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Rectangularization Revisited: Variability of Age at Death within Human Populations
Abstract
Rectangularization of human survival curves is associated with decreasing variability in the distribution of ages at death. This variability, as measured by the interquartile range of life table ages at death, has decreased from about 65 years to 15 years since 1751 in Sweden. Most of this decline occurred between the 1870s and the 1950s. Since then, variability in age at death has been nearly constant in Sweden, Japan, and the United States, defying predictions of a continuing rectangularization. The United States is characterized by a relatively high degree of variability, compared with both Sweden and Japan. We suggest that the historical compression of mortality may have had significant psychological and behavioral impacts.
... Since the 1980s, the development and application of inequality indices to lifespan distributions have proliferated (Goldman and Lord, 1986;Hanada, 1983;Myers and Manton, 1984;Silber, 1983Silber, , 1988Vaupel, 1986). Demographic research on lifespan inequality gained traction toward the early 2000s, when a series of seminal papers explored the properties of inequality measures in the context of length of life and demonstrated their theoretical and substantive significance (Edwards and Tuljapurkar 2005;Kannisto 2000;Shkolnikov, Andreev, and Begun 2003;Wilmoth and Horiuchi 1999). Over the following years, calls to monitor lifespan variability in addition to life expectancy grew (Edwards 2011;OECD 2007;Smits and Monden 2009; van Raalte, Sasson, and Martikainen 2018). ...
... It has an intuitive interpretation-the average lifespan given a fixed mortality schedule-and is a function of demography's most foundational tool, the life table. Yet, as many have noted over the past few decades, life expectancy is a central longevity indicator which overlooks inequalities in length of life within populations or social groups (Cheung et al. 2005;Wilmoth and Horiuchi 1999). Much like the gross national income per capita measures average economic performance, irrespective of how income is distributed in society, so does the life expectancy measures only how well a population is doing on average with no indication of how equal or unequal lifespans are distributed among individuals ( van Raalte, Sasson, and Martikainen 2018). ...
... Another approach was to develop new measures, motivated by the concept of life table entropy and its relationship with the survival curve, which could be expressed in years of life expectancy lost (Demetrius, 1978;Goldman and Lord, 1986;Keyfitz, 1977;Vaupel, 1986). However, demographic research on lifespan inequality became widespread only around the early 2000s, after a series of seminal papers provided further conceptual impetus for studying the dynamics of variability in age at death within and between populations (Edwards and Tuljapurkar 2005;Kannisto 2000;Shkolnikov, Andreev, and Begun 2003;Wilmoth and Horiuchi 1999). ...
Research on mortality inequalities has proliferated in demography in recent decades, documenting disparities between nations and within them across multiple social dimensions. Yet, this literature remains largely descriptive and atheoretical. In this paper , I identify three open questions in need of theoretical development. First, I identify a general shift from gender (and race) based mortality inequalities to class-based inequalities across low-mortality countries. I argue that this shift may be better understood by focusing on the structural determinants of population health, in addition to explanations grounded in individual behavior and risk factors. Second, a growing body of literature has called for moving beyond group differences in life expectancy and adopting the concept of lifespan inequality. However, the drivers of lifespan inequality are not well understood. I argue that a comprehensive framework is needed for clarifying the interplay of nature, nurture, and chance in shaping variability in individual lifespans. Third, I draw attention to the causal role that mortality inequalities may play in driving social change. I argue that differential exposure to death in one's network of social relations may give rise to different modes of thinking, feeling , and acting, and in turn lead to group differences in preferences, actions, and outcomes.
... We therefore provide a tentative account of how the present considerations might apply to human data. There has been a global trend of rectangularization of human survival curves in the past century 7 -in many countries, the median lifespan has increased and survival curve steepness has increased as well 63 (Fig. 6a). There is a tight correlation between life expectancy and lifespan equality (a measure of steepness) 64 . ...
... There is a tight correlation between life expectancy and lifespan equality (a measure of steepness) 64 . There are also exceptions in certain countries in which lifespan variability has increased or remained constant in recent decades 63 . ...
Longevity research aims to extend the healthspan while minimizing the duration of disability and morbidity, known as the sickspan. Most longevity interventions in model organisms extend healthspan, but it is not known whether they compress sickspan relative to the lifespan. Here, we present a theory that predicts which interventions compress relative sickspan, based on the shape of the survival curve. Interventions such as caloric restriction that extend mean lifespan while preserving the shape of the survival curve, are predicted to extend the sickspan proportionally, without compressing it. Conversely, a subset of interventions that extend lifespan and steepen the shape of the survival curve are predicted to compress the relative sickspan. We explain this based on the saturating-removal mathematical model of aging, and present evidence from longitudinal health data in mice, Caenorhabditis elegans and Drosophila melanogaster. We apply this theory to identify potential interventions for compressing the sickspan in mice, and to combinations of longevity interventions. This approach offers potential strategies for compressing morbidity and extending healthspan.
... By modelling the life-table death counts, we can understand a redistribution of life-table death counts, where deaths at younger ages gradually shift towards older ages. Lifespan variability can be assessed using measures like the interquartile range or the Gini coefficient (see Wilmoth & Horiuchi 1999, van Raalte & Caswell 2013, Debón et al. 2017. In life-table death counts, a Gini coefficient of 0 represents perfect inequality among ages, while 1 indicates perfect equality, with deaths occurring at the same age. ...
... In com par ing existing approaches and our novel method, we use the var i ance of the age-at-death dis tri bu tion as a mea sure of lifespan inequal ity, fre quently employed in appli ca tions (Aburto et al. 2023;Xu et al. 2021). Although other mea sures have been pro posed (van Raalte and Caswell 2013;Wilmoth and Horiuchi 1999), the var iance can be read ily decomposed into its com po nents (Shorrocks 1982), which is partic u larly use ful for our cross-cohort approach. For the sake of com pa ra bil ity, across all approaches con sid ered (period, CAL, overlapping cohorts), we eval u ate lifespan inequal ity exclu sively with the var i ance of the age-at-death dis tri bu tion. ...
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.
... Moreover, the spread of the distribution indicates lifespan variability. A decrease in variability over time can be observed directly and can be measured by the Gini coefficient (Wilmoth and Horiuchi, 1999;Debón et al., 2017). ...
This paper presents several forecasting methods to model and forecast subnational age distribution of death counts. The age distribution of death counts has many similarities to probability density functions, which are nonnegative and have a constrained integral, and thus live in a constrained nonlinear space. To address the nonlinear nature of objects, we implement a cumulative distribution function transformation that has an additional monotonicity. Using the Japanese subnational life-table death counts obtained from the Japanese Mortality Database (2025), we evaluate the forecast accuracy of the transformation and forecasting methods. The improved forecast accuracy of life-table death counts implemented here will be of great interest to demographers in estimating regional age-specific survival probabilities and life expectancy.
... We use the standard deviation of the age-at-death distribution to measure lifespan variation due to its simplicity for interpretation and the fact that it is in the same scale as life expectancy (years). Moreover, measures of lifespan variation are highly correlated with each other [45,47], so the choice of measure will likely not affect our main conclusions to a great extent. ...
Commonly used measures of socioeconomic inequalities in mortality, such as the slope and the relative index of inequality, are based on summary measures of the group-specific age-at-death distributions (e.g. standardized mortality rate or life expectancy). While this approach is informative, it ignores valuable information contained in the group-specific distributions. A recent approach applied a measure of distributional dissimilarity (the non-overlap index) to measure lifespan stratification. In this paper, we rigorously evaluate and further implement the multi-group extension of the non-overlap index () to measure socioeconomic inequalities in mortality across a number of groups, and assess whether differences across countries and over time are driven by mortality or compositional changes in two applications with different data availability: educational groups (Sweden and Denmark) and groups defined by an area-level deprivation measure (England). Our findings suggest that the multi-group is sensitive not only to changes in the means or variances, but also to broader mortality changes that affect distributional shapes. The method can be employed to any context where mortality rates by age are available by sub-groups. Furthermore, levels and trends in mortality inequalities computed with the multigroup often differ compared to other conventional summary-based measures. Moreover, we find that the contribution of mortality changes to changes in inequalities is generally greater than that of the changes in the population composition. Whereas levels and trends of inequalities may depend on whether life expectancy- or lifespan variation-based measures are employed, the multi-group provides a holistic perspective by capturing both dimensions simultaneously.
... For the purposes of this study, the upper quartile (75th percentile) indicates the age at which 75% of the hypothetical cohort of 100 000 are likely to exit, while the lower quartile (25th percentile) represents the age at which 25% of this cohort exit from the system. The interquartile range ( IR ), calculated from the upper and lower quartiles, indicates the number of years in which half of the hypothetical cohort will exit after one quarter of the generation has already exited [42]. ...
Background
The decrease in the number of healthcare workers and the resulting deterioration in healthcare quality and availability have been subjected to intensive discussion in Czechia in recent years. Estimating future healthcare worker capacities requires a detailed analysis of their “movement” within the healthcare system. This study focuses on exits of the primary care physicians from the healthcare system in Czechia.
Methods
Using anonymised data obtained from the largest Czech health insurance company (2012–2022), we constructed working life tables and calculated working life expectancy, which indicates the expected average number of remaining years of work at the exact age of the physician. The study focuses on primary care physicians, who are crucial for the effective functioning of the healthcare system.
Results
At age 50, working life expectancy was 20 years for female physicians and approximately 21 years for male physicians. Over the monitored period, working life expectancy decreased by 1 year for both genders. Gynaecologists had the longest working life expectancy, while dentists had the shortest.
Conclusions
The decrease in the working life expectancy and the length of tenure indicates the need to create favourable conditions for the extension of the working lives of physicians to avoid early exits from the system.
... Over the last decades, our understanding of human mortality patterns has grown significantly, particularly at older ages. These studies have largely been conducted in high-income countries about past, current and future trends of mortality and longevity (Oeppen & Vaupel, 2002;Tuljapurkar & Edwards, 2011;Vallin & Meslé, 2009;Wilmoth & Horiuchi, 1999;Wilmoth et al., 2000) with some country-specific and cross-country comparative analyses on older age mortality (Crimmins et al., 2010;Vaupel et al., 2011aVaupel et al., , 2011b. ...
Most studies on old age mortality and survival focus on high-income countries, leaving limited knowledge about these trajectories in low- and middle-income countries. We use the longest-run longitudinal study of aging in Latin America, the Mexican Health and Aging Study (MHAS), to assess mortality and survival in the Mexican older adult population. We examine the likely impact of survey attrition and missing date of death on estimates of age-specific death rates, life expectancy, and the link between sociodemographic characteristics and mortality risk among Mexican older adults. Results show attrition of less than 6% of the baseline sample in MHAS from 2001 to 2015. Being lost to follow-up (LFU) is associated with age, education, and place of residence. Age-specific death rates and life expectancy estimates in MHAS align with vital statistics suggesting minimal impact of survey attrition in these estimates at older ages. However, ignoring sample attrition produces statistically significant educational gradients in mortality among males (but not among females), but imputing attrition and/or death date deaths reverses this pattern. Thus, we recommend imputing vital status by, for example, assuming attrited respondents survived to the midpoint of their LFU interval and assessing mortality determinants, including imputed cases. We also found sizeable sex differences in life expectancy at age 50 favoring women with larger sex differences in more populous places. We conclude that MHAS reliably supports the study of older age mortality and survival in Mexico, offering a unique chance to enhance knowledge in a middle-income country in the Americas.
This paper proposes a novel approach for assessing changes in the expected present value of life annuities by adapting decomposition techniques traditionally applied in demographic research. Building upon Vaupel and Canudas–Romo’s method, a closed-form relationship is derived within the continuous framework to express changes in the expected present value of life annuities. By employing commutation functions, the concept of financially adjusted life-table entropy is introduced to capture the interplay between mortality and interest rate changes, offering a new and comprehensive measure for evaluating annuity price sensitivity. The approach is illustrated through a numerical application using life tables from Italy and the United Kingdom.
Introduction: Population Without Age.- The Life Table.- The Matrix Model Framework.- Mortality Comparisons The Male-Female Ratio.- Fixed Regime of Mortality and Fertility: The Uses of Stable Theory.- Birth and Population Increase from the Life Table.- Birth and Population Increase from Matrix Population Models.- Reproductive Value from the Life Table.- Reproductive Value from Matrix Models.- Understanding Population Characteristics.- Markov Chains for Individual Life Histories.- Projection and Forecasting.- Perturbation Analysis of Matrix Models.- Some Types of Instability.- The Demographic Theory of Kinship.- Microdemography.- The Multi-State Model.- Family Demography.- Heterogeneity and Selection in Population Analysis.- Epilogue: How Do We Know the Facts of Demography?.
Can death rates be reduced for octogenarians, nonagenarians, and even centenarians? It is widely assumed that mortality at advanced ages is attributable to old age per se and that death rates at advanced ages cannot be substantially reduced. Using a larger body of data than previously available, the authors find that developed countries have made progress in reducing death rates even at the highest ages. Furthermore, the pace of this progress has accelerated over the course of the twentieth century. In most developed countries outside Eastern Europe, average death rates at ages 80-99 have declined at a rate of 1 to 2 percent per year for females and 0.5 to 1.5 percent per year for males since the 1960s. For an aggregate of nine countries with reliable data through 1991, the annual average rate of improvement between 1982-86 and 1987-91 was 1.7 percent for male octogenarians and 2.5 percent for female octogenarians.
The authors examine some implications of current trends in life expectancy and morbidity in developed countries such as the United States. In particular they "demonstrate empirically that there is as yet no clear evidence supporting the idea of compression of mortality in either human populations or in experimental animal models; discuss statistical and mathematical modeling issues--including open research problems pertaining to the measurement of compression of mortality morbidity and disability; [and] relate debates about compression to population forecasting and health policy agendas." (EXCERPT)
Time series methods are used to make long-run forecasts, with confidence intervals, of age-specific mortality in the United States from 1990 to 2065. First, the logs of the age-specific death rates are modeled as a linear function of an unobserved period-specific intensity index, with parameters depending on age. This model is fit to the matrix of U.S. death rates, 1933 to 1987, using the singular value decomposition (SVD) method; it accounts for almost all the variance over time in age-specific death rates as a group. Whereas e0 has risen at a decreasing rate over the century and has decreasing variability, k(t) declines at a roughly constant rate and has roughly constant variability, facilitating forecasting. k(t), which indexes the intensity of mortality, is next modeled as a time series (specifically, a random walk with drift) and forecast. The method performs very well on within-sample forecasts, and the forecasts are insensitive to reductions in the length of the base period from 90 to 30 years; some instability appears for base periods of 10 or 20 years, however. Forecasts of age-specific rates are derived from the forecasts of k, and other life table variables are derived and presented. These imply an increase of 10.5 years in life expectancy to 86.05 in 2065 (sexes combined), with a confidence band of plus 3.9 or minus 5.6 years, including uncertainty concerning the estimated trend. Whereas 46% now survive to age 80, by 2065 46% will survive to age 90. Of the gains forecast for person-years lived over the life cycle from now until 2065, 74% will occur at age 65 and over. These life expectancy forecasts are substantially lower than direct time series forecasts of e0, and have far narrower confidence bands; however, they are substantially higher than the forecasts of the Social Security Administration's Office of the Actuary.