Volume 38 - Article 29 | Pages 773–842  

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

By Joel E. Cohen, Christina Bohk-Ewald, Roland Rau


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.

Author's Affiliation

Other articles by the same author/authors in Demographic Research

Taylor's power law in human mortality
Volume 33 - Article 21

Measuring the concentration of urban population in the negative exponential model using the Lorenz curve, Gini coefficient, Hoover dissimilarity index, and relative entropy
Volume 44 - Article 49

Is the fraction of people ever born who are currently alive rising or falling?
Volume 30 - Article 56

Minor gradient in mortality by education at the highest ages: An application of the Extinct-Cohort method
Volume 29 - Article 19

Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy
Volume 24 - Article 11

Life expectancy is the death-weighted average of the reciprocal of the survival-specific force of mortality
Volume 22 - Article 5

Constant global population with demographic heterogeneity
Volume 18 - Article 14

Seasonal mortality in Denmark: the role of sex and age
Volume 9 - Article 9

Most recent similar articles in Demographic Research

Two-dimensional contour decomposition: Decomposing mortality differences into initial difference and trend components by age and cause of death
Volume 50 - Article 41    | Keywords: decomposition methods, mortality

International completeness of death registration
Volume 50 - Article 38    | Keywords: data collection, death, mortality, statistics, sustainable development goals, vital registration

Incorporating subjective survival information in mortality and change in health status predictions: A Bayesian approach
Volume 50 - Article 36    | Keywords: Bayesian demography, health, mortality, self report, subjective mortality probabilities

Standardized mean age at death (MADstd): Exploring its potentials as a measure of human longevity
Volume 50 - Article 30    | Keywords: formal demography, life expectancy, mean age at death, mortality, standardization

Differences in mortality before retirement: The role of living arrangements and marital status in Denmark
Volume 50 - Article 20    | Keywords: inequalities, living arrangements, marital status, mortality, retirement