Unobserved population heterogeneity

Vaupel, James W.; Missov, Trifon I.

doi:10.4054/demres.2014.31.22

Cited by 45 publications

(25 citation statements)

References 39 publications

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“…Barker frailty augments the force opposing mortality decline with excess mortality among those who, having been exposed to adverse early-life conditions, manifest Barker effects upon attaining adult ages. Our formulation reconciles standard frailty with Barker frailty as survivors to critical ages become a “newly born” cohort at that age that experiences mortality rates with standard frailty and mortality multiplier R δ (rather than δ) (Aalen 1988; Steinsaltz and Wachter 2006; Vaupel and Missov 2014; Vaupel and Yashin 1987; Vaupel et al 1979). …”

Section: Summary Discussion and Extensionsmentioning

confidence: 99%

“…First, although there are multiple pathways through which early-life conditions can manifest themselves as delayed effects, the current model treats them all as if they share the same dominant features and ignores dissimilarities. The simplification is needed to make the model tractable as a simple generalization of the standard frailty model (Vaupel and Missov 2014; Vaupel and Yashin 1987; Vaupel et al 1979). Second, unlike the continuous version of the model, it assumes that early conditions can be treated as a 1/0 binary variable and that the population at birth can be classified into those who did and those who did not had adverse experiences.…”

Section: Introductionmentioning

confidence: 99%

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Discrete Barker Frailty and Warped Mortality Dynamics at Older Ages

Palloni

Beltrán‐Sánchez

2017

Demography

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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.

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Section: Summary Discussion and Extensionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Discrete Barker Frailty and Warped Mortality Dynamics at Older Ages

Palloni

Beltrán‐Sánchez

2017

Demography

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“…We suppressed those artificial fluctuations over age by having a strong penalty term λ for the P-spline smoothing. Death rates at ages below 80 y for contemporary Sweden , [1] which expresses the population hazard μ(x) as a mixture of Gompertz-distributed hazards with Gamma-distributed "frailty" among individuals (62). We used the same parametric model to estimate mortality for historic Sweden, contemporary Sweden, and Japan at ages 80 y and higher.…”

Section: Box 1 Demographic Distributions and Measuresmentioning

confidence: 99%

The emergence of longevous populations

Colchero

Rau

Jones

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

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139

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The human lifespan has traversed a long evolutionary and historical path, from short-lived primate ancestors to contemporary Japan, Sweden, and other longevity frontrunners. Analyzing this trajectory is crucial for understanding biological and sociocultural processes that determine the span of life. Here we reveal a fundamental regularity. Two straight lines describe the joint rise of life expectancy and lifespan equality: one for primates and the second one over the full range of human experience from average lifespans as low as 2 y during mortality crises to more than 87 y for Japanese women today. Across the primate order and across human populations, the lives of females tend to be longer and less variable than the lives of males, suggesting deep evolutionary roots to the male disadvantage. Our findings cast fresh light on primate evolution and human history, opening directions for research on inequality, sociality, and aging.biodemography | equality | lifespan | pace and shape | senescence L ongevous populations have two characteristics: The average length of life is long and relative variation in lifespans is low. For example, life tables for contemporary Sweden and Japan indicate that most deaths occur at ages between the late 70s and early 90s. Our primate relatives, in contrast, have lifespans that are highly variable in length but short on average and rarely longer than 30 y (Fig. 1). An association between the average length of life and its variability has been found for industrialized societies (1, 2). However, detailed knowledge is lacking about whether and how this association varies across species separated by millions of years of primate evolution or whether it has changed over the past several centuries of unprecedented social progress in human populations. Fuller comprehension of the relationship between rising lifespans and reduced lifespan variability across evolution and history holds potential insights that might illuminate past, current, and future longevity.We pose three related questions aimed at filling this knowledge gap: How long and variable are lifespans for humans compared with nonhuman primates, for humans today compared with the past, and for males compared with females? We provide answers to these questions by applying a powerful framework that simultaneously examines changes in both the average length of life in a population or species-the "pace" of life-and relative variation in the length of life, i.e., the "shape" of the distribution of ages at death (3-5). Studying variation in lifespan links to increasing interest in social, economic, and health inequalities and to key sociological findings that relate social factors-including high social status and social integration-to longer, healthier lifespans in human and animal societies (6-10).Estimating the average length of life (here measured by life expectancy, the mean age at death) and variation in lifespans relative to the average (measured here as "lifespan equality"; Box 1) requires data on the ages at death of individuals...

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“…Other models for the effects of frailty can be incorporated into the construction of the matrices U i , by replacing the proportional hazard formulation in (11) with an expression appropriate to the frailty effects. Vaupel and Yashin (2006), for example, briefly considered a model in which frailty acted to accelerate aging, with…”

Section: Models For the Effects Of Frailtymentioning

confidence: 99%

“…For example, Vaupel and Yashin (2006) considered a model with two frailty states, z 1 and z 2 . Individuals begin life with frailty z 1 and mortality µ 1 (x), and change from state one to state two at a rate λ(x).…”

Section: Models For the Dynamics Of Frailtymentioning

confidence: 99%

A matrix approach to the statistics of longevity in the gamma-Gompertz and related mortality models

Caswell¹

2014

DemRes

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BACKGROUNDThe gamma-Gompertz model is a fixed frailty model in which baseline mortality increases exponentially with age, frailty has a proportional effect on mortality, and frailty at birth follows a gamma distribution. Mortality selects against the more frail, so the marginal mortality rate decelerates, eventually reaching an asymptote. The gammaGompertz is one of a wider class of frailty models, characterized by the choice of baseline mortality, effects of frailty, distributions of frailty, and assumptions about the dynamics of frailty. OBJECTIVETo develop a matrix model to compute all the statistical properties of longevity from the gamma-Gompertz and related models. METHODSI use the vec-permutation matrix formulation to develop a model in which individuals are jointly classified by age and frailty. The matrix is used to project the age and frailty dynamics of a cohort and the fundamental matrix is used to obtain the statistics of longevity. RESULTSThe model permits calculation of the mean, variance, coefficient of variation, skewness and all moments of longevity, the marginal mortality and survivorship functions, the dynamics of the frailty distribution, and other quantities. The matrix formulation extends naturally to other frailty models. I apply the analysis to the gamma-Gompertz model (for humans and laboratory animals), the gamma-Makeham model, and the gamma-Siler model, and to a hypothetical dynamic frailty model characterized by diffusion of frailty with reflecting boundaries.The matrix model permits partitioning the variance in longevity into components due to heterogeneity and to individual stochasticity. In several published human data sets, Caswell: A matrix approach to the statistics of longevity in heterogeneous frailty models heterogeneity accounts for less than 10% of the variance in longevity. In laboratory populations of five invertebrate animal species, heterogeneity accounts for 46% to 83% of the total variance in longevity. IntroductionThe gamma-Gompertz (hereafter, G-G) model is one of a class of models that investigate the effects of hidden heterogeneity -heterogeneity that is either unobservable or unobserved -on mortality rates. It calculates various consequences of that heterogeneity for the dynamics of cohorts (e.g. Vaupel, Manton, and Stallard 1979;Vaupel and Yashin 1985;Yashin, Iachine, and Begun 2000;Vaupel 2010;Wienke 2010;Missov 2013). I refer to this heterogeneity as frailty, making no assumptions about its causes. The goal of this paper is to present a matrix formulation that permits easy computation of all the properties of the G-G and other frailty models. Frailty models can be characterized by their components:1. the baseline mortality rate, 2. the effects of frailty on the baseline mortality rate, 3. the dynamics of individual frailty over time, and 4. the initial distribution of frailty.The baseline mortality rate in the G-G model is the Gompertz model, in which mortality increases exponentially with age t, µ(t) = ae bt .Frailty affects mortality as a proportio...

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Unobserved population heterogeneity

Cited by 45 publications

References 39 publications

Discrete Barker Frailty and Warped Mortality Dynamics at Older Ages

Discrete Barker Frailty and Warped Mortality Dynamics at Older Ages

The emergence of longevous populations

A matrix approach to the statistics of longevity in the gamma-Gompertz and related mortality models

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