Mortality postponement and compression at older ages in human cohorts

Abstract A key but unresolved issue in the study of human mortality at older ages is whether mortality is being compressed (which implies that we may be approaching a maximum limit to the length of life) or postponed (which would imply that we are not). We analyze historical and current population mortality data between ages 50 and 100 by birth cohort in 19 currently-industrialized countries, using a Bayesian technique to surmount cohort censoring caused by survival, to show that while the dominant historical pattern has been one of mortality compression, there have been occasional episodes of mortality postponement. The pattern of postponement and compression across different birth cohorts explain why longevity records have been slow to increase in recent years: we find that cohorts born between around 1900 and 1950 are experiencing historically unprecedented mortality postponement, but are still too young to break longevity records. As these cohorts attain advanced ages in coming decades, longevity records may therefore increase significantly. Our results confirm prior work suggesting that if there is a maximum limit to the human lifespan, we are not yet approaching it.

Citation: McCarthy D, Wang P-L (2023) Mortality postponement and compression at older ages in human cohorts. PLoS ONE 18(3): e0281752. https://doi.org/10.1371/journal.pone.0281752 Editor: Calogero Caruso, University of Palermo, ITALY Received: April 21, 2022; Accepted: January 31, 2023; Published: March 29, 2023 Copyright: © 2023 McCarthy, Wang. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All data are publicly available at the Human Mortality Database (www.mortality.org). Code and data can be found on Zenodo at https://doi.org/10.5281/zenodo.7643750. Funding: The author(s) received no specific funding for this work. Competing interests: The authors have declared that no competing interests exist.

Introduction Whether or not there is a limit to the human lifespan has been a subject of debate for millennia. Historical estimates of the maximum possible lifespan strongly suggest that it has increased substantially over recorded history. The Hebrews of the late Bronze Age famously regarded 80 years as the maximum length of a human life. (Psalm 90:10; <500BC); around 1,000 years later, the ancient Romans set their official estimate of the maximum, the so-called saeculum naturale, at 100 or 110 years [1,2]. Modern longevity records are higher still: the current human longevity record is 122, but has remained unchanged since 1997. Scholarly interest in the topic has not abated [3] use the slow change in longevity records in recent years to argue that the human lifespan has reached an absolute limit, a finding supported by [4,5] use biological hypotheses to reach the same conclusion [6–8], on the other hand, interpret recent improvements in old-age mortality and the pattern of deaths at older ages to assert the opposite. An important recent study in this area is [9]. They use period data to show that the gap between the percentiles of the distribution of age at death of those older than 65 remained roughly constant between 1960 and 2010 in 20 countries. They conclude that period effects therefore appear to be driving the improvement in longevity at older ages and that lifespan does not appear to be approaching an upper limit. In this paper, like [9], we analyze mortality of older individuals in richer countries using the Human Mortality Database [10]. However, we analyze mortality by birth cohort, rather than by period. Using cohort data follows a fixed set of individuals over time, and is therefore most suited to clarifying the biological mechanisms underlying mortality. In particular, the use of cohort data may avoid the conflation in period data of changes in mortality rates over time and/or age with changes across cohorts. Cohort analysis is made difficult by cohort censoring due to survival, which we surmount using a Bayesian estimation approach [11–13] that improves the precision of estimates for non-extinct cohorts. Mortality rates at the highest ages are subject to a great deal of error because the number of people reaching these ages is very small. We therefore approximate mortality by fitting the Gompertz law, which posits that mortality rates in humans increase roughly exponentially with age after around age 50 [14], to each individual cohort between the ages of 50 and 100. As we will demonstrate, the Gompertz law fits cohort mortality data extremely well in this age range, especially in recently extinct cohorts. We use the Gompertz law to estimate the age (which we call the Gompertzian Maximum Age or GMA) at which individuals first reach an assumed mortality plateau, and test whether this age has changed across birth cohorts or not. If the GMA is found to be constant, this would provide strong evidence in favor of the existence of a maximum limit to human life. In this case, improvements in mortality rates at younger old ages will lead to a compression in the distribution of age at death, called mortality compression. If, however, the GMA increases across cohorts, mortality is being postponed. The presence of mortality postponement would suggest that if there is a maximum limit to human life, it has not yet been reached. A key parameter in the Gompertz law is the exponential rate of increase in the observed mortality rates with age, called the Rate of Demographic Aging (RDA). Because the RDA determines the distribution of age at death, changes in the RDA across cohorts are a key measure of whether mortality is being postponed or compressed. As shown in Fig 1, if lifespans are approaching an absolute limit, improvements in mortality at younger old ages will be associated with increases in the RDA and the distribution of age at death will be increasingly compressed at advanced ages below the limit. On the other hand if the RDA is constant, improvements in mortality at younger ages will simply shift the distribution of ages at death to the right, suggesting that old-age mortality is being postponed and that a limit to the human lifespan, if it exists, is still far away. PPT PowerPoint slide

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TIFF original image Download: Fig 1. Dividing changes in mortality rates between compression and postponement. https://doi.org/10.1371/journal.pone.0281752.g001 Despite the good fit of the Gompertz law between ages 50 and 100, there are strong theoretical reasons to expect the RDA to fall at extreme old ages. The most likely of these is differential frailty within each cohort [15], leading to selection effects as the frailest individuals in each cohort die earlier, on average, than the less frail. This would cause the RDA to fall gradually at extremely advanced ages, even if the rate of increase of mortality with age of each individual is identical. Such a fall has been observed empirically in the mortality of super-centenarians. In fact, as more high-quality data becomes available, the evidence in support of a levelling-off of the risk of dying has increased [16] found that at extreme ages, mortality probabilities appear to be independent of age and sex, and are around 1/2; [17] suggest that the mortality hazard rate among super-centenarians in Italy appears to level off at around 2/3 (and so annual mortality probabilities are around 1/2), a finding confirmed in other countries by [18]. For analytical convenience, and to be consistent with these findings, we therefore hypothesize the existence of a GMA–the youngest age at which the Gompertz law would predict the mortality hazard rate to be 2/3 –and estimate confidence intervals for this GMA and the age at death of the longest-lived person in each cohort. This approach allows us to disaggregate changes in remaining life expectancy at 50 in historical and current cohorts between postponement and compression explicitly using an analytical method. We emphasize, however, our conclusions on compression and postponement do not depend on either the existence of this plateau or on its assumed level. For instance, if we set the mortality plateau at 1, rather than 2/3, remaining theoretical life expectancy at age 50 would fall by around 5 days, with little variation across cohorts. Removing the plateau altogether causes a fall of a further 4 hours. These values are almost two orders of magnitude smaller than our smallest confidence intervals. Our analysis reveals that over much of our data, the GMA appears to have remained unchanged, in some countries for centuries. But we do find past and current episodes where mortality postponement has occurred and the GMA has risen. In particular, for cohorts born between 1910 and 1940, we project that the GMA will increase rapidly, confirming the finding of [9] that in recent data, longevity does not appear to be approaching an upper limit. In contrast to this work, however, we show that old-age mortality patterns can be well explained by cohort effects, rather than period effects. These cohort patterns show further why longevity records have not changed in recent decades despite the well-documented improvements in mortality at older ages across much of the industrialized world in recent years. We show that confidence intervals for the length of life of the longest-lived person in each cohort in each country derived using our approach fit historical data on extreme longevity very well. Extrapolating these results forward, we show that it is likely that longevity records will rise as cohorts born after 1910 reach advanced ages, although our projections of by how much they will rise depend on our modelling assumptions.

Materials and methods Although more complex approaches are possible, we define the base mortality of individuals aged x and born in year c to be μ x,c and use the Gompertz law to write: (1) where δ c is the RDA of cohort c and λ( 50c ) ≡ log(μ 50,c ). This approach makes the assumption that each cohort is endowed with δ c and λ 50c at age 50, and that these remain fixed thereafter. These fixed cohort parameters can be regarded as proxies for causes of death (such as heart disease, stroke, cancer and neurological disorders) that are the consequence of lifestyle and other factors operating over very long time scales. Other than Covid-19, these have become the predominant cause of death among the elderly. We disregard data above age 100 because of evidence that the Gompertz law ceases to apply at these ages, although data here is so sparse that including it in our estimation with appropriate credibility makes little difference. We include data between ages 50 and 80 to ensure that our estimates of the RDA are as precise as possible for each cohort. To account for shorter-latency causes of death, we allow a cohort’s observed mortality to fluctuate around base mortality due to calendar year effects (e.g. epidemics such as Covid-19 and the 1918 Spanish flu, climatic fluctuations, wars and famines). The error term also captures model misfit (e.g. the gradual fall in the RDA at extreme old age), data errors, and random variation. We therefore write the observed central rate of death at age x of an individual born in year c, defined as m x,c , as: (2) We estimated Eq (2) using population mortality data, organized by birth year, for males and females between the ages of 50 and 100 for a set of currently 19 rich industrialized countries (although in some cases our data precede industrialization) from the Human Mortality Database, described in Table 1, along with the set of cohorts used to estimate the three prior distributions we use to test robustness [10]. PPT PowerPoint slide

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TIFF original image Download: Table 1. Description of data. https://doi.org/10.1371/journal.pone.0281752.t001 Estimation technique Details are provided in appendix M1. Combining (1) and (2), the central rate of death at age 50 ≤ x ≤ 100 of an individual who was born in year c, written m x,c , as: (3) We use changes in the estimates of λ( 50c ) and δ c measured off extinct or nearly-extinct cohorts (where estimation errors are small; nearly-extinct cohorts have data up to age 90) to formulate a Bayesian prior for how these parameters will change over cohorts that are currently censored by survival, and use Bayes’ Theorem to calculate a joint posterior distribution of the parameters for both extinct and censored cohorts conditional on the data and the chosen prior. We allow for fixed-age effects to capture systematic deviations across age from the Gompertz law, such as the slow-down in mortality rates at very high ages. We then use the Metropolis-Hastings (MH) algorithm to draw a sample from this posterior distribution, denoted , with mode and . We then define the Gompertzian Maximum Age of cohort c (GMA, denoted Λ c ) as that age at which the base mortality hazard in Eq (1) first reaches 2/3 (so mortality hazards have likely plateaued and annual death probabilities are around 1/2): (4) Division between compression and postponement Details are provided in appendix M2. We used our MH sample to quantify the changes in remaining cohort base life expectancy at age 50 between cohorts born ten years apart due to compression and postponement, as well as 95% confidence intervals. As shown in Fig 1, the full change in λ 50,c and any consequent change in δ c needed to keep Λ c fixed were ascribed to compression, and any further changes in δ c to postponement. 95% confidence intervals for the GMA and the maximum length of life Details are provided in appendix M3. We then calculated 95% confidence intervals for Λ c directly using our sample and (4), and confidence intervals for M c , the age at death of the longest-lived person in each cohort conditional on at least one person in each cohort reaching age Λ c , under the assumption that the mortality hazard at ages older than Λ c (so after the plateau) are constant and equal to two thirds. Although Λ c only changes with underlying mortality rates, M c will increase as cohort size increases even if mortality rates remain the same.

Discussion We summarize historical mortality data in 19 currently-industrialized countries by birth cohort using a variant of the Gompertz mortality law, and find that it fits cohort mortality data extremely well. Using this law, we identify the youngest age at which individuals in each cohort reach an assumed mortality plateau, which we call the Gompertzian Maximum Age (GMA). We find that over much of the period covered by our data, there was no increase in the GMA. Historical improvements in life expectancy were therefore largely the result of mortality compression. We demonstrate, however, that there have been episodes where the GMA increased. The presence of these episodes of mortality postponement suggests that the maximum length of a human life is not, in fact, fixed. The first episode of mortality postponement that we identify occurred for cohorts born in the early part of the second half of the 19th century, and was more pronounced for females than for males. Over this period, the GMA increased by around 5 years. We can only speculate as to the causes of this increase, but as the first of these cohorts reached age 50 just after 1900 and the last reached age 100 in 1980, this may be related to a first wave of improvements in public health and medical technology. We identify a second, and much more significant, episode of mortality postponement, which is affecting cohorts born between 1910 and 1950 (so those currently aged between 70 and 110). We estimate that the GMA for these cohorts may increase by as much as 10 years, and remaining life expectancy at age 50 by as much as 8 years, depending on the country. The timing of these episodes of mortality postponement explain why longevity records have been so slow to increase in recent years–cohorts old enough to have broken longevity records were too old to experience the current bout of postponement–and identifies significant potential for longevity records to rise by the year 2060 as younger cohorts, who did experience it, reach advanced old age. Our results on the division of changes in remaining life expectancy at age 50 across cohorts between compression and postponement are robust to our modelling choices. Likewise, our conclusion that longevity records will likely be broken in the coming decades is also robust to a wide range of possible assumptions. But our predictions of precisely by how much these records will rise, and when, depend on our modelling assumptions, in particular on the maximum mortality rate we assume. We emphasise further that cohorts born before 1950 will only have the potential to break existing longevity records if policy choices continue to support the health and welfare of the elderly, and the political, environmental and economic environment remains stable. The emergence of Covid-19 and its outsize effect on the mortality of the elderly provides a salutary warning that none of this is certain. If, however, the GMA does increase as the current mortality experience of incomplete cohorts suggests is likely, the implications for human societies, national economies and individual lives will be profound.