The threshold age of the lifetable entropy

Abstract

BACKGROUND Indicators of relative variation of lifespans are markers of inequality at the population level and of uncertainty at the time of death at the individual level. In particular, the lifetable entropy H represents the elasticity of life expectancy to a change in mortality. However, it is unknown how this measure changes over time and whether a threshold age exists, as it does for other lifespan variation indicators. RESULTS The time derivative of H can be decomposed into changes in life disparity e1" and life expectancy at birth eo. Likewise, changes over time in H are a weighted average of agespecific rates of mortality improvements. These weights reflect the sensitivity of H and show how mortality improvements can increase (or decrease) the relative inequality of lifespans. Further, we prove that in the assumption that mortality is reduced at all ages, H, as well as 6†, has a threshold age below which saving lives reduces entropy, whereas improvements above that age increase entropy. CONTRIBUTION We give a formal expression for changes of H over time and provide a formal proof of the existence of a unique threshold age that separates reductions and increases in lifespan variation as a result age-specific mortality improvements.

Headnote

Abstract

BACKGROUND

Indicators of relative variation of lifespans are markers of inequality at the population level and of uncertainty at the time of death at the individual level. In particular, the lifetable entropy H represents the elasticity of life expectancy to a change in mortality. However, it is unknown how this measure changes over time and whether a threshold age exists, as it does for other lifespan variation indicators.

RESULTS

The time derivative of H can be decomposed into changes in life disparity e1" and life expectancy at birth eo. Likewise, changes over time in H are a weighted average of agespecific rates of mortality improvements. These weights reflect the sensitivity of H and show how mortality improvements can increase (or decrease) the relative inequality of lifespans. Further, we prove that in the assumption that mortality is reduced at all ages, H, as well as 6†, has a threshold age below which saving lives reduces entropy, whereas improvements above that age increase entropy.

CONTRIBUTION

We give a formal expression for changes of H over time and provide a formal proof of the existence of a unique threshold age that separates reductions and increases in lifespan variation as a result age-specific mortality improvements.

(ProQuest: ... denotes formulae omitted.)

1.Relationship

The lifetable entropy is a dimensionless indicator of the relative variation in the length of life compared to life expectancy at birth (Leser 1955; Keyfitz 1968, 1977; Demetrius 1974, 1978). It is usually defined as

...

where et(t) = - J0" ¿(a, t) ln ¿(a, t) da is the life disparity or number of life-years lost as a result of death (Vaupel and Canudas-Romo 2003), eo(t) = /0" ¿(a, t) da is the life expectancy at birth at time t, ¿(a, t) is the lifetable survival function, c(a, t) = ¿(a, t) / /0" ¿(x, t) dx is the population structure, and H (a, t) = /" p(x, t) dx is the cumulative hazard to age a, where p(x, t) is the force of mortality (hazard rate or risk of death) at age x at time t. Note that H(t) can be interpreted as an average value of H (a, t) in the population at time t.

Goldman and Lord (1986) and Vaupel (1986) proved that

...

where d(a, t) represents the distribution of deaths, and e(a, t) = JJ" ¿(x, t) dx / ¿(a, t) is the remaining life expectancy at age a at time t. This formulation provides an alternative expression for the lifetable entropy as

...

Let H denote the partial derivative of H with respect to time.5 We define p(x) = -fd(x) / p(x) as the age-specific rates of mortality improvements. Then, the relative derivative of H can be expressed as a weighted average of p(x),

(1) ...

with weights

...

Function H(x) is the lifetable entropy conditioned on surviving to age x, defined as

...

where e† (x) = f° d(a) e(a) da / t(x) refers to life disparity above age x, and e(x) is the remaining life expectancy at age x.

Note that the lifetable entropy H is a measure of relative lifespan variation. Thus, higher values represent more variation, whereas lower values denote less variation of lifespans. If mortality improvements over time occur at all ages, there exists a unique threshold age aH that separates positive from negative contributions to the lifetable entropy H resulting from those mortality improvements. This threshold age aH is reached when

(2) ...

2.Proof

Fernandez and Beltran-Sanchez (2015) showed that the relative derivative of H can be expressed as

(3) ...

This formula indicates that relative changes in H over time are given by the difference between relative changes in e1 (dispersion component) and relative changes in eo (translation component). We will first provide expressions for eo and e1 to prove that (1) and (3) are equivalent. Next, we will prove the existence of a threshold age for H and its uniqueness.

2.1Relative changes over time in H

Vaupel and Canudas-Romo (2003) showed that changes over time in life expectancy at birth are a weighted average of the total rates of mortality improvements, given by

(4) ...

where p(x) = -ji(x) / p(x) are the age-specific rates of mortality improvements, and w(x) = p(x) i(x) e(x) = d(x) e(x) is a measure of the importance of death at age x.

Since d(x) = ^(x) £(x) and £(x) e(x) = f° £(a) da, the partial derivative with respect to time of e1 = f° d(x) e(x) dx can be expressed as

...

where H(a) is the cumulative hazard to age a. By reversing the order of integration and doing some additional manipulations, we get

(5) ...

In Proposition 1 in the Appendix, we prove that

(6) ...

Replacing (6) in (5) yields

(7) ...

Finally, replacing the expressions of eo and ¿† from (4) and (7) in (3), we get

...

which proves (1) and shows that relative changes over time in the lifetable entropy H are the average of the rates of mortality improvements weighted by the product w(x) W(x). □

2.2The threshold age for H

Using (1), changes over time in the lifetable entropy H are given by

(8) ...

Whenever H > 0, lifespan inequality increases over time, whereas H < 0 implies that variation of lifespans decreases over time. Because l(x) is a positive function bounded between 0 and 1, we have that H > 0. Moreover, assuming age-specific death rates p(x) improve over time at all ages, then p(x) < 0 and p(x) > 0 at any age x. Therefore, (8) implies that

1. Those ages x in which w(x) W(x) > 0 will contribute positively to the lifetable entropy H and increase lifespan variation;

2. Those ages x in which w(x) W(x) < 0 will contribute negatively to the lifetable entropy H and favor lifespan equality;

3. Those ages x in which w(x) W(x) = 0 will have no effect on the variation over time of H.

Our goal is to prove that if mortality improvements occur (or all ages and p(x) > 0, there exists a unique threshold age aH such that w (aH) W (aH) = 0. That threshold age will separate positive from negative contributions to H resulting from mortality improvements.

The product w(x) W (x) can be re-expressed as

...

Since yu(x), t(x), e(x) and eT are all positive functions, the threshold age of H occurs whenever

(9) ...

When x is close to 0, g(x) takes negative values since

...

Likewise, g(x) takes positive values when x becomes arbitrarily large. Note that H does not depend on age, and therefore

...

because limæ ^TO H (x) = to. By definition, H(x) > 0 for all x, so regardless of the behavior of H(x), when x is arbitrarily large, the limit of g(x) tends to infinity. Hence, given that g(0) = -1 and limæ ^TO g(x) = to, in a continuous framework the intermediate value theorem guarantees the existence of at least one age aH at which g(aH) = 0.

Moreover, as shown in Proposition 2 in the Appendix, g(x) is a strictly increasing function, and therefore a one-to-one function assuming continuity. As a result, there is a unique threshold age aH that separates positive from negative contributions to the lifetable entropy H, and that threshold age is reached when

...

which proves (2). □

3.Related results

Demographers have developed a battery of indicators to measure how lifespans vary in populations (van Raalte and Caswell 2013; Colchero et al. 2016). The most common indexes are the variance (Edwards and Tuljapurkar 2005; Tuljapurkar and Edwards 2011), standard deviation (Alvarez, Aburto, and Canudas-Romo 2019; van Raalte, Sasson, and Martikainen 2018) and coefficient of variation (Aburto et al. 2018) of the age at death distribution, the Gini coefficient (Shkolnikov, Andreev, and Begun 2003; Gigliarano, Basellini, and Bonetti 2017; Archer et al. 2018), the Theil index (Smits and Monden 2009), and the years of life lost (Vaupel, Zhang, and van Raalte 2011; Aburto and van Raalte 2018), among others. However, only a few studies have analytically derived formulas for the threshold age below and above which mortality improvements respectively decrease and increase lifespan variation. Zhang and Vaupel (2009) showed that the threshold age (a1) for life disparity (e1) occurs when H(x) +H(x) = 1. Similarly, Gillespie, Trotter, and Tuljapurkar (2014) determined a threshold age for the variance of the age at death distribution. Van Raalte and Caswell (2013) also showed that it is possible to determine the threshold age by performing an empirical sensitivity analysis of lifespan variation indicators.

In this article, we contribute to the lifespan variation literature by deriving the threshold age aH for the lifetable entropy H. This age separates negative from positive contributions of age-specific mortality improvements. We analytically proved its existence and - in the assumption that mortality improves over time for all ages - also its uniqueness. In Section 4 we empirically show that it differs from the threshold age of e1.

4.Applications

The code and data to reproduce the results and graphs presented in this section are publicly available through the repository in the link https://bit.ly/2wqzOFp.

4.1 Numerical findings

Figure 1 depicts the threshold ages of the two related measures: life disparity e1 and lifetable entropy H. Calculations were performed using data from the Human Mortality Database (2018) for females in the United States and Italy in 2005. The blue line represents g(x) from Equation (9). The threshold age aH occurs when g(x) crosses zero. The red and grey lines display the same functions that Zhang and Vaupel (2009) used to find the threshold age for e1 rescaled to fit in the graph. The intersection of these two lines denotes the threshold age a1. Finally, the dashed black line depicts the life expectancy at birth. Vaupel, Zhang, and van Raalte (2011) noted that a† tends to fall just below eo. The threshold age for the lifetable entropy aH is greater than a† and is very close above life expectancy for these countries. Note the similarity between the formulas for a1, given by H(a1) + H(a1) = 1, and aH, given by H(aH) + H(aH) = 1 + H.

Panels a) and b) in Figure 2 illustrate the evolution over time of the threshold ages for e1 and H in French and Swedish females, respectively. We chose these countries because they portray large series of reliable data available at the Human Mortality Database (2018).

Values for a† are close to life expectancy throughout the period. However, around 1950 there is a crossover between a) and eo such that a) remained close to life expectancy, but below it most of the time. This result shows that values of a1" below eo is a modern feature of aging populations with high life expectancy. From the beginning of the period of observation to the 1950s, the threshold age for the lifetable entropy was above life expectancy for both countries. During some periods aH was roughly constant whereas eo trended upward. After the 1950s, aH converged toward life expectancy.

4.2The threshold age of the lifetable entropy within the Gompertz mortality model

We further analyze the relationship between aH and eo, assuming that the force of mortality follows a Gompertz distribution with hazard ^(x) = a eßx, where x > 0 denotes the age and a, ß > 0 are parameters. In Proposition 3 in the Appendix, we prove that in the Gompertz model, the threshold age aH of the lifetable entropy H is approximately proportional to eo by a factor S, which only depends on parameters a and ß, and the Euler-Mascheroni constant y « 0.57722. A value of S close to 1 indicates that mortality is roughly following a Gompertz model.

Figure 3 shows the evolution over time of factor S for French and Swedish females. The observation that this value converges toward 1 could be explanatory for the convergence of the threshold age and life expectancy at birth in modern mortality profiles. It also indicates that modern mortality schedules are roughly Gompertzian. Therefore, it can be speculated that differences between these two measures in earlier years are a consequence of a big proportion of deaths occurring in ages where the force of mortality does not follow a Gompertz, such as in infancy. This pattern is consistent with historical trends which suggest that, from hunter-gathers to modern populations, death rates have decreased at all ages, but mostly at younger ones (Burger, Baudisch, and Vaupel 2012).

4.3 Decomposition of the relative derivative of H

The relative derivative of H defined in Equation (1) can be decomposed between components before and after the threshold age aH as follows:

(10) ...

If mortality reductions occur at every age, the early life component in Equation (10) is always negative (contributing to reduce entropy) while the late life component is positive (contributing to increase entropy). Thus, it is clear that a negative relationship between life expectancy and entropy over time occurs if the early life component outpaces the late life component. This decomposition is based on the additive properties of the derivatives of life expectancy and e1", as previously shown by Vaupel and Canudas-Romo (2003) and Fernandez and Beltran-Sanchez (2015).

5.Conclusion

Several authors have been interested in decomposing changes over time in life expectancy (Arriaga 1984; Vaupel 1986; Pollard 1988; Vaupel and Canudas-Romo 2003; BeltranSanchez, Preston, and Canudas-Romo 2008; Beltran-Sanchez and Soneji 2011). Most recently, scholars have also investigated how life disparity fluctuations over time can be decomposed (Zhang and Vaupel 2009; Wagner 2010; Shkolnikov et al. 2011; Aburto and van Raalte 2018; Aburto and Beltran-Sanchez 2019). Here, we bring both perspectives together and shed light on the dynamics behind changes in the lifetable entropy.

Leser (1955) first derived the lifetable entropy as the elasticity of life expectancy. Keyfitz (1977) proposed H as a lifetable function "that measures the change in life expectancy at birth consequent on a proportional change in age-specific rates" (Keyfitz 1977: 413). Since then, several authors have been interested in this measure and its use (Demetrius 1978, 1979; Mitra 1978; Goldman and Lord 1986; Vaupel 1986; Hakkert 1987; Hill 1993; Fernandez and Beltran-Sanchez 2015). Even though the lifetable entropy and e1 are both measures of lifespan variation, their demographic interpretation differs. The former is defined as the elasticity of life expectancy due to changes in death rates (Keyfitz 1968) whereas the latter one refers to the average years lost due to death (Vaupel, Zhang, and van Raalte 2011). The life table entropy measures relative variability, while e1 measures absolute lifespan variation. Therefore, the lifetable entropy is appropriate to compare different shapes of age-at-death distributions across different species and over time (Baudisch 2011; Wrycza, Missov, and Baudisch 2015), while e1 has been used to obtain insights about lifespan variation in different countries and in subpopulation groups, for instance by occupational class or income (van Raalte, Martikainen, and Myrskyla 2014; Brønnum-Hansen 2017). Both measures are meaningful and complementary, but the calculation of their threshold ages should be performed accordingly to correctly interpret changes of age patterns of mortality.

In this article, we uncovered the mathematical regularities behind the changes over time in the lifetable entropy. In particular, this study contributes to the existing literature by showing that (1) the lifetable entropy can be decomposed as a weighted average of rates of mortality improvements, and (2) there exists a unique threshold age that separates positive from negative contributions to lifetable entropy as a result of reductions in mortality over time.

Footnote

1 Interdisciplinary Center on Population Dynamics, University of Southern Denmark, Odense, Denmark and Max Planck Institute for Demographic Research, Rostock, Germany. Email: [email protected].

2 Interdisciplinary Center on Population Dynamics, University of Southern Denmark, Odense, Denmark.

3 Department of International Health, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, MD, USA.

4 Interdisciplinary Center on Population Dynamics, University of Southern Denmark, Odense, Denmark, and Max Planck Institute for Demographic Research, Rostock, Germany.

5In the following, a dot over a function will denote its partial derivative with respect to time t, but variable t will be omitted for simplicity.

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(ProQuest: Appendix omitted.)