In a Gompertz mortality model with constant yearly improvements at all ages, linear increases in period life expectancy correspond to linear increases in the respective cohort life expectancy. The link between the two measures can be given by a simple approximate relationship.cohort life expectancy, Gompertz mortality, Linear Shift Models, period life expectancy

Adam Lenart

Trifon I. Missov

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
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DEMOGRAPHIC RESEARCH
VOLUME 24, ARTICLE 19, PAGES 455-468
PUBLISHED 15 MARCH 2011
http://www.demographic-research.org/Volumes/Vol24/19/
DOI: 10.4054/DemRes.2011.24.19
Formal Relationships 13
Linking period and cohort life-expectancy lin-
ear increases in Gompertz proportional haz-
ards models
Trifon I. Missov
Adam Lenart
This article is part of the Special Collection “Formal Relationships”.
Guest Editors are Joshua R. Goldstein and James W. Vaupel.
c ° 2011 Trifon I. Missov & Adam Lenart.
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 456
2 Proof of the relationship 456
3 History and related results 458
4 Applications 464
5 Acknowledgements 466
References 467Demographic Research: Volume 24, Article 19
Formal Relationships 13
Linking period and cohort life-expectancy linear increases in
Gompertz proportional hazards models
Trifon I. Missov 1
Adam Lenart 2
Abstract
In a Gompertz mortality model with constant yearly improvements at all ages, linear
increases in period life expectancy correspond to linear increases in the respective cohort
life expectancy. The link between the two measures can be given by a simple approximate
relationship.
1 Max Planck Institute for Demographic Research, Konrad-Zuse-Str. 1, 18057 Rostock, Germany
2 Max Planck Institute for Demographic Research, Konrad-Zuse-Str. 1, 18057 Rostock, Germany
http://www.demographic-research.org 455Missov & Lenart: Linking period and cohort life expectancy
1. Relationship
Suppose the mortality schedule of a population is described by
(1) ¹(x;y) = e¡½y a0 ebx
on a period basis, which corresponds to
(2) ¹C(x;y) = e¡½y a0 e(b¡½)x
on a cohort basis, i.e., by a Gompertz hazard that decreases with time y at a relative rate
½ at all ages x simultaneously. Denote by eP
0 (y) and eC
0 (y) the period and cohort life
expectancy in y. Vaupel (1986) proved that (1) results in a linearly increasing eP
0 (y) with
an approximate slope ½=b. We will prove that (2) implies linear increase in eC
0 (y), too.
Moreover, for y ! 1
(3)
eC
0 (y)
eP
0 (y)
¡!
1
1 ¡ _ eP
0 (y)
and
(4) _ eC
0 (y) ¡!
_ eP
0 (y)
1 ¡ _ eP
0 (y)
;
where _ eP
0 (y) = deP
0 (y)=dy and _ eC
0 (y) = deC
0 (y)=dy.
2. Proof of the relationship
The number of years lived in the next n by those who have survived to age x in year y is
(5) L(x;n;y) =
x+n Z
x
`(v;y)dv;
where
(6) `(x;y) = exp
8
<
:
¡
x Z
0
¹(v;y)dv
9
=
;
is the survival function of an individual aged x at time y (Keyﬁtz and Caswell 2005, 30).
456 http://www.demographic-research.orgDemographic Research: Volume 24, Article 19
Substituting (1) and (6) into (5), assuming `(0;y) = 1, setting x = 0 and letting
n ! +1, we get the following expression for period life expectancy at birth:
(7) eP
0 (y) =
1
b
exp
na0
b
e¡½y
o
E1
³a0
b
e¡½y
´
;
where
(8) E1(z) =
1 Z
z
e¡t
t
dt
is the exponential integral. A relationship, based on (2) and similar to (7), holds for eC
0 (y),
as well:
(9) eC
0 (y) =
1
b ¡ ½
exp
½
a0
b ¡ ½
e¡½y
¾
E1
µ
a0
b ¡ ½
e¡½y
¶
:
The limit of eC
0 (y)=eP
0 (y) for y ! 1 can be represented as
(10) lim
y!1
eC
0 (y)
eP
0 (y)
=
b
b ¡ ½
¢ lim
y!1
exp
½
¡a0½
b(b ¡ ½)
e¡½y
¾
¢ lim
y!1
E1
¡a0
b e¡½y¢
E1
³
a0
b¡½ e¡½y
´:
The ﬁrst limit on the right-hand side of (10) is equal to 1. The arguments of both ex-
ponential integrals in the second limit tend to 0 for y ! 1. This implies that both the
numerator and the denominator will tend to +1 as
(11) E1(z) = ¡° ¡ lnz ¡
1 X
n=1
zn
n ¢ n!
(see, for example, Abramowitz and Stegun 1964, 229), where ° ¼ 0:577 is the Euler-
Mascheroni constant. Applying L’Hôpital’s rule yields
(12) lim
y!1
E1
¡a0
b e¡½y¢
E1
³
a0
b¡½ e¡½y
´ = lim
y!1
exp
½
a0½
b(b ¡ ½)
e¡½y
¾
= 1
As a result, for y ! 1
(13)
eC
0 (y)
eP
0 (y)
¡!
b
b ¡ ½
=
1
1 ¡
½
b
=
1
1 ¡ _ eP
0 (y)
:
http://www.demographic-research.org 457Missov & Lenart: Linking period and cohort life expectancy
Finally, differentiating (9) with respect to y, we get
_ eC
0 (y) =
½
b ¡ ½
µ
exp
½
¡
2a0
b ¡ ½
e¡½y
¾
+ e¡½y a0 ec
0(y)
¶
; (14)
which implies for y ! 1
_ eC
0 (y) ¡!
½
b ¡ ½
=
½=b
1 ¡ ½=b
=
_ eP
0 (y)
1 ¡ _ eP
0 (y)
(15)
Q.E.D.
3. History and related results
Linearly increasing record period life expectancy from 1840 to present was detected by
Oeppen and Vaupel (2002). One of the possible models that yields constant increase per
time unit is (1), introduced by Vaupel (1986). Goldstein and Wachter (2006) discussed
the latter as a special case of Linear Shift Models (½=b = r), under which the hazard
rate at every age x in year y is given by the hazard rate at a younger age x ¡ ry, r > 0,
y years earlier: ¹(x;y) = ¹(x ¡ ry;0). Wilmoth (2005) discussed the relationship
between period and cohort mortality by comparing different mean life span measures
in general model settings. Without specifying any functional interrelationship, Schoen
and Canudas-Romo (2005) compare period and cohort life expectancy in the Gompertz
proportional hazards framework with different shapes expf¡f(y)g of mortality progress
over time y, including f(y) = expf¡ryg, speciﬁed in (1). Canudas-Romo and Schoen
(2005) study a Siler model with two different (constant) rates of mortality decline: one for
infant and one for non-infant mortality. This model converges with time to (1) as levels
of and improvements in infant mortality become negligibly small. Canudas-Romo and
Schoen (2005) quantify the gaps and lags between period and cohort life expectancy over
time in terms of the model parameters.
In this paper, within the framework of Gompertz proportional hazards (1), we express
in a simple functional form the relationship between period and cohort life expectancy
as well as between their (constant) rates of change. The two relationships are approx-
imate and do not depend on model parameters as it is the case in Canudas-Romo and
Schoen (2005). Eq. (13) suggests that we can link period to cohort life expectancy by the
following approximation
(16) eC
0 (y) ¼
eP
0 (y)
1 ¡ _ eP
0 (y)
;
458 http://www.demographic-research.orgDemographic Research: Volume 24, Article 19
whereas (15) implies that
(17) _ eC
0 (y) ¼
_ eP
0 (y)
1 ¡ _ eP
0 (y)
:
Note that for y ! 1
exp
½
¡a0½
b(b ¡ ½)
e¡½y
¾
= 1 ¡
a0½
b(b ¡ ½)
e¡½y + o(e¡½y)
and, following (11),
E1
¡a0
b e¡½y¢
E1
³
a0
b¡½ e¡½y
´ = 1 +
E1
¡a0
b e¡½y¢
¡ E1
³
a0
b¡½ e¡½y
´
E1
³
a0
b¡½ e¡½y
´
= 1 +
lnb ¡ ln(b ¡ ½) +
a0½
b(b¡½) e¡½y + o(e¡½y)
¡° ¡ lna + lnb + ½y ¡ a0
b¡½ e¡½y + o(e¡½y)
;
which implies that for a0 = 0:00001, b = 0:14, and ½ = 0:027, formula (16) overesti-
mates eC
0 (y) by 2:12%-2:39% for y = 1;2;:::;100. Relationship (16) is sensitive with
respect to mortality progress. For instance, if ½ = 0:01, keeping all other parameters
as before, we get an overestimate by 0:7%-0:8%. Lower a0 and higher b increase the
accuracy in (16).
Relationship (17), though underestimating cohort life-expectancy increase, is much
more accurate that (16). Note that for y ! 1
_ eC
0 =
½
b ¡ ½
µ
1 + a0 e¡½y
µ
¡
2
b ¡ ½
¡ ° ¡ lna + lnb + ½y
¶
+ o(e¡½y)
¶
For a0 = 0:00001, b = 0:14, and ½ = 0:027, the error in formula (17) is only between
4:05 £ 10¡6 and 8:47 £ 10¡5 for y = 1;2;:::;100.
Formulae (16) and (17) provide a simple approximate relationship between eP
0 (y)
and eC
0 (y), as well as their rates of change. The two major assumptions under which this
result has been derived are: 1) the same yearly mortality improvement at all ages, 2) the
Gompertz assumption about the baseline hazard.
The Gompertz assumption about the adult force of mortality can be easily veriﬁed by
looking at the logarithm of the unsmoothed mortality surface for Japan, the current life-
expectancy leader (whose almost linear increase in life expectancy in the last decades is
well pronounced). The mortality on a log-scale increases linearly with age in accordance
with the exponentially increasing force of mortality. (Fig. 1)
http://www.demographic-research.org 459Missov & Lenart: Linking period and cohort life expectancy
Figure 1: Logarithm of the observed force of mortality in Japan, ages 30-95,
years 1960-2008
Fig. 1 shows clearly the signiﬁcance of age-invariant period effects, especially at older
ages where the force of mortality is increasingly Gompertz-like. Only one signiﬁcant
cohort effect can be seen for the cohorts born during the Second World War.
The constant ½(x;y) = ½(y) at all ages in a single year y has been veriﬁed in numer-
ous studies including Kannisto (1994), Tuljapurkar, Li, and Boe (2000) and Bongaarts
and Feeney (2003), for the industrial countries in the second half of the 20th century. In
this paper we assume that ½ is constant not only over age x, but also with time y. Deﬁning
460 http://www.demographic-research.orgDemographic Research: Volume 24, Article 19
the observed rate of mortality change as
½(x;y + 1) = ¡log
¹(x;y + 1)
¹(x;y)
;
and plotting it in Fig. 2, using again the Japanese data, we do not observe any particular
age or period dependent pattern. The rate of mortality change appears rather to be a cloud
centered around the mean value 0:027. Life-expectancy increase in Japan has obviously
not been exactly the same each year: there have been years of decrease followed by years
of fast increase (negative and very high ½ values, respectively). Nevertheless a constant ½
is a good approximation to the observed mortality processes. The orange points on Fig. 2
show the interval of one standard deviation distance from the mean. It encompasses 72%
of the data. The purple points (two standard deviations distance from the mean) and the
orange points constitute together 95% of the observed mortality changes.
Figure 2: Observed rate of mortality change in Japan, ages 30-95, years
1960-2008
Notes: Orange band designates the ½ values that fall into the mean § standard deviation. The ½ values in the
purple band are in the mean § two standard deviations interval.
Link: Click here to watch the animated ﬁgure.
http://www.demographic-research.org 461Missov & Lenart: Linking period and cohort life expectancy
Fig. 3 shows the density estimate for the observed mortality change in Japan. As the
rates are tightly packed around the mean, they corroborate the notion of age and time-
invariant mortality change.
Figure 3: Density of observed rate of mortality change in Japan, ages 30-95,
years 1960-2008 pooled together
Using Maximum Likelihood Estimation to obtain the optimal a;b and ½ parameters
can help visualizing the error made by ﬁtting a Gompertz model to Japanese mortality
surface for ages 30-95 and years 1960-2008. Fig. 4 shows the Q-Q plot for year 2008. If
we compare the observed and expected distribution of ages at death, it can be seen that for
ages under 50 the Makeham term would still have a signiﬁcant effect and the Gompertz
model underestimates the number of deaths. However, for older adult ages the Gompertz
model predicts the age-speciﬁc number of deaths accurately even if we use a unique age
and time constant mortality decline parameter for the whole mortality surface.
462 http://www.demographic-research.orgDemographic Research: Volume 24, Article 19
Figure 4: Q-Q plot of observed and expected ages at death for Japanese ages
30-95 in 2008
We derived (16) by using (7). The latter can be used to calculate life expectancy gains
GLE(x;y) at every age x in each year y:
GLE(x;y) =
de0(y)
dy
=
a0 ½
b
e¡½y(ebx ¡ 1)exp
n
¡
a0
b
e¡½y(ebx ¡ 1)
o
: (18)
Moreover, the age at which the maximal gains in eP
0 (y) occur can be easily calculated as
x¤(y) =
dGLE(x;y)
dx
=
ln(b + e¡½ya0) + ½y ¡ lna0
b
: (19)
http://www.demographic-research.org 463Missov & Lenart: Linking period and cohort life expectancy
Note that the age x¤(y) at maximal gains in life expectancy shifts almost uniformly
(y ! 1) over time at a pace of
dx¤(y)
dy
=
½
b
: (20)
Thus the shift of x¤(y) by periods (see Fig. 5) asymptotically (y ! 1) equals the rate of
change in life expectancy at birth.
4. Applications
A proportionally changing force of mortality leads to a linearly shifting distribution of
person-years gained by periods. In case of cohorts, this linear shift expands over time.
Based on this relationship, cohort life expectancy at birth can be represented in terms of
period life expectancy at birth and its derivative. A similar representation applies to the
rate of change in cohort life expectancy at birth.
Relationships (16) and (17) provide an approximate estimate of how period gains in
life expectancy are transformed on a cohort basis. Consider individuals that were born in
2010 in a country with a period life expectancy at birth equal to 80 years. In this country,
assume that (i) there is a constant 2:5-year increase in life expectancy per decade (as in
Oeppen and Vaupel 2002) and (ii) the force of mortality follows a Gompertz curve with
constant yearly deceleration rate ½. In a period perspective, this would mean that during
their childhood and young adulthood these individuals gain additional six hours of life
every day (Vaupel 2009, 352). Throughout the life course this gain shrinks because there
are less remaining years, in which mortality can improve. In order to claim that individu-
als gain six hours a day in their infancy, one has to assume that they will experience the
same probability of death, when they turn ten in 2020, as the one for a 10-year-old child
in 2010. Similarly, when they turn 80 in 2090, they would have the same probability to
die as a 80-year-old person in 2010. This assumption should hold on every day (or birth-
day, depending on the measurement units) of their life. However, within our assumptions,
mortality improves in such a way that individuals aged 80 in 2090 would have the same
probability to die as a 60-year-old person in 2010.
In a cohort view, when we account for the ever changing mortality regime, we can
see that these life expectancy gains are not negligible. Individuals in this country will
have a (cohort) life expectancy of 103:92 years. If we do not correct for overestimation,
our result will be 80
1¡0:25 = 106:67 years. Individuals born ten years later will have life
expectancy, which will be higher by 2:5
1¡0:25 = 3:33 years. Thus 2:5-year life expectancy
gains per decade on a period basis correspond to approximately 3:3-year life expectancy
gains per decade on a cohort basis.
464 http://www.demographic-research.orgDemographic Research: Volume 24, Article 19
Fig. 6 shows an example of changes in person-years lived in an age interval between
two consecutive periods. Using the data from the Human Mortality Database, we chose
Japanese females for demonstration. As life expectancy at birth augmentation over time
is not constant even in Japan, we drew person-years-lived changes from one period to
another to the same scale and removed six years when life expectancy at birth was lower
than in the previous year. The resulting age distribution of person-years gained and the
movement of the curve resembles the one predicted by the model (see Fig. 5).
Figure 5: Change in the force of mortality and distribution by ages of years
gained by constant period life expectancy at birth increase
http://www.demographic-research.org 465Missov & Lenart: Linking period and cohort life expectancy
Figure 6: Distribution by ages of person-years gained
Life expectancy gains result from the continuous linear shift to older ages in the distri-
bution of person-years gained. Gompertz proportional hazards models stipulate reduction
in the late middle-age and old-age mortality. Mortality of oldest-old, though, stays un-
modiﬁed. However, as time advances, older and older ages beneﬁt more from mortality
reduction.
5. Acknowledgements
The authors express their gratitude to James W. Vaupel for providing deeper insight into
the subject, as well as to the three anonymous reviewers for pointing out missing refer-
ences and making thoughtful comments for improving the quality of the paper.
466 http://www.demographic-research.orgDemographic Research: Volume 24, Article 19
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Linking period and cohort life-expectancy linear increases in Gompertz proportional hazards models

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Linking period and cohort life-expectancy linear increases in Gompertz proportional hazards models

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