A modal age at death approach to forecasting adult mortality.

Recent studies have shown that there are some advantages to forecasting mortality with indicators other than age-specific death rates. The mean, median, and modal ages at death can be directly estimated from the age-at-death distribution, as can information on lifespan variation. The modal age at death has been increasing linearly since the second half of the twentieth century, providing a strong basis from which to extrapolate past trends. The aim of this paper is to develop a forecasting model that is based on the regularity of the modal age at death and that can also account for changes in lifespan variation. We forecast mortality at ages 40 and above in 10 West European countries. The model we introduce increases forecast accuracy compared with other forecasting models and provides consistent trends in life expectancy and lifespan variation at age 40 over time.

Using a penalized likelihood to detect mortality deceleration.

We suggest a novel method for detecting mortality deceleration by adding a penalty to the log-likelihood function in a gamma-Gompertz setting. This is an alternative to traditional likelihood inference and hypothesis testing. The main advantage of the proposed method is that it does not involve using a p-value, hypothesis testing, and asymptotic distributions. We evaluate the performance of our approach by comparing it with traditional likelihood inference on both simulated and real mortality data. Results have shown that our method is more accurate in detecting mortality deceleration and provides more reliable estimates of the underlying parameters. The proposed method is a significant contribution to the literature as it offers a powerful tool for analyzing mortality patterns.

Quantifying impacts of the COVID-19 pandemic through life-expectancy losses: a population-level study of 29 countries.

BACKGROUND: Variations in the age patterns and magnitudes of excess deaths, as well as differences in population sizes and age structures, make cross-national comparisons of the cumulative mortality impacts of the COVID-19 pandemic challenging. Life expectancy is a widely used indicator that provides a clear and cross-nationally comparable picture of the population-level impacts of the pandemic on mortality. METHODS: Life tables by sex were calculated for 29 countries, including most European countries, Chile and the USA, for 2015-2020. Life expectancy at birth and at age 60 years for 2020 were contextualized against recent trends between 2015 and 2019. Using decomposition techniques, we examined which specific age groups contributed to reductions in life expectancy in 2020 and to what extent reductions were attributable to official COVID-19 deaths. RESULTS: Life expectancy at birth declined from 2019 to 2020 in 27 out of 29 countries. Males in the USA and Lithuania experienced the largest losses in life expectancy at birth during 2020 (2.2 and 1.7 years, respectively), but reductions of more than an entire year were documented in 11 countries for males and 8 among females. Reductions were mostly attributable to increased mortality above age 60 years and to official COVID-19 deaths. CONCLUSIONS: The COVID-19 pandemic triggered significant mortality increases in 2020 of a magnitude not witnessed since World War II in Western Europe or the breakup of the Soviet Union in Eastern Europe. Females from 15 countries and males from 10 ended up with lower life expectancy at birth in 2020 than in 2015.

Adequate life-expectancy reconstruction for adult human mortality data.

Mortality information of populations is aggregated in life tables that serve as a basis for calculation of life expectancy and various life disparity measures. Conventional life-table methods address right-censoring inadequately by assuming a constant hazard in the last open-ended age group. As a result, life expectancy can be substantially distorted, especially in the case when the last age group in a life table contains a large proportion of the population. Previous research suggests addressing censoring in a gamma-Gompertz-Makeham model setting as this framework incorporates all major features of adult mortality. In this article, we quantify the difference between gamma-Gompertz-Makeham life expectancy values and those published in the largest publicly available high-quality life-table databases for human populations, drawing attention to populations for which life expectancy values should be reconsidered. We also advocate the use of gamma-Gompertz-Makeham life expectancy for three reasons. First, model-based life-expectancy calculation successfully handles the problem of data quality or availability, resulting in severe censoring due to the unification of a substantial number of deaths in the last open-end age group. Second, model-based life expectancies are preferable in the case of data scarcity, i.e. when data contain numerous age groups with zero death counts: here, we provide an example of hunter-gatherer populations. Third, gamma-Gompertz-Makeham-based life expectancy values are almost identical to the ones provided by the major high-quality human mortality databases that use more complicated procedures. Applying a gamma-Gompertz-Makeham model to adult mortality data can be used to revise life-expectancy trends for historical populations that usually serve as input for mortality forecasts.

Estimating the contribution of mortality selection to the East-West German mortality convergence.

BACKGROUND: Before German reunification, old-age mortality was considerably higher in East Germany than West Germany but converged quickly afterward. Previous studies attributed this rapid catch-up to improved living conditions. We add to this discussion by quantifying for the first time the impact of mortality selection. METHODS: We use a gamma-Gompertz mortality model to estimate the contribution of selection to the East-West German mortality convergence before and after reunification. RESULTS: We find that, compared to the West, frailer East Germans died earlier due to deteriorating mortality conditions leading to converging mortality rates for women and men already over age 70 before 1990. After 1990, the selection of frailer individuals played only a minor role in closing the East-west German mortality gap. However, our study suggests that, after reunification, old-age mortality improved quickly because the more robust population in the East benefitted greatly from ameliorating external factors such as health care and better living standards. CONCLUSION: Our results from a natural experiment show that selection of frail individuals plays an important role in population-level mortality dynamics. In the case of the German reunification, East German old-age mortality already converged before 1990 because of stronger selection pressure.

Stochasticity, heterogeneity, and variance in longevity in human populations.

Inter-individual variance in longevity (or any other demographic outcome) may arise from heterogeneity or from individual stochasticity. Heterogeneity refers to differences among individuals in the demographic rates experienced at a given age or stage. Stochasticity refers to variation due to the random outcome of demographic rates applied to individuals with the same properties. The variance due to individual stochasticity can be calculated from a Markov chain description of the life cycle. The variance due to heterogeneity can be calculated from a multistate model that incorporates the heterogeneity. We show how to use this approach to decompose the variance in longevity into contributions from stochasticity and heterogeneous frailty for male and female cohorts from Sweden (1751-1899), France (1816-1903), and Italy (1872-1899), and also for a selection of period data for the same countries. Heterogeneity in mortality is described by the gamma-Gompertz-Makeham model, in which a gamma distributed "frailty" modifies a baseline Gompertz-Makeham mortality schedule. Model parameters were estimated by maximum likelihood for a range of starting ages. The estimates were used to construct an age×frailty-classified matrix model, from which we compute the variance of longevity and its components due to heterogeneous frailty and to individual stochasticity. The estimated fraction of the variance in longevity due to heterogeneous frailty (averaged over time) is less than 10% for all countries and for both sexes. These results suggest that most of the variance in human longevity arises from stochasticity, rather than from heterogeneous frailty.

Evolutionary theory of ageing and the problem of correlated Gompertz parameters.

The Gompertz mortality model is often used to evaluate evolutionary theories of ageing, such as the Medawar-Williams' hypothesis that high extrinsic mortality leads to faster ageing. However, fits of the Gompertz mortality model to data often find the opposite result that mortality is negatively correlated with the rate of ageing. This negative correlation has been independently discovered in several taxa and is known in actuarial studies of ageing as the Strehler-Mildvan correlation. We examine the role of mortality selection in determining late-life variation in susceptibility to death, which has been suggested to be the cause of this negative correlation. We demonstrate that fixed-frailty models that account for heterogeneity in frailty do not remove the correlation and that the correlation is an inherent statistical property of the Gompertz distribution. Linking actuarial and biological rates of ageing will continue to be a pressing challenge, but the Strehler-Mildvan correlation itself should not be used to diagnose any biological, physiological, or evolutionary process. These findings resolve some key tensions between theory and data that affect evolutionary and biological studies of ageing and mortality. Tests of evolutionary theories of ageing should include direct measures of physiological performance or condition.

Quantifying the shape of aging.

In Biodemography, aging is typically measured and compared based on aging rates. We argue that this approach may be misleading, because it confounds the time aspect with the mere change aspect of aging. To disentangle these aspects, here we utilize a time-standardized framework and, instead of aging rates, suggest the shape of aging as a novel and valuable alternative concept for comparative aging research. The concept of shape captures the direction and degree of change in the force of mortality over age, which­on a demographic level­reflects aging. We 1) provide a list of shape properties that are desirable from a theoretical perspective, 2) suggest several demographically meaningful and non-parametric candidate measures to quantify shape, and 3) evaluate performance of these measures based on the list of properties as well as based on an illustrative analysis of a simple dataset. The shape measures suggested here aim to provide a general means to classify aging patterns independent of any particular mortality model and independent of any species-specific time-scale. Thereby they support systematic comparative aging research across different species or between populations of the same species under different conditions and constitute an extension of the toolbox available to comparative research in Biodemography.

Gompertz-Makeham life expectancies: expressions and applications.

In a population of individuals, whose mortality is governed by a Gompertz-Makeham hazard, we derive closed-form solutions to the life-expectancy integral, corresponding to the cases of homogeneous and gamma-heterogeneous populations, as well as in the presence/absence of the Makeham term. Derived expressions contain special functions that aid constructing high-accuracy approximations, which can be used to study the elasticity of life expectancy with respect to model parameters. Knowledge of Gompertz-Makeham life expectancies aids constructing life-table exposures.

Optimal semelparity.

Semelparous organisms have a simple life cycle characterized by immediate death after reproduction. We assume that semelparous life histories can be separated into a juvenile non-reproductive period followed by an adult period during which reproduction is possible. We derive formulae for the optimal age and size at reproduction and for the optimal size of the offspring (e.g., seeds). Our main contribution is to determine the conditions under which the optimal size of the offspring does not depend on the optimal size at reproduction and vice versa.

Admissible mixing distributions for a general class of mixture survival models with known asymptotics.

Statistical analysis of data on the longest living humans leaves room for speculation whether the human force of mortality is actually leveling off. Based on this uncertainty, we study a mixture failure model, introduced by Finkelstein and Esaulova (2006) that generalizes, among others, the proportional hazards and accelerated failure time models. In this paper we first, extend the Abelian theorem of these authors to mixing distributions, whose densities are functions of regular variation. In addition, taking into account the asymptotic behavior of the mixture hazard rate prescribed by this Abelian theorem, we prove three Tauberian-type theorems that describe the class of admissible mixing distributions. We illustrate our findings with examples of popular mixing distributions that are used to model unobserved heterogeneity.