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A matrix approach to the statistics of longevity in heterogeneous frailty models

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  • Hal Caswell

    (Universiteit van Amsterdam)

Abstract

Background: The 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 gamma-Gompertz 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. Objective: To develop a matrix model to compute all the statistical properties of longevity from the gamma-Gompertz and related models. Methods: I 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. Results: The 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, 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.

Suggested Citation

  • Hal Caswell, 2014. "A matrix approach to the statistics of longevity in heterogeneous frailty models," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 31(19), pages 553-592.
    • Handle: RePEc:dem:demres:v:31:y:2014:i:19
    DOI: 10.4054/DemRes.2014.31.19
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    References listed on IDEAS

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    9. Hal Caswell, 2011. "Beyond R0: Demographic Models for Variability of Lifetime Reproductive Output," PLOS ONE, Public Library of Science, vol. 6(6), pages 1-21, June.
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    Cited by:

    1. Silke van Daalen & Hal Caswell, 2015. "Lifetime reproduction and the second demographic transition: Stochasticity and individual variation," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 33(20), pages 561-588.
    2. Hartemink, Nienke & Missov, Trifon I. & Caswell, Hal, 2017. "Stochasticity, heterogeneity, and variance in longevity in human populations," Theoretical Population Biology, Elsevier, vol. 114(C), pages 107-116.
    3. Coste, Christophe F.D. & Pavard, Samuel, 2020. "Analysis of a multitrait population projection matrix reveals the evolutionary and demographic effects of a life history trade-off," Ecological Modelling, Elsevier, vol. 418(C).
    4. Coste, Christophe F.D. & Austerlitz, Frédéric & Pavard, Samuel, 2017. "Trait level analysis of multitrait population projection matrices," Theoretical Population Biology, Elsevier, vol. 116(C), pages 47-58.
    5. Hal Caswell & Silke van Daalen, 2021. "Healthy longevity from incidence-based models: More kinds of health than stars in the sky," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 45(13), pages 397-452.
    6. van Daalen, Silke & Caswell, Hal, 2020. "Variance as a life history outcome: Sensitivity analysis of the contributions of stochasticity and heterogeneity," Ecological Modelling, Elsevier, vol. 417(C).
    7. Hal Caswell, 2020. "The formal demography of kinship II: Multistate models, parity, and sibship," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 42(38), pages 1097-1146.

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    More about this item

    Keywords

    frailty; mortality; longevity; heterogeneity; Markov chains; individual stochasticity;
    All these keywords.

    JEL classification:

    • J1 - Labor and Demographic Economics - - Demographic Economics
    • Z0 - Other Special Topics - - General

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