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Unobserved population heterogeneity

Author

Listed:
  • James W. Vaupel

    (Syddansk Universitet)

  • Trifon Missov

    (Syddansk Universitet)

Abstract

Background: Survival models accounting for unobserved heterogeneity (frailty models) play an important role in mortality research, yet there is no article that concisely summarizes useful relationships. Objective: We present a list of important mathematical relationships that govern populations in which individuals differ from each other in unobserved ways. For some relationships we present proofs that, albeit formal, tend to be simple and intuitive. Methods: We organize the article in a progression, starting with general relationships and then turning to models with stronger and stronger assumptions. Results: We start with the general case, in which we do not assume any structure of the underlying baseline hazard, the frailty distribution, or their link to one another. Then we sequentially assume, first, a relative-risk model; second, a gamma distribution for frailty; and, finally, a Gompertz and Gompertz-Makeham specification for baseline mortality. Comments: The article might serve as a handy overall reference to frailty models, especially for mortality research.

Suggested Citation

  • James W. Vaupel & Trifon Missov, 2014. "Unobserved population heterogeneity," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 31(22), pages 659-686.
    • Handle: RePEc:dem:demres:v:31:y:2014:i:22
    DOI: 10.4054/DemRes.2014.31.22
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    References listed on IDEAS

    as
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    Citations

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    Cited by:

    1. Lucia Zanotto & Vladimir Canudas-Romo & Stefano Mazzuco, 2021. "A Mixture-Function Mortality Model: Illustration of the Evolution of Premature Mortality," European Journal of Population, Springer;European Association for Population Studies, vol. 37(1), pages 1-27, March.
    2. Feehan, Dennis & Wrigley-Field, Elizabeth, 2020. "How do populations aggregate?," SocArXiv 2fkw3, Center for Open Science.
    3. Lindholm, Mathias, 2017. "A note on the connection between some classical mortality laws and proportional frailty," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 76-82.
    4. Dennis Feehan & Elizabeth Wrigley-Field, 2021. "How do populations aggregate?," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 44(15), pages 363-378.
    5. Hui Zheng, 2020. "Unobserved population heterogeneity and dynamics of health disparities," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 43(34), pages 1009-1048.
    6. Andrea Verhulst & Hiram Beltran-Sanchez & Alberto Palloni, 2019. "Impact of delayed effects on human old-age mortality," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 40(41), pages 1167-1210.
    7. Alberto Palloni & Hiram Beltrán-Sánchez, 2017. "Discrete Barker Frailty and Warped Mortality Dynamics at Older Ages," Demography, Springer;Population Association of America (PAA), vol. 54(2), pages 655-671, April.
    8. Yifan Yang & Omer Karin & Avi Mayo & Xiaohu Song & Peipei Chen & Ana L. Santos & Ariel B. Lindner & Uri Alon, 2023. "Damage dynamics and the role of chance in the timing of E. coli cell death," Nature Communications, Nature, vol. 14(1), pages 1-11, December.
    9. 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.
    10. Anders Ledberg, 2020. "Exponential increase in mortality with age is a generic property of a simple model system of damage accumulation and death," PLOS ONE, Public Library of Science, vol. 15(6), pages 1-17, June.
    11. Alexander, Monica, 2018. "Deaths without denominators: using a matched dataset to study mortality patterns in the United States," SocArXiv q79ye, Center for Open Science.
    12. Giambattista Salinari & Gustavo De Santis, 2020. "One or more rates of ageing? The extended gamma-Gompertz model (EGG)," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(2), pages 211-236, June.

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    More about this item

    Keywords

    unobserved heterogeneity; relative-risk models; Gompertz–Makeham law of mortality; gamma-distributed frailty;
    All these keywords.

    JEL classification:

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

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