Search

This chapter examines the implications of mortality heterogeneity, that is the dispersion of longevity prospects within the population. It begins by discussing the extended Gompertz–Makeham model, as well as the compensation law of mortality, linking moments of the remaining lifetime random variable. It then introduces non-chronological measures of age, such as biological age and (especially) longevity risk-adjusted age to illustrate its dispersion. This chapter illustrates how true age can differ around the world and even within countries, based on wealth and income. The main computational implication of this chapter is that the human longevity random variable Tx, depends on (much) more than just chronological age x. Such heterogeneity must be accounted for in any intelligent drawdown methodology or pensionization scheme.

We analyze lung cancer mortality by age and sex in the United States, 1959-2013. It is already known that male lung cancer death rates exceed those of women and that tobacco use is the leading reason for the sex difference. We elaborate on this knowledge by showing that unlike most causes of death, lung cancer mortality patterns by age are a very good fit to a quadratic-Gompertz model, i.e., log mortality rates are quadratic by age, with the mode above age 70. With a little additional historical data on sex differences in tobacco use, the quadratic models help to paint a clear quantitative picture of behavior-led convergence in lung cancer mortality by sex. The shape of the sex-specific mortality curves have converged dramatically: since 1983, the sexes have had statistically indistinguishable shapes of their quadratic-Gompertz mortality curves. Female lung cancer mortality patterns have shown a transformation from a nonsmoking to a smoking pattern. The modal age of the quadratic-Gompertzian lung cancer death rates has potential application in countries at earlier stages of the global tobacco epidemic. Where the modal age is falling (year-on-year), we can infer that the local lung cancer epidemic is getting worse, and where it is rising, that it has begun to abate.

Building on our earlier research (Case and Deaton 2015), we find that mortality and morbidity among white non-Hispanic Americans in midlife since the turn of the century continued to climb through 2015. Additional increases in drug overdoses, suicides, and alcohol-related liver mortality—particularly among those with a high school degree or less—are responsible for an overall increase in all-cause mortality among whites. We find marked differences in mortality by race and education, with mortality among white non-Hispanics (males and females) rising for those without a college degree, and falling for those with a college degree. In contrast, mortality rates among blacks and Hispanics have continued to fall, irrespective of educational attainment. Mortality rates in comparably rich countries have continued their premillennial fall at the rates that used to characterize the United States. Contemporaneous levels of resources—particularly slowly growing, stagnant, and even declining incomes—cannot provide a comprehensive explanation for poor mortality outcomes. We propose a preliminary but plausible story in which cumulative disadvantage from one birth cohort to the next—in the labor market, in marriage and child outcomes, and in health—is triggered by progressively worsening labor market opportunities at the time of entry for whites with low levels of education. This account, which fits much of the data, has the profoundly negative implication that policies—even ones that successfully improve earnings and jobs, or redistribute income—will take many years to reverse the increase in mortality and morbidity, and that those in midlife now are likely to do worse in old age than the current elderly. This is in contrast to accounts in which resources affect health contemporaneously, so that those in midlife now can expect to do better in old age as they receive Social Security and Medicare. None of this, however, implies that there are no policy levers to be pulled. For instance, reducing the overprescription of opioids should be an obvious target for policymakers.

Mounting evidence suggests that early-life conditions have an enduring effect on an individual’s mortality risks as an adult. The contribution of improvements in early-life conditions to the overall decline in adult mortality, however, remains a debated issue. We provide an estimate of the contribution of improvements in early-life conditions to mortality decline after age 30 in Dutch cohorts born between 1812 and 1921. We used two proxies for early-life conditions: median height and early-childhood mortality. We estimate that improvements in early-life conditions contributed more than five years or about a third to the rise in women’s life expectancy at age 30. Improvements in early-life conditions contributed almost three years or more than a quarter to the rise in men’s life expectancy at age 30. Height appears to be the more important of the two proxies for early-life conditions.

Narrowing of the life expectancy gap In the United States, the rich can expect to enjoy better health and a longer life than the poor. Despite policies directed at improving the health of both the young and the poor, there is little evidence that this relationship has changed. Currie and Schwandt looked specifically at the life expectancy of present-day children and young adults, finding that mortality inequality has in fact declined over the past 25 years (see the Perspective by Bailey and Timpe). Science , this issue p. 708 ; see also p. 661

This paper develops a computational framework for inverting Gompertz-Makeham (GM) hazard rates, in a manner consistent with a compensation law of mortality for heterogenous populations, to define a Longevity-Risk-adjusted Global (LRaG) Age. To illustrate its salience and relative magnitude, the paper implements and calibrates this framework using rates from the Human Mortality Database (HMD). Among other things, this paper demonstrates that when properly benchmarked, the {\em longevity-risk-adjusted global age} of a 55-year-old Swedish male is 48, whereas a 55-year-old Russian male is closer in age to 67. The paper also discusses the connection between the LRaG age and the related concept of *biological* age. In a world of growing mortality heterogeneity, a new and distinct LRaG age could be used for pension and retirement policy. Perhaps a salient and transparent metric that adjusts chronological age for longevity risk might help capture the public's attention, educate them and induce them to take action to increase the odds of retirement success, by, for example, working longer and retiring later.

Who {\em values} life annuities more? Is it the healthy retiree who expects to live long and might become a centenarian, or is the unhealthy retiree with a short life expectancy more likely to appreciate the pooling of longevity risk? What if the unhealthy retiree is pooled with someone who is much healthier and thus forced to pay an implicit loading? To answer these and related questions this paper examines the empirical conditions under which retirees benefit (or may not) from longevity risk pooling by linking the {\em economics} of annuity equivalent wealth (AEW) to {\em actuarially} models of aging. I focus attention on the {\em Compensation Law of Mortality} which implies that individuals with higher relative mortality (e.g. lower income) age more slowly and experience greater longevity uncertainty. Ergo, they place higher utility value on the annuity. The impetus for this research today is the increasing evidence on the growing disparity in longevity expectations between rich and poor., 57 pages, 8 figures, 5 tables

Background: Due to the result of improving longevity the number of survivors at old ages increased in India. The annual growth rate of the population aged 60 and above is higher than the rate of growth of the total population. So the process population ageing and associate problems become serious issue in the country. This study examines the changes in old age mortality in India from 1971-2011 and also examined the age group contribution to the gain in e60. Data and Methods: Measure used here to describing the mortality change was life expectancy at age 60 (e60). Using Age Specific Death Rates from Sample Registration System, new life tables were constructed for the analysis. Pollardâ’s Decomposition analysis were applied for determining age group contribution. Results: Old age mortality in India shows a considerable decline during the period but the pace of mortality decline was slowdown after 1990. Most of the increase in life expectancy at age 60 came from the result of decreasing mortality at young old and oldest age for females. While among males decreasing mortality at young old and old-old ages was important for increase in life expectancy at age 60. Conclusion: Understanding old-age mortality trend and possible determinants are important for health care professionals for constructing health care strategies on possible mortality development in the future.

Background Reducing inequalities in health is a great challenge for public health, but how relative and absolute inequalities in mortality respond to changes in mortality by socioeconomic group is not well understood. Methods We derived arithmetically what combinations of changes and starting levels of mortality by socioeconomic group produce narrowing, and what combinations produce widening of relative and absolute inequalities in mortality. We then determined empirically how often these scenarios actually occur with data on inequalities in cause-specific mortality in five European countries spanning four decades (1970–2010). Results Changes in the rate ratio depend exclusively on the ratio of relative mortality change between socioeconomic groups, whereas changes in the rate difference depend on whether the ratio of relative mortality change between socioeconomic groups is larger or smaller than the rate ratio. This implies that, in case of declining mortality and faster relative mortality decline in higher socioeconomic groups, the rate difference will increase until the rate ratio becomes equal to the ratio of relative mortality decline, but will then start to decline. In the most common scenario in our data set (starting rate ratio above 1.00 and faster relative mortality decline in higher than lower socioeconomic groups), the rate ratio indeed always goes up but the rate difference goes down in about half of all cases, sometimes after a period of growth. Conclusions A narrowing of absolute inequalities occurs under a wider range of conditions than a narrowing of relative inequalities in mortality.

(ProQuest: ... denotes formulae omitted.)1. BackgroundHuman mortality has undergone remarkable declines over the years. The increase in life expectancy is probably the best expression for the dramatic mortality decline in the last 170 years (Oeppen and Vaupel 2002). Improvements in living conditions, nutrition and medicine are among the main reasons for this development (Riley 2001; Oeppen and Vaupel 2002). These changes in economic, social, and sanitary conditions first triggered an important decline in infant, child, and early adult mortality, which contributed to the reduction in lifespan disparities (Wilmoth and Horiuchi 1999; Edwards and Tuljapurkar 2005; Vaupel, Zhang, and van Raalte 2011). As individuals became more homogeneous in their ages at death, a compression of the distribution of deaths in a more narrow age-interval was observed in many low-mortality countries in the first half of the twentieth century (Fries 1980; Wilmoth and Horiuchi 1999; Kannisto 2000, 2001; Cheung et al. 2009). Fries (1980) hypothesized that this dynamic can be interpreted as a compression of deaths against the upper limit of the human lifespan. Assuming a nearly negligible role for premature mortality, he stated the limit of the average age at death as approxi-mately 85 years, with 95% of all deaths occurring in an age range of 4 years deviation (Fries 1980). The "compression of mortality hypothesis" motivated a rich discussion on the occurrence and interpretation of this development. Several studies provided evidence for a compression, but emphasized that the achieved mortality levels differ substantially from Fries' predictions (Nusselder and Mackenbach 1996; Wilmoth and Horiuchi 1999; Cheung et al. 2005).After the period of strong compression, low-mortality countries entered a new era of change. Since the second half of the twentieth century, the main contributions to the increase in average age at death shifted from infant and early adult ages to old and very old-ages (Christensen et al. 2009). This generated changes in the mechanisms behind the increase in life expectancy (Wilmoth and Horiuchi 1999; Edwards and Tuljapurkar 2005; Smits and Monden 2009). The new mechanism behind improvement in life expectancy is best illustrated by a shift in the distribution of death toward older ages with a shape remaining nearly constant (Yashin et al. 2001; Bongaarts 2005; Cheung et al. 2005; Cheung and Robine 2007; Canudas-Romo 2008). Vaupel (1986), Vaupel and Gowan (1986) and Bongaarts (2005) were among the first to articulate the idea of shifting mortality. Canudas-Romo (2008) deepens this idea by studying the variability around and the change of the modal age at death. He finds that over time mortality shifts to higher ages, with approximately constant variability in age at death. He concludes that the shifting mortality pattern might be the new dynamic behind mortality improvements, subsequent to the compression process.The ages at which mortality reductions occur tend to determine the dominating mortality dynamic: compression or shift. Compression is more pronounced when mortality reductions occur at very young and adult ages (Nusselder and Mackenbach 1996; Wilmoth and Horiuchi 1999; Kannisto 2000; Cheung et al. 2005). On the other hand, shifting mortality requires changes at old and very old-ages (Canudas-Romo 2008). Vaupel, Zhang, and van Raalte (2011) report relatively stable variability patterns for survivors beyond age 50 in the last 100 years. Engelman, Caswell, and Agree (2014) and Engelman, Canudas-Romo, and Agree (2010), however, provide evidence for a modest expansion of lifespan variability for survivors at older ages, resulting from mortality improvement at these same ages.The measurement of compression and shifting mortality is an important issue, as both dynamics translate differently into survival, mortality density and hazard distributions (Wilmoth and Horiuchi 1999). Alterations are, however, visible in all three functions due to their interrelation. …

Objective: This study aimed to investigate the relationship between education and different indicators of material wealth with mortality, and to analyze whether this relationship varies with the leading causes of death. Methods: All persons aged 65 and older residing in Spain in 2001 were followed up for 7 years to determine their vital status. The relationship between mortality and four indicators of socioeconomic position (education, number of rooms in home, surface area of home and number of vehicles) was estimated in three age groups: 65-74, 75-84 and 85 and older. Rate ratios and relative index of inequality (RII) were calculated for general mortality and for the leading causes of death by Poisson regression. Results: In women, the mortality rate ratio for low vs. high educational level was 1.48 for persons aged 65-74, 1.43 for those aged 75-84 and 1.40 for those aged 85 and older. The respective rates for men were 1.30, 1.25 and 1.29. For the indicators of material wealth, the differences between morality rates in the lower vs. the higher socioeconomic categories decline with age. Mortality differences by the leading causes of death decline with age, except in the case of cancer in women and cardiovascular and digestive mortality in men according to educational level. Conclusions: Relative socioeconomic differences in mortality in the older Spanish population decrease with age using indicators of material wealth but not using educational level. The variation in the pattern of mortality by cause of death by level of education may be responsible for these findings.

The article discusses long-term mortality trends (since 1956) from external causes of death in Russia. Russia has long lagged behind developed countries in this domain. The level of mortality from external causes of death remains high and its structure is still archaic with large contribution of homicides, alcohol poisoning and injuries of undetermined intent. Excess number of deaths from life tables of Russia and Western European countries is compared. It is shown that in Russia the greatest excess losses are associated with mortality from poisonings among both sexes, suicide among men and homicide among women. Mortality from external causes, along with mortality from diseases of the circulatory system, has had a significant impact on life expectancy. In general, over the period 1956-2012 the increase in mortality from external causes in the 15-64 age group reduced life expectancy by 2.6 years for males and 0.7 years for females. The decline, starting in 2003, of mortality from external causes of death has slightly reduced the gap between Russia and developed countries, bringing the current Russian level closer to those levels reached in Russia in the mid-1960s and 1980s. However, given the fluctuations of mortality from external causes, it is premature to say whether the current decline in mortality is robust.

Progressive gains in life expectancy have brought increasing aging of developed countries’ populations and stimulated researches with respect to the limit of lifespan, the increasing presence of centenarians and the possible determinants of their “successful” ageing. In the international framework the population of the region of Sardinia was characterized by exceptional longevity. Through the comparison of life tables’ rates and life expectancy estimates of birth cohorts born in the period 1872-1910, this paper reports analyses of differences in overall mortality between Italy and the long-living community of Sardinia. The adoption of a longitudinal approach allows us to explore whether the previously detected differences at advanced ages are also observable in other age groups. In addition, the study focuses on the occurrence of mortality crossover as an indicator of significant mortality differences between populations. Results show that differences between the two populations are not limited to oldest old ages. Crossover points in mortality rates are observed in the first years of life and at age 50 years. Both intersections represent the points where differences in life expectancy are the highest. At age 5 years differences are of 1.8 years for females and 3.6 years for males, while at age 50 years they are respectively 1.5 years for females and 3.7 years for males. The observed differences suggest that possible explanations for differential mortality and crossover might be related to selection process timing and dynamics which could be determined by genetics, and specific causes of death, as well as by behavioural and environmental factors.

This paper looks at the fertility and mortality experience of racial and ethnic groups in the United States from the early 20th century to the present. The first part consist of a description and critique of the racial and ethnic categories used in the federal census and in the published vital statistics. The second part looks at these two dimensions of demographic behavior. There has been both absolute and relative convergence of fertility across groups. It has been of relatively recent origin and has been due, in large part, to stable birth rates for the majority white population combined with declining birth rates for blacks and the Asian-origin, Hispanic-origin, and American Indian populations. This has not been true for mortality. The black population has experienced absolute convergence but relative deterioration in mortality (neonatal and infant mortality, maternal mortality, expectation of life at birth, and age-adjusted death rates), in contrast to the American Indian and Asian-origin populations. The Asian-origin population actually now has age-adjusted death rates significantly lower than those for the white population. The disadvantaged condition of the black population and the deteriorating social safety net are the likely origins of this outcome. This is a clear indication of relative inequality, as the black population is not sharing as much in the mortality improvements in recent decades.

Objectives The purpose of this study was to explore the pattern of mortality above the age of 100 years. In particular, we aimed to examine whether Scandinavian data support the theory that mortality reaches a plateau at particularly old ages. Whether the maximum length of life increases with time was also investigated. Methods The analyses were based on individual level data on all Swedish and Danish centenarians born from 1870 to 1901; in total 3006 men and 10 963 women were included. Birth cohort-specific probabilities of dying were calculated. Exact ages were used for calculations of maximum length of life. Whether maximum age changed over time was analysed taking into account increases in cohort size. Results The results confirm that there has not been any improvement in mortality among centenarians in the past 30 years and that the current rise in life expectancy is driven by reductions in mortality below the age of 100 years. The death risks seem to reach a plateau of around 50% at the age 103 years for men and 107 years for women. Despite the rising life expectancy, the maximum age does not appear to increase, in particular after accounting for the increasing number of individuals of advanced age. Conclusion Mortality among centenarians is not changing despite improvements at younger ages. An extension of the maximum life span and a sizeable extension of life expectancy both require reductions in mortality above the age of 100 years. This article is protected by copyright. All rights reserved.

Mortality is a crucial dimension of wellbeing and inequality in a population, and mortality trends have been at the core of public debates in many Western countries. In this paper, we provide the first analysis of mortality inequality in Canada and compare its development to trends in the U.S. We find strong reductions in mortality rates across both genders and at all ages, with the exception of middle ages which only experienced moderate improvements. Inequality in mortality, measured across Canadian Census Divisions, decreased for infants and small children, while it increased slightly at higher ages. In comparison to the U.S., mortality levels in Canada improved at a similar rate despite lower initial levels. Inequality at younger ges, however, fell more strongly in the U.S., implying converging mortality gradients between the two countries.

Recent scientific publications suggest that human longevity records stopped increasing. Our finding that the mortality of centenarians has not decreased noticeably in recent decades (despite a significant mortality decline in younger age groups) is consistent with this suggestion. However, there is no convincing evidence that we have reached the limit of human life span. The future of human longevity is not fixed and will depend on human efforts to extend life span.

Physiological senescence is characterized by the increasing limitation of capabilities of an organism resulting from the progressive accumulation of molecular damage, which at group (cohort) level translates into, among other things, an increase in mortality risks with age. Physiological senescence is generally thought to begin at birth, if not earlier, but models of demographic aging (i.e., an increase in mortality risks) normally start at considerably later ages. This apparent inconsistency can be solved by assuming the existence of two mortality regimes: “latent” and “manifest” aging. Up to a certain age, there is only latent aging: physiological senescence occurs, but its low level does not trigger any measurable increase in mortality. Past a certain level (and age), molecular damage is such that mortality risks start to increase. We first discuss why this transition from latent to manifest aging should exist at all, and then we turn to the empirical estimation of the corresponding threshold age by applying Bai’s approach to the estimation of breakpoints in time series. Our analysis, which covers several cohorts born between 1850 and 1938 in 14 of the countries included in the Human Mortality Database, indicates that an age at the onset of manifest aging can be identified. However, it has not remained constant: it has declined from about 43 and 47 years, respectively, for males and females at the beginning of the period (cohorts born in 1850–1869) to about 31 for both males and females toward its end (cohorts born in 1920–1938). A discussion of why this may have happened ensues.

This study probes into the evolution in young-adult mortality according to urbanisation degree in Belgium and moves beyond mere description through decomposing mortality trends into changes in educational distribution and in overall mortality. As most of young-adult deaths are preventable and an enormous cost and loss to society, this study addresses a highly relevant public-health topic. Individual record-linked data between the Belgian censuses of 1991 and 2001 and register data on death and emigrations are used. Age-standardized mortality rates (ASMR), directly standardized to the European Population of 2013 are calculated with 95% confidence intervals (CI), as well as a decomposition measure to pinpoint the proportion mortality change attributable to differences in educational composition over time. The young-adult population consists of 2,458,637 19–34 year-olds in 1991, with 11,898 deaths in a five-year period, and is slightly smaller in 2001 with 2,174,368 young adults and 8138 deaths. Overall, there is a positive evolution towards lower young-adult mortality, with the strongest declines in men living in large urban areas (ASMR from 149.0 [CI 142.1–155.8] in 1991–1996 to 94.6 [88.9–100.3] in 2001–2006). Decomposition analysis shows that the decrease in male mortality in non-urban areas over time is largely due to changes in the educational composition, while mortality in urban areas mainly decreases because of a decline in overall mortality. In urban areas all educational groups have benefitted over time. This clearly demonstrates that living and growing up in an urban area does not always have to imply a health penalty, but can have health advantages as well.

Age trajectories of total mortality represent an irreplaceable source of information about aging. In principle, age affects mortality from all diseases differently than it affects mortality from external causes. External causes (accidents) are excluded here from all causes, and the resultant category "all-diseases" is tested as a helpful tool to better understand the relationship between mortality and age. Age trajectories of all-diseases mortality are studied in the six most populated countries of the South America during 1996-2010. The numbers of deaths for specific causes of death are extracted from the database of WHO, where the ICD-10 revision is used. The all-diseases mortality shows a strong minimum, which is hidden in total mortality. Two simple deterministic models fit the age trajectories of all-diseases mortality. The inverse proportion between mortality and age fits the mortality decreases up to minimum value in all six countries. All previous models describing mortality decline after birth are discussed. Theoretical relationships are derived between the parameter in the first model and standard mortality indicators: Infant mortality, Neonatal mortality, and Postneonatal mortality. The Gompertz model extended with a small positive quadratic element fit the age trajectories of all-diseases mortality after the age of 10 years.

This study examines historical patterns of aging through the perspectives of cohort evolution and mortality selection, where the former emphasizes the correlation across cohorts in the age dependence of mortality rates, and the latter emphasizes cohort change in the acceleration of mortality over the life course. In the analysis of historical cohort mortality data, I find support for both perspectives. The rate of demographic aging, or the rate at which mortality accelerates past age 70, is not fixed across cohorts; rather, it is affected by the extent of mortality selection at young and late ages. This causes later cohorts to have higher rates of demographic aging than earlier cohorts. The rate of biological aging, approximating the rate of the senescence process, significantly declined between the mid- and late-nineteenth century birth cohorts and stabilized afterward. Unlike the rate of demographic aging, the rate of biological aging is not affected by mortality selection earlier in the life course but rather by cross-cohort changes in young-age mortality, which cause lower rates of biological aging in old age among later cohorts. These findings enrich theories of cohort evolution and have implications for the study of limits on the human lifespan and evolution of aging.

Unobserved heterogeneity in mortality risk is pervasive and consequential. Mortality deceleration—the slowing of mortality’s rise with age—has been considered an important window into heterogeneity that otherwise might be impossible to explore. In this article, I argue that deceleration patterns may reveal surprisingly little about the heterogeneity that putatively produces them. I show that even in a very simple model—one that is composed of just two subpopulations with Gompertz mortality—(1) aggregate mortality can decelerate even while a majority of the cohort is frail; (2) multiple decelerations are possible; and (3) mortality selection can produce acceleration as well as deceleration. Simulations show that these patterns are plausible in model cohorts that in the aggregate resemble cohorts in the Human Mortality Database. I argue that these results challenge some conventional heuristics for understanding the relationship between selection and deceleration; undermine certain inferences from deceleration timing to patterns of social inequality; and imply that standard parametric models, assumed to plateau at most once, may sometimes badly misestimate deceleration timing—even by decades.

BACKGROUNDThe rate of mortality increase with age among adults is typically used as a measure of the rate of functional decline associated with aging or senescence. While black and white populations differ in the level of mortality, mortality also rises less rapidly with age for blacks than for whites, leading to the well-known black/white mortality "crossover".OBJECTIVEThis paper investigates black/white differences in the rate of mortality increase with age for major causes of death in order to examine the factors responsible for the black/white crossover.METHODSThe analysis considers two explanations for the crossover: selective survival and age misreporting. Mortality is modeled using a Gompertz model for 11 causes of death from ages 50-84 among blacks and whites by sex.RESULTSMortality increases more rapidly with age for whites than for blacks for nearly all causes of death considered. The all-cause mortality rate of mortality increase is nearly two percentage points higher for whites. The analysis finds evidence for both selective survival and age misreporting, although age misreporting is a more prominent explanation among women.CONCLUSIONSThe black/white mortality crossover reflects large differences in the rate of age-related mortality increase. Instead of reflecting the impact of specific causes of death, this pattern exists across many disparate disease conditions, indicating the need for a broad explanation.(ProQuest: ... denotes formulae omitted.)1. IntroductionHuman mortality rates begin to increase with advancing age starting around age 20 in most populations. Especially at older ages, age-related increases in the observed mortality rate are thought to reflect functional decline and processes associated with physiological aging. Gompertz (1825) originally noted that adult mortality increases exponentially as a function of age with remarkable consistency. That is, age-specific death rates increase at a nearly constant exponential rate for each year of age. Gompertz' model has enjoyed success as the most heavily used model of adult mortality, and remains a quintessential tool for demographers and gerontologists. Among the many attractive characteristics of the Gompertz model is an estimate of the percentage increase in mortality for each additional year of age, the Rate of Mortality Increase (RMI). Although this measure is assumed to be a species-typical indicator of physiological aging, human population subgroups exhibit significant differences in the rate of mortality increase (Horiuchi et al. 2003). Group differences in the rate of age- related mortality increase may reflect differences in accumulated risk over the life course or in the timing of the onset of specific diseases (Carey and Liedo 1995; Horiuchi and Wilmoth 1997; Luder 1993). In examining life course trajectories of mortality, comparing group differences in the rate of increase with age can supplement our understanding of differences in the level of mortality. One important comparison is blacks and whites in the United States. Although the vast majority of research on this topic focuses on black/white differences in the level of mortality, blacks and whites also differ in the slope of age-related increase (Elo and Drevenstedt 2004; Geronimus et al. 1996; Hayward et al. 2000). This leads to categorically distinct trajectories of mortality in the adult age range.Mortality profiles for the black and white populations in the United States differ considerably; blacks and whites experiences differences in the level and age-pattern of mortality and in the relative contribution of specific diseases. Studies examining the black mortality disadvantage throughout the life course also find that mortality increases more rapidly with age for whites than for blacks. Death rates for the two populations converge at older ages and mortality for whites begins to exceed that of blacks at the oldest ages (Manton et al. …

BACKGROUND Regional mortality differences are an important public health indicator and can serve as the basis for population forecasts and local planning. Health-related migration at old age may distort observed regional mortality. OBJECTIVE We assess whether or not internal migration in late life has the potential for distortion of regional old-age mortality rates and differences therein. DATA AND METHODS Using data from the Dutch population register, we analyzed migration and death rates in the population aged 80 years and above across the Dutch NUTS 2 regions, NUTS 3 regions and municipalities in the years 2002 through 2006. Observed sex-specific age-standardized death rates-and regional differences therein-are compared to hypothetical mortality rates, which disregard internal migration in the three years prior to death. RESULTS Late-life internal migration intensities were highest at municipality level, for women, and for those who died. Almost half of the municipalities showed differences of more than +/-5% between the observed and hypothetical mortality rates. Many of the municipalities, whose observed mortality rates differed significantly from the Dutch average, displayed no significant difference from the average in the hypothetical scenario. Regional mortality variation across the municipalities decreased significantly if late-life migration was disregarded. Differences were larger for women than men. Minimal differences appeared between the observed situation and hypothetical scenario at NUTS 2 and NUTS 3 levels. CONCLUSION Late-life migration can significantly distort regional old-age mortality. Mortality risk factors therefore appear to be more equally distributed than the observed rates imply. The bias depends on age, regional level, migration intensity, and the role of nursing care.

Formal derivations and empirical evidence, in the framework of Fries-Kannisto's hypothesis, show that the indicators of mortality compression based on age-at-death distribution, left-censored at a fixed old age, may be subject to a bias toward showing mortality decompression in the case of a mortality decline. The previously reported increasing variance in ages at death above fixed old ages in developed countries was mainly the effect of a mortality shift, not decompression. When adjusted for this bias, the indicators of variance in ages at death show a compression of period mortality.

Deviations from the Gompertz law of exponential mortality increases in late-middle and early-old age are commonly neglected in overall mortality analyses. In this study, we examined mortality increase patterns between ages 40 and 85 in 16 low-mortality countries and demonstrated sex differences in these patterns, which also changed across period and cohort. These results suggest that the interaction between aging and death is more complicated than what is usually assumed from the Gompertz law and also challenge existing biodemographic hypotheses about the origin and mechanisms of sex differences in mortality. We propose a two-mortality model that explains these patterns as the change in the composition of intrinsic and extrinsic death rates with age. We show that the age pattern of overall mortality and the population heterogeneity therein are possibly generated by multiple dynamics specified by a two-mortality model instead of a uniform process throughout most adult ages.

Summary Life-history theory predicts a trade-off between early-life fitness and life span. While the focus traditionally has been on the fecundity-life span trade-off, there are strong reasons to expect trade-offs with growth rate and/or development time. We investigated the roles of growth rate and development time in the evolution of life span in two independent selection experiments in the outcrossing nematode Caenorhabditis remanei. First, we found that selection under heat-shock leads to the evolution of increased life span without fecundity costs, but at the cost of slower development. Thereafter, the putative evolutionary links between development time, growth rate, fecundity, heat-shock resistance and life span were independently assessed in the second experiment by directly selecting for fast or slow development. This experiment confirmed our initial findings, since selection for slow development resulted in the evolution of long life span and increased heat-shock resistance. Because there were no consistent trade-offs with growth rate or fecundity, our results highlight the key role of development rate – differentiation of the somatic cells per unit of time – in the evolution of life span. Since development time is under strong selection in nature, reduced somatic maintenance resulting in shorter life span may be a widespread cost of rapid development. A lay summary is available for this article.

We develop a discrete variant of a general model for adult mortality influenced by the delayed impact of early conditions on adult health and mortality. The discrete variant of the model builds on an intuitively appealing interpretation of conditions that induce delayed effects and is an extension of the discrete form of the standard frailty model with distinct implications. We show that introducing delayed effects is equivalent to perturbing adult mortality patterns with a particular class of time-/age-varying frailty. We emphasize two main results. First, populations with delayed effects could experience unchanging or increasing adult mortality even when background mortality has been declining for long periods of time. Although this phenomenon also occurs in a regime with standard frailty, the distortions can be more severe under a regime with Barker frailty. As a consequence, conventional interpretations of the observed rates of adult mortality decline in societies that experience Barker frailty may be inappropriate. Second, the observed rate of senescence (slope of adult mortality rates) in populations with delayed effects could increase, decrease, or remain steady over time and across adult ages even though the rate of senescence of the background age pattern of mortality is time- and age-invariant. This second result implies that standard interpretations of empirical estimates of the slope of adult mortality rates in populations with delayed effects may be misleading because they can reflect mechanisms other than those inducing senescence as conventionally understood in the literature.

Evolutionary explanations of senescence are based on the established fact that the selection gradients (the rate of change of fitness with respect to a trait) on age-specific survival and fertility decline with age. Improvements in survival or fertility at older ages have less of an effect than the same improvements at younger ages. As a result, traits with small positive effects at early ages may accumulate even if they are accompanied by larger negative effects at late ages. It is often claimed that these results imply that additional mortality should make it easier for senescence to evolve, a claim that has led to a great deal of empirical study. However, the actual effect of additional mortality depends on the age dependence of that mortality. We show in detail that age-independent mortality has no effect on the selection gradients in time-invariant, periodic, stochastic and density-dependent life cycles. When the additional mortality is age dependent, its effects on the evolution of senescence depend on where in the life cycle it acts. We develop indices Mp and Mf to measure the shape of the selection gradient on survival and fertility, respectively. These indices give the mean age calculated over the selection gradient; larger values indicate a life history that is more resistant to the evolution of senescence. For several model life tables and for data on human life tables from Sweden, we show that additional early mortality increases Mp and Mf, while mortality later in life has the opposite effect.

BACKGROUNDSocioeconomic mortality differentials are known to exist almost universally. Many studies show a trend towards convergence with increasing age. Information about the highest ages is very rare, though.OBJECTIVEWe want to find out whether socioeconomic factors determine the chance of death in the United States among the oldest people.METHODSBased on official death count records, we employ the extinct cohort method to estimate the age-specific probability of dying by level of education.RESULTSWe present evidence that socioeconomic differentials in mortality exist even at the highest ages (95+), although the gap is small.1. IntroductionFew relationships in demography are as well established as the one between socioeco- nomic status (SES) and mortality: people with lower SES suffer from higher mortal- ity (and vice versa). This finding is rather universal, irrespective how SES is measured (e.g., income, education, occupation) or for which countries the study is conducted (e.g., Kitagawa and Hauser 1973; Rogers, Hummer, and Nam 1995; Kunst 1997; Hum- mer, Rogers, and Eberstein 1998; Pappas et al. 1993; Mackenbach et al. 1999; Goldman 2001).4 Many studies show that excess mortality of disadvantaged groups decreases with increasing age (e.g., Doblhammer, Rau, and Kytir 2005). This tendency towards conver- gence is at least partly caused by compositional changes as a result of mortality selection. Other explanations typically refer to policies, such as Medicare, that reduce inequalities or to the larger influence of biological factors at older ages (see, for instance, Hoffmann (2008) for an overview of causal explanations).In this paper we analyze whether socioeconomic mortality differentials still exist at ages 95 and higher in the United States. Our goal is not to specify whether selection effects are stronger than policy or biological effects, or are potentially counteracted by an accumulation of detrimental effects acquired over the life-course (Ross and Wu 1996).Despite the fact that the number of people at the highest ages is increasing rapidly (e.g. Rau et al., 2008), little is known about whether socioeconomic mortality differences still persist at those ages. Using data from the "National Long Term Care Survey" (NLTCS) of the United States, Manton, Stallard, and Corder (1997) found differences in remaining life expectancy at age 95 of less than one year between people with high and low education. More recently Zhu and Xie (2007) found that education - in conjunction with urbanity - plays a major role in Chinese mortality at ages 90+ and 100+ based on data from the Chinese Longitudinal Healthy Longevity Survey.2. Data and methodFor this study, we used the "Multiple Cause of Death Data" from the National Vital Statis- tics System of the National Center for Health Statistics of the United States. These data can be downloaded from the website of the National Bureau of Economic Research (Na- tional Center for Health Statistics 2013). The data are currently available for the years 1959-2010 and list individual deaths by a multitude of covariates such as sex, age at death, cause of death, year of death, or state of residence. Since 1989, these data also contain information on education, which we used as our indicator for socioeconomic sta- tus. Besides this practical reason of data availability, education might be the best marker for SES as it can be expected to remain relatively stable throughout the adult life course, especially among the oldest-old. Additionally, from a causal perspective, it can be ex- pected to predate other measures of SES (such as occupation or income), which might be difficult to measure at ages 90+ at all.To estimate mortality from such right-truncated data, we employed the so-called Extinct-Cohort method pioneered by Vincent (1951) and Depoid (1973): Starting at the highest age the last surviving member of a birth cohort has reached, this method estimates the age-specific probability of dying, q (x), backwards using the number of people who died at a given age as the numerator and the cumulative number of people who died at that age or older as the denominator. …

BACKGROUND: A decrease in mortality across all ages causes a shift of the age pattern of mortality, or mortality delay, while differences in the rate of decrease across ages cause a change in the shape of the age-at-death distribution, mortality compression or expansion. Evidence exists for both compression and delay of mortality. Existing parametric models to describe the full age pattern of mortality are not able to capture mortality delay versus mortality compression. More recent models that assess delay versus compression mostly focused on the adult or old ages alone and did not distinguish mortality compression below and above the modal age at death, although they represent different mechanisms.METHODS: This paper presents a new parametric model that describes the full age pattern of mortality and assesses compression - at different stages of life - and delay of mortality: the CoDe model. The model includes 10 parameters, of which five are constant over time. The five time-varying parameters reflect delay of mortality and compression of mortality in infancy, adolescence, young adulthood, late adulthood, and old age. The model describes infant and background mortality by two simple functions, uses a mixed logistic model with different slopes in adult, middle, and old age, and includes the modal age at death as a parameter to account for the delay in mortality.RESULTS: Applying the CoDe model to age-specific probabilities of death for Japanese, French, American, and Danish men and women between 1950 and 2010 showed a very good fit of the full age pattern of mortality. Delay of mortality explained about two-thirds of the increase in life expectancy at birth, whereas compression of mortality due to mortality declines in young age explained about one-third. No strong compression of mortality in late adulthood age was observed. Mortality compression in old age has had a small negative impact on life expectancy.CONCLUSIONS: The CoDe model proved a valid instrument for describing the full age pattern of mortality and for disentangling the effects of mortality delay and compression - at different stages of life - on the increase in life expectancy.

Age affects mortality from diseases differently than it affects mortality from external causes, such as accidents. Exclusion of the latter leads to the “all-diseases” category. The age trajectories of mortality from all diseases are studied in the five most populated countries of the EU, and the shape of these 156 age trajectories is investigated in detail. The arithmetic mean of ages where mortality reaches a minimal value is 8.47 years with a 95% confidence interval of [8.08, 8.85] years. Two simple deterministic models fit the age trajectories on the two sides of the mortality minimum. The inverse relationship is valid in all cases prior to this mortality minimum and death rates exactly decreased to three thousandths of its original size during the first 3000 days. After the mortality minimum, the standard Gompertz model fits the data in 63 cases, and the Gompertz model extended by a small quadratic element fits the remaining 93 cases. This analysis indicates that the exponential increase begins before the age of 15 years and that it is overshadowed by non-biological causes. Therefore, the existence of a mechanism switching that would explain the exponential increase in mortality after the age of 35 years is unlikely.

Objective To demonstrate the benefits of relative survival analysis to historical cohort studies in occupational epidemiology. In historical cohort studies, it is never known whether each death observed during the follow-up is attributable to the risks specific to the cohort or to the general population risk. SMR analysis is the most widely employed technique for quantifying the excess risk in the cohort compared to an external reference population. SMR assumes that the rate in the cohort is proportional to the rate in the reference population, at any time within predefined strata. Methods As SMR analysis, relative survival analysis can be used to separate the excess mortality rate from the mortality due to the population risk. This approach requires the same data as SMR analysis (a reference life table and the death history of the cohort) and manages the death history much like right censored survival data. The time scaled used in the analysis is defined from a well-defined time origin (employment date, date of the first exposure, birthday) to the occurrence of the event (death day in mortality studies). Contrarily to SMR analysis, the excess rate is not supposed to be proportional to the population death rate but is a function of the time scale independently of the population death rate. Relative survival analysis is based on the estimation of the excess mortality rate function, the cumulative excess rate function and the net survival function. Patterns in these functions may reveal excess mortality during specific periods of the time scale. Results The proposed approach will be illustrated using data from several mortality studies. Conclusion Relative survival analysis is a relevant methodology for mortality analysis which allows for a dynamic analysis of the possible excess mortality rate.

Do all species experience the declines of old age? No. Do we currently have theories that accurately predict which species escape these declines? In PNAS, Warner et al. (1) demonstrate that the answer to this question is also “no.” Understanding the breadth and diversity of the patterns of aging across the tree of life is central to life history evolution, and recent studies demonstrate that there is considerable variation in the age trajectories of traits across species (2⇓–4). In general, the decline in function and increase in mortality that is associated with increasing age is expected, according to classic evolutionary theories, because of the decline in the strength of selection with age (5⇓–7). In other words, the alleles that predispose us to heart disease or dementia have effects primarily at late ages, after we have reproduced, thus their impact on Darwinian fitness is minimal and these deleterious traits—and many others like them—persist. In natural populations, as we look across species, a central premise of the evolutionary theories of aging is that the rate of aging in different species will depend on levels of environmentally imposed mortality (7). All living organisms are subject to a constant risk of mortality posed by accidents, predation, and disease. This environmental, or extrinsic, mortality will influence the probability that individuals will survive to a given age. The hypothesis is that species with the highest levels of environmental mortality should evolve high levels of early reproduction and rapid aging. Conversely, slower rates of aging … [↵][1]1Email: droach{at}virginia.edu. [1]: #xref-corresp-1-1

Life expectancy is increasing in most countries and has exceeded 80 in several, as low-mortality nations continue to make progress in averting deaths. The health and economic implications of mortality reduction have been given substantial attention, but the observed malleability of human mortality has not been placed in a broad evolutionary context. We quantify the rate and amount of mortality reduction by comparing a variety of human populations to the evolved human mortality profile, here estimated as the average mortality pattern for ethnographically observed hunter-gatherers. We show that human mortality has decreased so substantially that the difference between hunter-gatherers and today’s lowest mortality populations is greater than the difference between hunter-gatherers and wild chimpanzees. The bulk of this mortality reduction has occurred since 1900 and has been experienced by only about 4 of the roughly 8,000 human generations that have ever lived. Moreover, mortality improvement in humans is on par with or greater than the reductions in mortality in other species achieved by laboratory selection experiments and endocrine pathway mutations. This observed plasticity in age-specific risk of death is at odds with conventional theories of aging.

Mortality improvements, especially of the elderly, have been a common phenomenon since the end of World War II. The longevity risk becomes a major concern in many countries because of underestimating the scale and speed of prolonged life. In this study we explore the increasing life expectancy by examining the basic properties of survival curves. Specifically, we check if there are signs of mortality compression (i.e., rectangularization of the survival curve) and evaluate what it means to designing annuity products. Based on the raw mortality rates, we propose an approach to verify if there is mortality compression. We then apply the proposed method to the mortality rates of Japan, Sweden, and the United States, using the Human Mortality Database. Unlike previous results using the graduated mortality rates, we found no obvious signs that mortality improvements are slowing down. This indicates that human longevity is likely to increase, and longevity risk should be seriously considered in pricing...

Early environmental influences on later-life health and mortality are well recognized in the doubling of life expectancy since 1800. To further define these relationships, we analyzed the associations between early-life mortality and both the estimated mortality level at age 40 and the exponential acceleration in mortality rates with age characterized by the Gompertz model. Using mortality data from 630 cohorts born throughout the 19th and early 20th century in nine European countries, we developed a multilevel model that accounts for cohort and period effects in later-life mortality. We show that early-life mortality, which is linked to exposure to infection and poor nutrition, predicts both the estimated cohort mortality level at age 40 and the subsequent Gompertz rate of mortality acceleration during aging. After controlling for effects of country and period, the model accounts for the majority of variance in the Gompertz parameters (about 90% of variation in the estimated level of mortality at age 40 and about 78% of variation in the Gompertz slope). The gains in cohort survival to older ages are entirely due to large declines in adult mortality level, because the rates of mortality acceleration at older ages became faster. These findings apply to cohorts born in both the 19th century and the early 20th century. This analysis defines new links in the developmental origins of adult health and disease in which effects of early-life circumstances, such as exposure to infections or poor nutrition, persist into mid-adulthood and remain evident in the cohort mortality rates from ages 40 to 90.

Using a new distribution capable of exhibiting all the possible modes of accelerating and decelerating mortality, we conduct a systematic investigation of late-life mortality in humans. We check the insensitivity of the distribution to age cutoffs in the data relative to the logistic mortality model and propose a method to forecast evolution in the characteristic deceleration ages of the distribution. A number of data sets have been explored, with a particular emphasis on those originating from Scandinavia. Although those from Australia, Canada, and the USA are compatible with Gompertzian mortality, those from the other countries examined are not. We find in particular that the onset of mortality deceleration is being progressively delayed in Western societies but that there is evidence of mortality plateauing at earlier ages.

Humans exhibit considerable diversity in timing and rate of reproduction. Life-history theory (LHT) suggests that ecological cues of resource richness and survival probabilities shape human phenotypes across populations. Populations experiencing high extrinsic mortality due to uncertainty in resources should exhibit faster life histories. Here we use a path analytic (PA) approach informed by LHT to model the multiple pathways between resources, mortality rates, and reproductive behavior in 191 countries. Resources that account for the most variance in population mortality rates are predicted to explain the most variance in total fertility rates. Results indicate that resources (e.g., calories, sanitation, education, and health-care expenditures) influence fertility rates in paths through communicable and noncommnunicable diseases. Paths acting through communicable disease are more strongly associated with fertility than are paths through noncommunicable diseases. These results suggest that a PA approach may help disaggregate extrinsic and intrinsic mortality factors in cross-cultural analyses. Such knowledge may be useful in developing targeted policies to decrease teenage pregnancy, total fertility rates, and thus issues associated with overpopulation.

Causes underlying mortality disparities are often determined by causal decomposition. This method is based on the decomposition of differences in mortality or life expectancy into parameters representing the contribution of underlying causes. It quantifies disparities as differences in mortality rates and does not account for the fact that many underprivileged groups are more likely to die from nearly all causes. Results are driven by the frequency of cause of death. Alternatively, the cause deleted index quantifies the role of underlying causes in mortality disparities as the change in the relative risk of dying that is related to deleting a specific cause. The consistency between the methods in estimating cause of death contributions is analyzed using 2000 U.S. mortality data and simulated mortality profiles. The two methods often produce divergent results because causal decomposition relies on the prevalence of causes of death.

Many analyses of human populations have found that age-specific mortality rates increase faster across most of adulthood when overall mortality levels decline. This contradicts the relationship often expected from Williams' classic hypothesis about the effects of natural selection on the evolution of senescence. More likely, much of the within-species difference in actuarial aging is not due to variation in senescence, but to the strength of filters on the heterogeneity of frailty in older survivors. A challenge to this differential frailty hypothesis was recently posed by an analysis of life tables from historical European populations and traditional societies that reported variation in actuarial aging consistent with Williams' hypothesis after all. To investigate the challenge, we reconsidered those cases and aging measures. Here we show that the discrepancy depends on Ricklefs' aging rate measure, ω, which decreases as mortality levels drop because it is an index of mortality level itself, not the rate of increase in mortality with age. We also show unappreciated correspondence among the parameters of Gompertz-Makeham and Weibull survival models. Finally, we compare the relationships among mortality parameters of the traditional societies and the historical series, providing further suggestive evidence that differential heterogeneity has strong effects on actuarial aging.

The mortality rate dynamics between two related but different-sized populations are modeled consistently using a new stochastic mortality model that we call the gravity model. The larger population is modeled independently, and the smaller population is modeled in terms of spreads (or deviations) relative to the evolution of the former, but the spreads in the period and cohort effects between the larger and smaller populations depend on gravity or spread reversion parameters for the two effects. The larger the two gravity parameters, the more strongly the smaller population’s mortality rates move in line with those of the larger population in the long run. This is important where it is believed that the mortality rates between related populations should not diverge over time on grounds of biological reasonableness. The model is illustrated using an extension of the Age-Period-Cohort model and mortality rate data for English and Welsh males representing a large population and the Continuous Mortality Investigation assured male lives representing a smaller related population.

Mortality deceleration is the observed but yet to be understood phenomenon that the increase in the late-life death rate slows down after a certain species-related advanced age. Various definitions of onsets of mortality deceleration are examined. A new distribution based on the Strehler-Mildvan theory of aging takes on the required shapes. The application is done on mortality data from the 1892 cohort of Swedish women and on Mediterranean fruit flies.

The relative importance of cohorts' early-life conditions, compared to later period conditions, on adult and old-age mortality is not known. This article studies how cohort-level mortality depends on shocks in cohorts' early- and later-life (period) conditions. I use cohorts' own mortality as a proxy for the early-life conditions, and define shocks as deviations from trend. Using historical data for five European Countries i find that shocks in early-life conditions are only weakly associated with cohorts' later mortality. This may be because individual-level health is robust to early-life conditions, or because at the cohort level scarring, selection, and immunity cancel each other. Shocks in period conditions, measured as deviations from trend in period child mortality, are strongly and positively correlated with mortality at all older ages. The results suggest that at the cohort level changing period conditions drive mortality variation and change.

Extrinsic mortality is a key influence on organisms’ life history strategies, especially on age at maturity. This historical longitudinal study of 125 women in rural Domenica examines effects of extrinsic mortality on human age at maturity and pace of reproduction. Extrinsic mortality is indicated by local population infant mortality rates during infancy and at maturity between the years 1925 and 2000. Extrinsic mortality shows effects on age at first birth and pace of reproduction among these women. Parish death records show huge historical variation in age-specific mortality rates. The infant mortality rate (IMR) in the early 1920s was low, increased dramatically beginning in 1929, and reached a maximum in the 1950s, at which point IMR declined steadily to its present low rate. The mortality rate early in life showed a quadratic association with age at first birth. Women who experienced conditions of low IMR early in life reproduced relatively late in life. Those born into moderately high levels of infant mortality tended to reproduce earlier than those born at low levels. At very high infant mortality levels early in life, women went on to delay reproduction until relatively late, possibly as a result of somatic depletion and energetic stress associated with the conditions that lead to high IMR. Population mortality rates at age of maturity also showed a quadratic association with age at first birth. The pace of reproduction, estimated as number of surviving offspring controlled for maternal age, showed a similar quadratic effect. There were complex interactions between population mortality rates in infancy and at maturity. When extrinsic mortality was high during infancy, extrinsic mortality later in life had little effect on timing of first birth. When extrinsic mortality was low to moderate in infancy, extrinsic mortality later in life had significant effects on adult reproduction. I speculate that these effects are mediated through development of personality facets associated with reproduction.

I study how shocks in cohort-level early life conditions, as represented by deviations from trend in mortality before age 5, affect later mortality. I use data for six European countries and find that shocks that increase infant mortality decrease later life expectancy between ages 5-30. The effect is strong for England and Wales but small or insignificant for other countries. Shocks that increase mortality at ages 1-5 increase life expectancy between ages 5-30 and compress the mortality distribution. For both shocks the effects are weak at older ages. These results suggest that early life conditions have a transitory effect and potentially only little influence on old-age mortality.

The hazard of mortality is usually presented as a function of age, but can be defined as a function of the fraction of survivors. This definition enables us to derive new relationships for life expectancy. Specifically, in a life-table population with a positive age-specific force of mortality at all ages, the expectation of life at age x is the average of the reciprocal of the survival-specific force of mortality at ages after x, weighted by life-table deaths at each age after x, as shown in (6). Equivalently, the expectation of life when the surviving fraction in the life table is s is the average of the reciprocal of the survival-specific force of mortality over surviving proportions less than s, weighted by life-table deaths at surviving proportions less than s, as shown in (8). Application of these concepts to the 2004 life tables of the United States population and eight subpopulations shows that usually the younger the age at which survival falls to half (the median life length), the longer the life expectancy at that age, contrary to what would be expected from a negative exponential life table. (ProQuest: ... denotes formulae omitted.) 1. Background and relationships 1.1 Background The life table l(x), constant in time, with continuous age x, is the proportion of a cohort (whether a birth cohort or a synthetic period cohort) that survives to age x or longer. In probabilistic terms, l(x) is one minus the cumulative distribution function of length of life x. The maximum possible age w may be finite or infinite. If w = x, then some individuals may live longer than any finite bound. By definition, l(0) = 1 and l(w) = 0. Assume l(x) is a continuous, differentiable function of x, 0 ≤ x ≤ w, and assume life expectancy at age 0 is finite. The age-specific force of mortality at age x is, by definition, (1) ... Assume u(x) > 0 for all 0 ≤ x ≤ w. The life table l(x) is strictly decreasing from l(0) = 1 to l(w) = 0 so there is a one-to-one correspondence between age x in [0,w] and the proportion s in [0, 1] of the cohort that survives to age x or longer. One direction of this correspondence is given by the life table function s = l(x) (illustrated schematically in Figure 1 and for the United States population in 2004 in Figure 3A). There appears to be no standard demographic term for the inverse function that maps the proportion surviving s, 0 ≤ s ≤ 1, to the corresponding age x, so I propose to call it the age function a (illustrated schematically in Figure 2 and for the United States population in 2004 in Figure 3D). In words, the age a(s) at which the fraction s of the birth cohort survives is the age x at which the life table function l(x) is s. By definition, under the assumption u(x) > 0 for all 0 ≤ x ≤ 1, a(s) = x if and only if l(x) = s. Equivalently, by definition, for every s in 0 ≤ s ≤ 1 and every x in 0 ≤ x ≤ w, a(l(x)) = x and l(a(s)) = s. We define a(1/2) as the median life length, that is, the age by which half the cohort has died. For every s in 0 ≤ s ≤ 1, we define the survival-specific force of mortality λ(s) in terms of the age-specific force of mortality u(x) in (1) in three equivalent ways: (2) λ(s) = u(x) if s = l(x); or λ(s) = u(a(s)); or λ(l(x)) = u(x): In words, the survival-specific force of mortality λ(s) at surviving proportion s equals the age-specific force of mortality u(x) at the age x where the life table l(x) = s. The domain of the age-specific force of mortality u is 0 ≤ x ≤ w while the domain of the survival- specific force of mortality λ is 0 ≤ s ≤ 1. We give below an explicit formula (9) for the survival-specific force of mortality at surviving proportion s. This formula is analogous to (1) for the age-specific force of mortality. The complete expectation of life at age x, e(x), is the average number of years remaining to be lived by those who have attained age x: (3) ... Inserting the definition (1) in place of u(y) in (3) and integrating by parts gives (4) . …

Cold related mortality among people aged over 50 in England and Wales has decreased at a rate of 85 deaths per million population per year over the period 1976–2005. This trend is two orders of magnitude higher than the increase in heat-related mortality observed after 1976. Long term changes in temperature-related mortality may be linked to human activity, natural climatic forcings, or to adaptation of the population to a wider range of temperatures. Here we employ optimal detection, a formal statistical methodology, to carry out an end to end attribution analysis. We find that adaptation is a major influence on changing mortality rates. We also find that adaptation has prevented a significant increase in heat-related mortality and considerably enhanced a significant decrease in cold-related mortality. Our analysis suggests that in the absence of adaptation, the human influence on climate would have been the main contributor to increases in heat-related mortality and decreases in cold-related mortality.