mortality hazard during the Q-phase age range, and improvements in midlife survival could reverse the pattern observed for the 1900–1920 cohorts and resume a lengthening of the Q-phase. Given the relatively low rates of mortality at these ages, the impact of such changes on mortality may be modest, but they would increase the visibility of the Q-phase. However, if survival improvements for cohorts born later in the twentieth and twenty-first centuries are either absent, very small, or concentrated solely at older ages, the Q-phase length may remain constant. Alternatively, different rates of survival improvement throughout the Q-phase age range or a scenario of increasing mortality in young adulthood may cause cohort mortality trajectories to increasingly resemble their more peaked period counterparts.

Future research should examine the empirical evidence for these scenarios. Further investigations could also consider whether differential exposures to social and economic conditions across geographic settings and among national subpopulations differentiated by race/ethnicity, class, and other stratifying factors have an impact on the characteristics of the Q-phase as a population health indicator. Although opinions about the existence and morphology of the Q-phase may continue to vary, further investigation and discussion of the ages of least vulnerability will contribute to a richer understanding of the human mortality hazard trajectory.

Contrary to earlier placements of the onset of senescence around puberty, our analysis suggests that the end of the Q-phase corresponds more closely to the age range when the strictly biological reproductive program of evolution has been played out in the individual (Hawkes et al. 1998). Although demographers going back to Gompertz have recognized ages 30–40 as the starting point for analyses of adult mortality (Beltrán-Sánchez et al. 2012; Finch 2012; Olshansky and Carnes 1997), standard epidemiological and social studies of aging often choose a higher age cutoff in deference to convention or administratively constructed categories. To more fully understand health and mortality across the life course, we need to build in more leeway for understanding the factors that define the direction and path of mortality and health trajectories in midlife. In turn, a clearer delineation of the Q-phase might ultimately help to inform future studies using individual-level data (e.g., via biomarkers) and provide a framework for generating hypotheses about physiological dynamics associated with the processes of development and aging. A better understanding of individual-level patterns may also, in turn, help explain differences and changes in the Q-phase across populations and over time.

## Acknowledgments

Michal Engelman began work on this project while supported by a postdoctoral fellowship in the Epidemiology and Biostatistics of Aging at the Johns Hopkins Center on Aging and Health (NIA T32AG000247). She is now supported by the Center for Demography and Ecology (NICHD R24 HD047873) and Center for Demography of Health and Aging (NIA P30 AG17266) at the University of Wisconsin–Madison. Christopher L. Seplaki was supported by Mentored Research Scientist Development Award number K01AG031332 from the National Institute on Aging. Ravi Varadhan was a Brookdale Leadership in Aging Fellow. This work was also funded in part by the Older Americans’ Independence Center (OAIC) at the Johns Hopkins University. Previous versions of this article were presented at meetings of the Population Association of America and at the Berkeley Formal Demography Workshop. We thank John Wilmoth, Ron Lee, Joshua Goldstein, and Joshua Garoon for helpful comments and discussion. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute on Aging or the National Institutes of Health.

### Appendix: Parameter Estimation Methods

Although the hazard function and the survival probability function are mathematically equivalent—that is, there is a strict one-to-one correspondence between them—hazard functions are typically difficult to estimate from empirical data because they tend to be very noisy. Thus, in most survival analysis contexts, the analyst works with the survival probability model that corresponds to the hazard in order to estimate the parameters by maximizing the likelihood of the observed events. However, life tables are unique in containing relatively complete information (in terms of person-years of exposure and number of events) on the event history for the population of interest. This makes the direct computation and estimation of continuous hazard functions more tractable relative to other analytic contexts.

We can obtain the Siler model parameters in two different ways: (1) directly fitting the Siler hazard to the empirically observed hazards from life tables via nonlinear least squares (NLS) estimation, and (2) maximizing the likelihood function for the probability of death, an approach known as maximum likelihood estimation (MLE).

In the direct approach, we obtain the parameters by minimizing the sum of squared deviations of model-predicted and observed log of survival probability: that is, we minimize (Σxn[log(q(xn)) log(1 – eμ(xn))]2). This is the NLS approach. In the second approach, we write the binomial likelihood that corresponds to d(x) deaths of a total of l(x) individuals, where the probability of a single death is q(x) = 1 exp(–μ(x)). This likelihood is a function of the parameters involved in μ(x). We maximize this likelihood for the observed life table data (d(xn) and l(xn)) to obtain the Siler model parameters. This is MLE.

The two approaches for parameter estimation are asymptotically equivalent. Empirically, however, the parameter estimates and standard errors generated by each approach might differ slightly because of the finite sample size (especially for the oldest ages) and the discretization of continuous age into one-year intervals. Although maximum likelihood may intuitively be expected to yield parameter estimates with the best model fit based on the maximum likelihood estimator’s asymptotic properties, in specific (finite) samples representing whole populations where accounting for sampling is not a concern, the quality of parameter estimates from a directly fitted hazard may in fact be superior.

optimx (Nash and Varadhan 2011) is a unique tool that integrates several optimization algorithms available in R. More than a dozen algorithms including several Newton-type, gradient, and derivative-free algorithms are implemented under one simple call. optimx organizes the results of multiple algorithms according to the values of the objective function. The results are provided in a manner that facilitates the comparison of the performance of different algorithms in terms of objective function values, computational effort, and the quality of solution (i.e., whether the solution satisfies first- and second-order Kuhn-Karrush-Tucker conditions for local optimum). We employed optimx for both the NLS and MLE approaches.

We found that the NLS estimation approach is slightly better than the MLE approach in terms of better convergence of iterative algorithms for parameter estimation. Although the MLE estimates provided a better fit of the survival probability, the NLS approach provides a better fit to the observed mortality hazard (see Fig. 7), as determined by the closeness of the fitted values to the empirically observed values. Results and figure references in the main text therefore rely on the NLS estimates.

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