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Demographic Research a free, expedited, online journal

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in the population sciences published by the

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DEMOGRAPHIC RESEARCH

VOLUME 22, ARTICLE 5, PAGES 115-128

PUBLISHED 22 JANUARY 2010

http://www.demographic-research.org/Volumes/Vol22/5/

DOI: 10.4054/DemRes.2010.22.5

Formal Relationships 7

Life expectancy is the death-weighted average

of the reciprocal of the survival-specific force of

mortality

Joel E. Cohen

This article is part of the Special Collection “Formal Relationships”.

Guest Editors are Joshua R. Goldstein and James W. Vaupel.

c ? 2010 Joel E. Cohen.

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/

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Table of Contents

1

1.1

1.2

Background and relationships

Background

Relationships

116

116

118

2 Proofs119

3 History and related results120

4

4.1

4.2

4.3

4.4

Applications

Lower bound inequality

Lower bound in the exponential distribution

Discrete actuarial approximations

Example based on life tables of the United States in 2004

121

121

121

122

122

5 Acknowledgements126

References127

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Demographic Research: Volume 22, Article 5

Formal Relationships 7

Life expectancy is the death-weighted average of the reciprocal of the

survival-specific force of mortality

Joel E. Cohen1

Abstract

The hazard of mortality is usually presented as a function of age, but can be defined as a

function of the fraction of survivors. This definition enables us to derive new relationships

for life expectancy. Specifically, in a life-table population with a positive age-specific

forceofmortalityatallages, theexpectationoflifeatagexistheaverageofthereciprocal

of the survival-specific force of mortality at ages after x, weighted by life-table deaths at

each age after x, as shown in (6). Equivalently, the expectation of life when the surviving

fraction in the life table is s is the average of the reciprocal of the survival-specific force of

mortality over surviving proportions less than s, weighted by life-table deaths at surviving

proportions less than s, as shown in (8). Application of these concepts to the 2004 life

tables of the United States population and eight subpopulations shows that usually the

younger the age at which survival falls to half (the median life length), the longer the life

expectancy at that age, contrary to what would be expected from a negative exponential

life table.

1Laboratory of Populations, Rockefeller & Columbia Universities, 1230 York Avenue, Box 20, New York, NY

10065-6399, USA. E-mail: cohen@rockefeller.edu

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1.Background and relationships

1.1Background

The life table ?(x), constant in time, with continuous age x, is the proportion of a cohort

(whether a birth cohort or a synthetic period cohort) that survives to age x or longer. In

probabilistic terms, ?(x) is one minus the cumulative distribution function of length of

life x. The maximum possible age ω may be finite or infinite. If ω = ∞, then some

individuals may live longer than any finite bound. By definition, ?(0) = 1 and ?(ω) = 0.

Assume ?(x) is a continuous, differentiable function of x, 0 ≤ x ≤ ω, and assume life

expectancy at age 0 is finite. The age-specific force of mortality at age x is, by definition,

(1)

µ(x) = −

1

?(x)

d?(x)

dx

.

Assume µ(x) > 0 for all 0 ≤ x ≤ ω. The life table ?(x) is strictly decreasing from

?(0) = 1 to ?(ω) = 0 so there is a one-to-one correspondence between age x in [0,ω] and

the proportion s in [0,1] of the cohort that survives to age x or longer. One direction of

this correspondence is given by the life table function s = ?(x) (illustrated schematically

in Figure 1 and for the United States population in 2004 in Figure 3A).

Figure 1: When the force of mortality is positive at every age x, the pro-

portion surviving s, given by the life table according to s = ?(x),

strictly decreases as age x increases, so there is a one-to-one

correspondence between age x and the proportion surviving s.

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There appears to be no standard demographic term for the inverse function that maps

the proportion surviving s, 0 ≤ s ≤ 1, to the corresponding age x, so I propose to

call it the age function a (illustrated schematically in Figure 2 and for the United States

population in 2004 in Figure 3D). In words, the age a(s) at which the fraction s of the

birth cohort survives is the age x at which the life table function ?(x) is s. By definition,

under the assumption µ(x) > 0 for all 0 ≤ x ≤ ω, a(s) = x if and only if ?(x) = s.

Equivalently, by definition, for every s in 0 ≤ s ≤ 1 and every x in 0 ≤ x ≤ ω,

a(?(x)) = x and ?(a(s)) = s. We define a(1/2) as the median life length, that is, the age

by which half the cohort has died.

Figure 2:The function x = a(s) that expresses the age x at which a fraction

s of a birth cohort survives is the inverse of the life table function

s = ?(x) when the force of mortality is positive at every age. Apart

from a reflection across the diagonal line x = s, the curve in this

figure has the same relative shape as the curve in Figure 1 but the

rescaling of both axes makes the two curves look different.

For every s in 0 ≤ s ≤ 1, we define the survival-specific force of mortality λ(s) in

terms of the age-specific force of mortality µ(x) in (1) in three equivalent ways:

(2)

λ(s) = µ(x) if s = ?(x);

or

λ(s) = µ(a(s));

or

λ(?(x)) = µ(x).

In words, the survival-specific force of mortality λ(s) at surviving proportion s equals the

age-specificforceofmortalityµ(x)attheagexwherethelifetable?(x) = s. Thedomain

of the age-specific force of mortality µ is 0 ≤ x ≤ ω while the domain of the survival-

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specific force of mortality λ is 0 ≤ s ≤ 1. We give below an explicit formula (9) for the

survival-specific force of mortality at surviving proportion s. This formula is analogous

to (1) for the age-specific force of mortality.

The complete expectation of life at age x, e(x), is the average number of years

remaining to be lived by those who have attained age x:

(3)

e(x) =

1

?(x)

y=ω−x

?

y=x

(y − x)?(y)µ(y)dy.

Inserting the definition (1) in place of µ(y) in (3) and integrating by parts gives

(4)

e(x) =

1

?(x)

a=ω

?

a=x

?(a)da,

a standard formula for life expectancy at age x (Keyfitz 1968:6).

1.2 Relationships

It is well known that the age-specific force of mortality µ(x) equals a constant K > 0

at every age x if and only if the life table is negative exponential with parameter K, i.e.,

?(x) = exp(−Kx). In this case, the expectation of life at age x is the reciprocal of the

age-specific force of mortality:

(5)

e(x) =

1

K.

From the definition (2) of the survival-specific force of mortality, it is evident that

λ(s) = K > 0 at every surviving proportion s if and only if the life table is negative

exponential with parameter K. Thus K in (5) may be viewed as a constant force of

mortality, both age-specific and survival-specific.

Generalizations (6) and (8) extend (5) when the survival-specific force of mortality is

not constant. These generalizations seem to be new.

A first generalization of (5) states that

(6)

e(x) =

1

?(x)

s=?(x)

?

s=0

ds

λ(s).

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In words, the expectation of life at age x is the average reciprocal of the survival-specific

force of mortality weighted by the life-table deaths ds after age x. When λ(s) = K,

(6) simplifies to (5) because

(7)

1

?(x)

s=?(x)

?

s=0

ds = 1.

One can entirely eliminate age x (years of life lived in the past) from life expectancy

(average years of life to be lived in the future) by defining a survival-specific life

expectancy E(s) (analogous to the survival-specific force of mortality defined above) as

the life expectancy when the surviving proportion of the birth cohort is s. Thus by defini-

tion, E(s) = e(x) if s = ?(x) and equivalently E(s) = e(a(s)) and E(?(x)) = e(x).

A second generalization of (5) is to rewrite (6) as

(8)

E(s) =1

s

s?=s

?

s?=0

ds?

λ(s?).

Here s?is the running variable for s. In words, the life expectancy when the surviving

fraction is s is the death-weighted average of the reciprocal of the survival-specific force

of mortality over each survival proportion smaller than s. The substantive difference

between (6) and (8) is that age x appears as an argument on both sides of (6) and nowhere

in (8). An illustration of (8) using United States data will be discussed in section 4. Appli-

cations.

We also demonstrate survival-specific forms of the force of mortality:

(9)

λ(s) = −1

s

ds

da= −1

s

?da

ds

?−1

.

2. Proofs

Since a(s) and ?(x) are inverse functions, elementary calculus shows that

(10)

d?

dx=

?da

ds

?−1

=ds

da

and that it is permissible under an integral to write

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(11)

ds =

?da

ds

?−1

da.

Then, for s = ?(x) and running variables s?= l(x?) (and with the equalities numbered

for subsequent explanation), we have

E(s)

1= e(x)

2=

1

?(x)

x?=ω

?

x?=x

s?=s

?

s?=0

s?=s

?

s?=0

−1

µ(x?)

d?(x?)

dx?

dx?

3=1

s

+1

λ(s?)

?da

ds?

?−1

da

4=1

s

ds?

λ(s?).

(12)

Equality 1 in (12) holds by definition of E. Equality 2 takes (4) and replaces the integrand

?(a) in (4) with the result of exchanging ?(x) and µ(x) in (1). Equality 3 uses the defi-

nitions s = ?(x) and λ(s) = µ(x) from (2), changes the minus one to plus one because

of the reversal in the direction of integration, and uses (10) to replace one derivative with

another. Finally, equality 4 uses (11) to “cancel” the differentials da. This proves (8), and

using the definition s = ?(x) gives (6).

Finally, (9) follows immediately from the definitions (1) and (2) and the fact (10).

3. History and related results

I believe I was the first to state a special case of (6) in my first problem set dated 4 October

1971 for an undergraduate course on mathematical population models which I introduced

at Harvard University (Biology 150). Assuming ?(0) = 1, I asked the students to prove

that

(13)

e(0) =

1

?

l=0

dl

λ(l).

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The text for the course was Nathan Keyfitz’s then recent Introduction to the Mathematics

of Population (1968). When Keyfitz began teaching at Harvard in the fall of 1972,

I showed him (13) to find out if he had seen it before. He had not. I believe Keyfitz

subsequently published (13), but not (6) or (8), as an exercise in one of his books. I can-

not find the citation. To my knowledge, (6) and (8) and (9) have appeared nowhere before

and no proof of (6), (8) or (13) has been published previously. The inequality (14) below

also seems to be new.

4.Applications

4.1 Lower bound inequality

The expression (6) for e(x) yields a new lower bound on life expectancy at age x. The

reciprocal function that maps each positive real number x into 1/x is strictly convex.

Therefore, Jensen’s inequality for the average of a convex function applies to (6) and

yields

(14)

e(x) =

1

?(x)

l=?(x)

?

l=0

dl

λ(l)≥

1

1

?(x)

l=?(x)

?

l=0

λ(l)dl

=

?(x)

l=?(x)

?

l=0

λ(l)dl

.

This inequality is strict unless all deaths occur at a single age, that is, unless the life table

is rectangular (in which case the age-specific force of mortality is not positive at ages less

than ω), or unless the age-specific force of mortality is constant at all ages, in which case

the survival-specific force of mortality is also constant. In words, the expectation of life at

age x is at least as large as (and, apart from a rectangular life table or a constant force of

mortality, is greater than) the reciprocal of the average survival-specific force of mortality

weighted by life-table deaths dl at surviving proportions less than ?(x), that is, at ages

above x.

4.2 Lower bound in the exponential distribution

The special case when the (age-specific or survival-specific) force of mortality is a con-

stant K > 0 for 0 ≤ x ≤ ∞ and 0 ≤ s ≤ 1 verifies and illustrates (14). The left side of

(14) is then e(x) = 1/K. The numerator of the right side of (14) is ?(x) = exp(−Kx),

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and the denominator of the right side is

(15)

K

l=?(x)

?

l=0

dl = K ?(x) = K exp(−Kx).

Thus the right side of (14) equals ?(x)/(K ?(x)) = 1/K and equality holds in (14), as

expected in the case of a constant force of mortality.

4.3 Discrete actuarial approximations

If we are given the life-table proportions ?xsurviving at each exact age x, the life-table

probability qxof dying by age x + 1 given survival to exact age x, and the expecta-

tion of life exat exact age x, then the estimation of a(s), λ(s), and E(s) for given

surviving proportions s requires only linear (or other) interpolation. For example, the

Matlab command interp1(lx,x,s,‘spline’) uses piecewise cubic spline inter-

polation to produce a(s) from three arguments: the life table expressed as a vector lx, a

vector x of ages, and a vector s of proportions surviving. Similarly, Matlab command

interp1(lx,qx,s,‘spline’) estimates the survival-specific force of mortality

λ(s) and Matlab command interp1(lx,ex,s,‘spline’) estimates the survival-

specific life expectancy E(s). Both spline and linear interpolation were tried and the

results were very similar. Spline interpolation was preferred to linear interpolation be-

cause spline interpolation was smoother and took advantage of information outside the

local interval of age.

The ability to compute a(s), λ(s), and E(s) using existing actuarial methods plus

interpolation is a double advantage. It requires little retooling of methods or software,

and it sheds new light on, and raises fresh questions about, familiar data, as the following

example is intended to show.

4.4Example based on life tables of the United States in 2004

Arias (2007) tabulated ?x, qx, and exfor exact ages 0,1,2,...,99, and a terminal catch-

all group 100 years or older, for the 2004 United States population and eight subpopula-

tions. Figure 3 plots (A) ?(x) ≈ ?x, (B) µ(x) ≈ qx, and (C) e(x) ≈ ex, for exact ages

x = 0,1,...,99, ignoring the final age group 100 years or older. Figure 3 also plots

the corresponding (D) age function a(s), (E) survival-specific force of mortality λ(s),

and (F) survival-specific life expectancy E(s), as functions of the proportion surviving s,

for s = 0.04,0.05,...,0.99,1. The values s = 0.01,0.02,0.03 are omitted because the

fraction who survived to at least age 99 was ?99 = 0.03423 and estimates of functions

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for arguments s smaller than ?99would have required extrapolation rather than interpo-

lation. Just as Arias’s estimates for the age group 100 years or older required additional

assumptions, estimates for s < ?99would have required additional assumptions.

Figure 3:United States population in 2004 showing

(A) the life table ?(x),

(B) the age-specific force of mortality µ(x), and

(C) the expectation of remaining life e(x),

as functions of age x, based on Arias (2007:Table 1) and

(D) the age a(s) at which the proportion s survives,

(E) the survival-specific force of mortality λ(s), and

(F) the expectation of remaining life E(s)

as functions of the surviving proportion s.

The functions based on age differ from the corresponding functions based on the pro-

portion surviving. The shoulder on the right of the age function (Figure 3D) is much more

pronounced than that of the life table (Figure 3A), highlighting the rapid drop-off in age

associated with the last increases in the proportion surviving from about 0.95 to 1. While

the age-specific force of mortality (on this linear scale) (Figure 3B) rises notably only

after the first six decades, and then rises gradually, the survival-specific force of mortality

(Figure 3E) declines sharply for small values of s (corresponding to extreme old ages)

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and substantially across the entire range of s. Finally, while the age-specific expecta-

tion of life (Figure 3C) falls gradually, almost linearly, across the entire range of age, the

survival-specific expectation of life Figure 3F) rises slowly until s approaches 1 and then

rises quite sharply. Looked at another way, by moving s in a decreasing direction from

right to left in Figure 3F, the greatest losses in expectation of remaining life occur when

the first small fraction dies (at high s). Thereafter, as s decreases further, the decline in

expectation of remaining life is much more gradual.

This perspective provides new ways to compare populations. To illustrate, Table 1

compares E(1/2), the survival-specific life expectancy when half the cohort survives, as

defined in (8), for the total population and eight subpopulations of the United States in

2004 (Figure 4), based on data of Arias (2007:Tables 1-9). For example, for the total

population of the U.S. in 2004 (Arias 2007:Table 1), by exact age 81 the proportion sur-

viving was 0.50987 with remaining life expectancy of 8.6 years and by exact age 82 the

proportion surviving was 0.47940 with remaining life expectancy of 8.2 years. By linear

interpolation I estimated a life expectancy of 8.4704 years when the proportion surviving

was 0.5, and by spline interpolation I estimated a life expectancy of 8.4708, so I tabulated

E(1/2) = 8.5 years. For simplicity in this illustration, the median life length a(1/2),

that is, the age at which the proportion surviving s equaled 1/2, was approximated by the

whole number of years of life completed, that is, by the integer part of a(1/2). Males’

E(1/2) exceeded females’ E(1/2) by 0.8 year but half the male cohort had died by age

78, five years younger than half the female cohort had died, by age 83. Similarly, the

black population’s E(1/2) exceeded the white population’s E(1/2) by 2.3 years but half

the black cohort had died by age 76, five years younger than half the white cohort died,

at age 81. Of the subpopulations considered in Table 1, black males had the longest

E(1/2), 10.9 years, and the shortest median life length, reaching half survival youngest,

at 72 years. White females had the shortest E(1/2), 7.7 years, and the longest median

life length, 83 years, tied for oldest with all U.S. females. The younger half of a cohort

died, that is, the shorter the median life length, the longer its life expectancy at that age.

This relationship is the opposite of the relationship expected from the simplest model,

when the force of mortality is a constant K and the life table is negative exponential

?(x) = exp(−Kx). Then a(1/2) = ln(2)/K while the expectation of life (at any age

or any surviving proportion) is 1/K. Not surprisingly, the higher the force of mortality,

the sooner half the cohort dies and the shorter the life expectancy. Both a(1/2) and

E(1/2) = e(a(1/2)) are inversely proportional to the force of mortality K and are directly

proportionaltooneanotherinthefamilyofnegativeexponentiallifetableswithparameter

K, contrary to the observations of United States subpopulations.

Understanding why U.S. subpopulations with shorter median life length had longer

remaining expectation of life at that age is a topic for further theoretical and empirical

analysis. Theoretically consider a family of life tables of specified form indexed by a pa-

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Demographic Research: Volume 22, Article 5

rameter or several parameters (for example, the family of negative exponential life tables

is indexed by the parameter K). One problem is to find necessary and sufficient condi-

tions on the form of the life table and the values of the parameter(s) such that E(s) and

a(s) are positively (or, negatively) associated as a parameter increases within some range.

An empirical problem is to identify the demographic, economic, and cultural conditions

under which E(s) and a(s) are observed to be positively (or, negatively) associated, and

to interpret these conditions in terms of the theoretical conditions.

Table 1: Life expectancy E(1/2) and the year of age x to x + 1 in which half

the life-table cohort survived, in the United States’ 2004 total popu-

lation and selected subsets

Life expectancy E(1/2) Age when half the

when half the life-table

cohort survived (years)

8.5

8.7

7.9

8.3

8.5

7.7

10.6

10.9

9.7

life-table cohort

survived (years)Population

Total population

Males

Females

White population

White males

White females

Black population

Black males

Black females

81-82

78-79

83-84

81-82

79-80

83-84

76-77

72-73

79-80

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Figure 4:Integer part of the age a at which half a cohort survived (vertical

axis) as a function of the complete expectation of life at age a, for

the United States total population and eight subpopulations, 2004,

estimated by interpolation from data of Arias (2007)

US = total population, w = white, b = black, m = male, f = female

5.Acknowledgements

I acknowledge with thanks very helpful comments from two anonymous referees, the sup-

port of U.S. National Science Foundation grant DMS-0443803, the assistance of Priscilla

K. Rogerson, and the hospitality of William T. Golden and family during this work. This

work was completed during a visiting faculty appointment at the University of Montpel-

lier II, France, thanks to the hospitality of Michael Hochberg.

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Demographic Research: Volume 22, Article 5

References

Arias, E. (2007). United States Life Tables, 2004. National Vital Statistics Reports 56(9).

Hyattsville, MD: National Center for Health Statistics. http://www.cdc.gov/

nchs/data/nvsr/nvsr56/nvsr56_09.pdf.

Keyfitz, N. (1968). Introduction to the Mathematics of Population.

Addison-Wesley.

Reading, MA:

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