Artículo de revista: ABURTO, José Manuel; BASELLINI, Ugofilippo; BAUDISCH, Annette; VILLAVICENCIO; Francisco (2022) "Drewnowski's index to measure lifespan variation: Revisiting the Gini coefficient of the life table". Theoretical Population Biology, 148: 1-10 (ISSN: 0040-5809). https://doi.org/10.1016/j.tpb.2022.08.003 Theoretical Population Biology 148 (2022) 1–10 Contents lists available at ScienceDirect Theoretical Population Biology journal homepage: www.elsevier.com/locate/tpb Drewnowski’s index to measure lifespan variation: Revisiting the Gini coefficient of the life table José Manuel Aburto a,b,c , Ugofilippo Basellini d,e , Annette Baudisch c , ∗ Francisco Villavicencio c,f,g , a Department of Population Health, London School of Hygiene and Tropical Medicine, London WC1E 7HT, UK b Leverhulme Centre for Demographic Science, Department of Sociology and Nuffield College, University of Oxford, Oxford OX1 1JD, UK c Interdisciplinary Centre on Population Dynamics, University of Southern Denmark, 5230 Odense, Denmark d Laboratory of Digital and Computational Demography, Max Planck Institute for Demographic Research, 18057 Rostock, Germany e Mortality, Health and Epidemiology Unit, Institut national d’études démographiques (INED), 93322 Aubervilliers, France f Centre for Demographic Studies (CED), Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain g Department of International Health, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD 21205, USA article info a b s t r a c t Article history: The Gini coefficient of the life table is a concentration index that provides information on lifespan Received 17 November 2021 variation. Originally proposed by economists to measure income and wealth inequalities, it has Available online 6 September 2022 been widely used in population studies to investigate variation in ages at death. We focus on the complement of the Gini coefficient, Drewnowski’s index, which is a measure of equality. We study its Keywords: Concentration index mathematical properties and analyze how changes over time relate to changes in life expectancy. Gompertz Further, we identify the threshold age below which mortality improvements are translated into Life expectancy decreasing lifespan variation and above which these improvements translate into increasing lifespan Lifespan inequality inequality. We illustrate our theoretical findings simulating scenarios of mortality improvement in the Mortality Gompertz model, and showing an example of application to Swedish life table data. Our experiments Threshold age demonstrate how Drewnowski’s index can serve as an indicator of the shape of mortality patterns. These properties, along with our analytical findings, support studying lifespan variation alongside life expectancy trends in multiple species. © 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction uncertainty about the eventual age at death at the individual level, and measures how evenly mortality conditions are shared The life table is an essential tool in mortality studies. It rep- at the population level. There exist several indicators to measure resents the current mortality experience of a population and lifespan variation (for an overview, see Shkolnikov et al., 2003; it is usually summarized by life expectancy at birth (eo ). Life van Raalte and Caswell, 2013), such as the entropy of the life expectancy at birth is the average years a synthetic cohort of table (Leser, 1955; Keyfitz, 1977; Demetrius, 1978), the standard newborns is expected to live if individuals were to experience the deviation or variance of the age-at-death distribution (as applied current mortality conditions throughout their lives. It is, however, by Tuljapurkar and Edwards, 2011), the coefficient of variation (as applied by Aburto and van Raalte, 2018; Aburto et al., 2018), an average indicator that masks variability in ages at death, which years of life lost (e† ) (Goldman and Lord, 1986; Vaupel, 1986; can be substantial. This variability is often referred to as lifespan Hakkert, 1987; Vaupel and Canudas Romo, 2003), or the Gini variation or lifespan inequality, and has received increasing atten- coefficient (Hanada, 1983). tion over the past two decades (see, to mention a few, Wilmoth Here we study in greater detail the Gini coefficient of the life and Horiuchi, 1999; Edwards and Tuljapurkar, 2005; Smits and table (G) and its complement, Drewnowski’s index (D) Monden, 2009; Baudisch, 2011; Vaupel et al., 2011; Fernández (Drewnowski, 1982; Hanada, 1983), from a formal demographic and Beltrán-Sánchez, 2015; Colchero et al., 2016; Ebeling et al., perspective. While the properties of these two indicators in the 2018; van Raalte et al., 2018; Permanyer and Scholl, 2019; Aburto demographic sense are analogous, D directly provides an added et al., 2020; Vaupel et al., 2021). Lifespan variation reveals the meaningful interpretation: it reflects the proportion of life ex- pectancy that two newborns of a cohort are expected to live ∗ Corresponding author at: Centre for Demographic Studies (CED), Universitat together when the life table assumptions hold. We additionally Autònoma de Barcelona, 08193 Bellaterra, Spain. aim to understand how changes in age-specific mortality un- E-mail address: fvillavicencio@ced.uab.es (F. Villavicencio). derpin trends in lifespan variation. We focus on how changes https://doi.org/10.1016/j.tpb.2022.08.003 0040-5809/© 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by- nc-nd/4.0/). J.M. Aburto, U. Basellini, A. Baudisch et al. Theoretical Population Biology 148 (2022) 1–10 over time in D relate to changes in eo , and highlight a new each individual lifespan is changed by the same proportion (scale measure of absolute variation related to perturbation theory, independence); and (iii) it decreases if years of life are transferred named ϑ . This new indicator gives the value of life expectancy if from a longer to a shorter lived individual (Pigou-Dalton condi- the hazard is doubled at every age. We provide the mathematical tion). Note that property (i) enables straightforward comparison foundation of how Drewnowski’s index (and analogously the Gini between populations, including comparisons between different coefficient) evolves over time, and give analytical formulae to species (Wrycza et al., 2015). Furthermore, the coefficient is not find the threshold age below which mortality improvements are too sensitive to redistributions at early ages of life, and it well translated into decreasing lifespan variation and above which reflects changes at adult ages (Shkolnikov et al., 2003). these improvements translate into increasing lifespan inequality. Being bounded between 0 and 1, the Gini coefficient can be readily transformed from a measure of inequality into a measure 2. The Gini coefficient and Drewnowski’s index of equality of lifespans. From (1), its complement indicator of lifespan equality immediately derives, the Drewnowski index, The Gini coefficient is one of the most popular indices em- defined as ployed in social sciences to measure concentration in the dis- ∫∞ tribution of a non-negative random variable (Gini, 1912, 1914). ϑ ℓ(x, t)2 dx D=1−G= = ∫0 ∞ , (2) Originally proposed by economists to measure income or wealth eo 0 ℓ(x, t) dx inequality, this coefficient has been applied in demography and survival analysis to investigate within-group inequality in terms and sharing all the important properties of G. In the case of the of ages at death (see, for instance, Hanada, 1983; Shkolnikov et al., Pigou-Dalton condition, though, the effect is the opposite and 2003; Bonetti et al., 2009; Gigliarano et al., 2017; Barthold Jones D increases as years of life are transferred from a longer to a et al., 2018; Diaz et al., 2018; Basellini and Camarda, 2019; Aburto shorter lived individual without changing their relative ranks. et al., 2020). According to Hanada (1983), this index was first proposed by Jan Drewnowski on a working group on health indicators at 2.1. Definition the World Health Organization in the early 1980s (Drewnowski, 1982). As thoroughly discussed by Yitzhaki and Schechtman (2013), In addition to its interpretation as the proportion of life ex- there are several equivalent ways to define the Gini coefficient. pectancy that will be shared by two newborns (if mortality rates Let X be a non-negative random variable with probability density do not change over their lives), D is akin to already established function f (x) and expected value E[X ], a common definition is demographic measures. Specifically, D is similar both in its con- 1 ∫ ∞ ∫ ∞ struction and age pattern to e† (Vaupel and Canudas Romo, 2003), G= |x1 − x2 | f (x1 ) f (x2 ) dx1 dx2 . to the life table entropy (Leser, 1955; Keyfitz, 1977; Demetrius, 2 E[X ] 0 0 1978), and to the variance or standard deviation (Edwards and Accordingly, if X is a random variable of the ages at death in a Tuljapurkar, 2005). Moreover, following (2), and similarly to other population, the Gini coefficient expresses the average of absolute indicators, D has the advantage that it can be interpreted as the differences in individual lifespans relative to the mean length of weighted average of the survival function (with weights equal to life E[X ]. the survival function itself). Michetti and Dall’Aglio (1957), and later Hanada (1983), sug- gest a re-formulation of the Gini coefficient in terms of the life 3. Changes over time in Drewnowski’s index table functions, given by ∫∞ ℓ(x, t)2 dx ϑ In order to analyze changes over time in Drewnowski’s index – G = 1 − ∫0 ∞ =1− , (1) 0 ℓ(x, t) dx eo or, equivalently, the Gini coefficient – we aim to find an analytical expression for the time derivative of D. In the following, a dot ∫where ℓ(x, t) is the life table survival function at time t, eo = ∞ over a function will denote the partial derivative with respect to 0 ℓ(x, t) dx the life expectancy at birth at time t, and ϑ = ∫∞ time, but variable t will be omitted for simplicity. 0 ℓ(x, t)2 dx is the resulting life expectancy at birth of doubling the hazard at all ages. Barthold Jones et al. (2018) interpret ϑ as a measure of shared life expectancy, that is, the average time 3.1. Relative derivative of D two newborns at time t are expected to survive together. For the purposes of this article, the definition of the Gini coefficient in (1) Proposition ∫∞ 1. Let D∫ = ϑ / eo be Drewnowski’s index, where ∞ will be used in the following. ϑ = 0 ℓ(x)2 dx, eo = 0 ℓ(x) dx is the life expectancy at birth, and ℓ(x) the probability of surviving from birth to age x. Then, relative 2.2. Main properties changes over time in D are given by Ḋ ϑ̇ ėo The Gini coefficient takes values between 0 and 1, and can be = − . (3) interpreted as a measure of inequality. A value of 0 denotes equal- D ϑ eo ity in ages at death, i.e. when every individual in the population has the exact same length of life. The index increases approaching Proof. Note that D = ϑ / eo implies that D eo − ϑ = 0. 1 as lifespans become more spread and unequal in the population. Differentiating with respect to time yields This makes the interpretation clear and intuitive: higher values Ḋ eo + D ėo − ϑ̇ = 0 . correspond to greater within-group inequality in ages at death. An additional attractive feature of the Gini coefficient is that Solving for Ḋ and dividing both sides by D, we get (3). □ it fulfills three important properties for inequality indices (Sen, 1973; Anand, 1983): (i) it does not change if the number of Eq. (3) decomposes relative changes in D into relative changes individuals at each age at death is changed by the same pro- of the shared life expectancy between two individuals ϑ , and portion (population-size independence); (ii) it does not change if relative changes in the life expectancy at birth eo . 2 J.M. Aburto, U. Basellini, A. Baudisch et al. Theoretical Population Biology 148 (2022) 1–10 3.2. Time derivatives of eo and ϑ w(x) W (x), where w(x) = µ(x) ℓ(x) e(x) are the same weights as in (4) and Vaupel and Canudas Romo (2003) showed that changes over 2 ℓ(x) D(x) − D time in life expectancy at birth are a weighted average of the total W (x) = . (8) eo rates of mortality improvement, expressed as ∫ ∞ 4. The threshold age ėo = ρ (x) w (x) dx . (4) 0 4.1. Positive and negative contributions to lifespan equality Function ρ (x) = −µ̇(x) / µ(x) stands for the age-specific rates of mortality improvement, where µ(x) is the force of mortality Because Drewnowski’s index is a measure of equality, Ḋ > 0 (hazard rate) at age x. The weights w (x) = µ(x) ℓ(x) e(x) are indicates that lifespan variation decreases over time, whereas a ∫ ∞measure of the importance of death at age x, where e(x) = Ḋ < 0 implies that lifespan variation increases over time. Eq. (7) x ℓ(a) da / ℓ(x) is the remaining life expectancy at age x. Fol- can then be used to analyze the existence of a threshold age that lowing a similar approach, we aim to express the time derivative separates positive from negative contributions to lifespan equality of ϑ as a weighted average of mortality improvements, but with as a result of mortality improvements. different weights. In the assumption that mortality improvements occur at all ages, ρ (x) = −µ̇(x) / µ(x) > 0 is a strictly positive function. Definition 1. Let ℓ(x) be the probability of surviving from birth Therefore, from (7), to age x. A measure of lifespan equality at age x is Drewnowski’s 1. Those ages x for which w (x) W (x) > 0 will contribute index conditional upon survival up to age x, defined as ∫∞ positively to Drewnowski’s index D and increase lifespan 1 ℓ(a)2 da equality; D(x) = ∫x ∞ . (5) 2. Those ages x for which w (x) W (x) < 0 will contribute ℓ(x) x ℓ(a) da negatively to Drewnowski’s index D and favor lifespan in- ∫∞ equality; Proposition 2. Let ϑ = 0 ℓ(x)2 dx, where ℓ(x) is the probability of 3. Those ages x for which w (x) W (x) = 0 will have no effect surviving from birth to age x. Then, its partial derivative with respect on the variation over time of D. to time can be expressed as ∫ ∞ Any existing threshold age that separates positive from nega- ϑ̇ = ρ (x) w(x) 2 ℓ(x) D(x) dx , (6) tive contributions to lifespan equality will occur whenever w (x) 0 W (x) = 0. Since µ(x), ℓ(x), and e(x) are all positive functions, so where ρ (x) are the age-specific rates of mortality improvement, w (x) are w (x) and eo . Hence, the same weights defined in (4), and D(x) as defined in (5). w(x) W (x) = 0 ⇐⇒ 2 ℓ(x) D(x) − D = 0 . (9) Proof. Applying the chain rule, the time derivative of ϑ is simply 4.2. Existence and uniqueness of the threshold age ∫ ∞ ϑ̇ = 2 ℓ(x) ℓ̇(x) dx . By means of the following two propositions and one theorem, 0 ∫x we aim to prove that when mortality improvements occur at all Using that ℓ̇(x) = −ℓ(x) 0 µ̇(a) da, and reversing the order of ages and ρ (x) > 0 for all x ≥ 0, there exists a unique threshold integration, we get age aD that separates positive from negative contributions to ∫ ∞ ∫ x∫ ∞ ∫ ∞ lifespan equality (measured by D) as a result of those improve- ϑ̇ = −2 ℓ(x)2 µ̇(a) da dx = −2 µ̇(a) ℓ(x)2 dx da ments. The assumption of ρ (x) being a strictly positive function 0 0 ∫∞ 0 a is necessary to ensure that the existence of a threshold age and ℓ(a)2 da ∫ ∞ its uniqueness only depend on the weights and the ages at which =2 ρ (x) µ(x) ℓ(x) e(x) ∫x ∞ dx ω(x) W (x) = 0. The same assumption was made in previous 0 x ℓ(a) da ∫ ∞ works that identified the threshold age of other lifespan variation = ρ (x) w(x) 2 ℓ(x) D(x) dx , indicators (Zhang and Vaupel, 2009; Gillespie et al., 2014; Aburto 0 et al., 2019). If ρ (x) were to take both positive and negative where w (x) = µ(x) ℓ(x) e(x), which proves (6). □ values, the threshold age may not be unique. Following (7), the threshold age would also be unique if ρ (x) were strictly nega- tive and there were mortality increases for all ages over time. 3.3. Changes over time in D in terms of mortality improvements However, we did not explore that scenario. Eqs. (4) and (6) enable us to express changes over time in D Remark. Following (2), Drewnowski’s index D is bounded be- in terms of mortality improvements. Using (2), and subsequently tween 0 and 1, reaching a value of 1 at complete equality in the replacing (4) and (6) in (3), yields ages at death within a population. A score of 0 would express ( ϑ̇ ėo ) ϑ̇ − ėo D maximum inequality in the ages at death. By definition, however, Ḋ = D − = this value can never be reached: ϑ eo eo ∫∞ ℓ(x)2 dx ∞ ∫ ∞ 2 ℓ(x) D(x) − D ∫ = ρ (x) w (x) dx D = 0 ⇐⇒ ∫0 ∞ = 0 ⇐⇒ ℓ(x)2 dx = 0 eo 0 ℓ(x) dx 0 ∫0 ∞ ⇐⇒ ℓ(x) = 0 (10) = ρ (x) w (x) W (x) dx . (7) 0 for all ages x ≥ 0. This implies that the denominator in (10) equals This result shows that changes over time in D (and analogously 0 because ℓ(x) ≥ 0 is always positive and, consequently, D would in G) are a total average of mortality improvements weighted by be undefined. Hence, 0 < D ≤ 1. 3 J.M. Aburto, U. Basellini, A. Baudisch et al. Theoretical Population Biology 148 (2022) 1–10 [ ∫x Proposition 3. Let ℓ(x) be the probability of surviving from birth Note that ℓ(x) = exp − 0 µ(a) da for a given ] [ ∫age-specific x hazard function µ(x). Therefore, ℓ(x)2 = exp − 0 2 µ(a) da ] to age x, D Drewnowski’s index as defined in (2), and D(x) as defined in (5). Define the function g(x) := 2 ℓ(x) D(x) − D. Then, there exists can be interpreted as the survival schedule with doubling haz- at least one age aD such that g(aD ) = 0. ∫ ∞ 2 µ2(x) at all2 ages x ≥ 0. We can then define ẽ(x) = ard x ℓ(a) da / ℓ(x) as the remaining life expectancy at age x of Proof. At age x = 0, a population with survival schedule ℓ(x)2 and age-specific force of mortality 2 µ(x). Accordingly, g(0) = 2 ℓ(0) D(0) − D = 2 D − D = D > 0 (11) ∂ ∫ ∞ ∫ ∞ 1 1 by definition, since 0 < D ≤ 1. g(x) < 0 ⇐⇒ ℓ(a) da < 2 ℓ(a) da When ages become arbitrarily large, ∂x ℓ(x)2 x ℓ(x) x ⇐⇒ ẽ(x) < e(x) lim g(x) = lim (2 ℓ(x) D(x) − D) = 2 lim ℓ(x) D(x) − D , x→∞ x→∞ x→∞ for all x ≥ 0, which holds true since doubling the hazard corre- which only depends on the behavior of ℓ(x) D(x). Because ℓ(x) ∈ sponds to a lower remaining life expectancy, in the reasonable [0, 1] for all ages x ≥ 0, we have that 0 ≤ ℓ(x)2 ≤ ℓ(x) and assumption that µ(x) > 0 for all ages. □ ∫ ∞ ∫ ∞ ℓ(a) da ≤ lim ℓ(a) da . ∫ ∞ϑ / eo be Drewnowski’s index, where ϑ = 2 Theorem. Let D = 0 ≤ lim (12) ∫∞ x→∞ x x→∞ x 0 ℓ (x)2 dx, eo = 0 ℓ(x) dx is the life expectancy at birth, and Let e(x) be the remaining life expectancy at age x, since ℓ(x) the probability of surviving from birth to age x. Assume mor- ∫ ∞ tality improvements over time occur at all ages. Then, there exists lim ℓ(a) da = lim e(x) ℓ(x) = 0 , a unique threshold age aD that separates positive from negative x→∞ x x→∞ contributions to lifespan equality, measured by D, as a result of those both integrals in (12) tend to 0 as x approaches ∞. Consequently, improvements. ∫∞ ℓ(a)2 da lim ℓ(x) D(x) = lim ∫x ∞ Proof. Following (7), changes over time in D can be expressed as x→∞ x→∞ x ℓ(a) da a weighted average of mortality improvements, given by is indeterminate, but applying L’Hôpital’s rule, we get ∞ ∫ ∂ ∫∞ Ḋ = ρ (x) w(x) W (x) dx , −ℓ(x)2 ℓ(a) da 2 0 lim ℓ(x) D(x) = ∫∞ lim ∂ ∂x = lim x x→∞ x ℓ(a) da x→∞ x→∞ −ℓ(x) ∂x where ρ (x) are the age-specific rates of mortality improvement = lim ℓ(x) = 0 . over time, and w (x) W (x) the weights. By assumption, ρ (x) > 0 x→∞ for all ages x ≥ 0. Therefore, any threshold age that separates pos- As a result, itive from negative contributions to lifespan equality as a result of mortality improvements will occur whenever w (x) W (x) = 0. lim g(x) = 2 lim ℓ(x) D(x) − D = 0 − D < 0 . (13) From (9), x→∞ x→∞ Finally, using (11) and (13), on a continuous framework the w(x) W (x) = 0 ⇐⇒ 2 ℓ(x) D(x) − D = 0 , intermediate value theorem guarantees the existence of at least one positive age aD at which g(aD ) = 0. □ where D(x) is as defined in (5). Proposition 3 guarantees ( ) ( the existence of at least one positive age aD such that 2 ℓ aD D aD − ) Proposition 4. Let ℓ(x) be the probability of surviving from birth D = 0. In addition, from Proposition 4 the function g(x) := to age x, D Drewnowski’s index as defined in (2), and D(x) as defined 2 ℓ(x) D(x) − D is strictly decreasing. Hence, assuming continuity, in (5). Then, g(x) := 2 ℓ(x) D(x) − D is a strictly decreasing function. g(x) is a one-to-one function and therefore the threshold age aD is unique. □ Proof. In order to demonstrate that g(x) is a strictly decreasing function it suffices to show that its first derivative is negative for Corollary. Let G be the Gini coefficient as defined in (1). Provided all ages x ≥ 0. Note that since D does not depend on age, that G = 1 − D, following (7) ∞ ∂ ∂ ( ∫ g(x) < 0 ℓ(x) D(x) < 0 . ρ (x) w (x) W (x) dx . ) ⇐⇒ Ġ = −Ḋ = − ∂x ∂x 0 Applying the quotient rule together with the fundamental Hence, G and D have the same threshold age, which is unique theorem of calculus, we get in the assumption that mortality improvements occur at all ages. (∫∞ ℓ(a)2 da ) Improvements below the threshold age will reduce lifespan variation ∂ ( ∂ ℓ(x) D(x) = ∫x ∞ ) (lowering G, but enlarging D), and improvements above will increase ∂x ∂x x ℓ(a) da lifespan inequality (enlarging G, but lowering D). ∂ ℓ(a)2 da − x ℓ(a)2 da ∂∂x x ℓ(a) da ∫∞ (∫ ∞ ) ∫∞ (∫ ∞ ℓ(a) da ) x ∂x x = (∫ ∞ )2 x ℓ(a) da 5. Application ∫∞ ∞ ℓ(a) da −ℓ(x)2 − x ℓ(a)2 da (−ℓ(x)) ( ) ∫ = x (∫ ∞ )2 . x ℓ(a) da We illustrate our theoretical findings first by simulating sce- Hence, narios under the Gompertz mortality model, and next by showing an example of application to Swedish life table data from the 20th ∂ ∞ ∞ ∫ ∫ g(x) < 0 ⇐⇒ ℓ(x) ℓ(a)2 da − ℓ(x)2 ℓ(a) da < 0 century. The R code (R Core Team, 2021) and data to reproduce ∂x x x all the figures and results presented in the following are publicly ∞ ∞ available for research purposes from the GitHub repository https: ∫ ∫ 1 1 ⇐⇒ ℓ(a)2 da < ℓ(a) da . //github.com/panchoVG/Drewnowski. ℓ (x)2 x ℓ(x) x 4 J.M. Aburto, U. Basellini, A. Baudisch et al. Theoretical Population Biology 148 (2022) 1–10 Fig. 1. Gompertz mortality model with positive aging (β > 0) for different levels of baseline mortality α and a fixed rate of aging β = 0.1. 5.1. The Gompertz model and the threshold age Lower levels of baseline mortality α translate into vertical downward shifts in the log-hazard (Panel A). Correspondingly, We explore different scenarios of mortality improvement de- lifespan variation falls as deaths concentrate at older ages, as pending on level versus rate changes in the Gompertz mortality captured by larger values of D (Panel B). Lower levels of mortality model (Gompertz, 1825). This model captures an exponential raise the threshold age aD at which the weights W (x) switch sign change of the force of mortality µ(x) = α · eβ x over age x, from positive to negative (Panel C). Positive weights at some age where α is the baseline level of mortality and β the rate of imply that saving lives at that age increases equality, i.e. enlarging aging. As previously mentioned, for any[ hazard ∫x function µ(x) D; negative weights imply that saving lives decreases equality, the corresponding survival is ℓ(x) = exp − 0 µ(a) da , and the i.e. lessening D. The threshold age aD marks the boundary be- ] tween these positive and negative effects of life saving on lifespan distribution of deaths d(x) = µ(x) ℓ(x) follows immediately. The equality and is indicated by the dashed vertical lines. Panel C survival function of the Gompertz model is given by shows that under the assumption of mortality improvements α ( [ ] over all ages, this threshold age is unique and that the magnitude ℓ(x) = exp 1 − eβ x . ) (14) β of the weights W (x) over age changes depending on the different shapes of the distribution of deaths, supporting our theoretical Applying (14) to (2), (5), (8) and (9), and using different values of results. α and β , we calculate the corresponding Drewnowski indices D, Complementing these findings, Fig. 2 analyzes how slowing weights W (x), and threshold ages aD reported in Figs. 1 to 5. the rate of aging β affects lifespan variation. For a fixed baseline Specifically, we explore patterns of increasing, constant, and level of mortality α = e−6 ≈ 0.0025, Panel A illustrates how decreasing mortality with age. While our simulations are hypo- lower values of β translate, as expected, into a slower rise in the thetical, these patterns are found across the tree of life (Jones log-hazard. In contrast to the first scenario, Panel B reveals that et al., 2014). Humans and non-human primates experience in- slowing the rate of aging β reduces D (lighter colors) and widens creasing exponential mortality from early ages (Colchero et al., the distribution of ages at death. That is, extending lifespan by 2021), but constant patterns of mortality with unchanged risk of slowing the rate of aging increases lifespan inequality, which death over age are observed in organisms such as Hydra (Schaible is consistent with previous research by Colchero et al. (2021). et al., 2015). Some species like turtles may show declining rates The threshold age aD increases as lifespan equality decreases. of mortality in adulthood (Jones et al., 2014), something that While life saving has opposite effects on lifespan variability and is commonly referred to as ‘negative senescence’ (Vaupel et al., the level of the threshold age in the two scenarios, the general 2004; Baudisch, 2008; Cayuela et al., 2019). dynamics of the weights W (x) remain similar. Saving lives before the threshold age reduces inequality; saving lives later increases 5.1.1. Gompertz force of mortality with positive aging: Level vs. rate inequality. The threshold age aD remains unique, independent of change specific values of the mortality parameters, further supporting Our first scenario demonstrates how improvements in mor- our theoretical predictions. tality affect our outcome variables when progress results from reducing the level of mortality, but not from slowing the rate 5.1.2. Zero rate of aging of aging. Fig. 1 depicts the Gompertz mortality function on a Fig. 3 captures a scenario in which the rate of aging equals log-scale (Panel A), the corresponding age-at-death distribution zero. Under varying but constant levels of mortality (Panel A), (Panel B), and associated weights W (x) (Panel C), for a fixed rate lifespan variation remains unchanged at D = 0.5 and the largest of aging β = 0.1 and changing levels of baseline mortality α . number of deaths occur at the earliest ages (Panel B). The lower Values of Drewnowski’s index D and corresponding threshold the level of mortality, the longer the length of life, and the later ages aD are reported in Panels B and C, respectively. the threshold age (Panel C). 5 J.M. Aburto, U. Basellini, A. Baudisch et al. Theoretical Population Biology 148 (2022) 1–10 Fig. 2. Gompertz mortality model with positive aging for different rates of aging β > 0 and a fixed level of mortality α = e−6 . Fig. 3. Gompertz mortality model with zero rate of aging (β = 0) and different levels of baseline mortality α . This can also be verified analytically. When β = 0, the survival which implies that the threshold age is aD = ln(2) /α and ℓ(aD ) = function for the Gompertz mortality model simplifies to ℓ(x) = D e−α a = 0.5. This means that, when mortality is constant, the e−α x . Then, life expectancy as the integral of the survivorship threshold age and the median age at death are the same. over all ages becomes 1/α , which is the inverse of the force of mortality. Following Definition 1, in this scenario the conditional 5.1.3. Negative rate of aging Drewnowski index D(x) is To complete the analyses, Figs. 4 and 5 capture scenarios of de- 1 e−2 α x / 2 α creasing mortality, which are consistently marked by values of D D(x) = = 0.5 . e−α x e−α x / α below one half (Panels B). Different to the scenario with positive aging rates, here effects of both level and rate changes point in the In other words, when mortality is constant over age, D(x) = 0.5 same direction: higher mortality results in less lifespan variation for all ages and it is independent of the baseline mortality level (higher values of D in Panels B). Both Drewnowski’s index and α . Accordingly, from Proposition 3 the threshold ages barely change, and the latter remain close to g(x) = 0 ⇐⇒ 2 e−α x 0.5 − 0.5 = 0 ⇐⇒ x = ln(2) / α , the earliest age (Panels C). This is because in these scenarios most 6 J.M. Aburto, U. Basellini, A. Baudisch et al. Theoretical Population Biology 148 (2022) 1–10 Fig. 4. Gompertz mortality model with negative rate of aging (β < 0) for different levels of baseline mortality α and a fixed rate of aging β = −0.05. Fig. 5. Gompertz mortality model with negative rate of aging (β < 0) for different rates of aging β and a fixed level of mortality α = e−0.5 . individuals die young, and life expectancy is extremely short for 5.2. Rates of mortality improvement among Swedish females all cases. The two negative aging scenarios substantially differ, however, To show the practical usefulness of our framework, we analyze in the maximum age for the survivors (not shown). This reflects mortality patterns for Swedish females, whose life expectancy has the exceeding disparity among the majority of individuals who improved over the last decades. We apply the discrete approxi- die upon – or shortly after – birth, and those few surviving to mation suggested by Vaupel (1986) to estimate age-specific rates t manifold higher ages as they benefit from the improvements of mortality improvement. Let n qx1 denote the probability of death in mortality over age. These differences are also reflected in between ages x and x + n at time t1 in a given population, and t2 the density functions: while varying the rate of aging leaves no n qx the corresponding probability at time t2 > t1 . Then, the rate distinguishable effect on the density function (Fig. 5, Panel B), of morality improvement from age x to x + n between times t1 and t2 can be approximated as varying the baseline level of mortality α distinctly affects the t t shape of the density function with lower mortality levels yielding ln(− ln(1 − n qx1 )) − ln(− ln(1 − n qx2 )) an increasing rectangularization (Fig. 4, Panel B). n ρx = . (15) t2 − t1 7 J.M. Aburto, U. Basellini, A. Baudisch et al. Theoretical Population Biology 148 (2022) 1–10 Fig. 6. Age-specific rates of mortality improvement (5-year age groups), Swedish females. Panel A shows estimated improvements in 5-year time intervals (1975–1980 and 1995–2000), whereas Panel B shows estimated improvements in 20-year time intervals (1960–1980 and 1980–2000). Dashed horizontal lines mark corresponding threshold ages. Source: Human Mortality Database, 2022. Fig. 6 illustrates the results of applying (15) to life table data Drewnowski’s index D, the complement of the Gini coefficient of for Swedish females from the Human Mortality Database (2022). the life table, by means of formal demography. We contribute to Panel A shows estimated improvements in 5-year time intervals the literature by disentangling how changes in age patterns of (1975–1980 and 1995–2000), whereas Panel B shows estimated mortality affect lifespan variation and provide an analytical proof improvements for 20-year periods (1960–1980 and 1980–2000). for the existence of a threshold age aD below which mortality Although the assumption of ρ (x) being a strictly positive func- improvements are translated into increasing lifespan equality tion does not hold for the whole time period, in these time and above which these improvements translate into increasing windows the threshold ages fall within the age ranges in which lifespan inequality. Previous research determined such age for mortality improvements are observed. Over time, threshold ages the life table entropy (Aburto et al., 2019), the variance (Gillespie are postponed to higher ages. Aligned with these dynamics, life et al., 2014) and years of life lost (Zhang and Vaupel, 2009). expectancy for Swedish females increased from 74.88 in 1960 to We test and illustrate our results under multiple scenarios of 82.02 in 2000 (Human Mortality Database, 2022). The improve- mortality changes with age, including positive, zero, and negative ments depicted in Fig. 6 are responsible for these increments in rates of aging. This is relevant because shapes of mortality pat- life expectancy. During the same period, lifespan equality mea- terns across the tree of life vary substantially (Jones et al., 2014). sured by D increased from 0.899 to 0.921. This means that, on Our experiments demonstrate how Drewnowski’s index D can average, lifespans increased and also became more predictable for serve as an indicator of the ‘shape’ of mortality patterns (Baud- Swedish females. isch, 2011; Wrycza et al., 2015), distinguishing between increas- The example of Sweden can only serve as an illustration, and ing mortality (D > 0.5), constant mortality (D = 0.5), and results may differ for other countries. In particular, mortality decreasing mortality (D < 0.5) over age. These properties, along improvements may be more volatile in low- and middle-income countries, or during mortality crises such as the Covid-19 pan- with our analytical findings, support studying lifespan variation demic. Researchers should apply Eq. (15) to identify the age alongside life expectancy trends in multiple species. ranges in which the assumption of positive rates of mortality The existence of a unique threshold age, though, is conditioned improvement holds. on having positive rates of mortality improvement throughout the whole age range, which is certainly a strong assumption. Us- 6. Discussion ing Sweden as a benchmark, we hypothesize that among human populations this may be likely to hold at adult ages (60–95 years) In this article we uncovered how age-specific patterns of in which more deaths are concentrated, and where it may be mortality underpin trends in lifespan variation as measured by more meaningful to identify a (unique) threshold age. Above that 8 J.M. Aburto, U. Basellini, A. Baudisch et al. Theoretical Population Biology 148 (2022) 1–10 range, one could argue that the small fraction of the remaining Drewnowski, J., 1982. Social indicators and their application for measuring the deaths does not suffice to counterbalance the uniqueness of the progress of development. In: Working Group on Indicators for Monitoring threshold age, though we have not tested that explicitly. Our re- and Evaluation of Strategies for Health for All By the Year 2000. WHO Regional Office for the Western Pacific, Manila, Philippines, pp. 103–130, sults also show that our framework may be suitable for studying URL: https://apps.who.int/iris/handle/10665/207170, 2–9 March 1982. long term trends (20 years), so that mortality improvement can Ebeling, M., Rau, R., Baudisch, A., 2018. Rectangularization of the survival curve be substantiated across a wide age range. reconsidered: The maximum inner rectangle approach. Popul. Stud. 72 (3), This framework may also be applicable for non-human species 369–379. when comparing populations, not chronologically over time, but Edwards, R.D., Tuljapurkar, S., 2005. Inequality in life spans and a new perspec- ordered from harsher to milder conditions, so that improve- tive on mortality convergence across industrialized countries. Popul. Dev. Rev. 31 (4), 645–674. ments in survival can be studied across changing environments Fernández, O.E., Beltrán-Sánchez, H., 2015. The entropy of the life table: A (Colchero et al., 2019). Such analyses may aid understanding reappraisal. Theor. Popul. Biol. 104, 26–45. the principle limits of the plasticity of aging in different species Gigliarano, C., Basellini, U., Bonetti, M., 2017. Longevity and concentration in (Colchero et al., 2021) survival times: The log-scale-location family of failure time models. Lifetime Data Anal. 23 (2), 254–274. Declaration of competing interest Gillespie, D.O., Trotter, M.V., Tuljapurkar, S.D., 2014. Divergence in age patterns of mortality change drives international divergence in lifespan inequality. Demography 51 (3), 1003–1017. The authors declare that they have no known competing finan- Gini, C., 1912. Variabilità e mutabilità: Contributi allo studio delle distribuzioni cial interests or personal relationships that could have appeared delle relazioni statistiche. Stud. Econ.-Giuridici Univ. Cagliari 3. to influence the work reported in this paper. Gini, C., 1914. Sulla misura della concentrazione e della variabilità dei caratteri. Atti Real. Ist. Veneto Sci. Lett. Ed. Art. 73, 1203–1248. Acknowledgments Goldman, N., Lord, G., 1986. A new look at entropy and the life table. Demography 23 (2), 275–282. Gompertz, B., 1825. On the nature of the function expressive of the law of human This work was partially funded by the European Research mortality, and on a new mode of determining the value of life contingencies. Council (Grant 2019-AdG-884328) (J.M.A. and F.V.). J.M.A. also ac- In a letter to Francis Baily, Esq. F.R.S. &c. Philos. Trans. R. Soc. 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