1 Gompertz, Makeham, and Siler models explain Taylor's law in human - PDF document

1 Gompertz, Makeham, and Siler models explain Taylor's law in human mortality data 2 Research Article 3 Authors 4 Joel E. Cohen*, Christina Bohk-Ewald*, Roland Rau 5 * JEC and CBE contributed equally to the preparation of the manuscript. 6 7 Joel E. Cohen Ph.D., Dr.P.H. (Corresponding author) 8 Laboratory of Populations, The Rockefeller University and Columbia University; Earth Institute 9 and Department of Statistics, Columbia University; Department of Statistics, University of Chicago, 10 USA 11 Email: cohen@mail.rockefeller.edu 12 Phone: +1(212)327-8883 13 Fax: +1(212)327-8712 (please put my name on top; this is a shared fax) 14 Address: Laboratory of Populations 15 Rockefeller University & Columbia University 16 1230 York Avenue, Box 20 17 New York, NY 10065-6399 USA 18 19 Christina Bohk-Ewald Ph.D. 20 Max Planck Institute for Demographic Research, Germany 21 Email: BohkEwald@demogr.mpg.de 22 23 Roland Rau Ph.D. 24 University of Rostock and Max Planck Institute for Demographic Research, Germany 25 Email: roland.rau@uni-rostock.de and Rau@demogr.mpg.de 26

27 Abstract 28 BACKGROUND 29 Taylor’s law (TL) states a linear relationship on logarithmic scales between the variance and the 30 mean of a nonnegative quantity. TL has been observed in spatiotemporal contexts for the population 31 density of hundreds of species including humans. TL also describes temporal variation in human 32 mortality in developed countries but no explanation has been proposed. 33 OBJECTIVE 34 To understand why and to what extent TL describes temporal variation in human mortality, we 35 examine whether the mortality models of Gompertz, Makeham, and Siler are consistent with TL. 36 We also examine how strongly TL differs between observed and modeled mortality, between 37 women and men, and between countries. 38 METHOD 39 We analyze how well each mortality model explains TL fitted to observed occurrence-exposure 40 death rates by comparing three features: the log-log linearity, the age profile, and the slope of TL. 41 We support some empirical findings from the Human Mortality Database with mathematical proofs. 42 RESULTS 43 TL describes modeled mortality better than observed mortality and describes Gompertz mortality 44 best. The age profile of TL is closest between observed and Siler mortality. The slope of TL is 45 closest between observed and Makeham mortality. 46 The Gompertz model predicts TL with a slope of exactly 2 if the modal age at death increases 47 linearly with time and the parameter that specifies the growth rate of mortality with age is constant 48 in time. 49 CONCLUSIONS

50 TL describes human mortality well in developed countries because their mortality schedules are 51 approximated well by classical mortality models, which we have shown to obey TL. 52 CONTRIBUTION 53 We provide the first theoretical linkage between three classical demographic models of mortality 54 and TL. 55

56 1. Introduction 57 Taylor's law (TL) states that the logarithm of the variance of some nonnegative quantity is 58 approximately a linear function of the logarithm of the mean of that quantity, in multiple sets of 59 observations. TL describes the population densities of hundreds of species in ecology (Taylor 1961; 60 Kilpatrick and Ives 2003; Kendal 2004) and other nonnegative quantities in medicine (numbers of 61 metastases), epidemiology (cases of infectious diseases), genetics (numbers of single-nucleotide 62 polymorphisms), and many other fields (Eisler et al. 2008). In human demography, TL describes the 63 density (people per area) of Norway's population (Cohen, Xu, and Brunborg 2013) and the age- 64 specific force of mortality (henceforth, simply "mortality") in developed countries (Bohk, Rau, and 65 Cohen 2015). 66 As an empirical generalization, TL invites attempts at explanation. Why is there an approximately 67 linear relationship between the log of the temporal mean and the log of the temporal variance of 68 mortality in many developed countries? Is this empirical regularity in human mortality just a 69 coincidence or does it come from an underlying pattern or mechanism? Here we answer these 70 questions by comparing the completely empirical results of Bohk et al. (2015) with the results of 71 fitting three human mortality models: those of Gompertz (1825), Makeham (1860), and Siler (1979; 72 1983), which belong to the same family. We show that fitted mortality of these models obeys TL, 73 exactly or approximately. Hence these models provide a theoretical basis for TL in human 74 mortality. 75 Mortality models express mathematically the age schedule of mortality (that is, mortality as a 76 function of age) in a given year. Models differ in the number of parameters they use and in the age 77 ranges for which they model mortality well. The more parameters they use, the more flexibly they 78 can fit mortality at different ages, but the more difficult they are to analyze mathematically. 79 Gompertz' model (1825), with two parameters, is one of the most popular models in demography. It 80 assumes a linear increase in the logarithm of mortality with age. It usually describes well mortality

81 at ages 30 to 90 (so-called senescent mortality). The model of Makeham (1860) adds to Gompertz' 82 model an additional age-independent constant to represent non-senescent background mortality 83 effective at all ages. This improves the fit at some younger ages. The model of Siler (1979; 1983) 84 adds to the Makeham model an exponential decay in mortality to represent the decline in mortality 85 from infancy to childhood. 86 The main objectives of this study are to examine whether the models of Gompertz, Makeham, and 87 Siler are consistent with TL and can help to explain why and to what extent TL holds. In addition, 88 we examine how strongly TL differs between observed mortality and model mortality schedules, 89 between women and men, and between countries. 90 We base our analysis on empirical occurrence-exposure death rates, statistical analysis, and 91 theoretical explanations, and thus provide the first theoretical linkage between three classical 92 demographic models of mortality and TL: We show mathematically that the Gompertz model (the 93 simplest of the three models we considered) with linearly changing modal age at death and a 94 constant rate of growth of mortality with age predicts TL with a slope that is exactly equal to 2. As 95 the Gompertz model is a special case of the other two models, it is evident that the results from the 96 Gompertz model are also valid for certain parameter values of the more complex models of 97 Makeham and Siler. 98 Our analyses yield theoretical and empirical insights into the occurrence of TL in human mortality, 99 giving a comprehensive picture of the extent to which TL describes the temporal variance in age- 100 specific mortality in human populations. As we have confirmed TL to be a regular pattern (rather 101 than a coincidence) in human mortality, it can be validly applied in other demographic studies such 102 as generating and evaluating mortality forecasts. 103 In the remainder of this article, section 2 describes data and methods; section 3 presents results; and 104 section 4 summarizes the main findings. Appendices 1 and 2 give theorems, mathematical proofs

105 and approximations of TL for the Gompertz and Makeham models, respectively. Supplementary 106 material (online) includes estimates of model parameters, figures, and descriptive files. 107 2. Data and methods 108 We extracted deaths and life-years of exposures to the risk of death by single year of age, 0 to 100, 109 and calendar year, 1960 to 2009, for 12 countries of the Human Mortality Database (2015). Given 110 these data, we estimated observed mortality (or "observations") defined as deaths divided by 111 exposures by single years of age and calculated predicted mortality for each year separately with the 112 models of Gompertz, Makeham, and Siler. Henceforth the word "observations" means "occurrence- 113 exposure death rates." 114 2.1 Three theoretical models of mortality 115 We used mathematical expressions of the models of Gompertz, Makeham, and Siler that are based 116 on the (old-age) modal age at death (Horiuchi et al. 2013; Missov et al. 2015; Bergeron-Boucher, 117 Ebeling, and Canudas-Romo 2015). The modal age at death is the age (beyond infancy and 118 childhood) at which the probability density of life table deaths has a maximum. The conventional 119 formulas for the models of Gompertz and Makeham use a parameter for an initial level of mortality 120 instead of the modal age at death. The formulas based on the modal age at death, given below, are 121 numerically more stable, have a better demographic interpretation, and are more comparable across 122 populations and points of time (Missov et al. 2015; Horiuchi et al. 2013). 123 The Gompertz model expresses mortality μ at age x in year t as 𝜈 �,� = 𝛾 � 𝑓 � � (��� � ) , 𝛾 � > 0, 𝑁 � > 0, for 𝑢 = 1, … , 𝑈, 𝑦 = 0, … , 𝑌. 124 (1) 125 Here β t is the growth rate of mortality with age, M t the modal age at death in year t , T the number of 126 years of observations, and X the maximum observed age. The Gompertz model predicts, on a 127 logarithmic scale of mortality and a linear scale of age, a linear increase in mortality from some 128 young age to the oldest age X = 100 here.