Reconsidering mortality compression and deceleration: an alterative model of mortality rates

Trends in lifespan variation across the spectrum of rural and urban places in the United States, 1990-2017

Mortality disparities between urban and rural areas in the United States widened in recent decades as mortality improvements in rural areas slowed. Although the existence of a rural mortality penalty is well-documented, previous research in this area has focused almost exclusively on differences in average levels of mortality between rural and urban areas rather than differences in levels of lifespan variation within rural and urban areas. This oversight is important because monitoring trends in lifespan variation provides unique insights into levels of inequality in the age-at-death distribution within a population. Does the rural mortality penalty in life expectancy extend to lifespan variation? We used U.S. Multiple Cause of Death data files to measure life disparity at birth (e 0 † ) from 1990 to 2017. We found that the rural mortality penalty extends to lifespan variation as large metropolitan areas had greater improvements in life disparity than nonmetropolitan areas. Beginning around 2011, all areas began to show increased life disparity with the largest increases occurring in nonmetropolitan areas. Age decomposition results showed that the nonmetropolitan increases were due to rising working-age mortality. Greater variability in the age-at-death distribution represents an additional dimension of inequality for Americans living in rural places.

What have we learned about mortality patterns over the past 25 years?

In this paper, I examine progress in the field of mortality over the past 25 years. I argue that we have been most successful in taking advantage of an increasingly data-rich environment to improve aggregate mortality models and test pre-existing theories. Less progress has been made in relating our estimates of mortality risk at the individual level to broader mortality patterns at the population level while appropriately accounting for contextual differences and compositional change. Overall, I find that the field of mortality continues to be highly visible in demographic journals, including Population Studies. However much of what is published today in field journals could just as easily appear in neighbouring disciplinary journals, as disciplinary boundaries are shrinking.

A Mixture-Function Mortality Model: Illustration of the Evolution of Premature Mortality

Premature mortality is often a neglected component of overall deaths, and the most difficult to identify. However, it is important to estimate its prevalence. Following Pearson's theory about mortality components, a definition of premature deaths and a parametric model to study its transformations are introduced. The model is a mixture of three distributions: a Half Normal for the first part of the death curve and two Skew Normals to fit the remaining pieces. One advantage of the model is the possibility of obtaining an explicit equation to compute life expectancy at birth and to break it down into mortality components. We estimated the mixture model for Sweden, France, East Germany and Czech Republic. In addition, to the well-known reduction in infant deaths, and compression and shifting trend of adult mortality, we were able to study the trend of the central part of the distribution of deaths in detail. In general, a right shift of the modal age at death for young adults is observed; in some cases, it is also accompanied by an increase in the number of deaths at these ages: in particular for France, in the last twenty years, premature mortality increases.

Multidimensional Mortality Selection: Why Individual Dimensions of Frailty Don't Act Like Frailty

Theoretical models of mortality selection have great utility in explaining otherwise puzzling phenomena. The most famous example may be the Black-White mortality crossover: at old ages, Blacks outlive Whites, presumably because few frail Blacks survive to old ages while some frail Whites do. Yet theoretical models of unidimensional heterogeneity, or frailty, do not speak to the most common empirical situation for mortality researchers: the case in which some important population heterogeneity is observed and some is not. I show that, when one dimension of heterogeneity is observed and another is unobserved, neither the observed nor the unobserved dimension need behave as classic frailty models predict. For example, in a multidimensional model, mortality selection can increase the proportion of survivors who are disadvantaged, or "frail," and can lead Black survivors to be more frail than Whites, along some dimensions of disadvantage. Transferring theoretical results about unidimensional heterogeneity to settings with both observed and unobserved heterogeneity produces misleading inferences about mortality disparities. The unusually flexible behavior of individual dimensions of multidimensional heterogeneity creates previously unrecognized challenges for empirically testing selection models of disparities, such as models of mortality crossovers.

New Trend in Old-Age Mortality: Gompertzialization of Mortality Trajectory

There is great interest among gerontologists, demographers, and actuaries in the question concerning the limits to human longevity. Attempts at getting answers to this important question have stimulated many studies on late-life mortality trajectories, often with opposing conclusions. One group of researchers believes that mortality stops growing with age at extreme old ages, and that hence there is no fixed limit to the human life span. Other studies found that mortality continues to grow with age up to extreme old ages. Our study suggests a possible solution to this controversy. We found that mortality deceleration is best observed when older, less accurate life span data are analyzed, while in the case of more recent and reliable data there is a persistent mortality growth with age. We compared the performance (goodness of fit) of two competing mortality models - the Gompertz model and the Kannisto ("mortality deceleration") model - at ages of 80-105 years using data for 1880-1899 single-year birth cohorts of US men and women. The mortality modeling approach suggests a transition from mortality deceleration to the Gompertzian mortality pattern over time for both men and women. These results are consistent with the hypothesis about disappearing mortality deceleration over time due to improvement in the accuracy of age reporting. In the case of more recent data, mortality continues to grow with age even at very old ages. This observation may lead to more conservative estimates of future human longevity records.

Modelling and forecasting adult age-at-death distributions

Age-at-death distributions provide an informative description of the mortality pattern of a population but have generally been neglected for modelling and forecasting mortality. In this paper, we use the distribution of deaths to model and forecast adult mortality. Specifically, we introduce a relational model that relates a fixed 'standard' to a series of observed distributions by a transformation of the age axis. The proposed Segmented Transformation Age-at-death Distributions (STAD) model is parsimonious and efficient: using only three parameters, it captures and disentangles mortality developments in terms of shifting and compression dynamics. Additionally, mortality forecasts can be derived from parameter extrapolation using time-series models. We illustrate our method and compare it with the Lee-Carter model and variants for females in four high-longevity countries. We show that the STAD fits the observed mortality pattern very well, and that its forecasts are more accurate and optimistic than the Lee-Carter variants.

Separating the Signal From the Noise: Evidence for Deceleration in Old-Age Death Rates

Widespread population aging has made it critical to understand death rates at old ages. However, studying mortality at old ages is challenging because the data are sparse: numbers of survivors and deaths get smaller and smaller with age. I show how to address this challenge by using principled model selection techniques to empirically evaluate theoretical mortality models. I test nine models of old-age death rates by fitting them to 360 high-quality data sets on cohort mortality after age 80. Models that allow for the possibility of decelerating death rates tend to fit better than models that assume exponentially increasing death rates. No single model is capable of universally explaining observed old-age mortality patterns, but the log-quadratic model most consistently predicts well. Patterns of model fit differ by country and sex. I discuss possible mechanisms, including sample size, period effects, and regional or cultural factors that may be important keys to understanding patterns of old-age mortality. I introduce mortfit, a freely available R package that enables researchers to extend the analysis to other models, age ranges, and data sources.

Black-White Differences in Mortality Compression and Deceleration and the Mortality Crossover Reconsidered

Most studies investigating Black-White differences in mortality patterns have focused exclusively on the well-known crossover but have ignored other aspects of the mortality curves, such as deceleration and compression. Yet compression and deceleration are also important features of mortality curves that may vary by race. In this research, the authors developed models for data from 1972 to 1990 and estimated them using naive and more stringent assumptions about Black data quality. They found that mortality deceleration begins at older ages for Blacks than for Whites but that the ages of deceleration onset are converging. The authors also found that mortality compression is occurring for Blacks but not for Whites and that compression is more apparent for Blacks when data quality is considered. Finally, the authors found that a crossover exists, that the age at crossover is increasing across time, and that the age at crossover is later in adjusted data than in unadjusted data.

Phytohormone regulation of root growth triggered by P deficiency or Al toxicity

Phosphorus (P) deficiency and aluminum (Al) toxicity often coexist and limit plant growth on acid soils. It has been well documented that both P deficiency and Al toxicity alter root growth, including inhibition of primary roots and promotion of lateral roots. This suggests that plants adapt to both stresses through a common regulation pathway. Although an expanding set of results shows that phytohormones play vital roles in controlling root responses to Pi starvation and Al toxicity, it remains largely unknown whether P and Al coordinately regulate root growth through interacting phytohormone biosynthesis and signal transduction pathways. This review provides a summary of recent results concerning the influences of P deficiency and Al toxicity on root growth through the action of phytohormones, most notably auxin and ethylene. The objective is to facilitate increasing insights into complex responses of plants to adverse factors common on acid soils, which can spur development of 'smart' cultivars with better root growth and higher yield on these globally distributed marginal soils.

Subsistence-patterns, gender roles, effective temperature, and the evolutionary timing of a post reproductive life span

Evolutionary anthropologists explain menopause and the start of a post reproductive lifespan (PRLS), as beneficiary for older women who can now help contribute to their children/grandchildren's wellbeing. This paper presents a new model with the aim to elucidate when, where, and for whom, such benefits may have arisen. In foraging societies, women contribute nutrients to their social groups/family units to a greater degree as overall effective temperatures (ETs) rise. Where the ET is favorable for women's contributions (ETs between 15 and 20), selection does lengthen the PRLS of women because women contribute sufficiently to enhance their own inclusive fitness. Paleo-environment records suggest that the climate necessary to encourage an increase PRLS occurred shortly after the younger dryad in emerging subtropical settings. Subsistence patterns and gender roles may have played a role in the evolution of PRLS in human females.

Bayesian demography 250 years after Bayes

Bayesian statistics offers an alternative to classical (frequentist) statistics. It is distinguished by its use of probability distributions to describe uncertain quantities, which leads to elegant solutions to many difficult statistical problems. Although Bayesian demography, like Bayesian statistics more generally, is around 250 years old, only recently has it begun to flourish. The aim of this paper is to review the achievements of Bayesian demography, address some misconceptions, and make the case for wider use of Bayesian methods in population studies. We focus on three applications: demographic forecasts, limited data, and highly structured or complex models. The key advantages of Bayesian methods are the ability to integrate information from multiple sources and to describe uncertainty coherently. Bayesian methods also allow for including additional (prior) information next to the data sample. As such, Bayesian approaches are complementary to many traditional methods, which can be productively re-expressed in Bayesian terms.

Why lifespans are more variable among blacks than among whites in the United States

Lifespans are both shorter and more variable for blacks than for whites in the United States. Because their lifespans are more variable, there is greater inequality in length of life-and thus greater uncertainty about the future-among blacks. This study is the first to decompose the black-white difference in lifespan variability in America. Are lifespans more variable for blacks because they are more likely to die of causes that disproportionately strike the young and middle-aged, or because age at death varies more for blacks than for whites among those who succumb to the same cause? We find that it is primarily the latter. For almost all causes of death, age at death is more variable for blacks than it is for whites, especially among women. Although some youthful causes of death, such as homicide and HIV/AIDS, contribute to the black-white disparity in variance, those contributions are largely offset by the higher rates of suicide and drug poisoning deaths for whites. As a result, differences in the causes of death for blacks and whites account, on net, for only about one-eighth of the difference in lifespan variance.

An age-period-cohort analysis of cancer incidence among the oldest old, Utah 1973-2002

We used age-period-cohort (APC) analyses to describe the simultaneous effects of age, period, and cohort on cancer incidence rates in an attempt to understand the population dynamics underlying their patterns among those aged 85+. Data from the Utah Cancer Registry (UCR), the US Census, the National Center for Health Statistics (NCHS), and the National Cancer Institute's Surveillance, Epidemiology and End Results (SEER) programme were used to generate age-specific estimates of cancer incidence at ages 65-99 from 1973 to 2002 for Utah. Our results showed increasing cancer incidence rates up to the 85-89 age group followed by declines at ages 90-99 when not confounded by the separate influences of period and cohort effects. We found significant period and cohort effects, suggesting the role of environmental mechanisms in cancer incidence trends between the ages of 85 and 100.

SURVIVAL OF CENTENARIANS IN JAPAN

The objective of the study was to explore the survival trends of centenarians in Japan. A cohort of centenarians born between 1881 and 1900 was analysed based on national census data, and the average life expectancy at 100 years of age, risk of death and maximum age were estimated. An analysis of covariance and a Cox regression analysis were performed to explore the factors associated with life expectancy and risk of death. The death rates in centenarians tended to decrease with birth year, and the average life expectancy from the age of 100 slightly increased at a rate of 0.013 years (95% CI: 0.007-0.019) by birth year in men and 0.026 in women. Women had a longer life expectancy than men, with a difference of 0.174 years (95% CI: 0.071-0.277) at birth year 1881 and increasing by 0.013 years per year thereafter. The risk of death in both sexes decreased significantly by birth year over the course of the period analysed, and the risk of death in men was 1.16 (95% CI: 1.14-1.19) times that of women. In women, death rates at every age significantly decreased with birth year over the course of the period analysed until age 104. However, this trend did not hold true for ages 105 and older. The average life expectancy of centenarians at the age of 100 in Japan increased by birth year in the 1881-1900 birth cohort. In addition, Japanese centenarians had the lowest death rates among several countries.

Mortality deceleration and mortality selection: three unexpected implications of a simple model

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.

The Nicoya region of Costa Rica: a high longevity island for elderly males

Reliable data show that the Nicoyan region of Costa Rica is a hot spot of high longevity. A survival follow-up of 16,300 elderly Costa Ricans estimated a Nicoya death rate ratio (DRR) for males 1990-2011 of 0.80 (0.69-0.93 CI). For a 60-year-old Nicoyan male, the probability of becoming centenarian is seven times that of a Japanese male, and his life expectancy is 2.2 years greater. This Nicoya advantage does not occur in females, is independent of socio-economic conditions, disappears in out-migrants and comes from lower cardiovascular (CV) mortality (DRR = 0.65). Nicoyans have lower levels of biomarkers of CV risk; they are also leaner, taller and suffer fewer disabilities. Two markers of ageing and stress-telomere length and dehydroepiandrosterone sulphate-are also more favourable. The Nicoya diet is prosaic and abundant in traditional foods like rice, beans and animal protein, with low glycemic index and high fibre content.

A new method for determining why length of life is more unequal in some populations than in others

Why is there greater variability in individual longevity in some populations than in others? We propose a decomposition method designed to address that question by quantifying the effects of population differences in the spread, allocation, and timing of the principal causes of death. Applying the method to the United States and Sweden, we find that spread effects account for about two-thirds of the greater variance in age at death among American adults, meaning that two-thirds of the U.S.-Sweden difference would persist if the two countries differed only with respect to within-cause variance among adults. The remainder of the difference is due largely to allocation effects, with the greater incidence of homicides and fatal traffic accidents alone accounting for more than one-fourth of the greater variance in age at death among adults in the United States.

The significance of education for mortality compression in the United States

Recent studies of old-age mortality trends assess whether longevity improvements over time are linked to increasing compression of mortality at advanced ages. The historical backdrop of these studies is the long-term improvement in a population's socioeconomic resources that fueled longevity gains. We extend this line of inquiry by examining whether socioeconomic differences in longevity within a population are accompanied by old-age mortality compression. Specifically, we document educational differences in longevity and mortality compression for older men and women in the United States. Drawing on the fundamental cause of disease framework, we hypothesize that both longevity and compression increase with higher levels of education and that women with the highest levels of education will exhibit the greatest degree of longevity and compression. Results based on the Health and Retirement Study and the National Health Interview Survey Linked Mortality File confirm a strong educational gradient in both longevity and mortality compression. We also find that mortality is more compressed within educational groups among women than men. The results suggest that educational attainment in the United States maximizes life chances by delaying the biological aging process.

Heterogeneity in the Strehler-Mildvan general theory of mortality and aging

This study examines and further develops the classic Strehler-Mildvan (SM) general theory of mortality and aging. Three predictions from the SM theory are tested by examining the age dependence of mortality patterns for 42 countries (including developed and developing countries) over the period 1955-2003. By applying finite mixture regression models, principal component analysis, and random-effects panel regression models, we find that (1) the negative correlation between the initial adulthood mortality rate and the rate of increase in mortality with age derived in the SM theory exists but is not constant; (2) within the SM framework, the implied age of expected zero vitality (expected maximum survival age) also is variable over time; (3) longevity trajectories are not homogeneous among the countries; (4) Central American and Southeast Asian countries have higher expected age of zero vitality than other countries in spite of relatively disadvantageous national ecological systems; (5) within the group of Central American and Southeast Asian countries, a more disadvantageous national ecological system is associated with a higher expected age of zero vitality; and (6) larger agricultural and food productivities, higher labor participation rates, higher percentages of population living in urban areas, and larger GDP per capita and GDP per unit of energy use are important beneficial national ecological system factors that can promote survival. These findings indicate that the SM theory needs to be generalized to incorporate heterogeneity among human populations.

Quantitative Risk Stratification in Markov Chains with Limiting Conditional Distributions

Background . Many clinical decisions require patient risk stratification. The authors introduce the concept of limiting conditional distributions, which describe the equilibrium proportion of surviving patients occupying each disease state in a Markov chain with death. Such distributions can quantitatively describe risk stratification. Methods . The authors first establish conditions for the existence of a positive limiting conditional distribution in a general Markov chain and describe a framework for risk stratification using the limiting conditional distribution. They then apply their framework to a clinical example of a treatment indicated for high-risk patients, first to infer the risk of patients selected for treatment in clinical trials and then to predict the outcomes of expanding treatment to other populations of risk. Results . For the general chain, a positive limiting conditional distribution exists only if patients in the earliest state have the lowest combined risk of progression or death. The authors show that in their general framework, outcomes and population risk are interchangeable. For the clinical example, they estimate that previous clinical trials have selected the upper quintile of patient risk for this treatment, but they also show that expanded treatment would weakly dominate this degree of targeted treatment, and universal treatment may be cost-effective. Conclusions . Limiting conditional distributions exist in most Markov models of progressive diseases and are well suited to represent risk stratification quantitatively. This framework can characterize patient risk in clinical trials and predict outcomes for other populations of risk.

Sociodemographic Effects on the Dynamics of Task-specific ADL Functioning at the Oldest-old Ages: The Case of China

Studies that systematically examine the dynamics of task-specific ADL functioning and its associates are very rare. Using the first three waves of the Chinese Longitudinal Healthy Longevity Survey, this study examines the dynamics of each of the six ADL tasks (bathing, dressing, toileting, indoor transferring, eating, and continence) and their sociodemographic correlates among the oldest-old by including the ADL information both at the follow-up wave for survivors and at the time prior to death for those who died during survey intervals. Effects of age, gender, urban/rural residence, ethnicity, education, primary lifetime occupation, primary source of daily expenses, living alone, and marital status are examined in both the absence and presence of other various confounders. Our results show that each sociodemographic factor still plays some limited role in the dynamics of ADL functioning across tasks.

Modeling human mortality using mixtures of bathtub shaped failure distributions

Aging and mortality is usually modeled by the Gompertz-Makeham distribution, where the mortality rate accelerates with age in adult humans. The resulting parameters are interpreted as the frailty and decrease in vitality with age. This fits well to life data from 'westernized' societies, where the data are accurate, of high resolution, and show the effects of high quality post-natal care. We show, however, that when the data are of lower resolution, and contain considerable structure in the infant mortality, the fit can be poor. Moreover, the Gompertz-Makeham distribution is consistent with neither the force of natural selection, nor the recently identified 'late life mortality deceleration'. Although actuarial models such as the Heligman-Pollard distribution can, in theory, achieve an improved fit, the lack of a closed form for the survival function makes fitting extremely arduous, and the biological interpretation can be lacking. We show, that a mixture, assigning mortality to exogenous or endogenous causes, using the reduced additive and flexible Weibull distributions, models well human mortality over the entire life span. The components of the mixture are asymptotically consistent with the reliability and biological theories of aging. The relative simplicity of the mixture distribution makes feasible a technique where the curvature functions of the corresponding survival and hazard rate functions are used to identify the beginning and the end of various life phases, such as infant mortality, the end of the force of natural selection, and late life mortality deceleration. We illustrate our results with a comparative analysis of Canadian and Indonesian mortality data.

A model with long-term survivors for the analysis of current-status nuptiality data

The model proposed in this paper combines a logistic regression model and a Weibull regression model for the analysis of current-status data. This joint model allows a simultaneous estimation of two sets of effects on the covariates: one on the probability that the event occurs (also known as quantum) and the other on the timing of the event. Thus, the model can be seen either as an extension of a survival-analysis model for use with current-status data where a survival fraction is added in order explicitly to take into account the possibility that the event may never occur, or as an extension of survival analysis with long-term survivors to the analysis of current-status data. As an illustrative application we apply our model to a study of nuptiality in seventeenth-century Italy.

Markov mortality models: implications of quasistationarity and varying initial distributions

This paper explains some implications of Markov-process theory for models of mortality. We show that an important qualitative feature common to empirical mortality data, and which has been found in certain models-the convergence to a "mortality plateau"-is, in fact, a generic consequence of the models' convergence to a "quasistationary distribution". This phenomenon has been explored extensively in the mathematical literature. Not only does this generalization free important results from specifics of the models, it also suggests a new explanation for the convergence to constant mortality. At the same time that we show that the late behavior of these models-convergence to a finite asymptote-is almost logically immutable, we also point out that the early behavior of the mortality rates can be more flexible than has been generally acknowledged. We show, in particular, that an appropriate choice of initial conditions enables one popular model to approximate any reasonable hazard-rate data. This illustrates how precarious it can be to read a model's vindication from its consilience with a favored hazard-rate function, such as the Gompertz exponential.

The emergence of extremely old people: the case of Japan

It is acknowledged today that the number of oldest old persons, nonagenarians and centenarians, is increasing rapidly. Analysis of data from Japan, where female life expectancy at birth approaches the assumed limit of 85 years, can provide unique information on whether the rate of these demographic changes is accelerating. Adjusted for the size of the birth cohorts, the centenarian doubling time (CDT i.e. the number of years needed to double the number of centenarians), halved in 29 years. Moreover death rates at 100 years and above show a clear decrease if we exclude males at age 105 and over.