An Overlapping Cohorts Perspective of Lifespan Inequality Open Access

Abstract

A growing literature investigates the levels, trends, causes, and effects of lifespan inequality. This work is typically based on measures that combine partial cohort histories into a synthetic cohort, most frequently in a period life table, or focus on single (completed) cohort analysis. We introduce a new cohort-based method—the overlapping cohorts perspective—that preserves individual cohort histories and aggregates them in a population-level measure. We apply these new methods to describe levels and trends in lifespan inequality and to assess temporary and permanent mortality changes in several case studies, including the surge of violent deaths in Colombia in the 1990s and 2000s and cause-deleted exercises for top mortality causes such as cardiovascular diseases and cancer. The results from our approach differ from those of existing methods in the timing, trends, and levels of the impact of these mortality developments on lifespan inequality, bringing new insights to the study of lifespan inequality.

Introduction

Lifespan inequality, or the variation in ages at death, has emerged as a central demographic measure of population mortality, together with the widely used life expectancy. Interest in the levels and trends of lifespan inequality is based on its behavioral (Barro and Friedman 1977; Picone et al. 2004), biological (Aburto et al. 2020), and ethical (Seligman et al. 2016) implications, with the potential to inform policy recommendations. Alongside the applied literature on lifespan inequality, a growing body of work has continued to develop its theoretical underpinnings (Aburto et al. 2019; Gillespie et al. 2014; Nau and Firebaugh 2012; van Raalte and Caswell 2013); this piece is a contribution to the latter.

Most current applications use period measures (Aburto et al. 2023; Seaman et al. 2016; Xu et al. 2021), with only a few exceptions focusing on cohort measures (e.g., Myrskylä 2010; van Raalte et al. 2023). In addition, recent work by Nepomuceno et al. (2022) has introduced a lifespan indicator, the cross-sectional average inequality in lifespan (CAL^†), which incorporates some of cohort diversity. Interpretation of a single-cohort measure is straightforward, but the typical single-cohort approach may not represent the entire population. The dominant period approach has several advantages. Chief among these is that often the applied interest lies in what is happening now, or in a specific narrow time window—and in this sense, the period provides a seemingly reasonable answer. In addition, the data requirements for the period approach are modest compared to the cohort approach, which may require data over decades or centuries.

The key advantage of the period indicators—they answer the question of what is happening now—is, however, also a key disadvantage. By definition, period indicators, such as period lifespan inequality, are based on a synthetic or hypothetical cohort that experiences the rates of the period. Consequently, inference and interpretation of such period measures are inferences about a hypothetical cohort. For example, a period measure calculated during a pandemic will capture the lifespan inequality of the hypothetical cohort subject to the unusually high mortality patterns throughout their entire life, which is not accurate of any cohort's experience. More generally, any variation in rates over time creates a mismatch between the synthetic period and true cohort experiences. This sensitivity to temporary variations in rates is particularly widely recognized in the context of the analysis of fertility rates, as fertility delays may create a wedge between period and cohort measures (Bongaarts and Feeney 1998), but this also plays a role in the study of mortality patterns (Bongaarts 2005; Guillot 2006).

The synthetic cohort assumptions that come with the period measures—the stationarity assumptions—are well-known by all demographers. These assumptions are rarely met, which is also well-known. As a result, the interpretation of the period measures tends to focus on the synthetic cohort, which avoids explicit reference to true cohorts, and often also avoids reference to the true population composition. The period approach has become the industry standard, and the profession has developed ways to interpret the period measures without misleading references to true cohorts and innovative formal approaches to analyze the period metrics, such as novel decomposition techniques. However, there has been comparatively less effort in developing approaches that break away from the synthetic cohort model.

We argue that the (period) synthetic cohort representation of population-level patterns is founded on stringent assumptions, which can lead to misleading assessments when they are not met. In a stationary population, all cohorts share the same age-specific mortality rates, hence the period- and cohort-based perspectives coincide. In this context, measures derived from a period life table are identical to their cohort equivalents. For this reason, the period-based synthetic cohort provides a sufficient single-cohort representation of the population's mortality patterns. That is, lifespan inequality measured from the period life table coincides with that of all of the single cohorts present in the population. However, the reality is that actual populations are composed of cohorts that have experienced a diversity of living conditions, leading to differences in their mortality profiles. Hence, this challenges the interpretation of single-cohort-based measures as a sufficient statistic for those of the aggregate population, or of any of the individual cohorts that compose it. While this is well understood, the implications of how the period measures should be interpreted are not always clear.

We propose an overlapping cohorts perspective of lifespan inequality, fully incorporating cohort trajectories. To that effect, we begin by considering the mortality experience of all the cohorts alive in the period of interest. To do so in applications, we must rely on a combination of historical data (or data preceding the period of focus) and mortality projections (or data following the period of focus). The cohorts’ remaining lifespan inequality (from its current age forward) would be suitable for the study of the effects of short-term mortality fluctuations. Long-run or permanent mortality changes, such as the analysis of the importance of specific causes of death, would be better studied by considering the full cohort mortality experience. Finally, an individual cohort's mortality is aggregated by using a set of weights, which depend on the application. Notably, if the stationary population conditions are met, our method exactly coincides with existing synthetic cohort approaches and the standard real cohort approach. However, our analysis also illustrates the effects of departing from them and provides an alternative measurement on those occasions.

We develop the overlapping cohorts perspective on lifespan inequality and demonstrate how results differ from those from current methods across several applications. We do not aim to provide a numerical comparison of one method to another, as the approaches and assumptions differ to the extent that such strict numerical comparisons would be challenging. Instead, our question of comparison focuses on what we think matters more and asks whether the qualitative conclusions or possible policy advice would differ between the existing and the proposed novel approach. In the applications, we first report the trend and level differences across methods for a historical population covering the period 1901 through 2015. Then, we study the impact of temporary and permanent mortality changes on lifespan inequality through historical cases. We analyze French war- and flu-related mortality spikes in the first half of the nineteenth century, as well as the more recent surge in violent deaths during the armed conflict in Colombia (1984–2015). An additional counterfactual exercise investigates the effect of permanent mortality changes in 1951 to 2010 in Sweden. We also evaluate the impact of cancer and cardiovascular diseases on lifespan inequality through a cause-deleted exercise. Finally, we address data requirement considerations, among other issues, in the Discussion.

Methods

Synthetic Life Tables

We denote as synthetic the life tables that are built by combining subsets of the mortality rates from multiple cohorts into a single life table. While the most common synthetic approach is the period life table, alternative methods that use more comprehensive cohort mortality trajectories have recently been suggested, inspired by the cross-sectional average (CAL) approach pioneered by Brouard (1986). These methods tie together subsets of the age-specific mortality rate profiles from different cohorts into a single composite or synthetic cohort. These approaches aim to provide an adaptation of the life table, a method designed to analyze a single cohort, to population-level analysis that encompasses multiple cohorts.

Our aim is also to provide a method that encompasses multiple cohorts. In contrast to existing methods, our approach takes into account cohorts’ mortality both before (as in CAL) and after the period that is in focus. Instead of creating a synthetic cohort combining pieces of the mortality profiles of different cohorts, we preserve the overlapping cohort structure of the life table by aggregating full cohort profiles.

In comparing existing approaches and our novel method, we use the variance of the age-at-death distribution as a measure of lifespan inequality, frequently employed in applications (Aburto et al. 2023; Xu et al. 2021). Although other measures have been proposed (van Raalte and Caswell 2013; Wilmoth and Horiuchi 1999), the variance can be readily decomposed into its components (Shorrocks 1982), which is particularly useful for our cross-cohort approach. For the sake of comparability, across all approaches considered (period, CAL, overlapping cohorts), we evaluate lifespan inequality exclusively with the variance of the age-at-death distribution.

Period Life Table

The variance of the ages-at-death distribution is expressed as

where $t$ refers to a particular period, $a$ denotes the earliest age recorded in the life table, and $la$ is the radix of the population at age $a$⁠. $Ma$ is the average age at death after age $a$ (i.e., $Ma=e(a)+a$⁠). $dx$ refers to the number of deaths of the life table, and $x$ and $ω$ are the age at death and the terminal age in the life table, respectively. In its application to the period approach, this measure is based on the age-at-death distribution of the period life table, which is built from the period age-specific mortality rates. The same equation applies for the single-cohort approach, as well as for the overlapping cohorts approach.

Cross-sectional Average Life Table

The CAL family of measures is an alternative to period life tables indicators that is based on the past mortality experience of all the birth cohorts in a population. That is, unlike conventional period measures, it is based on the survival up to the period under analysis, and not exclusively period mortality rates. Typically, life table–based measures are then constructed as a function of the survival curve of these cohorts (Guillot 2003). For the case of lifespan inequality as measured by e^†, the cross-sectional average version (Nepomuceno et al. 2022), CAL^†, is defined as follows:

where $t$ is the period at which the measure is calculated, and $la$ represents the radix of the population at age $a$⁠. $lx,t−xi$ is the number of survivors recorded in the cohort life table at age $x$ for the cohort $i$ born in the year $t−x$⁠.

Alternatively, it is possible to interpret this approach as defining a new type of synthetic life table: the life table such that $lx=li$⁠, which we call the cross-sectional average life table. This interpretation is useful because it allows us to compute other lifespan measures from its age-at-death distribution. In our case, for comparability, we use the variance of the CAL-based life table instead of CAL^†.¹ Note, however, that beyond facilitating the implementation of alternative measures, an interpretation in terms of an individual cohort in a strict sense is not possible (Guillot 2003). This is because combining past cohort survival in this manner may imply negative age-specific mortality rates.² Nonetheless, this will not play a role in our analysis.

Overlapping Cohorts

We begin with the premise that, in any given period, the lifespan inequality of a population is an aggregation of this measure for each of the overlapping cohorts (OC) that compose it. In a cross section, each cohort is at a different current age, and thus we must decide what part of their age-at-death distribution will be considered in the lifespan inequality calculation for the current period. As we discuss below, depending on the goal of the analysis, we may consider either their past or future (with respect to the period of focus), or their entire mortality history. Finally, an aggregation method is required to condense all the cohort-specific information for a population-level measure. This section fully develops this intuition.

Perspectives

In any period, for each of the multiple cohorts in a population, we may consider lifespan inequality according to three perspectives: (1) from birth to its maximal age (full), (2) from its current age onward (forward), or (3) up to its current age (backward). This is similar but not the same as conditional lifespan inequality (i.e., above a threshold age); conditional measures are frequently used, for example, in studies that examine lifespan inequality trends net of the influence of infant mortality (Edwards and Tuljapurkar 2005). Instead, our approach considers each of the three perspectives from a specific period, and thus forward and backward measures capture the lifespan inequality of every cohort for different age ranges. For the forward perspective, instead of considering lifespan inequality from a unique age onward for all cohorts, this threshold is cohort-specific and centered around the reference period. That is, for each cohort and given a reference period, the forward perspective considers the lifespan inequality of their remaining life.

Each perspective is best suited for specific analysis, as we illustrate with our applications. We may be interested in full lifespan inequality when assessing permanent mortality changes that will eventually affect all cohorts. In turn, the other two perspectives may be better suited for temporary mortality changes. A forward measure captures the effects on the lifespan inequality that will affect the remaining cohort members. Thus, it is well-suited to assess the effects of short-term and unexpected mortality fluctuations, which we may care about because the survivors will actually experience the event. Finally, backward-looking perspectives capture the past lifespan inequality experienced by cohorts. Present cohorts may look back to inform their subjective survival expectations and possible lifespan inequality (Nepomuceno et al. 2022), a possibility that we do not explore in this work.

Aggregation

After settling on a particular perspective, the basic object of interest is the (cross-sectional) age-at-death distribution of a given population. Here, instead of using the age-at-death distribution of a single (synthetic or true) cohort, our approach is based on the mixture of the individual cohort's distribution. The idea is that this is a reflection of the lifespan inequality of the cohorts present in the population. The population's age-at-death distribution, of all cohorts $i$ alive in period $t$⁠, is constructed as follows:

where $dt$ and $di$ are the density function for the entire population and for the cohort of age $i$ at $t$⁠, respectively. $n$ is the total number of cohorts present in $t,and$$s0i$ is the weight in the period's population of cohort $i$⁠. We next discuss the choice of weights.

What weights are appropriate for this purpose? Whether age structures are a nuisance or a meaningful factor to consider is not a straightforward question. In some instances, weighting according to the contemporary age structure could make sense (Pifarré i Arolas et al. 2023). In other applications, we may be interested in assessing lifespan inequality differences across populations, independent of the influence of age structures. In this work, we remain agnostic on this issue and leave it to the analyst to decide whether to remove the influence of age structure from the results, as we view this as an application-dependent choice. The overlapping cohorts perspective can accommodate both approaches through the choice of cohort weights.

Current practices suggest several possible options to remove the influence of age structures. Direct standardization may be the most immediate, whereby cohorts may be weighted based on their relative weight in a standard population, such as the European (Pace et al. 2013) or the American standard populations (Klein and Schoenborn 2001). Critics of this procedure argue that the choice of standard population is arbitrary and may affect the resulting evaluation (Ahmad et al. 2001). To avoid this, one may follow a strategy similar to that of a period life table. The comparability of the measures derived from a period life table is grounded on the fact that the age structure of the stationary population it represents is fully determined by mortality (and not fertility). While the implicit age structure of two populations with different mortality rates is not the same, were their mortality rates to converge, so too would their age structures and mean ages at death.

In that vein, it is possible to determine the age structure of a population under the overlapping cohorts approach that is also fully determined by mortality. As for a standard stationary population, we assume that equal-sized cohorts satisfy the no-growth requirement. In turn, the age structure is given by the relative size of the survivors at each age, which is determined by their respective cohort life tables. That is, for each cohort in the population, we determine their relative size on the basis of the survivors at their age in the reference period. The resulting weights are then used in the aggregation procedure described in Eq. (3).

Finally, beyond theoretical considerations, do the results in our applications vary qualitatively depending on the weights considered? Our conclusion is that, by and large, the main source of differences across perspectives reported in the manuscript is not a consequence of the choice of weights. For each of the main figures, we also report the results under alternative weights in the online supplementary materials.

Lifespan Inequality From an Overlapping Cohorts Perspective

Once the age-at-death distribution is constructed, we simply compute the variance V_t(a) of this distribution to measure lifespan inequality. Alternatively, this may be calculated from cohort life tables using the well-known decomposition of the variance (Shorrocks 1982):

where $n$ is the total alive cohorts present in time period $t.$$sai$ represents the weight of each cohort $i$⁠. $Mai$ is the average age at death after age a in cohort $i$⁠, $Mt$ denotes the average age at death of individuals in all cohorts who survived to age a in time period $t,$ and $Vai$ is the variance of age at death of individuals present in a cohort $i$ that survived to age $a$⁠. That is, the overall lifespan in the population is composed of the weighted sum of the individual cohort variances and the between-cohort component. In applications, however, the within component accounts for nearly all of the total variance (see section 1 in the online supplementary materials).

Comparison Between Approaches

Figure 1 depicts the Lexis diagram of the cohorts present in the period $t=$ 1915. Each approach, CAL, period, and OC uses parts of the mortality rates located within that parallelogram to compute lifespan inequality. Thus, all approaches use some of the mortality experience of the cohorts alive in 1915. Period life table lifespan inequality is based on the aggregation of the contemporary mortality rates (the vertical red lines). CAL, instead, incorporates the mortality history up to 1915, represented by the green triangle to the left of (and up to) 1915. In other words, CAL accounts for the mortality conditions of the surviving cohorts in 1915. Finally, our measure uses different parts of the parallelogram depending on the perspective. Both past (the dotted blue triangle on the left) and future (dashed blue triangle on the right) mortality rates for all cohorts alive in 1915 are included in the full perspective (solid blue parallelogram). In turn, the left-hand side is used in the backward-based calculation (dotted blue triangle on the left), and the right-hand side is used for forward measures (dashed blue triangle on the right). Note that although the backward perspective is based on the cohort histories as CAL, there are differences in their respective calculations.

The differences in the information employed in each approach come into play only when evaluating real-world population experiences. Although there are analytical differences across methods, on top of the different mortality rates considered, in a constant mortality steady state all approaches yield the same result. That is, under the two conditions of a stationary population—equal cohort sizes (no growth) and constant mortality—lifespan inequality is identical for all three perspectives considered. This result is immediately apparent from the overlapping cohorts perspective, given that it is a mixture of cohort mortalities, which collapse to a single profile under constant mortality. In Figure 1, this is illustrated by the fact that the mortality rates in the vertical (cross section) cut of the Lexis diagram coincide with those in the diagonals (both right and left sides). Thus, it is when we recognize the diversity in mortality experiences across real cohorts that differences between approaches emerge. CAL-based lifespan inequality measures acknowledge this and incorporate the past mortality histories of current cohorts. However, CAL does not attempt to reflect the actual lifespan inequality of the current population. Instead, it has been justified from a behavioral perspective. The premise is that current cohorts may inform their subjective expectations on their own future mortality trajectories on the basis of the past experiences of their kin or those socially close to them (Nepomuceno et al. 2022). Thus, the gap we intend to fill with this approach is that of a measure that preserves true cohort mortality experiences but that also reflects the overall lifespan inequality of the population.

Panel a in Figure 2 is the empirical counterpart of Figure 1 for the year 1915 in Sweden, displaying age-specific mortality rates. These are aggregated into their respective implied mortality rate distributions in panel b. The main differences across methods concentrate in both ends of the age distribution, with particularly substantial discrepancies between CAL and the rest of the approaches at older ages. Note the implicit negative mortality rates for the CAL-based distribution, for example, in the range 40 through 50, and at older ages. These differences manifest in substantial disparities in the respective age-at-death distributions, as shown in panel c. Specifically, panel c depicts the disparities in survivors (⁠$lx$⁠) as a way to deal with negative mortality rates from CAL life tables. In later sections, we will illustrate how the large differences in ages-at-death influence lifespan inequality calculations in different contexts.

Counterfactuals

Counterfactual calculations for lifespan inequality may be used to study the effect of short-term mortality fluctuations (Aburto et al. 2021; García and Aburto 2019; Vigezzi et al. 2022) and permanent mortality changes, or in exercises that assess the contribution of specific causes of death to lifespan inequality (Aburto et al. 2023; Seligman et al. 2016). Typically, these calculations for mortality changes are based on period lifespan inequality measures. While there exist differences in the analytical strategies, at their core, these approaches evaluate the effect of mortality variations based on stationary population assumption. There are two fundamental assumptions in these exercises, which also disregard considerations related to the transition period, that is, until the entire population has been born under the new mortality regime. First, any mortality change is presumed to be permanent, such that current period mortality rates will be constant and thus match, in the long run, cohort mortality rates. Second, the current shares of deaths by cause will be constant and shared across cohorts. Related to this, decomposition approaches often (Aburto et al. 2023) but not always (Seligman et al. 2016) rely on the independence of mortality across causes.

Temporary Mortality Fluctuations

What happens when these assumptions are challenged? We illustrate the consequences of replacing some of these assumptions with real data in an effort to better match the actual mortality developments of the population in applications. We use case studies that cover a range of applications. In the first two applications, we analyze the impact of short-lived periods of elevated mortality. We drop the permanent mortality change assumption and allow for mortality to decline after the mortality-increasing shock, following its actual (or forecasted) patterns.

To create the counterfactual mortality rates (absent of the shock), we interpolate mortality for those periods. The first set of historical mortality shocks are the relatively short peaks of mortality in 1914–1918 and 1940–1945 in France, owing to a combination of the Spanish flu epidemic and World War I and II casualties. Thus, we replace mortality rates during the peak mortality years with a cubic spline interpolation that fits a third-degree polynomial to the surrounding periods’ age-specific mortality rates. In this case, we work with all-cause mortality rates because we lack data on cause-specific mortality. Then, we explore the implications of our approach in the context of a longer lived mortality crisis, the surge of violent deaths in Colombia in the period 1984–2015. For this example, we generate a counterfactual that specifically eliminates the surge of violent deaths. In this case, a linear interpolation performs the best, given the gaps in the data for some of the adjacent years.

Permanent Mortality Changes

We also investigate the trends in the contribution of cardiovascular and neoplasm mortality to lifespan inequality in Sweden for the period 1951–2018. In this exercise, we deviate minimally from existing period-based approaches in the implementation of the OC approach. Cohort-specific counterfactual mortality profiles are created using standard cause-deleted approaches (Preston et al. 2001), and cause-specific death shares are assumed to remain constant at the period's level for all cohorts.

The main difference with respect to the traditional period analysis is that we do not assume equal mortality profiles across cohorts. Instead, we keep existing age-specific mortality rate differences between cohorts constant in relative terms. In other words, our alternate assumption is that the rate of mortality improvements over time will keep their pace, and thus so will cohort relative mortality positions. This assumption has been chosen for simplicity; alternative (and perhaps empirically founded) scenarios for future cohort mortality differences are left for future exercises.

Employing this idea, we proceed as follows. First, using the last-born cohort (age 0 in the year under study) as a baseline, we calculate relative age-specific mortality rates across cohorts. For example, in 2000, the age 0 mortality rate for the oldest cohort (born in 1910) is 54 times that of the youngest cohort. Once relative cohort mortality is established, cohort-specific mortality rates depend on the level of mortality to which relative mortality is anchored. We use the period's mortality rates to set the mortality level. For each period, contemporary mortality rates are the assumed mortality for the last-born cohort, and all other cohorts are scaled accordingly by their relative mortality.

Data

Mortality rates and population exposures by single year of age are retrieved from the Human Mortality Database ([HMD] 2023). To increase the range of our period of analysis, HMD data are complemented with World Population Prospects (United Nations 2022) mortality and population projections (medium mortality scenario) from the last year available in the HMD and up to 2100. Period and cohort life tables are constructed following standard demographic techniques (extrapolation), considering the 85+ and 90+ open-ended age intervals.

Causes of death data for Colombia and Sweden were obtained from the World Health Organization (2020) mortality database. For Colombia, all causes of death associated with violence are considered (ICD-10: X85–Y09). In the case of Sweden, the ICD-10 codes I00–I99 are used to capture cardiovascular diseases (CVD) and C00–C97 and D00–D48 for malignant neoplasms. Causes of death categories were harmonized across versions of the International Classification of Diseases. Age-specific deaths are ungrouped into single ages using a univariate penalized composite link model (Pascariu et al. 2018).

Results

Trends

An examination of lifespan inequality across both synthetic approaches and the OC perspective reveals differences in both levels and trends. Panel a in Figure 3 displays the trends in lifespan inequality for France by method, as well as the underlying cohort-specific lifespan inequality. Given their data requirements, CAL and (full) OC measures are available from 1901 (the central vertical dashed line) to 2015. Throughout the years considered, period measures report the lowest lifespan inequality, followed by CAL and OC; the exception is the surge in mortality during World War II (which we will revisit in the next section). In addition, while period and CAL lifespan inequality trends downward after the end of the 1800s, OC measures remain higher (even growing moderately) up to the 1950s, when they rapidly decline.

The reason for this divergence across methods is that full OC indicators encompass more past cohort information than period methods, and also future developments (contrary to CAL indicators). In the case of France, the cohorts in the early 1900s will experience the two world wars and the flu pandemic. Thus, OC measures do not follow the contemporary declines in mortality rates, which dictate period trends, and are also pushed upward by the future mortality spikes. As individual cohorts’ lifespan inequality starts declining monotonically in the 1950s, so do OC indicators. We will examine in detail these effects in a case study covering the upsurge in violent deaths in Colombia starting in the mid-1980s. The extent to which CAL and period approaches deviate from the experience of contemporary cohorts can be seen more clearly in panel b. The figure displays the lifespan inequality of the cohorts alive in 1901 (blue dots), as well as the population-level indicator according to the three methods. CAL and period indicators are below the experience of any one cohort in 1901, whereas OC is a (cohort size) weighted average of the individual lifespan inequality.

The relevance of monitoring trends in lifespan inequality is often discussed in two main ways: as a measure of population-level mortality inequality and from an individual perspective, as an indicator of uncertainty (van Raalte et al. 2018). From an aggregate population perspective, the OC approach provides a measure of inequality both within and across cohorts in a population. This approach may be of interest for policies that consider intergenerational equity, such as pension design. In these contexts, supplementing results with measures stratified by socioeconomic variables is possible, but it is beyond the scope of this article. In turn, uncertainty about the length of life is considered consequential because of its demonstrated behavioral implications (Picone et al. 2004). In this sense, on a cross-cohort average, the OC measure more accurately captures lifespan inequality than the period indicator (panel b). Therefore, if the objective is to characterize aggregate behavior, the OC measure may provide a more reasonable representation. However, this depends on how individuals perceive this uncertainty, a topic that remains underexplored.

Temporary Mortality Changes

High-Mortality Episodes—France War Mortality

In the first half of the twentieth century, France experienced two temporary, high-mortality episodes from the two world wars, in addition to the mortality from the Spanish flu epidemic in 1918. In this exercise, we assess the impact of these events on lifespan inequality from the perspectives of the cohorts in the year 1940 in a counterfactual exercise.

Panel a in Figure 4 displays both series (real and counterfactual) of period crude mortality rates in France from 1855 to 2025, with elevated mortality in the years 1914–1918 and 1940–1945. The age pattern of elevated mortality is shown in panel b for all cohorts present in 1940. We can see that while mortality was relatively higher for most age groups during these peaks, the age group from 15 to 40 experienced the worst effects on both occasions.

Panel c displays the overall effects of these events on the cohorts in 1940 according to each method, as well as the cohort specific effects. Three noteworthy differences are apparent. First, period-based measures have a higher magnitude than either of the cohort-based approaches. As we know, period methods assume that the unusually elevated mortality will remain constant, overestimating the mortality effect of the event.

Second, we obtain opposite messages when considering the forward and full perspectives. According to the forward perspective, lifespan inequality increased, whereas the full perspective indicates a small decline. That is, for the survivors of the cohorts alive in 1940, the lifespan inequality for the remaining part of their lives increased as a result of these high-mortality events. Instead, it declined when considered for these cohorts from birth. Third, there is substantial heterogeneity across the effects these events had on cohorts, depending on their age in 1940. In the case of the forward measure, the extent to which cohorts experience increases in lifespan inequality varies, peaking for those aged 21 in 1940. These will be the cohorts that reach the ages with highest mortality throughout those years. In turn, the full measure shows a crossover around age 45. Those older than 45 will experience a decline in lifespan inequality as a result of these high-mortality events, whereas the later born cohorts will face an increase. Overall, full lifespan inequality declines, as the effect for older cohorts dominates.

High-Mortality Episodes—Colombia's Armed Conflict

During the last years of the twentieth century and the beginning of the twenty-first, Colombia experienced one of the most violent periods in postcolonial South America, with high rates of homicide. The violence was driven mainly by the drug wars in the 1980s and guerrilla warfare in the 1990s (Coatsworth 2003; Luna 2019). The war between the Colombian Armed Forces, paramilitary groups, and guerrilla groups lasted until 2016, when Colombia approved the peace agreement referendum, leading to the termination of hostilities between the Colombian government and guerrilla groups (Luna 2019). Our exercise illustrates the extent of the impact of high and large mortality episodes on lifespan inequality, based on a counterfactual exercise.

Panel a in Figure 5 depicts the trends of the real and counterfactual mortality rates in Colombia from 1951 to 2015, with high mortality occurring during the period from 1984 to 2015. The resulting age-specific excess mortality is shown in panel b, as the ratio of the real and counterfactual mortality. This high-mortality episode disproportionately affected individuals between the ages of 15 and 40. The resulting cohort-specific increase in forward lifespan inequality is illustrated in panel c. Cohorts born between 1970 and 1985 experienced the violent episode at the ages worst affected (15 through 40) and, as a result, had the greatest increase in their lifespan inequality. Older and younger cohorts progressively lived fewer of their more vulnerable years during the period of elevated mortality and hence suffered lower lifespan inequality increases.

Panel d displays the overall effect of the violent episode on lifespan inequality over time and across different methods. The most striking difference is in the peak years in which the population's lifespan inequality is most affected. While the impact peaks in years when mortality was the highest owing to violence (e.g., 2002, 1996, and 1992), for period measures, the cohort-based forward perspective peaks at the onset of the violent episode and gradually decreases over time. This result builds on the intuition from panel c: the beginning of the violence spike concentrates the largest number of highly affected cohorts, that is, cohorts that will experience the violence in their most vulnerable ages. In contrast, period-based measures track contemporary mortality rates. The results in this and the next application rely heavily on forecasted mortality rates, given the reference period and the age of the cohorts involved; to assess forecasting uncertainty, findings under alternative mortality projections are included in the online supplementary materials.

Permanent Mortality Changes

The next case involves permanent mortality changes, in the form of cause-specific assessments, for which we interpret the sign and magnitude of the influence on lifespan inequality. From an analytical standpoint, the direction of the effect is dictated by whether age-specific mortality changes fall below or above a specific threshold age. When mortality declines below the threshold age, it results in a decrease in lifespan inequality. Conversely, if mortality decreases above the threshold age, it leads to an increase in lifespan inequality (Seligman et al. 2016; van Raalte and Caswell 2013). This relationship will play an important role in the cause-deleted exercise, because the sign of the effect on lifespan inequality will depend on the age distribution of the cause of death in relation to the method-specific threshold ages. The magnitude of the effect is influenced by the distance from the threshold for the ages impacted by the mortality change, as well as the overall size of the change.

We examine the contribution of top causes of death to lifespan inequality. In particular, we investigate the contribution of cardiovascular diseases (CVD) and neoplasms in Sweden for the period 1951–2010. To identify the cause-specific impact, we generate a counterfactual through a standard cause-deleted life table analysis.³ Panels a and d in Figure 6 display the age-specific share of deaths for CVD and neoplasms by age in the year 1982. For CVD, the share grows steadily with age, concentrating the majority of CVD-related deaths among the elderly population. On the other hand, the proportions of neoplasms are higher within the age range of 30 to 80. These patterns remain stable throughout the entire analyzed period.

Panels b and e in Figure 6 depict the baseline and cause-deleted mortality rates, and corresponding threshold ages (vertical lines), for both period-based and cohort-based methods, also for 1982. Remarkably, the threshold age for the overlapping cohorts method is about half that of the period mortality distribution (33.8 vs. 61.8). This will influence the contribution these causes have to their respective lifespan inequalities, given that, for the period method, a higher proportion of the mortality declines will occur above the threshold age.

These differences in the threshold age result in substantial discrepancies across methods, as illustrated by panels c and f, which display the impact of each cause across the period (the ratio of real and counterfactual lifespan inequality). We find that, for most of the period, CVD is a positive contributor to lifespan variation in both methods (panel c). However, practically all mortality changes are located well above threshold in our approach, but less so for the period-based measure. This explains the differences in levels in the contribution, which are up to 1.2 times as large for our measure (in 1981). However, as mortality declines, increasing the threshold age, the sign of the contribution reverses for the period measure, and CVD reduces lifespan inequality from the 1990s onward. Threshold differences manifest in a more extreme manner for cancer (panel f). According to the period measure, across all years, neoplasms have a negative contribution to lifespan inequality. In contrast, up to the early 1990s, neoplasms lower lifespan inequality in our approach.

Discussion

We introduced a new perspective on lifespan inequality that fully incorporates cohort trajectories of all cohorts alive in the period of consideration, in contrast with current measures that utilize only partial cohort histories to create a synthetic cohort or focus on single cohorts. We applied it to several case studies to illustrate its implications for the assessment of temporary and permanent mortality developments. Relatively short-lived, high-mortality episodes were examined in the context of the early to mid-1900s war mortality in France and the surge in violent deaths in Colombia (1984–2015). Our method differs from existing approaches in the magnitude, timing, and direction of the effects of these episodes on lifespan inequality. Finally, we performed a counterfactual exercise to assess the impact of permanent mortality variations—specifically, cause-specific contributions to lifespan inequality. We demonstrated that the extent to which our results differ from those of current methods depends on the role played by the underlying discrepancies in the age-at-death distributions across approaches. These mortality differences affect the position of the critical threshold ages, with consequential effects on the results obtained by each method in our applications. We found that CVD had a substantially larger impact on lifespan inequality than seen with existing methodologies, and that cancer diminished lifespan inequality from a period perspective but increased it, for most periods, under our approach.

Unlike for period methods, requirements to deploy our approach to measure lifespan inequality go beyond the commonly available mortality data. For the most part, we were able to apply our method by relying on countries with high-quality historical data and on long-term mortality projections. Strong data requirements, however, are inherent to the measurement of lifespan inequality, a longitudinal measure that is highly sensitive to the shape of future mortality. Period measures address these inherent data requirements with strong assumptions on future mortality developments. Specifically, a critical assumption is that of equal mortality across cohorts. In the current work, we have shown how failure to meet this and other assumptions can result in potentially misleading assessments. In other words, either data and projections or strong assumptions will always be needed to assess lifespan inequality; there is no such thing as a free lunch.

In applications, forecasts are an integral part of our perspective. Thus, a natural extension is to evaluate the role of forecasting uncertainty on the results. We have considered the impact of different forecasting scenarios on the results in our applications. However, it is also possible to incorporate forecasting uncertainty considerations stemming from probabilistic methods, such as the widely utilized Lee–Carter approach (Koissi et al. 2006; Li et al. 2009). A possible strategy that follows previous work would be to evaluate lifespan inequality on the predicted rates, then generate confidence intervals through a bootstrapping approach (Efron 1992). Alternatively, methods tailored toward cohort mortality completion (Rizzi et al. 2021) could be particularly helpful in incorporating richer forecasting approaches into the overlapping cohorts perspective.

Beyond data-related considerations, there remain challenges to the rationale behind the period perspective on longitudinal measures. In the case of measures such as crude mortality rates, the period justification is clear: they aim to reflect purely contemporary mortality conditions. However, for inherently longitudinal measures, such as lifespan inequality, they are more difficult to interpret outside of the stationary population equivalence framework. As we know, if mortality is not equal across cohorts, this is not representative of the population's experience. A common approach is to refer to them as measuring the experience of a hypothetical cohort. However, given that period lifespan inequality measures can differ substantially from those of any one cohort, this may limit the interpretability of the synthetic cohort results. Thus, it is useful to consider alternatives. We believe that the overlapping cohorts approach presented in this article is one such useful alternative.

We are not alone in recognizing the limitations of period measures. A well-established body of work challenges the notion that period mortality rates truly reflect contemporary conditions, the argument being that past behavioral and contextual influences shape cohort mortality (Barker 2007; Myrskylä 2010; Wang and Preston 2009). A strand of work that is closer to our contribution also recognizes that, with nonconstant mortality, period measures are difficult to interpret given the resulting heterogeneous cohorts. CAL and related measures provide an alternative, based on single-cohort experiences (Bongaarts 2005), on combinations of cohorts (Guillot 2003; Nepomuceno et al. 2022), and using historical and projection-based mortality data for cohorts (Guillot and Payne 2019). Our proposed measures follow in this tradition.

Ultimately, analysts can choose between standard period approaches that rely on strong assumptions or alternative cohort-based perspectives that require more data or forecasting but depend less on assumptions. Period measures are well understood, straightforward to implement, and comparable across populations, and they have relatively low data requirements. However, this ease of implementation requires accepting stringent assumptions that may deviate substantially from reality. New cohort-based approaches such as ours require additional steps and more data or forecasting, but by incorporating more data, they depend less on assumptions. Additionally, a further consideration is that since period measures have long been an industry standard, a wealth of analysis tools, such as decomposition techniques, have been developed specifically for period approaches. While it is possible in principle to use all life table methodologies in our framework, the specifics remain unexplored.

We believe that a new mixture of revised assumptions, together with a larger role for mortality projections, is needed moving forward. In our cause-deleted exercise, we introduced a new assumption that preserves the overlapping cohort perspective and alleviates data requirements. At the same time, promising new efforts to complete cohort fertility and mortality open the door for more accurate forecasts (Basellini et al. 2020; Bohk-Ewald et al. 2018; Goldstein et al. 2023). Thus, there exist promising alternatives to the equal cohort mortality assumption. We view our work as a methodological innovation that provides a benchmark for lifespan inequality measurement, in a data-rich context. Ultimately, the best mixture between assumptions, data, and projections is likely to continue shifting in the future as new data become available and further developments in mortality forecasting arrive.

Notwithstanding the necessary caution regarding the policy impact of basic demographic research, previous contributions have argued for the importance of considering lifespan inequality in policy decisions (van Raalte et al. 2018). For example, when designing health policies, tracking their effects on lifespan inequality can provide better insights into their impact on health equity. As Seligman et al. (2016) illustrated, allocating resources to address the causes that lead to the greatest gains in life expectancy may not necessarily promote health equity, as measured by lifespan inequality. We have shown that across various common applications, our approach offers a different and often contrasting perspective on lifespan inequality. For instance, in several periods, declining cancer or CVD mortality decreases period lifespan variation but increases it from an overlapping cohorts perspective. Therefore, if a hypothetical decision-maker considers lifespan inequality as one of the measures informing policy decisions, our results often diverge from the recommendations derived from the period perspective.

Conclusion

This study introduced a new methodological framework to measure lifespan inequality by adopting an overlapping cohorts conception of the population. Our contribution is twofold. We propose a different approach to aggregate cross-cohort information that does not rely on a synthetic cohort. In addition, we discuss which fragments of cohort mortality to utilize in applications, beyond conditional lifespan inequality considerations. Significant challenges remain in applying this new method, particularly in low-data contexts. However, we believe valuable new insights can be gained by applying this framework to other life table and adjacent demographic measures.

The longitudinal interpretation of lifespan variability makes it particularly well-suited for our cohort-based approach, but our framework is not inherently limited to this measure. In principle, all demographic indices based on life tables can be calculated from an overlapping cohorts perspective. However, a key challenge for our work and related research will remain the data requirements intrinsic to cohort approaches. In this sense, the current developments in cohort-completion mortality forecasts (Basellini et al. 2020; Rizzi et al. 2021) are promising, as they could provide better suited predictions of future cohort mortality. Although there are significant challenges in developing the overlapping cohorts perspective, our results are encouraging, as they illustrate the potential of our approach to challenge existing narratives in applied work based on life table–derived measures.

Data Availability

All data and code to fully reproduce the analyses are available at the Open Science Framework: https://osf.io/t6qns/?view_only=64d64dc7ebe74312a8eb5dc9e6e28133.

Acknowledgments

We thank Jason Fletcher, Alberto Palloni, Marcus Ebeling, and Alyson van Raalte for their insightful comments and suggestions in early versions of this work. M.M. was supported by the Strategic Research Council, FLUX consortium, decision numbers 345130 and 345131; by the National Institute on Aging (R01AG075208); by grants to the Max Planck–University of Helsinki Center from the Max Planck Society (decision number 5714240218), Jane and Aatos Erkko Foundation, Faculty of Social Sciences at the University of Helsinki, and Cities of Helsinki, Vantaa and Espoo; and the European Union (ERC Synergy, BIOSFER, 101071773). Views and opinions expressed are, however, those of the author only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Support for this research was provided to H.P.A. by the University of Wisconsin–Madison, Office of the Vice Chancellor for Research and Graduate Education, with funding from the Wisconsin Alumni Research Foundation; by the Center for Demography of Health and Aging at the University of Wisconsin–Madison, which is supported by the National Institute on Aging grant P30 AG017266; and by the Herb Kohl Public Service Research Competition through the La Follette School of Public Affairs, University of Wisconsin–Madison.

Notes

References