There is a unique threshold age separating early deaths from late deaths such that averting an early death decreases life disparity, but averting a late death increases inequality in lifespans.life disparity, mortality

James W. Vaupel

Zhen Zhang

English

Research Papers in Economics

Demographic Research a free, expedited, online journal
of peer-reviewed research and commentary
in the population sciences published by the
Max Planck Institute for Demographic Research
Konrad-Zuse Str. 1, D-18057 Rostock ¢ GERMANY
www.demographic-research.org
DEMOGRAPHIC RESEARCH
VOLUME 20, ARTICLE 29, PAGES 721-730
PUBLISHED 18 JUNE 2009
http://www.demographic-research.org/Volumes/Vol20/29/
DOI: 10.4054/DemRes.2009.20.29
Formal Relationships 4
The age separating early deaths
from late deaths
Zhen Zhang
James W. Vaupel
This article is part of the Special Collection “Formal Relationships”.
Guest Editors are Joshua R. Goldstein and James W. Vaupel.
c ° 2009 Zhen Zhang & James W. Vaupel.
This open-access work is published under the terms of the Creative
Commons Attribution NonCommercial License 2.0 Germany, which permits
use, reproduction & distribution in any medium for non-commercial
purposes, provided the original author(s) and source are given credit.
See http://creativecommons.org/licenses/by-nc/2.0/de/Table of Contents
1 Relationship 721
2 Proof 722
3 Numerical ﬁndings 723
4 History and related results 725
5 Applications 726
References 727
Appendix 729Demographic Research: Volume 20, Article 29
Formal Relationships 4
The age separating early deaths from late deaths
Zhen Zhang 1; 3
James W. Vaupel 2; 3
Abstract
Averting deaths may either increase or reduce life disparity, as measured by ey. The sign
of the effect depends on the age at which the deaths are averted. We show that there may,
depending on the entropy of the life table, be an age such that averting deaths before that
age reduces disparity, but averting deaths after that age increases disparity. We say that
this age separates “early” from “late” deaths (in terms of their effect on disparity) and
investigate the factors determining this threshold age. If life table entropy is less than
one, a unique threshold age separating early and late deaths always exists. If life table
entropy is greater than one, averting deaths at any age increases disparity. If entropy
equals one, averting deaths at age zero has no effect, but averting a death at any age after
zero increases disparity.
1. Relationship
We measure life disparity by life expectancy lost due to death, ey =
R 1
0 e(a)d(a)da,
where e(a) =
¡R 1
a `(x)dx
¢
=`(a) is remaining life expectancy at age a and d(a) =
`(a) ¹(a) is the life table distribution of deaths. Note that `(a) = exp(¡
R a
0 ¹(x)dx),
with `(0) = 1, gives the probability of survival to age a, that ¹(a) denotes the age-
speciﬁc hazard of death, and that H(a) =
R a
0 ¹(x)dx is the cumulative hazard function
with H(0) = 0.
Consider a perturbation that alters the cumulative hazard function by a step function
with a single negative step of size s. Let ey
s be the corresponding value of life disparity
with the perturbation. To study the impact on ey of a concentrated decrease in mortality
at age a, it is useful to consider the function
(1) k(a) =
1
`(a)
dey
s
ds
¯ ¯
¯ ¯
s=0
= ey(a) + e(a)
¡
H(a) ¡ 1
¢
;
1 Max Planck Institute for Demographic Research. E-mail: zhang@demogr.mpg.de
2 Max Planck Institute for Demographic Research. E-mail: jwv@demogr.mpg.de
3 We thank Kenneth Wachter, Hal Caswell, Trifon Missov, Peter Wagner and two anonymous reviewers.
http://www.demographic-research.org 721Zhang & Vaupel: The age separating early deaths from late deaths
whereey(a) =
¡R 1
a e(x)d(x)dx
¢
=`(a)islifeexpectancylostduetodeathamongpeople
surviving to age a. Mortality reductions at ages with k < 0 decrease life disparity, while
those at ages with k > 0 increase life disparity.
We prove below that if ey=e(0) < 1, then there exists an age ay greater than zero
such that k > 0 for a greater than ay and k < 0 for values of a less than ay. We
also prove that if ey=e(0) = 1, then averting a death at age zero does not change life
disparity, but averting a death at any higher age increases disparity. Finally we prove that
if ey=e(0) > 1, then averting a death at any age increases life disparity. In the proof we
make use of the fact that k(0) = ey¡e(0). Note that the “entropy of the life table” equals
ey=e(0) (Keyﬁtz 1977; Mitra 1978; Goldman and Lord 1986; Vaupel 1986), which differs
from the standard statistical deﬁnition by a term equal to lne(0). If life table entropy is
less than one, then k(0) < 0; if entropy is one, then k(0) = 0; and if entropy is greater
than one, then k(0) > 0.
2. Proof
We consider three cases.
Case 1: k(0) < 0(entropy < 1)
Because life spans are ﬁnite, `(a) approaches zero and H(a) approaches inﬁnity.
Therefore at an old enough age x, H(x) > 1 and k(x) > 0. Since k(0) < 0 and
k(a) is continuous, the intermediate value theorem implies that there exists at least one
point, say z, in [0;1] such that k(z) = 0. We prove that at any such root the derivative
of k is positive. This implies the root is unique, as shown in the Appendix.
The derivative of (1) is given by
dk(a)
da
=
dey(a)
da
+
³
H(a) ¡ 1
´de(a)
da
+ e(a)
d
¡
H(a) ¡ 1
¢
da
= ¡¹(a)e(a) + ¹(a)ey(a) +
¡
H(a) ¡ 1)
¢¡
¹(a)e(a) ¡ 1
¢
+ e(a)¹(a) (2)
= ¹(a)ey(a) +
¡
H(a) ¡ 1)
¢¡
¹(a)e(a) ¡ 1
¢
:
When k(a) = 0 it follows from (1) that H(a) ¡ 1 = ¡ey=e(a). Substituting this
into (2) yields
dk(a)
da
¯
¯ ¯
¯
a=z
= ¹(a)ey(a) ¡
ey(a)
e(a)
¡
¹(a)e(a) ¡ 1
¢
=
ey(a)
e(a)
> 0: (3)
722 http://www.demographic-research.orgDemographic Research: Volume 20, Article 29
Case 2: k(0) = 0(entropy = 1)
In this case, z = 0. We need to show that this is the only value of z, i.e., the only age
when k = 0. It follows from (2) that if k(0) = 0 then
dk(a)
da
¯ ¯
¯ ¯
a=0
= 1:
Hence k becomes positive as age increases from zero. If there were an age above
zero when k(z) = 0, then the derivative of k at this age would have to be zero or neg-
ative. But as shown in (3), the derivative has to be positive at any age when k(z) = 0.
This contradiction implies that the value of ay equal to zero is unique in the case when
k(0) = 0.
Case 3: k(0) > 0(entropy > 1)
As noted above in Case 2, if there were an age when k(a) = 0 then the derivative of
k at this age would have to be zero or negative. But as shown in (3), the derivative has
to be positive at any age when k(a) = 0. This contradiction implies that there is no age
that separates early from late deaths when k(0) > 0: averting a death at any age would
increase life disparity.
Q.E.D.
3. Numerical ﬁndings
In Case 1, when entropy is less than one, to calculate ay it is necessary to ﬁnd the age
at which ey(a) = e(a)
¡
1 ¡ H(a)
¢
. Using interpolation routines widely available in
statistical software, ay can be readily obtained, as illustrated in Figure 1. Changes in ay
over time are shown in Figure 2. The value of ay in Japan increased from 65 in 1950 to
84 in 2005. Note that a decline in mortality at age 75 before 1972 increased life disparity
while such a decline after 1972 decreased life disparity.
We have computed the value of k(0) for all 5830 life tables since 1840 in the (Human
Mortality Database 2009). We have also computed the value of k(0) for the 3404 life
tables in the (Human Life-Table Database 2009). In every case k(0) < 0. The closest
approach to zero was found for females in 1911-1921 in India: for this population e(0) =
23:33 and ey = 23:08, so that k(0) = ¡0:22 and life table entropy = 0:99. Goldman
and Lord (1986), however, provide two examples of life tables for which k(0) is positive
because entropy exceeds one. Both pertain to selected populations in rural areas of China
in the period 1929-31. One is for females (Barclay et al. 1976) and the other is for males
http://www.demographic-research.org 723Zhang & Vaupel: The age separating early deaths from late deaths
(Coale, Demeny, and Vaughan 1983). For the Chinese women e(0) = 21:00, ey = 21:73,
and entropy = 1:03. For the Chinese males e(0) = 17:43, ey = 22:17, and entropy =
1:27. These cases may be artifacts of bad data.
It would be interesting to determine the conditions such that the life table entropy is
greater than, equal to, or less than one. Although life table entropy may rarely exceed one
for human populations, this may be common for some non-human species or for some
kind of machines or equipment.
Figure 1: A graphical depiction of the calculation of the threshold age (ay),
for US females, 2005
0 20 40 60 80 100
0
2
0
4
0
6
0
8
0
Age
e
†
(
x
)
 
a
n
d
 
e
(
x
)
(
1
−
H
(
x
)
)
 
i
n
 
Y
e
a
r
s
a
†
e
†(x)
e(x)(1−H(x))
Note: ey(0) = 11:3 and ey(100) = 2:2
724 http://www.demographic-research.orgDemographic Research: Volume 20, Article 29
Figure 2: The threshold age (ay) over time in four countries,
females 1950-2005
1950 1960 1970 1980 1990 2000
6
0
6
5
7
0
7
5
8
0
8
5
Year
a
†
Japan
USA
Denmark
France
4. History and related results
Based on the notion of life table entropy (Keyﬁtz 1977), Mitra (1978), Goldman and Lord
(1986) and Vaupel (1986) independently derived the mathematical expression for ey and
showed that life table entropy equals ey=e(0). Vaupel and Canudas-Romo (2003) later
showed that the derivative of life expectancy over time is given by the product of ey and
the rate of progress in reducing age-speciﬁc death rates.
Vaupel (1986) found that if the force of mortality follows a Gompertz curve,
¹(a) = ¹(0)eba, then ey ¼ 1=b. The parameter b has traditionally been interpreted
as the rate of aging, but this result indicates that an alternative interpretation of b is that it
is an inverse function of life disparity.
In addition to ey, several other measures of lifespan variation in a life table have been
http://www.demographic-research.org 725Zhang & Vaupel: The age separating early deaths from late deaths
proposed (Cheung et al. 2005). These include the variance of the age at death, the stan-
dard deviation, the standard deviation above age 10 (Edwards and Tuljapurkar 2005), the
inter-quartile range (Wilmoth and Horiuchi 1999), the Gini coefﬁcient (Shkolnikov, An-
dreev, and Begun 2003), and the entropy of the life table (Keyﬁtz 1977). These measures
are highly correlated with each other. In particular, the correlation of ey with the other
measures is never less than 0.952, according to our calculations based on 5830 period
life tables from 1840 to 2007 available from the Human Mortality Database (2009) (See
table in Vaupel, Zhang, and van Raalte 2009). Hence ey can be viewed as a surrogate for
the other measures. Moreover, it turns out that the derivatives of some other measures
(e.g., standard deviation of age at death, standard statistical entropy of age at death) with
respect to the change in mortality have a similar form and all imply the existence of an
age separating early and late deaths (Caswell, personal communication). We prefer ey
because of its desirable mathematical properties, used above and below, and because it
can be readily explained and interpreted.
5. Applications
At the threshold age ay, the change in ey resulting from mortality decline can be decom-
posed into two components
_ ey(t) =
dey(t)
dt
=
Z a
y(t)
0
k(a;t)d(a;t)½(a;t) da +
Z !
ay(t)
k(a;t)d(a;t)½(a;t) da; (4)
where ½(a;t) is the rate of progress in reducing death rates. The ﬁrst term on the right
side of (4) represents the compression of mortality at younger ages, and the second term
the expansion of mortality at older ages. The balance of the two components determines
whether the population experiences mortality compression or expansion (Zhang and Vau-
pel 2008).
Analogously, the increase in life expectancy at birth _ ey(t) can be broken into two
parts,
_ e(0;t) =
Z a
y(t)
0
e(a;t)d(a;t)½(a;t) da +
Z !
ay(t)
e(a;t)d(a;t)½(a;t) da; (5)
where the ﬁrst term captures the contribution of averting early deaths to increases in e(0),
while the second that of decreasing mortality among the elderly or very elderly. In a
recent study, Vaupel, Zhang, and van Raalte (2009) showed that the countries beneﬁting
from the longest life expectancies are those that have succeeded in reducing disparities in
how long individuals live by averting early deaths.
726 http://www.demographic-research.orgDemographic Research: Volume 20, Article 29
References
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Stable Populations. New York: Academic Press, 2nd ed.
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Paper presented at Population Association of America 2009 Annual Meeting, Detroit,
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728 http://www.demographic-research.orgDemographic Research: Volume 20, Article 29
Appendix
A simple lemma in calculus 4
Letf beareal-valueddifferentiablefunctionon[0;1)withderivativef0 andtheproperty
that f0(x) > 0 whenever f(x) = 0. Then f has at most one root.
Proof. Suppose to the contrary that there are at least two roots of f. First, we will show
that between any two distinct roots of f there is another root of f. Pick any two roots of
f, say a and b with a < b. Since f is differentiable at a,
lim
x!a;x>a
f(x) ¡ f(a)
x ¡ a
exists and equals f0(a). Since f0(a) > 0, there is an ²a > 0 such that f(x) ¡ f(a) > 0,
for all x 2 (a;a+²a); that is, f is positive on the interval (a;a+²a). Analogously, there
is an ²b > 0 such that f is negative on the interval (b ¡ ²b;b). Clearly, a + ²a · b ¡ ²b.
Thus a + ²a=2 < b ¡ ²b=2, and f(a + ²a=2) > 0 and f(b ¡ ²b=2) < 0. Note that, since
f0 exists, f is continuous on [0;1). By the intermediate value theorem applied to f on
[a + ²a=2;b ¡ ²b=2], f vanishes at some c 2 (a + ²a=2;b ¡ ²b=2). Hence f(c) = 0, for
some c 2 (a;b).
Having shown that between any two distinct roots of f there is another root of f, we
will use the above setup with a;b;²a and ²b to derive a contradiction. Let
X = fx 2 (a;b) : f(x) = 0g be the set of all roots of f between a and b. Clearly,
X 6= ; and X ½ [a + ²a;b ¡ ²b]. So X has a supremum s with s 2 [a + ²a;b ¡ ²b].
Now s = limn!1 xn, for some xn 2 X. Since f is continuous at s, we have f(s) =
limn!1 f(xn) = limn!1 0 = 0. Since s < b, f has a root t 2 (s;b), so that t 2 X.
This contradicts the fact that s is the supremum of X and ﬁnishes the proof. Q.E.D.
4 by Peter Wagner. E-mail: Wagner@demogr.mpg.de.
http://www.demographic-research.org 729Zhang & Vaupel: The age separating early deaths from late deaths
730 http://www.demographic-research.org

A reassessment of the demography of traditional rural China.

A short note on the Taeuber paradox.

Applied Mathematical Demography.

Database (2009). [electronic resource].

Decomposing changes in life expectancy: A bouquet of formulas in honor of Nathan Keyﬁtz’s 90th birthday.

Gini coefﬁcient as a life table function: Computation from discrete data, decomposition of differences and empirical examples.

How change in age-speciﬁc mortality affects life expectancy.

http://www.demographic-research.org 727Zhang & Vaupel: The age separating early deaths from late deaths

Human Life-Table Database (2009). [electronic resource].

Inequality in life spans and a new perspective on mortality convergence across industrialized countries. Population and Development Review

Paper presented at Population Association of America

Rectangularization revisited: Variability of age at death with human populations.

Regional Model Life Tables and Stable Populations.

The Threshold between Compression and Expansion of Mortality. Paper presented at Population Association of America

Three dimensions of the survival curve: Horizontalization, verticalization, and longevity extension.

The age separating early deaths from late deaths

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The age separating early deaths from late deaths

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