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

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

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

best. The age profile of TL is closest between observed and Siler mortality. The slope of TL is

44 closest between observed and Makeham mortality. 45 The Gompertz model predicts TL with a slope of exactly 2 if the modal age at death increases 46 linearly with time and the parameter that specifies the growth rate of mortality with age is constant 47 in time. 48 CONCLUSIONS 49

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

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1. Introduction

56 Taylor's law (TL) states that the logarithm of the variance of some nonnegative quantity is 57 approximately a linear function of the logarithm of the mean of that quantity, in multiple sets of 58

bservations. TL describes the population densities of hundreds of species in ecology (Taylor 1961;

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

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

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and approximations of TL for the Gompertz and Makeham models, respectively. Supplementary 105 material (online) includes estimates of model parameters, figures, and descriptive files. 106

2. Data and methods

107 We extracted deaths and life-years of exposures to the risk of death by single year of age, 0 to 100, 108 and calendar year, 1960 to 2009, for 12 countries of the Human Mortality Database (2015). Given 109 these data, we estimated observed mortality (or "observations") defined as deaths divided by 110 exposures by single years of age and calculated predicted mortality for each year separately with the 111 models of Gompertz, Makeham, and Siler. Henceforth the word "observations" means "occurrence- 112 exposure death rates." 113 2.1 Three theoretical models of mortality 114 We used mathematical expressions of the models of Gompertz, Makeham, and Siler that are based 115

n the (old-age) modal age at death (Horiuchi et al. 2013; Missov et al. 2015; Bergeron-Boucher,

116 Ebeling, and Canudas-Romo 2015). The modal age at death is the age (beyond infancy and 117 childhood) at which the probability density of life table deaths has a maximum. The conventional 118 formulas for the models of Gompertz and Makeham use a parameter for an initial level of mortality 119 instead of the modal age at death. The formulas based on the modal age at death, given below, are 120 numerically more stable, have a better demographic interpretation, and are more comparable across 121 populations and points of time (Missov et al. 2015; Horiuchi et al. 2013). 122 The Gompertz model expresses mortality μ at age x in year t as 123 𝜈, = 𝛾𝑓(), 𝛾 > 0, 𝑁 > 0, for 𝑢 = 1, … , 𝑈, 𝑦 = 0, … , 𝑌. (1) 124 Here βt is the growth rate of mortality with age, Mt the modal age at death in year t, T the number of 125 years of observations, and X the maximum observed age. The Gompertz model predicts, on a 126 logarithmic scale of mortality and a linear scale of age, a linear increase in mortality from some 127 young age to the oldest age X = 100 here. 128

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The Makeham model expresses mortality μ at age x in year t as 129 𝜈, = 𝑑 + 𝛾𝑓(), 𝑑 > 0. (2) 130 The parameter βt is the same as in the Gompertz model. Makeham added ct to represent background 131 mortality in year t, assumed to be the same at all ages x. The Makeham model predicts slowly 132 increasing mortality from infancy through childhood to young adulthood; thereafter, it models a 133 nearly linear increase in log mortality with increasing age. 134 The Siler model expresses mortality μ at age x in year t as 135 𝜈, = 𝛽𝑓, + 𝑑 + 𝛾,𝑓,(), 𝛽 > 0, 𝛾, > 0. (3) 136 Here 𝛾, = 𝛾 of the Makeham and Gompertz models. Siler adds two additional parameters: αt is 137 infant mortality, and β1,t is the rate of decline with increasing age x of childhood mortality in year t. 138 The Siler model predicts decreasing mortality from infancy to childhood, slowly increasing 139 mortality from childhood to young adulthood, and nearly linearly increasing log mortality with 140 increasing age throughout adulthood. 141 2.2 Taylor's law, mean and variance of mortality 142 TL describes a linear relationship of log10 of temporal variance (variance over time) to log10 of 143 temporal mean (mean over time) of mortality μ at age x: 144 𝑚𝑝𝑕𝑊𝑏𝑠(𝜈) = 𝑏 + 𝑐 ⋅ 𝑚𝑝𝑕𝐹(𝜈) (4) 145 where a is the intercept and b is the slope. With T years of observations or theoretical (fitted) values 146

f mortality, the temporal mean of mortality μ at age x is

147 𝐹(𝜈) =

∑

𝜈,

(5)

148 and the temporal variance is 149 𝑊𝑏𝑠(𝜈) =

∑

𝜈, − 𝐹(𝜈)

.

(6) 150

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We plot (on log-log coordinates) temporal variances and means of mortality for each age in one 151 national population at a time, separately for different national populations. These plots depict so- 152 called cross-age-scenarios of TL. 153 2.3 Statistical methods and visualization 154 2.3.1 Parameter estimation 155 We estimated the values of the parameters of the models of Gompertz, Makeham, and Siler from 156 deaths and exposures using the method of maximum likelihood. Specifically, we maximized a 157 Poisson log-likelihood in R (2015) with the function DEoptim (Mullen et al. 2011). 158 2.3.2 Analysis of log-log linearity 159 To analyze how closely TL approximates the log temporal mean and log temporal variance of 160

bserved and/or modeled mortality, we used the linear correlation coefficient r2. The closer r2 is to

161

ne, the better TL describes the relation of log temporal variance to log temporal mean of mortality.

162 2.3.3 Visualizing the age profile of TL 163 To compare the age profiles between observed and modeled mortality, we augmented the statistical 164 analysis and visualized the log10 temporal mean and log10 temporal variance of mortality by single- 165 year age groups using yellow and orange to represent children (0 to 20 years of age), red and 166 magenta for adults (21 to 60 years of age), and blue and green for older ages (61 to 100 years of 167 age). 168 2.3.4 Analysis of covariance 169 To analyze how well a theoretical model predicts the slope of TL fitted to observed mortality, we 170 used the analysis of covariance. 171

1. To determine whether the slopes of TL differ between observed and modeled mortality (by

172 sex, country, and model), we estimated a linear regression with an interaction term between the 173

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variables log10E(μx) and a dichotomous variable model (with values "observed" and "model") using 174 the lm() function in R: 175 𝑚𝑝𝑕𝑊𝑏𝑠(𝜈) = 𝑑 + 𝑑𝑚𝑝𝑕𝐹(𝜈) + 𝑑𝑛𝑝𝑒𝑓𝑚 + 𝑑(𝑚𝑝𝑕𝐹(𝜈) × 𝑛𝑝𝑒𝑓𝑚). (7) 176 We use observed mortality as the reference level for the categorical variable model. The null 177 hypothesis was that 𝑑 = 0, i.e., that there was no difference between the model and the 178

bservations in the slope of 𝑚𝑝𝑕𝑊𝑏𝑠(𝜈) as a linear function of 𝑚𝑝𝑕𝐹(𝜈). A very low p-value

179 indicated that the coefficient of the interaction term is non-zero, and that the slope of TL of a model 180 life table is not equal to the slope of TL of observed mortality data. 181

2a. We also used analysis of covariance to determine whether the slopes of TL differ between

182 males and females (by country and model). To analyze if the slopes of TL differed between 183 males and females, we estimated a linear regression with an interaction term between the variables 184 log10E(μx) and a dichotomous variable sex (with values "male" and "female"): 185 𝑚𝑝𝑕𝑊𝑏𝑠(𝜈) = 𝑒 + 𝑒𝑚𝑝𝑕𝐹(𝜈) + 𝑒𝑡𝑓𝑦 + 𝑒(𝑚𝑝𝑕𝐹(𝜈) × 𝑡𝑓𝑦). (8) 186 We used female mortality as the reference level. For the null hypothesis that 𝑒 = 0, a very low p- 187 value indicated that the coefficient 𝑒 of the interaction term is non-zero, and that the slope of TL is 188 different between males and females. 189

2b. To analyze if sex differences in the slope of TL differ between observed and modeled

190 mortality (by country and model), we estimated a linear regression with pairwise and three-way 191 interaction terms among the variables log10E(μx), sex, and model. Here, unlike equation (7) above, 192 the variable model had four values: 0 (observed), 1 (Gompertz), 2 (Makeham), 3 (Siler). Observed 193 mortality was the reference level for model, as in equation (7). For example, in testing the null 194 hypothesis that 𝑔

= 0, p = 0.00506 for the Gompertz model (model = 1) in relation to observed

195 mortality (model = 0) for all countries and p = 0.34987 for Siler (model = 3) in relation to observed 196 (model = 0) for all countries. We used female mortality as the reference level for sex. 197

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𝑚𝑝𝑕𝑊𝑏𝑠(𝜈) = 𝑔

+ 𝑔 𝑚𝑝𝑕𝐹(𝜈) + 𝑔 𝑡𝑓𝑦 + 𝑔 𝑛𝑝𝑒𝑓𝑚

198 +𝑔

(𝑚𝑝𝑕𝐹(𝜈) × 𝑡𝑓𝑦) + 𝑔 (𝑚𝑝𝑕𝐹(𝜈) × 𝑛𝑝𝑒𝑓𝑚) + 𝑔 (𝑡𝑓𝑦 × 𝑛𝑝𝑒𝑓𝑚)

199 +𝑔

(𝑚𝑝𝑕𝐹(𝜈) × 𝑡𝑓𝑦 × 𝑛𝑝𝑒𝑓𝑚).

(9) 200 The null hypothesis is that 𝑔

= 0, i.e., that model has no influence on the interaction between sex

201 and 𝑚𝑝𝑕𝐹(𝜈). 202 2.4 Availability of data and supplementary information 203 In addition to the data on mortality, which are publicly available from the Human Mortality 204 Database (2015), supplementary information deposited with this paper includes a spreadsheet 205 (TLinMortalityModels.csv) with the values of 41 variables and a text file (TLinMortalityModels- 206 Documentation-Variables.txt) that defines these variables. The spreadsheet gives the parameter 207 estimates of the models of Gompertz, Makeham, and Siler for observed female and male mortality 208 from 1960 to 2009 in 12 countries of the Human Mortality Database (2015). This information is 209 graphed in figures 1-17 and supplementary figures A1-A20. 210

3. Results

211 3.1 Fitted mortality of Gompertz, Makeham, and Siler models 212 Before we analyze how consistent the models of Gompertz, Makeham, and Siler are with TL, we 213 first show how well they fit observed mortality. Figs. 1 and 2 display the observed age-specific 214 mortality (on a logarithmic scale) as a function of age from 0 to 100, for women and men, 215 respectively, from 1960 (light gray) to 2009 (black) in 12 countries of the Human Mortality 216 Database (2015). Fitted mortality is depicted in green, blue and red for the models of Gompertz, 217 Makeham, and Siler, respectively, in 2009 for each country. 218

Figs. 1 and 2 here.

219

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The observed mortality shows a typical age pattern: a fall from infancy to around age 10, an 220 "accident bump" around age 20 (often more pronounced for men than for women), and a roughly 221 linear rise on the log scale beyond age 30. Mortality at each age declined with time in many 222 developed countries since 1960. The declines occurred mainly at younger ages before they spread 223 towards higher ages (Christensen et al. 2009; Rau et al. 2008; Vaupel 2010). 224 The Gompertz model fitted observed mortality better at adult and older ages than at younger ages, 225 where the predicted mortality was systematically and substantially too low. The Makeham model 226 fitted better than the Gompertz model, particularly for younger ages, but the modeled mortality 227 increased monotonically with age, unlike the observations. The Siler model fitted the age profile of 228 mortality better than the Gompertz and Makeham models. None of the models reproduced the 229

bserved "accident bump" of excess mortality of young adult ages. The findings in this paragraph

230 confirmed prior observations by others about the fit between human mortality data and the 231 Gompertz, Makeham, and Siler models (e.g., Bongaarts 2005; Canudas-Romo 2008; Horiuchi et al. 232 2013; Bergeron-Boucher, Ebeling, and Canudas-Romo 2015). 233 3.2 Statistical and visual tests of TL in observed and fitted mortality 234 This section reports the statistical analysis and visual tests proposed in section 2. Figs. 3 to 10 235 display the cross-age-scenarios of TL for observed and fitted mortality of women and men, 236 respectively, in 12 countries of the Human Mortality Database (2015). Odd-numbered figures are 237 for females, even-numbered for males. 238

Figs. 3-10 here.

239 3.2.1 Log-log linearity and r2 values 240

Figs. 1-2 compare the observed mortality with the three modeled mortality age schedules. Figs. 3-4

241 compare the temporal means and temporal variances of observed mortality with TL (the fitted least 242 squares straight line), on log-log coordinates. In these figures, r2 measures the linearity of observed 243

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log temporal variance as a function of observed log temporal mean. In Figs. 5-10, r2 measures the 244 linearity of the log temporal variance of modeled mortality as a function of the log temporal mean 245

f modeled mortality.

246 TL describes well the observed mortality (Figs. 3, 4) and the mortality of the models of Gompertz 247 (Figs. 5, 6), Makeham (Figs. 7, 8), and Siler (Figs. 9, 10). For observations on women, 0.96 ≤ r2 ≤ 248 0.99, and for men, 0.95 ≤ r2 ≤ 0.99 in the selected countries. Where the mortality models had r2 ≥ 249 0.999, sometimes there was excellent agreement with the strictly linear relationship posited by TL 250 (e.g., the Gompertz model for East Germany in Fig. 5 and the Makeham model for France and 251 Japan in Fig. 7). In some cases the models, and particularly the Gompertz model, predicted a 252 relation of log variance to log mean that was closer to linear than was the relation of log variance to 253 log mean based on observed mortality. The Gompertz model had r2 ≥ 0.99 more often than the other 254 two models. 255 According to the r2 values, TL describes Gompertz mortality best among these three models. 256 Appendix 1 gives a mathematical proof that TL describes Gompertz mortality exactly if the modal 257 age at death Mt increases linearly in time and if the rate of growth of mortality with age βt is 258 constant in time. The first assumption is close to reality, as we shall see. The second assumption is 259 not: βt increased slightly over time within a narrow range, even though βt is hypothesized to be 260 constant across individuals and over time (Vaupel 2010). Nevertheless, the excellent agreement 261 between TL and Gompertz mortality is at least partially explained by this mathematical analysis. 262 3.2.2 Visually comparing age profiles between observed and modeled mortality 263 In this section, we visually compare the age pattern of TL between observed and modeled mortality. 264 TL in observed mortality data (Figs. 3-4) has a typical pattern that is similar for women and men in 265 many populations. Both the log10 temporal variance and the log10 temporal mean of mortality 266 increase linearly from young adulthood (red) to the elderly (green), and they decrease from infancy 267

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(yellow) to childhood (orange). The changes in the log10 temporal mean are expected from the 268 increasing mortality with age at older ages and the decreasing mortality with age from infancy to 269

childhood. The corresponding linear changes in the log10 temporal variance were not known prior to

270 the work of Bohk, Rau and Cohen (2015). 271 TL of Gompertz mortality (Figs. 5-6) mirrors the pattern of TL of the observations well at adult 272 and old ages. However, both the log10 temporal variance and the log10 temporal mean of mortality 273

f infants and children (yellow to orange) are modeled to be smaller than those of young adults

274 (red), unlike the observations. This major difference between observed mortality and the Gompertz 275 model arises because the Gompertz model assumes a log-linear increase in mortality from the 276 youngest to the oldest age. The Gompertz model captures neither declining mortality from infancy 277 to childhood nor its related effect on TL. 278 TL of Makeham mortality (Figs. 7-8) mirrors the pattern of TL of the observations well at adult 279 and old ages. However, both the log10 temporal variance and the log10 temporal mean of mortality 280 are modeled to be almost equal for infants and children (yellow to orange), on the one side, and 281 young adults (red), on the other side, unlike the observed mortality and Gompertz mortality. These 282 major differences arise because the Makeham model assumes that mortality increases slowly from 283 infancy to young adulthood and increases exponentially thereafter. As a consequence, the Makeham 284 model captures neither the decline in observed mortality from infancy to childhood nor its related 285 effect on TL. As expected, TL of Makeham mortality is closer than TL of Gompertz mortality to TL 286

f observations.

287 TL of Siler mortality (Figs. 9-10) mirrors reasonably well the pattern of TL of the observations for 288 all ages in most of the 12 countries. The term in the Siler model that models declining mortality 289 from infancy to childhood captures the related effect on TL. 290 From visually comparing the age profiles, we conclude that the TL of the Siler model (fitted to 291

bserved mortality) is closest to the TL of observed mortality.

292

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3.2.3 Slopes of TL 293 In this section, we compare the slopes of TL between observed and modeled mortality. 294

A. Visual overview

295

Fig. 11 displays the slope of TL for women on the horizontal axis and the slope of TL for men on

296 the vertical axis for 12 countries of the Human Mortality Database (2015). Slopes are estimated 297 from observed mortality (black) and from the fitted models of Gompertz (green), Makeham (blue), 298 and Siler (red). Fig. 11 summarizes 96 slopes (96 = 12 × 4 × 2). Only two slopes exceeded 2 (for 299 TL fitted to the Gompertz model for Russian males, b = 2.02; and for TL fitted to the Siler model 300 for French females, b = 2.1). We regard these two slopes as outliers. All slopes exceeded 1. 301

Fig. 11 here.

302 Slopes of TL estimated from observed mortality are closer to slopes estimated from the Makeham 303 model than they are to the slopes estimated from the other two models. Women and men have 304 greater slopes according to the Siler model than are estimated from observed mortality. Hence the 305 Siler model assumes more rapid increases in the variance of mortality with increasing mean 306 mortality than is observed. Women and men have substantially smaller slopes according to the 307 Gompertz model than are estimated from observed mortality. The Gompertz model assumes slower 308 increases in the variance of mortality with increasing mean mortality than is observed. 309 The diagonal line in Fig. 11 represents equal TL slopes for women and men. Sex differences in 310 slopes of TL are given in Fig. 11 by vertical deviations above or below the diagonal. Sex 311 differences appear to be slightly smaller for observed mortality and the Makeham model than for 312 the Gompertz and Siler models. Exceptional outliers are the slopes of TL of Gompertz mortality for 313 countries with relatively high mortality such as Russia, Hungary, and Poland (Grigoriev, 314 Doblhammer-Reiter, and Shkolnikov 2013; Grigoriev et al. 2010; Shkolnikov et al. 2013). 315

B. Covariance analysis

316

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We examine differences in the slopes of TL between sexes and models using covariance analysis. 317

B1. Does the slope of TL for observed mortality differ from the slopes of TL for models?

318 The p-values of the test for the significance of the interaction term c3 in the analysis of covariance, 319

eq. (7), are given by sex, country, and model (Gompertz, Makeham, Siler) in Table 1. This analysis

320 confirms the previous findings of Fig. 11. Specifically, the slope of TL of the Makeham model is 321 not significantly different from the slope of TL of observed mortality for women and men in almost 322 all of the 12 selected countries. By contrast, the slope of TL of the Siler model is significantly 323 different from the slope of TL of observed mortality for women and men in almost all 12 countries. 324 An exception is, for example, Poland. The slope of TL of the Gompertz model is significantly 325 different from the slope of TL of observed mortality for women and men in almost all 12 countries. 326

B2a. Does the slope of TL differ between males and females for observed mortality and for

327 models? 328 The p-values of the test for the significance of the interaction term d3 in eq. (8) are given by country 329 for observed mortality and models in Table 2. 330 Slopes of TL fitted to observed mortality are not significantly different between males and females 331 in almost all 12 countries. With p = 0.001, West Germany is the only exception. However, the 332 slopes of TL differ between males and females almost as strongly in countries like France, the UK, 333 Japan, and Sweden. The vertical deviations from the diagonal in Fig. 11 are similar for these four 334 countries. 335 The slopes of TL of modeled mortality differ significantly between males and females for many 336

countries. These sex differences are slightly more pronounced in the models of Gompertz and Siler

337 than in the Makeham model. This supports the previous finding of Fig. 11. 338 East Germany is the only country for which the slopes of TL of observed and modeled mortality do 339 not differ between males and females. All the points for this country are almost on the diagonal in 340

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Fig. 11. We do not know if this agreement of male and female slopes indicates a problem in the

341 mortality data or a statistical fluctuation or some special feature of East Germany. 342

B2b. Do the sex differences in the slope of TL differ between observed mortality and models?

343 Table 3 lists the p-values of the test for the significance of the interaction term f7 between 344 log10E(μx), sex, and model of eq. (9). Consistent with the previous findings of Fig. 11 and Table 2, 345 the sex differences in the slope of TL do not differ much between the observed mortality and the 346 models of Gompertz, Makeham, and Siler. Specifically, the sex differences in the slope of TL are 347 not significantly different between observed mortality and the Makeham model in each of the 12 348

countries. The sex differences in the slope of TL differ significantly between observed mortality

349 and the Gompertz model in only three countries: Hungary, Russia, and the USA; and, though less 350 significantly, in Sweden, Japan, West Germany, France and Denmark. The sex differences in the 351 slope of TL differ significantly between observed mortality and the Siler model in only two 352 countries: Sweden and the USA; and, though less significantly, in Russia, France and Denmark. 353 That the sex differences in the slope of TL are significant for Hungary, Russia and Japan could be 354 explained by the increasing sex gap in life expectancy at birth in those countries in the 1980s and 355

1990s. By contrast, other European countries experienced a decline in the female-male differences

356 in mortality (Oksuzyan et al. 2008). 357 3.3 Mathematical proof and theoretical explanations for TL in Gompertz mortality 358 In this section, we prove mathematically that the Gompertz model can explain the form of TL under 359 certain conditions. We investigate theoretically whether the Gompertz model can explain the 360

bserved parameter values of TL.

361 3.3.1 Gompertz mortality predicts TL with slope 2 under certain conditions 362 We prove mathematically (in Appendix 1) that the Gompertz model eq. (1) with modal age at death 363 𝑁 increasing linearly in time and 𝛾 = 𝛾 > 0 obeys a cross-age-scenario of TL exactly with slope 364

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b = 2. Appendix 1 gives an explicit form for the intercept of TL and a detailed proof. This theorem 365 gives analytically the exact relation between the parameters of the Gompertz model and the 366 parameters of TL in one simple case. 367 3.3.2 Temporal trend of 𝛾 alters the slope of TL fitted to Gompertz mortality 368 The theorem's assumptions that βt is constant over time and that the modal age at death Mt increases 369 linearly with time t cannot describe the reality of many countries since, empirically, the slope b of 370 TL fitted to the temporal mean and the temporal variance of observed mortality was always less 371 than 2 (Figs. 3-4), ranging from 1.65 to 1.87. 372 The numerical estimates of the parameters of all three mortality models, separately for females and 373 males, for all countries and years, along with the parameters of linear regressions of these 374 parameters as functions of time, are given in the Supplementary spreadsheet and graphed in 375 Supplementary Figs. A1-A20. 376 In the Gompertz model, even if the mode 𝑁 increases approximately linearly with time (as shown 377 in Figs. A3-A4), the coefficient 𝛾 must change in time so that, with increasing age, the variance of 378 mortality does not increase as fast as the square of the mean mortality. Empirically, the annual 379 estimates of 𝛾 increase approximately linearly over time for females (Fig. A1) and males (Fig. A2) 380 for all 12 countries. 381

A. Theoretical analysis

382 We now analyze the impact of a temporal trend in 𝛾 on the estimated slope b of a cross-age- 383 scenario of TL fitted to Gompertz mortality rates. In the Gompertz model eq. (1), 𝛾 appears twice, 384 as a linear coefficient and in the exponent. We introduce separate notation for these two 385 appearances of 𝛾 so that we may analyze separately the two different effects of the temporal trend 386 in 𝛾: 387 𝜈, = 𝛾,𝑓,(). 388

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We examine two cases. 389 Case 1 If 𝛾, is constant over time and 𝛾, changes linearly over time, then 𝜈, may be 390 factored into one factor 𝑓, that depends on age x only, not on time t, and another factor 391 𝛾,𝑓, that depends on time t only, and not on age x. In this case, the analysis used to 392 prove the theorem in Appendix 1 applies immediately (with slightly different expressions for the 393 intercept to allow for the temporal trend in 𝛾,). It follows from that analysis that TL describes 394 Gompertz mortality exactly with slope b = 2. Hence a temporal trend in 𝛾, cannot explain why 395 the empirical estimates of b are strictly less than 2. 396 Case 2 If 𝛾, changes linearly in time and 𝛾, is constant over time, then 𝛾, has no effect 397

n the slope b of TL (though 𝛾, does affect the intercept a of TL) because 𝛾, simply

398 rescales the values of 𝜈,. If 𝛾, = 𝑡 + 𝑡𝑢 > 0, 𝑡 ≠ 0 and, as in the theorem, if 𝑁 = 𝑤 + 𝑥 ⋅ 399 𝑢 > 0, 𝑤 > 0, 𝑥 ≠ 0, for 𝑢 = 1, … , 𝑈, then 400 𝛾,(𝑦 − 𝑁) = (𝑡 + 𝑡𝑢)(𝑦 − {𝑤 + 𝑥 ⋅ 𝑢}) = 𝑦(𝑡 + 𝑡𝑢) − (𝑡 + 𝑡𝑢){𝑤 + 𝑥 ⋅ 𝑢} 401 ≡ 𝑦𝑔(𝑢) + 𝑕(𝑢). 402 Here 𝑔(𝑢) ≡ 𝑡 + 𝑡𝑢 is linear in time t and 𝑕(𝑢) ≡ (𝑡 + 𝑡𝑢){𝑤 + 𝑥 ⋅ 𝑢} is quadratic in time t. 403 Then 404 𝐹(𝜈) = 1 𝑈 𝜈,

= 𝛾,

𝑈 exp𝑦𝑔(𝑢) + 𝑕(𝑢)

,

405 𝑊𝑏𝑠(𝜈) = 1 𝑈 𝜈,

− 𝐹(𝜈)

= 𝛾,

𝑈

exp 2𝑦𝑔(𝑢) + 𝑕(𝑢)

− 𝐹(𝜈)

.

406 In this case, we are not able to express log 𝑊𝑏𝑠(𝜈) explicitly as a function of log 𝐹(𝜈). 407

B. Numerical experiment

408

SLIDE 19

Instead, we conducted a numerical experiment for women and men of 12 countries of the HMD. 409 This numerical experiment provides concrete answers conditional on the observed mortality, and 410 may guide possible future mathematical analysis. As an example, we describe our analysis of the 411

bserved mortality for Denmark's females from 1960 to 2009.

412 Input data A Gompertz model fitted by maximum likelihood to each year's mortality as a function 413

f age yielded time series of estimates of 𝛾 (Fig. A1) and of 𝑁 (Fig. A3) The supplementary

414 spreadsheet gives numerical values. These time series are summarized by the least-squares linear 415 approximations (shown to fewer significant digits in Figs. A1 and A3) 416 𝛾 = 0.085679264 + 0.00027542975 ⋅ 𝑢 , for 𝑢 = 1, … , 50, 417 𝑁 = 𝑤 + 𝑥 ⋅ 𝑢 = 80.97560676 + 0.098866512 ⋅ 𝑢 , for 𝑢 = 1, … , 50. 418 It seems helpful to appreciate the practical meaning of these two equations. The second equation 419 asserts that Danish females had modal age at death of approximately 81 years in 1960, and that 420 every year thereafter their modal age at death increased by just less than 0.1 year of age per 421 calendar year. According to this regression, in 2009, 50 years after 1960, Danish females had a 422 modal age of death of approximately 86 years (≈ 81 + 50 × 0.1). If the modal age at death increases, 423 why is the rate of increase of mortality with age, 𝛾, increasing (albeit very slowly)? In the 424 framework of the Gompertz model, the age that matters for mortality (the "effective age") is not the 425 chronological age x but the excess of the chronological age over the modal age at death, x - Mt. As 426 the modal age at death increases 0.1 year of age per calendar year, for each given chronological age 427 x, the effective age x - Mt gets younger by 0.1 year of age per calendar year. Deaths occur at 428 progressively later ages and (because of increasing 𝛾) mortality rises (slightly) more rapidly 429 (Canudas-Romo 2010; 2008). 430

SLIDE 20

Experimental design In a computational experiment, we put 𝑁 = 𝑤 + 𝑥 ⋅ 𝑢 as assumed above. 431 Then we calculated numerically three sets of values of 𝜈, for each age x = 0, ..., 100 and each year 432 t = 1, …, 50. In the first set of values, for the standard Gompertz model with 𝛾, = 𝛾, = 𝛾, 433 𝜈, = 𝛾𝑓(). 434 In the second set of values, for the Gompertz model with 𝛾, = 𝛾 and 𝛾, = 𝛾, 435 𝜈,

= 𝛾𝑓().

436 In the third set of values, for the Gompertz model with 𝛾, = 𝛾 and 𝛾, = 𝛾, 437 𝜈,

= 𝛾𝑓().

438 From each set of values, we calculated the corresponding mean and variance of mortality over time 439 for each age x. 440 Results Figs. 16 and 17 show the log temporal variance as a function of the log temporal mean for 441 these three hypothetical mortality schedules; in addition, Figs. 12 and 13 show the results if 442 𝛾, = 𝛾; Figs. 14 and 15 show the results if 𝛾, = 𝛾. The results are similar for both sexes 443 in the 12 HMD countries. As an example, we describe the results for Danish women. 444

Figs. 12-17 here.

445 For the standard Gompertz model 𝜈, (blue solid line) of Danish women in Fig. 16, the relation of 446 log temporal variance to log temporal mean is close to linear (as predicted by TL) except for the 447 large values of the mean and variance of old-age mortality in the upper right corner of the graph. A 448 fitted TL (dark blue solid line, log10 variance = -2.35 + 1.60 log10 mean) approximates the 449 Gompertz log temporal variance and log temporal mean closely over most of their range. 450 When 𝛾, changes linearly over time and 𝜸𝒖,𝒆𝒑𝒙𝒐 is constant over time, 𝜈,

gives a variance-

451 mean relationship (red solid line) that approximates the Gompertz log variance and log mean 452

SLIDE 21

closely but is slightly concave (on log-log coordinates). The average slope of this curve, estimated 453 by 454 log

/ log
,

455 is 1.64, close to the slope b = 1.60 of the fitted TL. Thus the second case considered above gives an 456 approximate explanation of the form and slope of the fitted TL. 457 By contrast, when 𝜸𝒖,𝒗𝒒 is constant over time and 𝛾, changes linearly in time, the relationship 458

f log variance of 𝜈,

to log mean of 𝜈, , calculated numerically (green solid line) from

459 𝜈,

, is visually indistinguishable from linear and has a numerically estimated slope

460 indistinguishable from 2 (to a precision of at least 5 decimal places). These results confirm 461 numerically the above mathematical analysis of Case 1. This case cannot explain the slope of the 462 TL fitted either to observed mortality or to Gompertz model mortality. 463 In conclusion, in this example, the linear trend in 𝑁 and the linear trend in 𝛾, in combination 464 explain qualitatively and quantitatively why TL for Gompertz modeled mortality has slope notably 465 less than 2, unlike the slope of exactly 2 that would be expected theoretically if 𝛾, were constant 466 (regardless of whether 𝛾, is constant or changing in time). 467

4. Conclusion

468 4.1 Summary 469 For females and males in 12 developed countries, the temporal means and temporal variances from 470 1960 to 2009 of observed age-specific mortality for each age from 0 to 100 years, when plotted on 471 log-log coordinates, were approximately linearly related, according to the data of the Human 472 Mortality Database (2015) (Bohk, Rau, and Cohen 2015). This approximate linearity was consistent 473 with Taylor's law. Here we sought to explain this pattern. 474

SLIDE 22

We compared TL fitted to temporal means and temporal variances of observed mortality with TL 475 fitted to mortality in the models of Gompertz (1825), Makeham (1860), and Siler (1979, 1983). 476 These models have progressively more parameters and, in the same order, fit the age profile of 477

bserved mortality progressively more closely.

478 We analyzed how well each mortality model's TL matched TL fitted to observed mortality by 479 comparisons of three features: the log-log linearity of the means and variances of the modeled 480 mortality, the age profile (defined as the set of pairs of log(temporal mean mortality at age x) and 481 log(temporal variance of mortality at age x), for all ages x), and the slope. 482 For log-log linearity, we found that TL approximated mortality in the fitted models of Gompertz, 483 Makeham, and Siler more closely than TL approximated observed mortality. As a consequence, r2 484 values of TL of the Gompertz model were very close, and rounded, to 1. Compared to the Gompertz 485 model, the models of Makeham and Siler resulted in closer fits to observed mortality and to the TL 486

f observed mortality. Consequently, the r2 values of TL of the models of Makeham and Siler were

487

ften slightly smaller than those of Gompertz, but were also often closer to those of the observed

488 mortality. 489 For the age profile of TL, we found that the TL of the Siler model fitted to observed mortality had 490 an age profile that was closer to the age profile of TL of observed mortality than were the age 491 profiles of TL of the fitted models of Makeham and Gompertz. 492 For the slopes of TL, we found that the TL of the Makeham model fitted to observed mortality had 493 a slope that was closest to the slope of TL fitted directly to observed mortality, among the three 494

models. Differences in the slope of TL between males and females in the fitted Makeham models

495 were also closest to the differences in the slope of TL between males and females of observed 496 mortality. 497

SLIDE 23

In addition to these empirical and statistical insights, we demonstrated mathematically that the log 498 temporal means and log temporal variances of mortality in the Gompertz model satisfy TL exactly 499 with slope b = 2 and an explicitly determined intercept when the modal age at death in the 500 Gompertz model increases linearly with time and the βt parameter for the increase of mortality with 501 age is constant in time t or 𝛾, is constant in time and 𝛾, changes linearly in time. As the 502 Gompertz model is a special case of the more complex models of Makeham and Siler, these 503 theoretical findings also apply to certain parameter values of the other two models. 504 Empirically, however, the slopes of TL fitted to observed mortality ranged from 1.65 to 1.87 and 505 the slopes of TL for the mortality models were all at least 1.28 and smaller than 2 (apart from the 506 two exceptions noted above, for Gompertz model mortality of Russian males and Siler model 507 mortality of French females). To explain why the slopes of TL fitted to observed mortality and the 508 fitted models were notably smaller than 2 (with the two exceptions just noted), our computational 509 experiments showed that, in the presence of an increasing modal age at death, it was necessary and 510 sufficient to take into account in the Gompertz model a linear trend in 𝛾,. When 𝛾, increased 511 linearly in time, the slope of TL based on Gompertz mortality was less than 2, and when 𝛾, was 512 constant, the slope of TL based on Gompertz mortality was numerically (and mathematically) 513 indistinguishable from 2. We tested numerically and confirmed this explanation for women and 514 men in 12 countries of the HMD. These numerical results indicate that, as long as 𝛾,, the growth 515 rate of mortality with age, increases linearly with time, TL fitted to mortality will have a slope that 516 is not equal to 2. 517 To conclude, our empirical, statistical, mathematical, and numerical findings confirm that the 518 temporal TL is a regular pattern with deep roots in widely recognized models of the age pattern and 519 temporal evolution of human mortality. 520 4.2 Future research 521

SLIDE 24

These results raise further theoretical and empirical questions. 522 Our mathematical analysis of the Gompertz model remains incomplete when both parameters (the 523 modal age at death and the growth rate of mortality with age) change in time. Our computational 524 experiment gave clear results about this case, but we have not proved these results mathematically. 525 Mathematical analysis is needed to reveal the necessary and sufficient conditions for TL fitted to 526 Gompertz mortality to have a slope less than 2 (not merely different from 2). 527 It would be desirable to complete the mathematical analysis of the Gompertz model and to extend it 528 to the Makeham, Siler, and other more complex models, e.g., those of Heligman and Pollard (1980) 529 and Thiele (1872), and piecewise constant mortality models of, e.g., Brouhns et al. (2002) and 530 Currie et al. (2009). These models may provide more precise approximations to empirical age- 531 profiles of mortality. However, their larger number of parameters and their greater mathematical 532 complexity make them more difficult to analyze mathematically and to understand. Since our goal 533 here was to understand an empirical pattern in a transparent way, we focused on simpler mortality 534 models. 535 Future research may extend the analysis to still more complex models. A potentially productive 536 approach to analyzing temporal trends and variations in mortality would be to construct a 537 generalized linear model (GLM) of all the observed mortality rates simultaneously, as in Brouhns et 538

al. (2002), Currie et al. (2009), and Renshaw and Haberman (2006), and reviewed by Booth and

539 Tickle (2008). In a GLM approach, the dependent variable would be 𝜈, for all ages x, all years t, 540 both sexes, and all countries. The independent variables (predictors) would be age x, year t, sex 541 (female or male), country, and various higher-order (e.g., x2 and t2 to model curvature) and 542 interaction terms to be determined in the course of the analysis. The link function would be logit or 543 probit since 𝜈, is a fraction between 0 and 1. The coefficients of predictor t and t2 would quantify 544 the importance of systematic trends, linear and nonlinear, respectively. A GLM can estimate the 545 mean and the variance of 𝜈, simultaneously (for example, by using quasi-likelihood techniques for 546

SLIDE 25

the variance). With the estimates of means and variances of 𝜈, from a GLM, it would be possible 547 to test TL with finer resolution than has been possible with the traditional approach used here, in 548 which temporal means and temporal variances are computed independently for each age, sex, and 549 country. 550 A GLM could also be used to analyze mortality from each of our three models, and the structure 551 and coefficients of the GLM for modeled mortality could be compared with the structure and 552 coefficients of the GLM for observed mortality. This comparison would permit an evaluation of the 553 models with finer resolution than has been possible with the traditional approach used here. 554 Another empirical question and approach prompted by a reviewer's question is this. For any fixed 555 age x, observed mortality 𝜈, over the 50 years t=1960, …, 2009 may have a systematic trend, 556 fluctuations from this trend in each year t, and an interaction between the trend and the fluctuations 557 (e.g., temporal heteroscedasticity or temporal changes in the skewness of fluctuations). In future 558 research, it would be interesting to decompose the temporal mean and the temporal variance of 559

bserved mortality at a given age into the contributions due to a systematic trend in time,

560 fluctuations, and their interaction; and to decompose the overall temporal TL of mortality into 561 components arising from trend, fluctuations, and interaction. A parallel theoretical analysis could 562 decompose age-specific mortality from models that explicitly incorporated stochastic fluctuations in 563 mortality, unlike the Gompertz, Makeham and Siler models. This empirical investigation could 564 provide a foundation for new theory generalizing the Gompertz and other models to allow for 565 stochastic fluctuations in mortality over time. 566 Reviewer Hal Caswell posed a more general theoretical question that is also related to variation in 567

mortality. Temporal fluctuations in mortality rates are a component of a demographic model in a

568 stochastic environment. What are the consequences for stochastic population growth of greater 569 temporal variance in mortality at (older) ages where the mean mortality is also higher? This 570

SLIDE 26

question shows the potential use of TL applied to mortality in modeling and simulating stochastic 571 age-structured populations. 572 The above outlines of potential applications of TL in human mortality indicate TL's possible 573 usefulness and relevance in formal and empirical demographic research. 574

5. Acknowledgments

575 We thank three reviewers (including Hal Caswell, who identified himself) and the Associate Editor 576 Jakub Bijak for helpful comments and suggestions. [Remainder of acknowledgments to be restored 577 after blinding is removed.] 578 579

SLIDE 27

References 580 Bergeron-Boucher, M. P., and Ebeling, M. and Canudas-Romo, V. (2015). Decomposing changes 581 in life expectancy: Compression versus shifting mortality. Demographic Research 33(14): 391–424. 582 Bohk, C., Rau, R., and Cohen, J. E. (2015). Taylor's power law in human mortality. Demographic 583 Research, 33(21): 589–610. 584 Bongaarts, J, (2005). Long-range trends in adult mortality: models and projection methods. 585 Demography, 42(1): 23–49. 586 Booth, H. and Tickle, L. (2008). Mortality modelling and forecasting: A review of methods. Annals 587

f Actuarial Science, 3(1-2): 3–43.

588 Brouhns, N., Denuit, M., and Vermunt, J. K. (2002). A Poisson log-bilinear regression approach to 589 the construction of projected lifetables. Insurance: Mathematics and Economics, 31(3): 373–393. 590 Cairns, A. J. G., Blake, D., Dowd, K., Coughlan, G. D., Epstein, D., Ong, A., and Balevich, I. 591 (2009). A quantitative comparison of stochastic mortality models using data from England and 592 Wales and the United States. North American Actuarial Journal, 13(1): 1–35. 593 Canudas-Romo, V. (2010). Three measures of longevity: time trends and record values. 594 Demography 47(2): 299-312. 595 Canudas-Romo, V. (2008). The modal age at death and the shifting mortality hypothesis. 596 Demographic Research 19(30): 1179-1204. 597 Christensen, K., Doblhammer, G., Rau, R., and Vaupel, J. W. (2009). Ageing populations: the 598 challenges ahead. The Lancet 374(9696): 1196-1208. 599 Cohen, J. E., Xu, M., and Brunborg, H. (2013). Taylor’s law applies to spatial variation in a human 600

population. Genus 69(1): 25–60.

601

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Gompertz, B. (1825). On the nature of the function of the law of human mortality, and on a new 602 mode of determining the value of life contingencies. Philosophical Transactions of the Royal 603 Society of London 115: 513–583. 604 Grigoriev, P., Doblhammer-Reiter, G., and Shkolnikov, V. (2013). Trends, patterns, and 605 determinants of regional mortality in Belarus, 1990-2007. Population Studies: A Journal of 606 Demography 67(1): 61-81. 607 Grigoriev, P., Vladimir S., Andreev, E. M., Jasilionis, D., Jdanov, D., Meslé, F., and Vallin, J. 608 (2010). Mortality in Belarus, Lithuania, and Russia: divergence in recent trends and possible 609

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626 102. 627 R Core Team (2015). R: A language and environment for statistical computing. R Foundation for 628 Statistical Computing,Vienna, Austria. http://www.R-project.org/ 629 Rau, R., Jasilionis, D., Soroko, E. L., and Vaupel, J. W. (2008). Continued reductions in mortality 630 at advanced ages. Population and Development Review 34(4): 747-768. 631 Renshaw, A. E. and Haberman, S. (2006). A cohort-based extension to the Lee-Carter model for 632 mortality reduction factors. Insurance: Mathematics and Economics 38(3): 556-570. 633 Shkolnikov, V. M., Andreev, E. M., McKee, M., and Leon, D. A. (2013). Components and possible 634 determinants of the decrease in Russian mortality in 2004–2010. Demographic Research 28(32): 635 917–950. 636 Siler, W. (1983). Parameters of mortality in human populations with widely varying life spans. 637 Statistics in Medicine 2: 373–380. 638 Siler, W. (1979). A competing-risk model for animal mortality. Ecology 60(4): 750–757. 639 Taylor, L. R. (1961). Aggregation, variance and the mean. Nature 189: 732–735. 640 Thiele, T. N. (1872). On a mathematical formula to express the rate mortality throughout the whole 641

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642 University of California, Berkeley (USA) and Max Planck Institute for Demographic Research, 643 Rostock (Germany) (2015). Human Mortality Database. www.mortality.org 644 Vaupel, J. W. (2010). Biodemography of human ageing. Nature 464: 536-542. 645 646

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Appendix 1. Taylor's law with slope 2 describes the Gompertz model 647

Theorem. The Gompertz mortality model with modal age at death increasing linearly in time obeys

648 a cross-age-scenario of Taylor's law (TL) exactly with slope b = 2. Explicitly, assuming the 649 Gompertz model 𝜈, at age x and time t, 650 𝜈, = 𝛾𝑓(), 𝛾 > 0, 𝑁 > 0, for 𝑢 = 1, … , 𝑈, 𝑦 = 1, … , 𝑌, 651 with a linear change (increase or decrease) over time in the modal age at death 652 𝑁 = 𝑤 + 𝑥 ⋅ 𝑢 > 0, 𝑤 > 0, 𝑥 ≠ 0, for 𝑢 = 1, … , 𝑈, 653 and an exponential rate increase of mortality with age x that is constant over time t 654 𝛾 = 𝛾 > 0, for 𝑢 = 1, … , 𝑈, 655 then TL holds with slope 2 and intercept log(𝐿 − 𝐿

) − 2 log 𝐿 on log-log coordinates:

656 log 𝑊𝑏𝑠(𝜈) = log

+ 2 ⋅ log 𝐹(𝜈), for 𝑦 = 1, … , 𝑌,

657 where the positive constants 𝐿, 𝐿 are defined below. 658

Proof. From the assumptions,

659 𝜈, = 𝛾𝑓() = 𝛾𝑓({⋅}) = 𝛾𝑓()𝑓 660 which implies that 𝜈, is an exponentially increasing function of age x for every time t. It also 661 implies that 𝜈, is an exponentially decreasing function of time t for every age x if w > 0, and is an 662 increasing function of time t for every age x if w < 0. Then, using the definitions in the main text of 663 𝐹(𝜈) as the temporal mean and 𝑊𝑏𝑠(𝜈) as the temporal variance of mortality at age x, 664 𝐹(𝜈) = 1 𝑈 𝜈,

= 1

𝑈 𝛾𝑓() 𝑓

= 𝑓 𝛾𝑓

𝑈 𝑓

,

665

SLIDE 31

where the first factor 𝑓 varies with age x only and the bracketed second factor varies with time t 666 and T only. Define 667 𝐿 ≡ 𝛾𝑓 𝑈 𝑓

, 𝑟 ≡ 𝑓 .

668 𝐿 does not depend on age x. Then since 𝛾 > 0, 𝑥 ≠ 0, we have 𝑟 < 1 if w > 0 and q > 1 if w < 0 669 and in both cases 670 𝐿 = 𝛾𝑓 𝑈 (𝑟 + 𝑟 + ⋯ + 𝑟) = 𝛾𝑓 𝑈 𝑟(1 + 𝑟 + ⋯ + 𝑟) = 𝛾𝑓 𝑈 ⋅ 𝑟(1 − 𝑟) 1 − 𝑟 . 671 Since 𝐿 > 0, 𝐹(𝜈) = 𝐿𝑓 increases exponentially at rate β with increasing x. Also 672 𝑊𝑏𝑠(𝜈) = 1 𝑈 𝜈, − 𝐹(𝜈)

= 1

𝑈 𝜈,

− 𝐹(𝜈)
673

= 1 𝑈 𝛾𝑓()𝑓

− 𝐿𝑓

= 𝑓 𝛾𝑓

𝑈 𝑓

− 𝐿

𝑓.

674 Define 675 𝐿 ≡ 𝛾𝑓 𝑈 𝑓

.

676 𝐿 does not depend on age x. Then 677 𝐿 = 𝛾𝑓 𝑈 (𝑟⋅ + 𝑟⋅ + ⋯ + 𝑟⋅) = 𝛾𝑓 𝑈 𝑟(1 − 𝑟⋅()) 1 − 𝑟 . 678 Also 679 𝑊𝑏𝑠(𝜈) = (𝐿 − 𝐿

)𝑓.

680 By Cauchy's inequality, 𝐿 − 𝐿

> 0. Therefore 𝑊𝑏𝑠(𝜈) increases exponentially at rate 2β with

681 increasing age x. Thus, we showed that 682

SLIDE 32

𝐹(𝜈) = 𝐿𝑓, 683 𝑊𝑏𝑠(𝜈) = (𝐿 − 𝐿

)𝑓 .

684 Therefore 685 𝑊𝑏𝑠(𝜈) =

𝐹(𝜈)

.

□ 686 Appendix 2. Taylor's law with slope less than 2 describes the model of Makeham 687 In the Makeham model, eq. (2), the additivity of expectations yields 688 𝐹(𝜈) = 𝐹(𝑑) + 𝐹𝛾𝑓() = 𝐹(𝑑) + 𝐹(𝜈). 689 The subscript M denotes the Makeham model and the subscript G denotes the Gompertz model. 690 Calculating the variance in the Makeham model requires specifying the relation between 𝑑 and 691 𝛾𝑓(). Based on the parameter estimates of the Makeham model in Figs. A5-A10, we 692 examine this empirically plausible special case: 693 𝑑 = 𝑑𝑓, 𝑑 > 0, 𝑒 > 0, 694 𝛾 = 𝛾 > 0, 695 𝑁 = 𝑤 + 𝑥 ⋅ 𝑢 > 0, 𝑤 > 0, 𝑥 > 0. 696 Thus 697 𝜈, = 𝑑𝑓 + 𝑓𝛾𝑓{⋅}. 698 On the right side, the first term depends only on t, not on x, and the second term factors into one 699 factor 𝑓 that depends on x only and another factor that depends on t only. By summing geometric 700 series as in Appendix 1, we can get explicit expressions for the constant 𝐷 (and 𝐿 is identical to 701 that constant in the Gompertz model) in the expression 702 𝐹(𝜈) = 𝐷 + 𝐿𝑓. 703

SLIDE 33

With increasing age x, 𝐹(𝜈) grows as the factor 𝑓. Also, 704 𝜈,

= 𝑑𝑓 + 𝑓𝛾𝑓{⋅} + 2𝑑𝑓𝛾𝑓{⋅} .

705 Therefore 706 𝐹𝜈,

= 𝐷 + 𝐿𝑓 + 𝐿𝑓,

707 𝐹

(𝜈) = 𝐷 + 𝐿 𝑓 + 2𝐷𝐿𝑓,

708 𝑊𝑏𝑠

(𝜈) = 𝐹𝜈, − 𝐹 (𝜈) = 𝐷 − 𝐷 + (𝐿 − 𝐿 )𝑓 + (𝐿 − 2𝐷𝐿)𝑓.

709 The same elementary methods used in Appendix 1 can determine 𝐿, 𝐿 explicitly. 710 From Figs. A7-A8, 𝛾 ≈ 0.1 for both females and males. Hence as x increases from 0 to 100, 𝑓 711 increases from 𝑓 = 1 to approximately 𝑓 ≈ 2.2 × 10. Hence the term of 𝑊𝑏𝑠

(𝜈) that

712 contains the factor 𝑓, which is approximately 𝑓 ≈ 4.8 × 10 when x = 100, increasingly 713 dominates the term of 𝑊𝑏𝑠

(𝜈) that contains the factor 𝑓. So, to a first approximation,

714 neglecting all but the dominant terms, 𝑊𝑏𝑠

(𝜈) scales with increasing age x as the square of

715 𝐹(𝜈). Thus, asymptotically for increasing age x, TL holds approximately with slope 𝑐 ≈ 2. 716 This is only a first approximation. 𝐹(𝜈) and 𝑊𝑏𝑠

(𝜈) each contains a constant term, and

717 𝑊𝑏𝑠

(𝜈) contains a term with factor 𝑓 which scales more slowly than the dominant term of

718 𝑊𝑏𝑠

(𝜈) which contains the square of 𝑓. Thus 𝑊𝑏𝑠 (𝜈) scales with increasing x more slowly

719 than the square of 𝐹(𝜈), i.e., TL is expected to hold approximately with a slope less than 2. In all 720 12 countries, the estimated values of the TL slope never exceeded the Japanese record of b = 1.86 721 for females (Fig. 7) and never exceeded the Japanese and French records of b = 1.82 for males (Fig. 722 8). Both record values were substantially less than 2. Thus, we have given, in this special case, an 723 argument to explain why TL with slope less than 2 approximates the mortality of the Makeham 724 model reasonably well. 725

SLIDE 34

Tables 726

SLIDE 35

Table 1 727 Equal to TL of

bserved data?

Gompertz Makeham Siler Women Men Both Women Men Both Women Men Both All countries 0.073200 0.284200 0.032000 0 Denmark 0.005358 0 0.022461 0.231000 0.284805 0.000312 0.208000 0.000392 France 0.748900 0.076500 0.486783 0 East Germany 0.000006 0 0.000580 0.329880 0.011300 0 0.000120 0 West Germany 0.351730 0.986726 0.432000 0 0.000014 0 Hungary 0.000007 0.001720 0.005730 0.296356 0.078070 0.106000 0.000661 0.008880 0.000077 Italy 0.003660 0.000132 0.000004 0.000001 0.001448 0 Japan 0.035169 0.001420 0.001870 0 Poland 0.032000 0 0.021900 0.811000 0.141000 0.233800 0.177000 0.153 Russia 0.009401 0 0.789410 0.000337 0.028860 0.002170 0.213779 0.000048 0.01896

SLIDE 36

Sweden 0.000002 0.532277 0.001853 0 0.000214 0 UK 0.002140 0.009960 0.000722 0.000380 0.000650 0.000003 USA 0.051616 0.114000 0.970258 0.000395 0 728 Table 1: P-values to test the null hypothesis of no differences in TL slope between observed data and the fitted models of Gompertz, Makeham, 729 and Siler, for women and men in 12 countries of the Human Mortality Database (2015). A p-value below 0.001 indicates that the coefficient of 730 the interaction term is statistically significantly non-zero. An entry of 0 means that the rounded value of p is 0.000000. 731

SLIDE 37

Table 2 732 733 Sex differences in TL? Observed data Gompertz Makeham Siler All countries 0.023190 0.087000 0.199000 0.000290 Denmark 0.644000 0.000844 0.000736 France 0.031800 0.204590 0.069400 0.000017 East Germany 0.306800 0.187000 0.943370 0.510000 West Germany 0.000968 0.336000 0.000136 Hungary 0.426000 0.920000 0.106000 Italy 0.126570 0.582600 0.000200 0.000514

SLIDE 38

Japan 0.003760 0.068200 0.000128 Poland 0.827000 0.000021 0.068100 0.335000 Russia 0.936100 0.095200 0.000035 Sweden 0.005600 UK 0.042600 0.000009 USA 0.171000 0.001040 0.001610 734 Table 2: P-values to test the null hypothesis of no differences between 735 females and males in the slope of TL fitted to observed death rates and 736 in the slope of TL fitted to the models of Gompertz, Makeham, and 737 Siler, in 12 countries of the Human Mortality Database (2015). A p- 738 value below 0.001 indicates that the coefficient of the interaction term 739 is statistically significantly non-zero. An entry of 0 means that the 740 rounded value of p is 0.000000. 741 742

SLIDE 39

Table 3 743 744 Sex differences in TL equal to

bserved data?

Gompertz Makeham Siler All countries 0.005060 0.581010 0.349870 Denmark 0.025680 0.011860 0.045340 France 0.008070 0.164250 0.056650 East Germany 0.447548 0.392768 0.586554 West Germany 0.003030 0.527880 0.237640 Hungary 0.648766 0.515661 Italy 0.226781 0.329177 0.414148 Japan 0.029082 0.165447 0.817667

SLIDE 40

Poland 0.00917 0.128900 0.650510 Russia 0.303197 0.046573 Sweden 0.004344 0.006243 0.000250 UK 0.401217 0.643469 0.354857 USA 0.000003 0.013959 0.000247 745 Table 3: P-values to test the null hypothesis of no differences in the 746 sex differences in TL slope between observed mortality rates and 747 fitted models of Gompertz, Makeham, and Siler, in 12 countries of the 748 Human Mortality Database (2015). A p-value below 0.001 indicates 749 that the coefficient of the interaction term is non-zero. An entry of 0 750 means that the rounded value of p is 0.000000. 751 752

SLIDE 41

Figure captions 753

1. Observed female mortality from 1960 (light gray) to 2009

754 (black), and fitted female mortality with models of Siler (red), 755 Gompertz (green) and Makeham (blue) in 2009 for 12 756 countries of the Human Mortality Database (2015) 757

2. Observed male mortality from 1960 (light gray) to 2009

758 (black), and fitted male mortality with models of Siler (red), 759 Gompertz (green) and Makeham (blue) in 2009 for 12 760 countries of the Human Mortality Database (2015) 761

3. TL in observed female mortality for the ages 0 (yellow) to 100

762 (green) from 1960 1 to 2009 on a logarithmic scale (base=10) 763 for 12 countries of the Human Mortality Database (2015) 764

4. TL in observed male mortality for the ages 0 (yellow) to 100

765 (green) from 1960 to 2009 on a logarithmic scale (base=10) for 766 12 countries of the Human Mortality Database (2015) 767

5. TL in fitted female mortality of the Gompertz model for the

768 ages 0 (yellow) to 100 (green) from 1960 to 2009 on a 769

SLIDE 42

logarithmic scale (base=10) for 12 countries of the Human 770 Mortality Database (2015) 771

6. TL in fitted male mortality of the Gompertz model for the ages

772 0 (yellow) to 100 (green) from 1960 to 2009 on a logarithmic 773 scale (base=10) for 12 countries of the Human Mortality 774 Database (2015) 775

7. TL in fitted female mortality of the model of Makeham for the

776 ages 0 (yellow) to 100 (green) from 1960 to 2009 on a 777 logarithmic scale (base=10) for 12 countries of the Human 778 Mortality Database (2015) 779

8. TL in fitted male mortality of the model of Makeham for the

780 ages 0 (yellow) to 100 (green) from 1960 to 2009 on a 781 logarithmic scale (base=10) for 12 countries of the Human 782 Mortality Database (2015) 783

9. TL in fitted female mortality of the model of Siler for the ages

784 0 (yellow) to 100 (green) from 1960 to 2009 on a logarithmic 785 scale (base=10) for 12 countries of the Human Mortality 786 Database (2015) 787

SLIDE 43

10. TL in fitted male mortality of the model of Siler for the ages 0

788 (yellow) to 100 (green) from 1960 to 2009 on a logarithmic 789 scale (base=10) for 12 countries of the Human Mortality 790 Database (2015) 791

11. Scatterplot of slope of TL for women (horizontal axis) and

792 men (vertical axis) for 12 countries of the Human Mortality 793 Database (2015). Observations are in black, fitted data of the 794 models of Gompertz, Makeham, and Siler are depicted in red, 795 green, blue, and red, respectively. 796

12. TL (solid black line) in fitted female mortality of the model of

797 Gompertz with βt,up = constant for the ages 0 (yellow) to 100 798 (green) from 1960 to 2009 on a logarithmic scale (base=10) for 799 12 countries of the Human Mortality Database (2015) 800

13. TL (solid black line) in fitted male mortality of the model of

801 Gompertz with βt,up = constant for the ages 0 (yellow) to 100 802 (green) from 1960 to 2009 on a logarithmic scale (base=10) for 803 12 countries of the Human Mortality Database (2015) 804

SLIDE 44

14. TL (solid black line) in fitted female mortality of the model of

805 Gompertz with βt,down = constant for the ages 0 (yellow) to 100 806 (green) from 1960 to 2009 on a logarithmic scale (base=10) for 807 12 countries of the Human Mortality Database (2015) 808

15. TL (solid black line) in fitted male mortality of the model of

809 Gompertz with βt,down = constant for the ages 0 (yellow) to 100 810 (green) from 1960 to 2009 on a logarithmic scale (base=10) for 811 12 countries of the Human Mortality Database (2015) 812

16. TL (solid dark blue line) in fitted female mortality of the

813 model of Gompertz (solid blue line), of the model of Gompertz 814 with βt,up = constant (solid green line) and of the model of 815 Gompertz with βt,down = constant (solid red line) for the ages 0 816 to 100 from 1960 to 2009 on a logarithmic scale (base=10) for 817 12 countries of the Human Mortality Database (2015) 818

17. TL (solid dark blue line) in fitted male mortality of the model

819

f Gompertz (solid blue line), of the model of Gompertz with

820 βt,up = constant (solid green line) and of the model of Gompertz 821 with βt,down = constant (solid red line) for the ages 0 to 100 822

SLIDE 45

from 1960 to 2009 on a logarithmic scale (base=10) for 12 823 countries of the Human Mortality Database (2015) 824

SLIDE 46

❋✐❣✉r❡ ✶✿ ❖❜s❡r✈❡❞ ❢❡♠❛❧❡ ♠♦rt❛❧✐t② ❢r♦♠ ✶✾✻✵ ✭❧✐❣❤t ❣r❛②✮ t♦ ✷✵✵✾ ✭❜❧❛❝❦✮✱ ❛♥❞ ✜tt❡❞ ❢❡♠❛❧❡ ♠♦rt❛❧✐t② ✇✐t❤ ♠♦❞❡❧s ♦❢ ●♦♠♣❡rt③ ✭❣r❡❡♥✮✱ ▼❛❦❡❤❛♠ ✭❜❧✉❡✮✱ ❛♥❞ ❙✐❧❡r ✭r❡❞✮ ✐♥ ✷✵✵✾ ❢♦r ✶✷ ❝♦✉♥tr✐❡s ♦❢ t❤❡ ❍✉♠❛♥ ▼♦rt❛❧✐t② ❉❛t❛❜❛s❡ ✭✷✵✶✺✮

Denmark

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

France

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

East Germany

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

West Germany

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

Hungary

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

Italy

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

Japan

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

Poland

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

Russia

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

Sweden

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

United Kingdom

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

USA

Age 20 40 60 80 100 1e−05 1e−04 0.001 0.01 0.1 1

Women Observed, 1960−2009 Gompertz, 2009 Makeham, 2009 Siler, 2009

✶

SLIDE 47