Abstract

What is the average lifespan in a stationary population viewed at a single moment in time? Even though periods and cohorts are identical in a stationary population, we show that the answer to this question is not life expectancy but a length-biased version of life expectancy. That is, the distribution of lifespans of the people alive at a single moment is a self-weighted distribution of cohort lifespans, such that longer lifespans have proportionally greater representation. One implication is that if death rates are unchanging, the average lifespan of the current population always exceeds period life expectancy. This result connects stationary population lifespan measures to a well-developed body of statistical results; provides new intuition for established demographic results; generates new insights into the relationship between periods, cohorts, and prevalent cohorts; and offers a framework for thinking about mortality selection more broadly than the concept of demographic frailty.

Introduction

How long, on average, will currently living people live? In a stationary population, this “average lifespan of the living” (ALL) can be interpreted as a length-biased life expectancy: a transformation of life expectancy in which lifespans are reweighted by their own values. We will show that this fact has several important implications—for example, that period life expectancy is always less than the average lifespan of the people alive in that period—and provides intuition for a number of previous results about stationary populations.

Length bias, also known as size bias or a self-weighted distribution, occurs when units are observed in proportion to their size (e.g., Correa and Wolfson 1999; Lann and Falk 2005; Patil 2005). Demographers are likely to be the most familiar with length-biased sampling in the context of Preston's (1976) analysis of the family sizes of children versus women. A survey that asks gestational parents how many children they have will receive one report per parent; by contrast, a survey that asks children how many children are in their family will receive one report from each one-child family, six reports from each six-child family, and so forth (Bytheway 1974; Patil and Rao 1978; Preston 1976; Ruggles 2012; Song 2021; Song and Mare 2015). Similarly, a length-biased lifespan distribution is a distribution of lifespans that is weighted by lifespan, such that the longer-lived are more likely to be observed in proportion to their longer lifespans. We will show that in a stationary population, the ALL is such a length-biased lifespan. In what follows, we formally prove this result; provide an intuitive explanation that also clarifies relationships between the lifespans of periods, cohorts, and prevalent cohorts; offer several empirical illustrations; and discuss how this result provides intuitions and implications for other demographic results and frameworks.

Proof

Background: Length Bias

Consider a nonnegative real-valued random variable, Y0—lifespans or family sizes, for example—with probability density function f(y) and a mean of E[Y]>0. The length-biased distribution arising from f(y) has the following density:
flb(y)=yf(y)E[Y].
(1)
The mean of this length-biased distribution is
Elb[Y]=E[Y2]E[Y].
(2)
This length-biased mean can be reexpressed by using the definition of the variance, σY2=E[Y2]E[Y]2, as a function of the squared coefficient of variation, cvY2=σY2E[Y]2:
Elb[Y]=σY2+E[Y]2E[Y]=E[Y]+σY2E[Y]=E[Y](1+cvY2).
(3)

These equations describing length-biased samples are well known in the sampling statistics literatures (e.g., Cochran 1977:249–255; deCarvalho 2016).

Proof: Length-Biased Life Expectancy

We will show that in a stationary population—one with constant birth and death rates, zero net migration at every age, and zero population growth—the living population's distribution of age at death, denoted X, is a length-biased sample of the cohort distribution of X. Denote the number of survivors at any age x as l(x), such that the cohort survivorship is l(x)/l(0) and the population at any moment in time has size 0ωl(x)dx, where ω is the oldest attainable age. Let the number of deaths at each age be denoted d(x). In a stationary population, the number of births at each instant equals the number of deaths, such that l(0)=0ωd(x)dx. Thus, the density of ages at death is given by f(x)=d(x)/l(0). The average lifespan (life expectancy at birth) of this cohort, e(0),is as follows:
e(0)=0ωl(x)dxl(0)=0ωxd(x)dx0ωd(x)dx,
(4)

where 0ωl(x)dx=0ωxd(x)dx because both express the total lifespan of the cohort by alternatively summing the incremental lifetime lived at each age, or the full lifespan of those dying at each age.

In this population, the lifespan distribution of the living, fl(x), is a length-biased transformation of the lifespan distribution of each cohort (or period), f(x):

fl(x)=xd(x)0ωl(x)dx
=xf(x)/l(0)e(0)/l(0)
=xf(x)Ef(x)[X].
(5)

The first line of Eq. (5) reflects that among the people currently alive, the number of people who will die at exact age x is the product of the per-cohort deaths at age x, d(x), and the x living cohorts that have future deaths at age x; and the density that will die at exact age x is the ratio of those xd(x) deaths to the size of the full living population, which is 0ωl(x)dx. Life expectancy, e(0), is the expected age at death in the cohort, denoted Ef(x)[X] (with the f(x) subscript to underscore that this expectation applies to the cohort, not the living population). Because Eq. (5) instantiates Eq. (1), Eq. (5) establishes that the distribution of lifespans in the living population is a length-biased sample of the lifespans in the stationary population's cohort or period.

Similarly, Eq. (5), along with Eqs. (2) and (3), implies that the average lifespan of the living is

ALL=0ωx2d(x)dx0ωxd(x)dx
=Ef(x)[X2]Ef(x)[X]
=e(0)+σx2e(0).
(6)

An alternative derivation of these expressions for the ALL, not relying on Eqs. (2) and (3) but derived directly from the life table functions, is given in the online appendix.1

Length bias is not the only perspective through which to view these results. In the remainder of this article, though, we use the length-biased interpretation of the ALL to develop new insights and connect these results to other concepts in demography and other fields.

Intuition for Length-Biased Life Expectancy

The results in Eqs. (5) and (6) connect lifespan demography to well-developed areas of statistics and help to clarify other results about stationary populations, discussed later in the article. The intuition for these results is simple. By definition, the living population dies at an age older than its current age. Thus, older ages of death are disproportionately likely among the living population compared with its original birth cohorts. This selective sample of lifespans is illustrated in the Lexis diagram shown in Figure 1, where the living population is defined by a vertical line and their future deaths are only those deaths included in the triangle on the right of this line. In epidemiology, the upper triangle in Figure 1—representing the deaths that contribute to the lifespan of the living—is called a prevalent cohort (e.g., Törner et al. 2011). The two horizontal bands reflect two example ages at death, illustrating that older ages at death will be experienced by more currently living cohorts than younger ages at death will. This concordance between age at death and the number of cohorts that can still reach that age at death produces length bias.

The results illustrated here offer a new perspective on the relationship between cohorts, periods, and prevalent cohorts. Cohort life expectancy takes everyone who starts their lives at a single moment in time and follows that unselected sample across time. Period perspectives take everyone alive in a moment, regardless of when they began, but stay within that single moment in time. Prevalent cohorts, whose lifespan is the ALL, are distinct from both the cohort and the period perspectives. Like periods, they first select people on their longevity; then, like cohorts, they follow these people through their own lifespans. Thus, the length bias arises because the ALL is asking a cohort question about a period slice of the data.

An alternative formulation underscores what is unique about prevalent cohorts. Cohort life expectancy and period mean age at death are the average lifespans of a birth cohort or “death cohort” (e.g., Riffe et al. 2017), respectively. Thus, cohort and period lifespans constrain birth and death time (respectively) to occur at a single instant, neither earlier nor later, while leaving the time of the other vital event (death or birth) unconstrained. In a stationary population, these alternative constraints generate the same distribution. Prevalent cohorts, in contrast, impose a one-sided constraint on each of birth time and death time: the ALL summarizes the lifespans of people who were born before time t and die after t. This pair of constraints produces the triangle shape in the Lexis diagram because for young ages at death, few combinations of birth and death cohort satisfy these constraints, but for old ages at death, many combinations of birth and death cohort satisfy these constraints.

A practical application of this result is that if age-specific death rates are unchanging over time, the lifespan of the population alive in a period will necessarily be greater than the period life expectancy.2 More broadly, other contexts that define a cohort through cross-sectional membership in some category will produce similarly length-biased estimates of the duration of category membership. For example, epidemiological research designs that begin by selecting people with some incurable medical condition at a particular time point and then follow those people as a cohort to see how long they live—a prevalent cohort study design—generate length-biased lifespan estimates analogous to the bias in the ALL (Zelen and Feinleib 1969). The results here underscore that demographers can conceptualize living populations and their relationship to underlying birth cohorts with the same analytical tools that epidemiologists use to represent prevalent cohort samples and their relationship to disease incidence (e.g., Addona et al. 2009; Alho 1992; Carone et al. 2012; Keiding et al. 2019). Finally, a conceptual implication—recalling the familiar family size example—is that lifespans cluster time the same way that families cluster children.

Empirical Illustration: Life Expectancy, Variation in Age at Death, and the ALL

The expression for the ALL given in the last line of Eq. (6) generates useful insights into the relationship between the ALL, life expectancy, and variance in the age at death. It implies that the ALL is always greater than life expectancy (except in the special case that everyone dies at the same age, in which case the ALL equals life expectancy). More generally, Eq. (6) shows that when life expectancy is held fixed, a population with more variable ages at death will have an older ALL that diverges more from life expectancy. Conversely, when the variation in the ages at death is held fixed, a population with a larger life expectancy will have a smaller ALL that is more similar to life expectancy.

We illustrate the relationships between life expectancy, variation in the age at death, and the ALL by calculating the ALL implied by model life tables.3 As a simple illustration, Table 1 summarizes four such life tables, which form two pairs. The first pair comprises two life tables with identical life expectancy (60 years) but divergent variance (represented in the table as the standard deviation, for ease of interpretation). The one with the smaller variance (the UN model life table for Far East Asian males) has an ALL that is 8 years above its life expectancy, whereas the one with the larger variance (the Coale–Demeny life table for females in the South) has an ALL that is more than 13 years above its life expectancy. The second pair comprises two life tables with nearly identical variance but radically different life expectancy. In this case, the life table with the larger life expectancy (the UN Latin American female life table at a life expectancy of age 67) generates an ALL that is less than 10 years above life expectancy, whereas the table with the low lifespan (the Coale–Demeny life table for males in the North at a life expectancy of age 21) generates an ALL that is more than 30 years above life expectancy. In this last population, the combination of relatively large variance and very small life expectancy generates an extreme discrepancy between the lifespan of the population alive in cross section and the lifespan of a cohort.

To illustrate how this divergence between life expectancy and the ALL occurs, Figure 2 shows the age-at-death distribution for the living population (hollow bars) and for birth cohorts (solid bars) implied by these four model life tables. As is well known, deaths at young ages have an outsized effect on life expectancy. Figure 2 shows that these deaths are dramatically underrepresented in the living population's lifespans. For example, someone who dies at 1 month old would need to appear in the living population at age less than 1 month, and the population contains few people in this narrow age band. More generally, the distribution of ages at death is systematically pulled upward among the living, compared with the cohorts from which they spring.

Figure 3 takes a more global view of these relationships across the full set of UN extended model life tables. (The life tables marked with Xs are the four example life tables just discussed.) Panels a and b show that the ALL diverges widely from life expectancy at low levels of life expectancy, but it diverges very little at high levels. This relationship between the ALL and life expectancy is driven not only directly by the level of life expectancy but also indirectly, by life expectancy's historical relationship with variance in the age at death. Panel c illustrates that historical relationship for these model life tables. This relationship between life expectancy and variation in the age at death can be roughly periodized into three phases. First, the earliest increases in life expectancy (away from the levels experienced by early human societies) may also have increased variation in the age at death, relative to a context in which most deaths occurred during childhood. Second, in the past few centuries, large increases in life expectancy were driven by declining mortality at young ages when such deaths were less common than in early human societies, and these changes tended to substantially reduce variation in the age at death. Third, in high-longevity populations, more recent increases in longevity are concentrated at older ages and may tend to reduce variation in the age at death by only a small amount and sometimesincrease it (Engelman et al. 2010; Tuljapurkar 2010). Because the ALL will diverge from life expectancy the most when the variance is high and life expectancy is low (Eq. (6)), these relationships help to explain why (as shown in panels a and b of Figure 3) the ALL is maximally divergent from life expectancy when life expectancy is roughly ages 20–40 and converges rapidly toward life expectancy once life expectancy reaches roughly age 50. Panel d of Figure 3 shows the resulting relationship between the variance in the age at death and the ALL. Even though, with life expectancy held fixed, a larger variance in the age at death means a larger divergence of the ALL from life expectancy, the historical relationship between life expectancy and variance complicates the overall relationship between variance and the ALL. As life expectancy increases across these model life tables, the ALL's divergence from life expectancy travels along the arcs shown in panel d, from the uppermost values (representing contexts with low life expectancy), then out to the right, and then down toward the bottom left.

A few examples help to calibrate expectations about how much longer the living population will live compared with birth cohorts. As Figure 3 implies, even in stationary populations, this additional lifespan can differ dramatically across populations. In a comprehensive review of hunter-gatherer societies, Gurven and Kaplan (2007) suggested that life expectancy at birth in such societies ranges from roughly 21 to 37; the model life tables in that range of life expectancy estimate average lifespans of the living that are 18–35 years older than life expectancy. On the other hand, for low-mortality populations, the divergence is much smaller. Recent U.S. life expectancy values (before the COVID-19 pandemic) were 76 for men and 81 for women. Model life tables at those values generate ALLs that are, respectively, 2.9–5.7 years and 2.1–4.3 years older than life expectancy. For hypothetical populations with a life expectancy of 95 or older, the ALL in these models diverges from life expectancy by less than 1 year.

The ALL, the Average Age, and the Average Remaining Life

The last line of Eq. (6), which represents the ALL as a function of life expectancy and the variance in the cohort age at death, also provides a core intuition for a well-known but somewhat inscrutable formula for the mean age of a stationary population, denoted A. The formula, given by Ryder (1975:8) and Preston et al. (2001:112), is as follows:
A=e(0)+σX2e(0)2.
(7)

Thus, from Eqs. (6) and (7), ALL = 2A: in a stationary population, the average lifespan of the living is twice the average age.

One way to understand this formula is to see that it follows from two properties of stationary populations. The first is the property shown here: that the average lifespan of a stationary population's members alive in a snapshot is a length-biased life expectancy. The second distinctive feature of stationary populations is that the distribution of lifetimes lived so far (i.e., age) equals the distribution of lifetimes still to come (i.e., remaining life expectancy) (Müller et al. 2004, 2007; Vaupel 2009; Villavicencio and Riffe 2016).4 This property can be considered a form of time symmetry in stationary populations, implying that in a stationary population observed at a specific moment in time, a randomly selected individual is equally likely to be observed at any point in their lives and, on average, is observed halfway through their lifespan (Goldstein 2009; Kim and Aron 1989). Thus, the average age of the population and the average remaining lifespan are each half of the ALL. In combination, these two properties imply that the average age and average remaining lifespan in a stationary population each equal one half of the length-biased life expectancy, generating Eq. (7).5

This perspective similarly provides intuition for Finkelstein's (2008:268) proof that a stationary population with a larger life expectancy than a second stationary population need not also have a larger average remaining lifespan: thus, it is possible for e1(0)>e2(0) while A1<A2 (where A, which we used to denote mean age, also equals mean remaining lifespan). Finkelstein's clever proof, which involves making a succession of changes to a survival curve, does not (to our mind) provide a clear intuition as to the mechanism by which a cohort with a larger life expectancy can have a smaller remaining lifespan. Given that the average remaining lifespan equals the average age in a stationary population, Eqs. (6) and (7) clarify that this possibility will come to pass if the variation in the age at death in the second cohort is large enough to offset its smaller life expectancy. In that circumstance (and only that circumstance), a randomly selected member of this second cohort at a single moment in time will have a longer lifespan than a randomly selected member of the reference cohort, even though a randomly selected member at birth will have a shorter lifespan.

Connections to Other Demographic Concepts

Mortality Selection Without Frailty

By highlighting the inherent bias in lifespan measures that begin with a population defined by a period, the length-biased lifespan in Eq. (5) also gives a somewhat different perspective on mortality selection than demographers' and biostatisticians' traditional perspective based on frailty modeling. Frailty is a construct representing systematic and sustained individual differences in longevity (Manton et al. 1979; Vaupel and Missov 2014; Vaupel and Yashin 1985). It implies that if the frailer people who die young instead lived to old age, their old-age mortality would be higher than that of the robust people who actually survive to old age. In other words, the concept of frailty implies that there are stable differences in individuals' expected longevity and, therefore, that differences in actual longevity select the population by continually removing those with the lowest expected longevity (Feehan and Wrigley-Field 2021:369–371).

Yet, recent theoretical and empirical investigations (Caswell 2014; Hartemink et al. 2017; Steiner and Tuljapurkar 2012) suggest that the vast bulk of heterogeneity in lifespans may be due to stochastic differences in individual luck rather than stable frailty. The results here highlight the importance of lifespan variability in generating mortality selection, whether or not there is any such frailty in the population. Indeed, no concept of frailty is needed to generate any of the results in this article. Imagine that everyone in the population has exactly the same age-specific hazards and that lifespans differ purely by chance. Even in this situation, the ALL reflects a length-biasing selection on individual differences in actual longevity—even though those differences in actual longevity do not reflect differences in individuals' expected longevity. Viewing the population in a cross section intrinsically means viewing individuals in proportion to their lifespan. The cross section selects cohort members not on frailty (expected longevity) but on actual longevity.

Longevity of the Living Population Versus Longevity of Living Cohorts: Comparison With the Cross-sectional Average Lifespan

The ALL is part of “a family of mortality indicators that make use of cohort information but refer to only one period” (Guillot and Payne 2019:418). One member of that family deserves special mention because, like the ALL, it uses cohort mortality to describe the longevity of cohorts alive during a particular period. That measure is the cross-sectional average lifespan (CAL) proposed by Brouard (1986; see Riffe and Brouard 2018) and developed by Guillot (2003). A comparison between the ALL and CAL is informative about both measures.

The CAL at time t is defined in terms of the cohort-specific survivorship pc(x,tx)=lc(x)/lc(0), the survivorship of cohort c at age x, as
CAL(t)=0ωpc(x,tx)dx.
(8)

Thus, “CAL(t) is the sum of proportions of survivors among the various cohorts present in the population at time t” (Guillot 2003:42). This definition implies that the CAL summarizes lifespans using only the deaths included in the left-hand triangle of our Figure 1 (past deaths of living cohorts), in contrast to the ALL, which uses only the deaths included in the right-hand triangle (future deaths of living people).6

This key contrast between the CAL and the ALL—whether past or future deaths of living cohorts comprise the lifespans that the measure summarizes—also means that the two measures describe different population units: the ALL describes the (future) longevity of living people, whereas the CAL describes the (past) longevity of living cohorts. In the CAL, that relationship of living people to their current cohort survivorships—reflecting the mortality that such people did not succumb to—may or may not proxy meaningful aspects of their own lives. In many contexts, being a rare survivor of a cohort with high past mortality, compared with a survivor of a cohort with lesser mortality, might capture something quite meaningful about the survivor (such as the intensity of the disease exposures they may have endured). On the other hand, in highly unequal and segregated contexts, high past mortality concentrated in a particular (disadvantaged) subpopulation might not be a meaningful descriptor of the experiences of survivors drawn largely from a different (advantaged) subpopulation within the same national population. In contrast to most other longevity measures, the ALL describes living populations directly rather than the cohorts into which they were born.

Conclusion

We have shown that in a stationary population, the lifespans of the living population form a length-biased sample of the cohort (and period) lifespan distribution. This insight connects demographic lifespan measures with a well-developed body of statistics (e.g., Cochran 1977:249–255; deCarvalho 2016), including many epidemiological applications (e.g., Asgharian et al. 2006; Hill et al. 2003; Törner et al. 2011), and provides new intuition for existing demographic results. It implies that when death rates are unchanging, the average lifespan of the living always exceeds period life expectancy, exceeding it by more years when variation in the age at death is large and life expectancy is low.

These results are relevant whenever the population of interest is the population that currently experiences a state rather than the population that ever experiences a state. Here, we considered the state of being alive, comparing the lifespan of the living with the lifespan of a cohort. The same results would apply when characterizing the expected length of current marriages (Alho 2016), disease durations among those who currently have a disease (Zelen and Feinleib 1969), or spells of incarceration (Patterson and Preston 2008) or unemployment (Beach and Kaliski 1983) among those currently incarcerated or unemployed. Each of these applications involves what we termed “asking a cohort question about a period slice of the data,” introducing length bias.

Acknowledgments

The authors gratefully acknowledge support from the Eunice Kennedy Shriver Institute for Child Health and Human Development via the Minnesota Population Center (P2C HD041023) and Berkeley Population Center (P2C HD073964); from the National Institute of Aging via the Life Course Center for the Demography and Economics of Aging (P30 AG066613); and from the Fesler-Lampert Chair in Aging Studies at the University of Minnesota; as well as helpful comments from Felix Elwert, Michelle Niemann, James Vaupel, and several anonymous reviewers.

Notes

1

Other derivations are also possible. For example, our Eq. (6) can be recovered from a decomposition of life expectancy provided by Cohen (2015), which divides life expectancy into three components defined by any arbitrary age x: the lifespan lived by those who die before age x, the first x years lived by those living beyond x, and the remaining life years lived beyond x—each weighted by the proportion of the original birth cohort that experiences them. The ALL in a stationary population can then be understood as the survivorship-weighted sum, over all ages, of Cohen’s second and third decomposition components, representing the past and future lifespans of those who survive to each successive age. Here, our goal is simply to provide a single lens through which some key demographic lifespan measures can be related to one another.

2

This implication assumes that not everybody dies at the same age, as we explain in the next section.

3

We downloaded UN extended model life tables from https://www.un.org/en/development/desa/population/publications/mortality/model-life-tables.asp on October 22, 2020. Specifically, we used the complete (unabridged) life tables in one-year age increments.

4

Specifically, Eq. (7) draws on the fact that the average age in a stationary population equals the average expected lifespan. Kim and Aron (1989) and Goldstein (2009) showed this result in a demographic context, and Goldstein (2012) elaborated on the result’s historical context. Vaupel (2009) broadened it to the more general claim about stationary populations referenced here—that the distribution of ages equals the distribution of past lifespans. Brouard (1989) and Carey (Müller et al. 2004, 2007) are generally considered to have independently discovered it, and Villavicencio and Riffe (2016) subsequently dubbed it the “Carey–Brouard equality.” Rao and Carey (2015) presented an alternative proof of the broader claim. The details of the Rao and Carey proof are controversial (Rao 2021; Villavicencio and Riffe 2016), but the claim is not. Finally, exploration of these time symmetries has been extended to other measures in stationary populations (Riffe 2015) and modified for stable populations (Vaupel and Villavicencio 2018). The intuition we provide here, based on length bias, is only one way of gaining insight into the mean age of a stationary population based on the symmetry of past and future lifespans.

5

Ryder (1975:8–11) discusses the implications of Eq. (7) for the relationship between life expectancy, the coefficient of variation in the age at death, and the mean age of a stationary population. His discussion is an intellectual forerunner to our empirical illustration presented earlier.

6

This restriction to past deaths of living cohorts generates one of the key practical advantages of the CAL as a period longevity indicator: although it is based on genuine cohort mortality rather than synthetic cohort mortality (unlike period life expectancy), it requires no data on the future (unlike cohort life expectancy or the ALL).

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