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Gompertz, Makeham, and Siler models explain Taylor's law in human mortality data

Author

Listed:
  • Joel E. Cohen

    (Rockefeller University)

  • Christina Bohk-Ewald

    (Limbus Medical Technologies GmbH)

  • Roland Rau

    (Universität Rostock)

Abstract

Background: Taylor’s law (TL) states a linear relationship on logarithmic scales between the variance and the mean of a nonnegative quantity. TL has been observed in spatiotemporal contexts for the population density of hundreds of species including humans. TL also describes temporal variation in human mortality in developed countries, but no explanation has been proposed. Objective: To understand why and to what extent TL describes temporal variation in human mortality, we examine whether the mortality models of Gompertz, Makeham, and Siler are consistent with TL. We also examine how strongly TL differs between observed and modeled mortality, between women and men, and among countries. Methods: We analyze how well each mortality model explains TL fitted to observed occurrence–exposure death rates by comparing three features: the log–log linearity of the temporal variance as a function of the temporal mean, the age profile, and the slope of TL. We support some empirical findings from the Human Mortality Database with mathematical proofs. Results: TL describes modeled mortality better than observed mortality and describes Gompertz mortality best. The age profile of TL is closest between observed and Siler mortality. The slope of TL is closest between observed and Makeham mortality. The Gompertz model predicts TL with a slope of exactly 2 if the modal age at death increases linearly with time and the parameter that specifies the growth rate of mortality with age is constant in time. Observed mortality obeys TL with a slope generally less than 2. An explanation is that, when the parameters of the Gompertz model are estimated from observed mortality year by year, both the modal age at death and the growth rate of mortality with age change over time. Conclusions: TL describes human mortality well in developed countries because their mortality schedules are approximated well by classical mortality models, which we have shown to obey TL. Contribution: We provide the first theoretical linkage between three classical demographic models of mortality and TL.

Suggested Citation

  • Joel E. Cohen & Christina Bohk-Ewald & Roland Rau, 2018. "Gompertz, Makeham, and Siler models explain Taylor's law in human mortality data," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 38(29), pages 773-842.
    • Handle: RePEc:dem:demres:v:38:y:2018:i:29
    DOI: 10.4054/DemRes.2018.38.29
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    References listed on IDEAS

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    Cited by:

    1. Alois Pichler & Dana Uhlig, 2021. "Mortality in Germany during the Covid-19 pandemic," Papers 2107.12899, arXiv.org, revised Apr 2022.
    2. Yang Yang & Han Lin Shang & Joel E. Cohen, 2022. "Temporal and spatial Taylor's law: Application to Japanese subnational mortality rates," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(4), pages 1979-2006, October.
    3. Jonas Šiaulys & Rokas Puišys, 2022. "Survival with Random Effect," Mathematics, MDPI, vol. 10(7), pages 1-17, March.
    4. María-Dolores Huete-Morales & Esteban Navarrete-Álvarez & María-Jesús Rosales-Moreno & María-José Del-Moral-Ávila & José-Manuel Quesada-Rubio, 2020. "Modelling the survival function of the Spanish population by the Wong–Tsui model with the incorporation of frailty and covariates," Letters in Spatial and Resource Sciences, Springer, vol. 13(2), pages 151-163, August.
    5. Mathias Voigt & Antonio Abellán & Julio Pérez & Diego Ramiro, 2020. "The effects of socioeconomic conditions on old-age mortality within shared disability pathways," PLOS ONE, Public Library of Science, vol. 15(9), pages 1-17, September.
    6. Alois Pichler & Dana Uhlig, 2023. "Mortality in Germany during the COVID-19 Pandemic," IJERPH, MDPI, vol. 20(20), pages 1-11, October.
    7. Tim Riffe & Jose Manuel Aburto, 2020. "Lexis fields," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 42(24), pages 713-726.

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    More about this item

    Keywords

    Taylor's law; mortality; mortality model; Gompertz mortality; Makeham; Siler model; mean mortality;
    All these keywords.

    JEL classification:

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

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