Copyright 1996 by The Gerontological Society of America
Journal of Gerontology: BIOLOGICAL SCIENCES
1996. Vol. 5IA, No. 5. B362-B375
Longevity in the United States:
Age and Sex-Specific Evidence on Life Span
Limits From Mortality Patterns 1960-1990
Kenneth G. Manton and Eric Stallard
Center for Demographic Studies, Duke University.
Determining the biological limits to human longevity is more difficult than for most other species because humans
are long-lived. Consequently, mortality data, such as from the U.S. vital statistics system, which have been available
for a long time (relative to most epidemiological studies) and have large numbers of cases, including deaths reported
to advanced ages, are important in studying human longevity — though care must be exercised in dealing with error
in age reporting. Furthermore, it is unlikely that free-living humans can realize as much of their biological
endowmentfor longevity as animals living in a highly controlled experimental environment. We examined changes,
1960 to 1990, in U.S. White male and female extinct cohort life tables and age at death distributions to (a) examine
evidence for the effects of a biological life span limit in current U.S. mortality patterns and (b) produce lower bound
estimates of that limit.
HPESTING whether U.S. mortality currently reflects bioA logical limits is difficult because of the length of human
life spans and because the environment of human populations is under less control than animals in experimental
studies, i.e., realized human life expectancy will be a
smaller proportion of their biological potential because of
environmental heterogeneity. Estimating human life span
limits is also complicated by uncertainty about ages reported
at death.
Consequently, biological life span limits are often studied
using animal models. Carey et al. (1992) found for 1.1
million fruit flies that mortality reached a high but constant
value after 90% had died — possibly due to the attrition of
frail individuals. Curtsinger et al. (1992) found mortality
reached a high constant value in Drosophila in genetically
homogenous groups. Brooks et al. (1994) found the nematode C. elegans manifested a high constant mortality at late
ages in wild populations. Kenyon et al. (1993) found modifying two genes doubled the life span in C. elegans, while
leaving the variance of age at death unchanged.
Studies of human mortality produce similar results. In one
(IPSEN, 1991; Vallin, 1993), male mortality increased only
2.2% per year from ages 100 to 109; 3.0% for females.
Another found male mortality increases of 8.8% per year of
age from 75 to 84 dropped to 3.2% from 95 to 104; female
mortality increases of 10.5% per year of age from 75 to 84
dropped to 2.5% from 100 to 109 (Lew and Garfinkel, 1990).
Analyses of Swedish and U.S. Medicare cohorts showed
mortality increases slowing at late ages (Manton et al., 1981,
1986) as did charter Social Security beneficiaries, U.S. vital
statistics (Bayo and Faber, 1985), and insurance data used to
form group annuity tables (Society of Actuaries, 1994).
Mortality rates rising exponentially to late ages suggest
the operation of life span limits. However, neither animal
nor human studies show such exponential, Gompertzian
increases in mortality. A possible reason is mortality selecB362
tion. Data suggest selection strongly affects the distribution
of health and function among survivors to age 85 + , i.e.,
ages where the Gompertz does not describe mortality.
Marenberg et al. (1994) found, in male and female twins
born 1886 to 1925, relative risks from coronary heart disease
declined from 13-15 to 1 in middle age to 1.0 to 1 above 85.
Decreased prevalences of genetically determined lung cancer (Sellers et al., 1990), thyroid auto-antibodies (Mariotti et
al., 1992), apolipoprotein E-4 allele frequency (Louhija et
al., 1994), and the C4B*Q0 gene (Kramer et al., 1991,
1994) were also found at late ages.
To directly analyze selection, longitudinal data with covariates are required. Analyses of the 34-year Framingham
Heart Study follow-up, and the 9.5-year follow-up of the
National Long Term Care Survey, showed mortality approaching a constant level above age 95 with deterioration of
population risk factor values and function slowing about the
same age. Thus, even with stochastically evolving state
variables, selection slowed age increases in survivors' average mortality risk and health deterioration. Selection on
clinical attributes affecting homeostasis to age 80+ were
found by Bild et al. (1993), Campbell et al. (1993), and Perls
etal. (1993).
To test for life span limits without using a specific model of
the age dependence of mortality requires large populations. In
U.S. data there are concerns about the accuracy of ages
reported at death. Recent studies, however, suggest data
quality improved because of Social Security (starting in 1937)
and Medicare (starting in 1966) requirements for age documentation to qualify for benefits (Kestenbaum, 1992). Age
reporting also likely improved as the education of elderly
U.S. cohorts rose, e.g., the proportion of persons aged 85 to
89 with less than 8 years of schooling is projected to decline
from 60%+ in 1980 to 10-20% in 2015 (Preston, 1992).
A final difficulty is mathematically defining a test of
whether human survival curves are becoming "rectangu-
B363
LONGEVITY IN THE U.S.
lar" (e.g., Fries, 1980; Rothenberg et al., 1991). Manton
and Tolley (1991) reviewed the conditions necessary to
identify "hard" curve squaring, produced by a fixed maximum life span, and "soft" curve squaring produced by
probabilistic limits.
This article evaluates possible effects of "soft" and
"hard" life span limits on U.S. mortality five ways. First,
extinct cohort life tables were calculated using 1960 to 1990
mortality data. Second, those data were reexamined to
identify changes in the ages by which certain proportions of
death occurred. Third, those analyses were refined by adjusting the age at death distribution for estimates of cohort size
differences. Fourth, changes in the age distribution of 8
causes of death above age 85 were examined. Finally, the
effects of cohort size on the highest age at death expected for
different life tables were evaluated.
Extinct Cohort Life Tables
There are insufficient U.S. data to calculate cohort life
tables. However, deaths occurring past a given age can be
summed to calculate "extinct cohort" life tables (Vincent,
1951; Depoid, 1973). Death certificate age reports are recorded continuously; because medical record information
is often available, they are generally viewed as more reliable
than U.S. census data (Hambright, 1969; Rosenwaike and
Logue, 1983), being comparable in quality in recent years,
at least for U.S. Whites, to Medicare data (e.g., Kestenbaum, 1992). Additionally, extinct cohort life table numerators and denominators are based on the same data, so age
reporting errors compensate over the ages examined (Manton and Stallard, 1994). Thus, to analyze late age mortality,
we first estimated extinct cohort life tables from U.S. White
male and female deaths occurring 1960 to 1990 (i.e., starting
significantly after Social Security's start in 1935, shortly
before Medicare's start in 1965, and the computerization of
death records in 1962).
Four data adjustments were made. In 1962-1963, race
was not recorded on New Jersey death certificates. Racespecific estimates of deaths in New Jersey were added to
mortality counts for the rest of the U . S . to make national
estimates for 1962-1963. Second, to extend the mortality
data back to 1960 (which balanced the better quality of
recent data with the need for sufficient cross-temporal data to
characterize the late age mortality of multiple cohorts), we
used published U.S. death counts for 1960-1961, interpolated to single years of age. Third, since the highest documented human age is now 121.5, to be conservative we
excluded the few deaths reported for age 119 + . Fourth, the
mortality experience of younger cohorts at late ages was
completed by estimating their death counts from the experience of older cohorts 1986-1990, adjusted for the percentage increase in the number of deaths 1981-1985 to 1986—
1990. Above 105 a pooled estimate was used.
Extinct cohort life tables were calculated for U.S. cohorts
born 1870-1874, 1880-1884, and 1890-1894 that reach
ages 90, 80, and 70 by 1960-1964. The 1870-1874 cohort,
reaching an average age of 115 in 1985-1989, is the only
"extinct" group. This group, aged 63 to 67 in 1937, had to
document their age because Social Security benefits could
not be earned after age 65 in 1937-1938 (Bayo and Faber,
1985). Accuracy in age reporting at death at late ages is
poorer for U.S. cohorts born before 1870 (Manton and
Stallard, 1995; Stallard and Manton, 1995).
Ungraduated cohort age-specific mortality probabilities
were computed,
q,, c =
Dt,c
where D x c denotes deaths at age x for cohort c = 18701874, 1880-1884, or 1890-1894. Ungraduated estimates
for age x were smoothed by a five-year moving average.
Thus, accounting for the five birth years in each cohort and
the five-year moving average used in smoothing, each graduated estimate of q xc is a function of 25 single year-of-age
and time death counts. Graduated estimates for U.S. Whites
are shown in Figure 1.
The qx,c for the 1870-1874 cohorts are 34.2% at 100,
38.6% at 105, and 41.8% at 110. The 1880-1884 qXiC
involve 1870-1874 cohort data above age 106. The 18901894 q xc involve data from the two older cohorts above age
96. Thus, q xc at late ages for the 1890-1894 cohort are
biased upward by the older cohorts' experience. This should
produce conservative life expectancy estimates for the
youngest cohort. At age 90, where there is little bias, the qIC
declined 24%, i.e., from 20.8% (1870-1874) to 15.8%
(1890-1894).
In Table 1 are gender-specific estimates of q xc and e xc (life
expectancy) at ages 85, 90, and 95, and the average q xc for
10-year age intervals.
The e85 between the two younger male cohorts increased
0.4 years; 0.9 years for females. The e^ increased 0.5 years
( + 1 5 . 5 % ) across the three male cohorts; 0.9 years
( + 24.5%) for females. These increases are lower bound
estimates of e xc changes because the 1890-1894 cohort uses
the older cohorts' experience above age 96.
The annual percent increase in mortality estimated for 10year age intervals from 70-79 to 100-109 (age 110 in 1990
is reached only by the 1870-1874 cohort) is similar to the
Gompertz exponential parameter. If this parameter is constant, or increases, a fixed life span limit is implied. If it
decreases with age, a constant mortality level may be
reached, implying that there is no fixed life span limit. For
the 1890-1894 male cohort, mortality increased 6.8% per
year from ages 70 to 89; 8.5% for females. For the 1880 to
1884 male cohort, mortality increased 6.2% from 80 to 99;
7.0% for females. For the 1870-1874 male cohort, mortality
increased 2 . 8 % from 90 to 109; 4 . 0 % for females. If a
common Gompertz determined gender-specific mortality at
late ages in all three cohorts, the percent increases would be
constant. Declines in the age rate of increase of mortality for
male and female cohorts are inconsistent with a fixed life
span limit.
A stochastic life span limit might be defined as the age by
which e xc drops below, say, a year. For the stochastic limit to
be at age x, assuming qx is constant at later ages, requires qx
2* 0.632. Coale and Kisker (1986, 1990) suggest a limit of
110 years for cohorts born before 1870. No estimate in Table
2 reaches the stochastic limit, i.e., q,10 never exceeds 0.632.
In Table 2 the most reliable estimates are for both sexes.
The 1979-1981 National Center for Health Statistics
M ANTON AND STALLARD
B364
U.3U
-
0.45 -
P&^
0.40 -
0.35 -
„
Rat
0.30
£
1
S
/
0.25
0.20
0.15 -^-1870-74
-a-1880-84
0.10
-A-1890-94
0.05
4
0.00 • - H
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \—H—I—I—I—I—I—i—I—I—I—I—I—I—I—I—I—I—I—I—I—
Age
Figure 1. Mortality rates for three cohorts, U.S. White population, both sexes.
(NCHS) q110 is lowest — 38.2%. The 1980 Social Security
Administration (SSA) estimate is highest, 55.3%. Kannisto's (1994) estimate for 14 countries in 1980-1990, and
Kestenbaum's (1992) 1987 Medicare estimate, are both
49.4%. In the 1990-1992 IPSEN study (779 centenarians;
about 600 deaths), q110 was 48.9%. Cross-sectional q,'s are
higher than cohort estimates if mortality is declining. They
may also be elevated if a year with influenza activity is
selected. The qno for the longitudinal Cancer Prevention
Study (CPS) 1960-1987 (969 deaths above 100) was 40.8%.
In the ungraduated group annuity experience for 1986 to 1990
(based on data from 11 large insurance companies; age
reporting for insured persons is considered excellent) used to
construct the 1994 Group Annuity Mortality Table, both male
and female rates reached 25% about age 95 and then fluctuated about that value (Society of Actuaries, 1994).
For simulations we heed "consensus" q, estimates. Thus,
we averaged the qx for both sexes for every fifth year, from
age 65 to 90, in the 1969-1971 (NCHS, 1975) and 19791981 (NCHS, 1985) U.S. life tables and fit a function to
them. This produced a q100 (estimator A, Table 2) of 30.0%
and a q110 of 47.0%, with mortality increasing 4.6% per year.
The average of 7 q100 estimates (SSA [9, 10] and NCHS [1])
estimates were excluded because they were mathematically
smoothed at late ages) was 33.3%, and 6 estimates of q110,
45.3%, with a smaller annual increase of 3.1% (estimator B).
The higher annual percent increase for estimator A produces
lower estimates of life span limits than estimator B. Estimator
A was used to extrapolate qxs to age 110 + .
Cross-sectional Changes in Age at Death Distributions
The 1960 to 1990 U.S. mortality data were also analyzed
by examining changes in the age at death distribution (Myers
and Manton, 1984a, 1984b; Rothenberg et al., 1991). This
involves different assumptions, i.e., instead of estimating
the late mortality experience of younger cohorts we assume
differences in birth cohort size have little effect on late age
mortality.
We determined the age by which a fixed proportion of
deaths occurs from U.S. age at death distributions for each
year 1960 to 1990. Though those ages are affected by birth
cohort size and age-specific mortality, mortality effects
dominate at late ages if cohort size differences are modest. In
Denmark, Norway, and Sweden, 0.0 to 15.6% of centenarian growth was due to cohort size differences, i.e., 84.4 to
100% was due to mortality changes; 51 to 75% of centenarian growth was due to mortality changes from age 80 to 100
(Vaupel and Jeune, 1994). In the U.S. the cohort born in
1990 was 121.3% larger than the cohort born in 1900. The
proportion surviving from 65 to 100 increased 627% from
the 1900 to the 1990 male cohort life tables; 390% for
females (Social Security Administration, 1992). Thus, mor-
B365
LONGEVITY IN THE U.S.
Table 1. A Comparison of Selected Survival Parameters From Three Cohorts Evaluated Using Extinct Cohort Life Table Procedures
White Males
(0
1890-1894
Age at which
e, is estimated
85
90
95
10-year age interval
for which q, is estimated*
70-79
80-89
90-99
Relative Change
(2)
1880-1884
(l)-(3)
(3)
(3)
1870-1874
Cohort
(1)
1890-1894
Life expectancy
5.04
3.80
2.86
Age at which
q, is estimated
85
90
95
White Females
Cohort
4.59
3.53
2.74
0.1573
0.2130
0.2793
(3)
1870-1874
(3)
Life expectancy
—
3.29
2.55
15.5%
12.2%
6.37
4.67
3.39
One year probability of death
0.1391
0.1915
0.2651
(2)
1880-1884
Relative Change
(l)-(3)
5.54
4.20
3.19
—
3.75
2.86
24.5%
18.5%
One year probability of death
—
0.2286
0.3038
Average one year
Ijrobability of death"
—
—
0.0712
—
0.1374
0.1527
0.2591
0.2755
0.2960
-16.2%
-12.7%
0.0974
0.1449
0.2161
—
0.1969
0.2695
-26.4%
-19.8%
-12.5%
Average one year
Ijrobability of death'
—
—
0.0426
—
0.0964
0.1178
0.2317
0.2611
0.2118
-18.9%
0.1213
0.1733
0.2362
% Change per year
q(70-79) to q(80-89)
6.80%
—
—
8.51%
—
—
% Change per year
q(80-89) to q(90-99)
—
6.08%
—
—
7.00%
—
% Change per year
q(90-99) to q( 100-109)
—
—
2.80%
—
—
3.95%
•q(x,x+10)= l -
tality changes from 65 to 100 had 5.2 times more effect on
the number of males surviving to 100 than initial cohort size;
for females, mortality effects were 3.2 times larger.
Figure 2 depicts the 25th to the 99.99th percentiles of
the 1960 to 1990 age at death distributions for deaths above
age 65.
Increases are similar in the upper tail, e.g., the 75th
percentile age increased 3.3 years; the 99.9th percentile 3.0
years. This can be compared to estimates of life endurancy
(99.999th percentile of deaths at all ages in life tables
calculated from Medicare data) which increased 4 years
(from 109 to 113) from 1960 to 1990 for females; 3 years
(108 to 111) for males (Social Security Administration,
1992).
Table 3 compares changes 1960 to 1990 in the percentiles
of the age at death distribution to the changes required to
reach life span limits of 120 and 130 years.
The observed change is a fraction of that required to reach
either limit: 4-19% for 120 years, 3-12% for 130 years.
Thus, if the mean life span potential was 120 (or 130) years,
limits on U.S. mortality improvements would currently be
negligible if individual life span potential is narrowly distributed around 120 years, e.g., from 110 to 130. The life span
limit must be at least as high as the highest documented age
achieved, i.e., 121.5 years (a French female). In the U.S.,
ages of 126 (male; SSA records; female; Kautzky, 1995),
124 (male; Bortz, 1991), and 123 (female; Allegood, 1994)
have been reported. Though these reports are incompletely
documented, if only one were accurate the empirical lower
bound to the life span limit would be 2 to 5 years higher.
Limits may be gender specific. For both genders the ages
for all percentiles increased 1960 to 1990. Female change is
larger (3.3 to 4.0 years from the 50th to 99.5th percentiles),
and to higher ages — even though White females live longer
than White males. The largest female increase (4.0 years)
was for the 90th percentile. For males the increase was the
same (2.5 years) for the 95th, 99th, and 99.5th percentiles.
The 1960-1962 and 1988-1990 data in Table 4 show
changes in the age distribution of deaths above 85, e.g., a
12.9% decline in the proportion of deaths occurring at age
85-89, and proportionately more deaths at age 90 + . To test
whether changes at late ages (e.g., 105 to 109) are significant, death counts were treated as random variables (Deming
and Stephan, 1941; Cassel et al., 1977), and Mests calculated for the proportions in an age class at two times.
Shifts from 85-89 to latter ages are significant. The mean
age for deaths over 85 increased significantly, 1.2 years —
0.8 years for males and 1.4 year for females. The standard
deviation also increased significantly, 0.64 years — 0.47
years for males and 0.65 years for females, i.e., there is no
compression of the upper tail of the age at death distribution
from 1960 to 1990 for either gender. The decline for males at
110+ (from 29 deaths in 1960-1962 to 10 in 1988-1990) is
due to improvements in the accuracy of reported late ages,
B366
MANTON AND STALLARD
Table 2. Estimates of Mortality Rates and Their Increase at Ages 100 and 110, By Age and Sex
Mortality Rate (%)
100 Years
Annual Rate of Increase (%)
110 Years
Source of Estimate
Male
Female
Both
Male
Female
Both
Male
Female
Both
1
2
3
4
5
6
7
8
9
10
32.8
32.0
34.3
35.6
36.9
43.6
36.8
42.1
34.2
34.2
29.1
29.4
30.2
31.1
33.3
34.6
32.0
36.8
30.5
30.8
30.1
30.1
31.1
32.1
34.2
35.3
32.4
37.8
31.2
31.4
40.0
50.3
—
33.5
36.9
—
—
60.2
55.2
55.7
37.6
38.8
—
43. L
43.5
50.8
—
48.8
54.6
55.2
38.2
40.8
—
41.3
41.8
48.9
49.4
49.4
54.8
55.3
2.0
4.6
2.6
2.8
—
3.3
2.7
3.9
—
2.9
6.0 +
6.0 +
2.4
3.1
—
2.6
2.0
3.3
4.3
2.7
5.8 +
5.8 +
—
—
—
—
30.0
33.3
—
—
—
—
47.0
45.3
U.S. 1979-81'
CPS 1960-87"
Cohort 1890-94'
Cohort 1880-84"
Cohort 1870-74'
IPSEN 1990-92'
Medicare 1987"
Kannisto 1980-90"
SSA 1990s
SSA 1980J
Estimators
A Quadratic function*
B Average of empirical estimates'
—
-0.6
0.0
—
—
3.6
5.0 +
5.0 +
—
—
—
—
4.6
3.1
Notes on Sources:
'NCHS (1985) U.S. decennial life tables for 1979-81, U.S. White population — age 100 as reported; age 110 extrapolated from age 109 using NCHS's
extrapolation formula.
"Lew and Garfinkel (1990), Cancer prevention study 1960-1987 — age 100 interpolated from 95-99 and 100-104; age 110 extrapolated from 100-104
and 105-109. Approximately 97% of the study population was White.
'Data in Figure 1 for 1890-1894 cohort, U.S. White population.
"Data in Figure 1 for 1880-1884 cohort, U.S. White population.
'Data in Figure 1 for 1870-1874 cohort, U.S. White population.
' Vallin (1993), data from IPSEN study 1990-92 — age 100 as reported; age 110 extrapolated from 100-104 and 105-108.
eKestenbaum (1992), Medicare mortality 1987 — age 100 is average of 99.5 and 100.5; age 110 is estimated from age 107.5-112.5 assuming reported
value for 109.5 + is constant above age 109.5 — figures by sex at age 100 are for U.S. Whites; figures for both sexes at ages 100 and 110 are for total U.S.
population.
h
Kannisto (1994), Centenarian life table, 1980-1990, composite of 14 countries' data (Japan plus 13 European countries) — age 100 as reported; age 110
estimated from 109-111 (males) and 108-112 (females; both sexes).
'JSSA(1992), Life tables for 1980 and 1990 (includes both U.S. White and non-White populations) — age 100 and 110 as reported for males and females;
both sexes generated as weighted average of male and female q,'s using /,'s as weights. Mortality is extrapolated ( + ) by SSA above age 95 at a fixed 5%
(male) or 6% (female) per year increase in q,.
'Data from eq. [1] for q\ at age 100 and 110.
'Average of empirical, non-model based estimates for age 100 (Sources 2, 3, 4, 5, 6, 7, 8) and age 110 (Sources 2, 4, 5, 6, 7, 8).
— = not available.
i.e., deaths in 1960-1962 are from pre-1870 cohorts (Stallard and Manton, 1995). The female change at 110+ is not
significant.
Adjusting Age at Death Distributions for Cohort Size
To refine analyses of the 1960 to 1990 age at death
distributions, we adjusted for cohort size differences using
estimates of U.S. population growth. This eliminates the
assumption that cohort size effects are small relative to
mortality effects and substitutes the assumption that we can
accurately estimate the initial size of elderly cohorts. From
1960 to 1990, the U.S. elderly population increased an
average of 1.5% per year (U.S. Bureau of the Census, 1974,
1993), i.e., in year y, the cohort population surviving to age
65, /«(y), is 1.5% smaller than in year y + 1. Cohort size at
age x (in 1975, the midpoint of 1960-1990) was calculated as
c, = c,., x .985°+<*-«vio)
(10 is a scale factor estimated from the data). This suggests
the U.S. centenarian population grew 7% per year, e.g.,
cjcm
= 1.070; c,03/clw = 1.075. Vaupel and Jeune (1994)
found centenarian populations grew 7.4% per year 1960 to
1990 in countries with reliable data. Kestenbaum's (1992)
1987 estimate of 22,600 centenarians and Siegel and Passel's (1976) estimate of 3,300 centenarians for 1960 implies
an annual growth of 7.2% for the U.S. centenarian population 1960 to 1987. A growth rate of 7.0% was estimated for
the 1870-1874 and 1880-1884 cohorts (Whites, both sexes)
in Section II. Thus, the c,s are consistent with two U.S.
estimates (Siegel and Passel, 1976; Kestenbaum, 1992; and
the three extinct cohorts) and estimates of centenarian population growth in other developed countries (Vaupel and
Jeune, 1994).
The life table survival function is adjusted for population
growth,
4*(0) = f, x c,,
where y is set to 0 for 1975. The !xs were calculated from
estimator A qxs (Table 2) for ages 65 + , using a quadratic
(i.e., a linear first-difference) function,
q*+. = q* + .0004 ( x - 6 5 ) + 0.0012.
(1)
Since the qxs used to estimate this function were not the
lowest in Table 2, the qxs are conservative. The function,
estimated with the maximum life span assumed to be 132.5
(i.e.,q133 = 1.00 [100%]), produced ane65of 16.0 years; 9.7
LONGEVITY IN THE U.S.
B367
115 -r
110 99.99%
99.9%
65
Year
Figure 2. Selected percentiles of distribution of ages at death for deaths at age 65 and above, U.S. White population, both sexes.
Table 3. Ages by Which X Percent of U.S. White Deaths at Age 65 and Above Have Occurred,
and the Rate of Change Relative to Theoretical Maximum Life Spans of 120 and 130 Years
Percent of
Deaths
25
50
75
90
95
99
99.5
99.9
99.99
1960-1962
1988-1990
Years of Age
Change
1960-62 to 1988-90
71.6
77.3
83.3
88.3
91.0
95.9
97.6
101.0
105.8
73.5
80.1
86.6
91.9
94.7
99.3
100.9
104.0
107.8
1.9
2.8
3.3
3.6
3.7
3.5
3.3
3.0
2.1
Years
Remaining in
1990 to Limit
of 120
Years
Remaining in
1990 to Limit
of 130
Percent
Change
Relative to
Limit of 120
Percent
Change
Relative to
Limit of 130
46.5
39.9
33.4
28.1
25.3
20.7
19.1
16.0
12.2
56.5
49.9
43.4
38.1
35.3
30.7
29.1
26.0
22.2
4.1
7.0
9.8
12.9
14.5
16.8
17.1
18.7
17.1
3.4
5.6
7.6
9.5
10.4
11.4
11.2
11.5
9.4
Note: All values independently rounded.
years at age 75; 5.6 years at 85; and 3.3 years at 95 — within
± . 2 years of e, averages for the NCHS 1969-1971 and
1979-1981 life tables.
To examine the behavior of qx as it progressed from the
current empirical mortality schedule (qx) to the theoretical
limiting mortality schedule (say, q j , we needed to (a) gener-
ate q,s, and (b) select the number of years it takes q, to reach
qx. Setting these two factors will determine if, for these
assumptions, we should observe certain trends in the q, as
they move to the assumed limit.
The qxs were calculated by multiplying q, by the fraction
of life lived by age x (relative to the assumed maximum life
B368
MANTON AND STALLARD
Table 4. Percent Distribution of Deaths Above Age 85 in 1960-62 and 1988-90 for U.S. White Population
Both Sexes
Age
1960-62
85-89
90-94
95-99
100-104
105-109
110 +
65.54
27.26
6.322
0.799
0.0611
0.0133
1988-90
52.63
32.51
12.450
2.225
0.1788
0.0095
Mean age at death
Standard deviation
Total number
89.28
3.43
543,797
90.50
4.07
1,273,957
Males
Females
Change
1960-62
1988-90
Change
1960-62
1988-90
Change
-12.92**
5.25**
6.127**
1.426**
0.1177**
-0.0038*
68.57
25.51
5.290
0.569
0.0523
0.0130
60.02
29.45
9.111
1.327
0.0846
0.0046
-8.55**
3.94**
3.821**
0.758**
0.0324**
-0.0084**
63.43
28.49
7.045
0.960
0.0673
0.0135
49.06
33.99
14.061
2.659
0.2243
0.0119
-14.37**
5.50**
7.016**
1.698**
0.1569**
-0.0016
1.22**
0.64**
730,160
89.02
3.27
223,872
89.80
3.75
414,662
0.77**
0.47**
190,790
89.46
3.52
319,925
90.84
4.17
895,295
1.38**
0.65**
539,370
*p< .05;**p< .01.
span) raised to a power k which determines the degree of
change at each age as the assumed maximum life span is
approached, i.e.,
q. = q*
x
I.132.5
(2)
This function assumes that, as the distribution of life spans
for individuals is approached by the empirical age at death
distribution, smaller proportions of deaths occur at young
ages; larger proportions at late ages. If there is a fixed
distribution of life spans, the proportion of a cohort surviving to x is limited, with decreasing proportions living to
x + 1 , etc. The larger the value of k the higher the life
expectancy produced by the qx. For k = 2, e^ = 24.6 years;
fork = 3, egj = 28.5 years; and for k = 4, e^ = 32.0years.
In the evaluation below, to be conservative, we selected the
lowest value of k consistent with current mortality patterns,
assuming that the limit is achieved in 90 years. Other
estimates of k could have been used if different age-specific
mortality rate declines were assumed, or changes were
allowed to occur over a longer period of time.
With k set at 2, qxs at age 111-115 are not much lower
than qxs for the 1880-1884 cohort, i.e., the 1880-1884 q110
(41.3%) is 27.5% higher than q110 (32.4%). The q1I0 (38.2%)
for 1979-1981 U.S. life tables is only 17.9% higher than
q,,0- The &a calculated from qx is 24.6 years; 16.2 years at 75;
9.8 years at 85; and 5.6 years at 95. The differences
between the exs calculated from qx (with k = 2) and qx are 8.6,
6.5, 4.2, and 2.3 years at ages 65, 75, 85, and 95. Another
estimate of U.S. life expectancy limits is 85 years; 90 years
eliminating all circulatory diseases, diabetes, and cancer
deaths (Olshansky et al., 1990). Those estimates assume a
reduction of 70% in qxs at all ages. The form of (2) does not
require qx to be a constant proportion of qx — an assumption
we felt was unrealistic. With k = 2, q75 is 32.0% of q75
(68.0% less); q95 is 51.4% of q95 (48.6% less); and q110 is
68.9% of q110 (31.1% less). Thus, for k = 2 an e« of 24.6
years (implying an e0 near 89 years) is generated with less
than a 70% reduction in qxs above age 72 (e.g., 31.1% at
110) using a conservative estimate of the annual age change
(4.6%) in mortality (the empirical average was 3.1% in
Table 2; it was assumed to be 0.0 at age 112 in 1994 actuarial
estimates; Society of Actuaries, 1994).
The trajectory of the qx with k = 2 is consistent with Lee
and Carter's (1992) estimates that U.S. mortality declined
1.0-1.2% per year for ages 65 to 85 from 1900 to 1989.
Ahlburg and Vaupel (1990) report declines of 1% to 2% per
year at most ages 1968 to 1982; Kannisto et al. (1993)
reported declines of 0.5% per year for centenarians in 14
countries 1960 to 1989; Vaupel and Lundstrom (1994)
suggest a decline of 0.52% per annum for Swedish female
centenarians 1960 to 1989. Convergence of qx with qx, with
k = 2 assuming convergence requires 90 years, generates qx
declines of 1.58% per year at 65; 1.26% at 75; 0.99% at 85;
0.74% at 95; and 0.52% at 105, i.e., age-specific reductions
consistent with the cited studies.
By varying the time to convergence (from 90 years) and k
we can examine how the age at death distribution, fx,
changes as qx approaches qx. The estimate of qx at date y is
designated q*(y). Assuming qx is reached in 90 years, and
k = 2, qx*(y) was calculated,
q?(y) =
132.5
x
132.5
y>45k
To adjust fx for cohort size, death counts are needed for each
x and y. Calculating the cross-sectional age at death distribution, ff(y), requires renormalizing d*(y) to 1.0, i.e.,
133
= dx*(y)/Sd*(y),
where dx*(y) = /x*(y) q*(y). This is analogous to calculating
the life table distribution of deaths (from age 65) from clx
(which is determined by qx),
or from qx:
f, = dx//"«.
LONGEVITY IN THE U.S.
Because 4, = q*(0), the starting distribution (y = 0) is
133
which differs from r\ only by the cohort reduction factors, cx.
Using these relations, we first examined changes in the
life table age at death distributions for qx (i"x) and qx (fx). f*(0)
is calculated from q*(0) = qx. Differences between fx*(0)
and ?x reflect both initial cohort size (cxs) and survival
differences. If all cohort sizes and prior mortality were the
same, then f*(0) = ?x and the upper limit to the age at death
distribution is fx. Differences between ?x and fx are summarized by differences in exs, e.g., eM - eM = 8.6 years. For
every million deaths above 65, f\ yields 3 above 115 and
none above 118; fx yields 1,033 above 115, 39 above 120, 3
above 123, and none above 125. Thus, no one, out of a
million deaths, has much chance of reaching the absolute
limit of 132.5 with qx where k = 2. There is a likelihood that
someone reaches the stochastic "limit" of 123.
Setting k = 0 illustrates the effect of biological limits on
the age at death distribution, i.e., q*(y) = qx = qx, for all y
5= 0. Here, differences in f*(y) are due to the initially larger
6xs for older cohorts which cause the age by which a proportion of deaths occurs to increase linearly for 30 years before
slowing.
If mortality is improving, age differences between pairs of
percentiles are constant. Age increases for the 90th (or
above) percentile of fx are not linear after 30 years due to
differences in the cx between older (up to 7%) and younger
cohorts (1.5%). If mortality stops improving, it takes 20
years for the age for each percentile in fx to stop increasing;
nonlinear changes begin in 10 years. Thus, if U.S. mortality
reached a biological limit, changes in fx would take 10 years
to show nonlinearity — even if cohort sizes increased.
To affect fx, the distributions of individual life spans have
to overlap more than ?x and fx where the mean life span is 8 or
9 years above the mean of the age at death distribution. A life
span limit requires that 50 to 100% of deaths above age 85
could not be delayed, i.e., most persons above 85 realize
their longevity potential; fx implies 1 in a million persons
lives to age 124 (in 1990 there are 1.6 million U.S. deaths
above 65). The overlap of fx and fx is smaller than 50% —
implying life span limits do not currently affect mortality.
This lack of effect is for k = 2 and a life expectancy of about
89 years. If k = 3, the theoretical limit to life expectancy is
about 93 years and the effects of the restrictions take longer
to emerge.
Changes in Cause-Specific Age at Death Distributions
If there is a fixed life span distribution, there should be
rapid age increases in the proportions of deaths from causes
occurring at ages far from the limit — with little age change
in percentiles for diseases causing deaths close to the limit. If
the limit is due to senescence, and the mean age at death for
each cause increases, cause-specific distributions should
become increasingly similar, i.e., a general process of senescence reduces homeostasis so the triggering threshold for
any cause declines with age. When homeostasis is sufficiently weak, any biological insult causes death, forcing
cause-specific age of death distributions to converge, i.e.,
B369
the average age at death for each cause nears the life span
limit. Examining specific causes of death is also useful
because, if there is error in age reporting, unless that error is
correlated with specific causes, it will tend to raise the mean
age at death from all causes, and be biased toward expressing a senescent effect. If causes operate independently (i.e.,
are not governed by a single limiting process), or are operating far from the limiting distribution, then changes in their
distributions will not be restricted.
Changes in the U.S. cause of death distributions at late
ages can be examined relative to past mortality trends. Agestandardized heart disease and stroke mortality rates declined
53.0% and 70.4%, respectively, from 1950 to 1992; agestandardized cancer mortality rates increased 6.2%. This
shifted the proportion of deaths in the age-standardized distribution of deaths from 36.5% to 28.6% for heart disease; from
10.5% to 5.2% for stroke; from 11.3 to 23.6% for cancer.
The effects of these changes at later ages are of interest
because cancer and senescence may depend on similar mechanisms (e.g., Cutler and Semsei, 1989). Above 85, heart
disease and stroke mortality rates declined 28.8% and
47.6%, respectively — cancer mortality rates increased
23.2%. Above 85, the proportion of deaths due to heart
disease declined from 45.3 to 43.5%; for stroke from 14.8 to
10.5%. For cancer it increases from 7.2 to 11.9%. Thus,
cause-specific changes for younger ages hold for ages 85 + .
For 50,000 persons aged 75 + followed 1960 to 1987 (Lew
and Garfinkel, 1990), cancer caused 16% of deaths at age
75-79 but only 7.1% (male) and 4.6% (female) at 95-99.
This shows the decline in cancer mortality with age — but
not the persistence of the age decline over time. Thus, we
examined changes 1962 to 1990 in the distributions of ages
at death for eight causes above age 85. In addition to heart
disease, cancer, and stroke, we selected conditions (e.g.,
diabetes; nephritis and nephrosis; lung disease) that may
trigger lethal events, or acute morbid sequelae of other agerelated failures (e.g., changes in immunological response
[pneumonia, septicemia]). Distributions of death counts for
each cause for five-year age categories from 85-89 to 110 +
for 1962-1964 and 1988-1990 are in Table 5. Mests were
calculated for differences in the proportion of deaths in each
age category and for changes in the mean and standard
deviation of the cause-specific distributions.
The largest declines in the proportion of deaths occurring
at age 85-89 (and the greatest increases above 90) are for
septicemia (-17.7%) and diabetes (-16.0%). Cause-specific
mean ages at death increased 0.5 to 1.6 years. Standard
deviations of cause-specific age distributions increased 7.1%
(chronic lung disease) to 29.5% (diabetes). Thus, all causespecific distributions showed increases in the mean and
standard deviation of ages at death — neither would be
expected if all causes were limited by a common process.
For age 100-104 the proportion of deaths due to heart
disease changed little (50.5 vs 50.3%). Stroke declined from
15.7 to 9.3%. Cancer increased from 2.5 to 3.9%. Thus, for
age 100-104 declines in stroke, and increases in cancer,
mortality continue.
In Tables 6 and 7 are cause-specific results for males and
females.
Male cause-specific means (Table 6) increased 0.6 to 1.3
MANTON AND STALLARD
B370
Table 5. Percent Distribution of Deaths Above Age 85 in 1962-64 and 1988-90 for Eight Conditions Listed
as Underlying Causes of Death for U.S. White Population (Males and Females)
Septicemia
Cancer
Change
1962-64 1988-90
1962-64 1988-90
85-89
90-94
95-99
100-104
105-109
110 +
69.96
24.35
4.928
0.870
0.0000
0.0000
-17.71**
52.14
9.13**
33.48
7.349**
12.277
1.079
1.949
0.1411
0.1411
0.0134
0.0134
Mean age at death
Standard deviation
Total number
88.92
3.24
345
90.51
3.97
14,882
Age
1962-64 1988-90
85-89
90-94
95-99
100-104
105-109
100 +
65.84
27.25
6.218
0.642
0.0415
0.0028
-13.53**
52.31
6.25**
33.50
12.146
5.928**
1.274**
1.916
0.0874**
0.1289
0.0001
0.0029
59.40
30.86
8.248
1.306
0.1574
0.0286
45.33
-14.07**
35.47
4.62**
7.443**
15.691
1.917**
3.223
0.2692
0.1118**
0.0148 -0.0138
Mean age at death
Standard deviation
Total number
89.27
3.35
106,014
90.48
3.96
138,075
89.83
3.71
27,958
91.20
4.27
94,358
74.50
21.43
3.793
0.260
0.0167
0.0048
64.74
27.25
7.170
0.794
0.0442
0.0014
-9.75**
5.82**
3.377**
0.534**
0.0275**
-0.0033
1.59**
0.73**
14,537
88.56
2.97
41,892
89.36
3.49
138,001
0.80**
0.52**
96,109
Change
1962-64 1988-90
1962-64 1988-90
1.21*
0.61**
32,061
Change
1.37**
0.55**
66,400
Change
1962-64 1988-90
Change
76.53
20.12
3.126
0.210
0.0175
0.0000
-16.01**
60.51
29.77
9.65**
8.502
5.375**
1.145
0.935**
0.0680
0.0505
0.0000
0.0000
64.96
27.58
6.591
0.806
0.0529
0.0088
50.68
-14.27**
33.21
5.63**
13.389
6.798**
2.493
1.686**
0.2080
0.1552**
0.0105
0.0017
88.37
2.83
5,726
89.73
3.67
19,126
1.37**
0.84**
13,400
89.35
3.44
272,449
90.69
4.14
572,466
Change
1962-64 1988-90
Influenza/Pneumonia
Stroke
:Heart Disease
Diabetes
Change
Age
Nephritis/Nephrosis
Lung Disease
1962-64 1988-90
1.34**
0.70**
300,017
70.28
23.98
5.108
0.541
0.0601
0.0300
64.63
26.99
7.408
0.894
0.0740
0.0000
-5.65**
3.01**
2.299**
0.353*
0.0140
-0.0300
88.85
3.31
3,328
89.38
3.55
35,113
0.53**
0.23**
31,785
Change
68.03
24.84
6.749
0.317
0.0634
0.0000
52.95
-15.08**
33.23
8.38**
11.770
5.021**
1.594**
1.911
0.1432
0.0798
0.0000
0.0000
89.15
3.33
3,156
90.43
3.96
16,066
1.28**
0.63**
12,910
*p < .05; **p < .01.
Table 6. Percent Distribution of Deaths Above Age 85 in 1962-64 and 1988-90 for Eight Conditions Listed
as Underlying Causes of Death for U.S. White Males
Septicemiii
Cancer
Diabetes
Age
1962-64 1988-90
85-89
90-94
95-99
100-104
105-109
110 +
72.51
22.81
4.678
0.000
0.0000
0.0000
58.42
-14.10**
30.17
7.37*
9.775
5.096*
1.518
1.518
0.0920
0.0920
0.0230
0.0230
76.55
20.01
3.207
0.213
0.0152
0.0102
68.78
25.08
5.613
0.501
0.0221
0.0017
-7.77**
5.08**
2.406**
0.288**
0.0068
-0.0085
Mean age at death
Standard deviation
Total number
88.63
2.89
171
89.96
3.82
4,348
88.42
2.86
19,675
89.01
3.28
58,877
0.59**
0.42**
39,202
Change
1.33**
0.93**
4,177
Stroke
1962-64 1988-90
Change
Heart Disease
1962-64 1988-90
Change
65.48
-11.26**
26.78
6.76**
6.930
3.779**
0.813
.720**
0.0000
0.0000
0.0000
0.0000
67.99
25.89
5.499
0.580
0.0367
0.0082
58.46
30.13
9.803
1.505
0.0991
0.0046
-9.53**
4.25**
4.304**
0.925**
0.0625**
-0.0037
88.35
2.78
2,158
89.29
3.47
5,411
89.08
3.28
109,130
89.94
3.82
174,544
0.86**
0.54**
65,414
0.94**
0.68**
3,253
Lung Disease
Nephritis/Nephrosis
Age
1962-64 1988-90
85-89
90-94
95-99
100-104
105-109
100 +
68.94
25.42
5.129
0.490
0.0254
0.0025
60.03
29.95
8.734
1.232
0.0527
0.0000
-8.91**
4.53**
3.605**
0.742**
0.0273
-0.0025
63.00
28.96
6.951
0.924
0.1472
0.0245
-11.11**
51.89
33.51
4.56**
12.312
5.362**
2.183
1.259**
0.0916 -0.0556
0.0092 -0.0154
74.38
21.22
3.998
0.363
0.0454
0.0000
67.69
25.54
6.064
0.648
0.0535
0.0000
-6.68**
4.32**
2.066**
0.284
0.0081
0.0000
Mean age at death
Standard deviation
Total number
89.02
3.21
39,383
89.77
3.66
36,031
0.75**
0.46**
-3,352
89.50
3.56
12,229
90.53
4.00
32,756
88.51
3.09
2,201
89.11
3.37
18,683
0.60**
0.28**
16,482
*p< .05;**p< .01.
1962-64 1988-90
Change
1.03**
0.44**
20,527
Change
76.74
20.02
3.151
0.093
0.0000
0.0000
Influenza/Pneumonia
Change
1962-64 1988-90
1962-64 1988-90
Change
1962-64 1988-90
Change
69.11
23.97
6.401
0.457
0.0653
0.0000
56.10
-13.01**
32.15
8.18**
10.138
3.737**
1.473
1.016**
0.1410
0.0757
0.0000
0.0000
89.06
3.31
1,531
90.11
3.81
6,382
1.05**
0.50**
4,851
B371
LONGEVITY IN THE U.S.
Table 7. Percent Distribution of Deaths Above Age 85 in 1962-64 and 1988-90 for Eight Conditions
Listed as Underlying Causes of Death for U.S. White Females
Septicemia
Cancer
Diabetes
Age
1962-64 1988-90
85-89
90-94
95-99
100-104
105-109
110 +
67.24
25.86
5.172
1.724
0.0000
0.0000
49.55
-17.69**
34.84
8.98*
13.309
8.137**
2.126
0.402
0.1614
0.1614
0.0095
0.0095
72.68
22.69
4.312
0.302
0.0180
0.0000
-10.94**
61.74
6.17**
28.86
4.017**
8.329
1.012
0.711**
0.0607
0.0427**
0.0013
0.0013
Mean age at death
Standard deviation
Total number
89.20
3.54
174
90.74
4.01
10,534
88.69
3.06
22,217
89.63
3.62
79,124
Change
1.54**
0.47
10,360
1962-64 1988-90
0.94**
0.56**
56,907
62.93
28.72
7.321
0.958
0.0637
0.0092
47.28
-15.65**
34.57
5.85**
7.641**
14.962
2.926
1.968**
0.2558
0.1922**
0.0131
0.0039
88.38
2.87
3,568
89.91
3.73
13,715
1.53**
0.87**
10,147
89.52
3.53
163,319
91.02
4.22
397,922
1962-64 1988-90
Change
1962-64 1988-90
Lung Disease
85-89
90-94
95-99
100-104
105-109
100 +
64.01
28.34
6.862
0.732
0.0510
0.0030
49.58
-14.43**
6.41**
34.75
13.350
6.488**
2.158
1.426**
0.1558
0.1048**
0.0039
0.0009
56.61
32.34
9.257
1.602
0.1653
0.0318
41.84
-14.77**
36.52
4.18**
17.488
8.231**
2.174**
3.776
0.3636
0.1983**
0.0179 -0.0139
62.29
29.37
7.276
0.887
0.0887
0.0887
61.15
28.64
8.935
1.175
0.0974
0.0000
-1.14
-0.73
1.659
0.287
0.0087
-0.0887
Mean age at death
Standard deviation
Total number
89.41
3.42
66,631
90.73
4.03
102,044
90.08
3.81
15,729
91.56
4.36
61,602
89.50
3.62
1,127
89.69
3.71
16,430
0.19*
0.09
15,303
Change
1.48**
0.55**
45,873
1.50**
0.70**
234,603
Nephritis/Nephrosis
1962-64 1988-90
1.31**
0.60**
35,413
1962-64 1988-90
Change
58.56
-17.85**
30.95
10.77**
6.010**
9.121
1.276
0.996**
0.0948
0.0668
0.0000
0.0000
Age
Change
1962-64 1988-90
Change
76.40
20.18
3.111
0.0280
0.0948
0.0000
Influenza/Pneumonia
Stroke
Heart Disease
1962-64 1988-90
Change
Change
67.02
25.66
7.077
0.185
0.0615
0.0000
50.88
-16.14**
33.93
8.27**
12.846
5.769**
2.200
2.015**
0.1446
0.0830
0.0000
0.0000
89.23
3.34
1,625
90.64
4.05
9,684
1.41**
0.70**
8,059
*p < .05; **p < .01.
years; female means 0.2 to 1.5 years. Male standard deviations increased 9.1% (lung disease) to 32.1% (septicemia).
Female standard deviations increased 2.5% (nonsignificant)
for lung disease to 30.3% for diabetes (significant; with a
1.5-year increase in the mean). Female declines in the
proportion of deaths at age 85-89 are generally larger than
for males. Thus, the pattern of increases in all means and
standard deviations shown for all deaths is found for males
and females separately.
At age 100-104 for males the proportion of heart disease
(49.7 to 47.7%) and stroke (15.2 to 8.1%) deaths declined,
whereas cancer (3.3 to 5.4%) increased. For females heart
disease deaths did not change (50.9% at both times). Stroke
deaths declined significantly (from 15.9 to 9.6%) while
cancer (2.2 to 3.5%) increased. Thus, the male and female
cause-specific distributions of deaths at age 100-104 have a
similar structure.
Cohort Size and Mortality Effects on the Maximum
Expected Age at Death
Empirical analyses of life span are limited because we
only observe survival outcomes for small cohorts born far in
the past. To estimate the maximum age at death expected in a
population, x ^ , we must combine the population's size at
birth and a life table describing its mortality. Below we
identify 25 combinations used to assess the dependence of
xm,, on those two factors.
Table 8A. Assumptions Used in Construction of Table 8B
Mortality assumptions:
Cohort Size
(Both Sexes)
1. Small: 61,000
(Swedish birth cohort 1910)
2. Intermediate: 774,000
(French birth cohort 1910)
3. Large: 2.8 million
(U.S. birth cohort 1910)
4. Historically large: 4.3
million (U.S. birth cohort
1961)
5. Cohort groups: 13,814,000
(U.S. birth cohorts
1908-1912)
d
c
e
Modified Modified Modified
U.S.
U.S.
U.S.
1910
1960
1990
Cohort
Cohort
Cohort
Life
Life
Life
Table
Table
Table
(Modified (Modified (Modified
Medium
High
Low
Mortality) Mortality) Mortality)
a
b
U.S.
1910
Cohort
Life
Table
(High
Mortality)
U.S.
I960
Cohort
Life
Table
(Medium
Mortality)
la
lb
lc
Id
le
2a
2b
2c
2d
2e
3a
3b
3c
3d
3e
4a
4b
4c
4d
4e
5a
5b
5c
5d
5e
The letters in Table 8A indicate which of five cohort (or
modified cohort) life tables, and numbers which of five
cohort sizes, were used in calculations. The first four sizes
represent actual birth cohorts, e.g., 4.3 million persons were
born in the U.S. in 1961 — 70.5 times more than the 61,000
MANTON AND STALLARD
B372
Swedes born in 1910. If both cohorts are subject to the same
life table, 70.5 times as many persons in the U.S. cohort
would survive to a given age. Furthermore, the age for
which there is a 50% chance of observing at least one
survivor would be higher, holding mortality constant, for the
larger birth cohort. We determined the age at which the last
death in a population is expected (at a given probability) for
these cohort sizes and life tables.
The /xs used were from 1910, 1960, and 1990 U.S. cohort
life tables (SSA, 1992). Estimates of late age survival are
conservative for conditions a or b because SSA's qll0 estimates are the highest in Table 2. The e0 for the 1910 U.S.
cohort is only 56.2 years for males; 63.8 years for females; at
65, 14.2 and 18.9 years. For the 1960 U.S. cohort, e0 is 72.3
and 79.9 years; at 65, 17.0 and 21.3 years. In the 1990 U.S.
cohort, e0 is 76.4 for males; 83.3 for females; at 65, 18.2 and
22.7 years. These values can be compared with period e0 and
e^ estimates;
France (1991)
Male
Female
Japan(1992)
Male
Female
Sweden (1990)
Male
Female
U.S. (1993; Whites)
Male
Female
73.6
82.0
16.2
20.9
76.3
83.0
16.6
21.1
74.9
80.6
15.4
73.5
80.1
15.7
19.6
19.2
In Table 8B are age-specific numbers of survivors expected for cohort sizes 1-4 using 1910 and 1960 U.S. cohort
life tables (a,b). The expected number of survivors to age x
is equal to the expected number of deaths at or above x. For a
Poisson distribution, \ma is the age at which the expected
number of survivors is 0.693 with a probability of 50% that
the last death occurs above this age. If mortality is improving. XmM m Table 8B should exceed the highest ages yet
observed because the 1910 cohorts are only aged 85 in 1995
(e.g., age 110 is reached in 2020). Currently observed
maximum ages at death come from smaller cohorts with
higher early mortality.
The expected number of male deaths at 110 + in the 1910
Swedish cohort with the 1910 U.S. cohort life table (a) is
0.6, the probability of one or more deaths at 110 + is 45%.
Using the 50% rule, *„,„ is 109. This can be compared to 7.7
male deaths at 110 + expected in the 1910 French cohort, 27
for the 1910 U.S. cohort, and 43 for the 1960 U.S. cohort.
Since the oldest Swede observed to date was 111, this
suggests that these estimates (which apply to future dates)
are conservative. We can also determine the uncertainty of
the probability that at least one person reaches age x. The 14
male deaths at 110+ expected for the 1910 Swedish cohort
with the 1960 U.S. cohort life table (b) has a standard error
of 3.75, i.e., between 6.5 and 21.5 deaths are expected with
95% confidence. The Mest of the hypothesis that no deaths
occur above 110 is 3.74; i.e., less than a 1 in 1,000 chance.
Table 8B shows that x,,^ is a function of survival, cohort
size, and gender. The 1910 U.S. cohort life table (a) suggests a 2% chance of a male death occurring at 115 + for the
1910 Swedish cohort compared to 56% for the 1961 U.S.
cohort (i.e., x,,^ is 115).
For the modified 1910 U.S. cohort life table (c), q, is
assumed to reach a maximum of 50% at age 110+. It is
assumed to reach 50% at age 115+ in the modified U.S.
1960 cohort life table (d). In the 1994 Group Annuity
Mortality Table, q, is set to 50% for age 112 + . Setting qx at
Table 8B. Number of Persons Expected To Survive
to Given Ages From Cohorts of Four Different Sizes
Using 1910 and 1960 U.S. Cohort Life Tables
Life Table Used in Calculations
a. 1910 Cohort
Age and Cohort Sizes
Males
Females
b. 1960 Cohort
Males
Females
95(/95)
0.01901
Expected number of survivors
Sweden 1910
579.8
France 1910
7,356.9
25,663.5
U.S. 1910
40,879.0
U.S. 1961
2,309.2
29,299.8
102,208.5
162,806.8
100(/10o)
Expected number of survivors
Sweden 1910
France 1910
U.S. 1910
U.S. 1961
0.00359
0.01888
0.01833
109.5
1,389.3
4,846.5
7,719.9
573.4
7,275.6
25,380.0
40,427.5
559.1
2,020.3
7,093.7 25,634.9
24,745.5 89,424.0
39,416.8 142,442.5
105(/1O5)
Expected number of survivors
0.00040
0.00276
0.00369
12.2
154.8
540.0
860.2
84.2
1,068.1
3,726.0
5,935.1
112.5
528.9
1,428.0 6,710.6
4,981.5 23,409.0
7,935.0 37,287.9
0.00002
0.00019
0.00046
0.00279
0.6
7.7
27.0
43.0
5.8
73.5
256.5
408.6
14.0
178.0
621.0
989.2
85.1
1,079.7
3,766.5
5,999.6
H5(/115)
3.8X10"7 5.92xlO" 6
Expected number of survivors
0.00003
0.00022
0.181
2.214
7.992
12.728
0.9
11.6
40.5
64.5
6.71
85.1
297.0
473.1
Sweden 1910
France 1910
U.S. 1910
U.S. 1961
H0(/ ll0 )
Expected number of survivors
Sweden 1910
France 1910
U.S. 1910
U.S. 1961
Sweden 1910
France 1910
0.02
0.142
U.S. 1910
U.S. 1961
0.513
0.817
l.OxlO" 9
120(/l2O)
Expected number of survivors
Sweden 1910
France 1910
U.S. 1910
U.S. 1961
1.5 x 10" 4
1.9X10" 3
0.007
0.011
0.07571
4.0 x l O " 8
4.9 x l O " 7
U.S. 1910
1.8X10" 7
U.S. 1961
2.8xlO- 7
0.17677
1,917.5
5,391.5
24,330.7 68,409.9
84,901.5 238,639.5
135,238.7 380,126.2
0.06624
0.01734
1.9X10"8 6.7X10"7 4.3X10" 6
0.00058
0.0071
0.026
0.041
1.3 xlO 1 2 2.3x10-"
125(/125)
Expected number of survivors
Sweden 1910
France 1910
0.06289
0.020
0.259
0.90
1.44
0.13
1.66
6.00
9.25
9xl0"» 1.4X10"7
7xlO- 7 2.7XIO"4
8.6x10-* 3.5X10" 3
3.2X10-*
0.012
4.95X10" 5
0.019
0.0043
0.524
0.189
0.301
LONGEVITY IN THE U.S.
50% means there is no fixed life span limit and no stochastic
limit, i.e., no age for which ex is less than a year. Even so, in
any finite population with a qx of 50%, there is an age at
which less than one survivor is expected. For the modified
1990 U.S. cohort life table (e), qx is set to 34.2% (males) and
32.2% (females; the best single year birth cohort estimates)
for age 110 + . The x ^ s for all 25 combinations of life tables
and cohort sizes are in Table 9.
Applying the modified 1910 U.S. cohort life table to the
1910 Swedish cohort (lc), x ^ is 109 for males and 113 for
females. For the 1961 U.S. cohort (4c), xmax increased to 116
for males ( + 7 years); 119 for females ( + 6 years). Applying
the modified 1960 U.S. cohort life table to the 1910 Swedish
cohort (Id), xMa is 115 for males; 118 for females. For the
1961 U.S. cohort (4d), x ^ is 121 for males ( + 6 years) and
124 for females (+ 6 years). Thus, for a cohort the size of the
1961 U.S. cohort, x,,^ is 6-7 years higher than for a cohort
the size of the 1910 Swedish cohort — assuming identical
mortality.
For the modified 1990 U.S. cohort life table applied to the
1961 U.S. cohort (4e), x^ is 127 for males and 133 for
females. Larger cohort sizes are simulated by considering
births in multiple years. The 1908 to 1912 U.S. cohorts
represent 13,814,000 births, with an average birth year of
1910. For this group the modified 1990 U.S. cohort life table
(5e) produces an x ^ of 130; 136 for females.
The range of cohort sizes (61,000 to 13.8 million; 226 to
1), under conservative mortality conditions (la vs 5a),
produces male differences of 8 years in x ^ ; 6 years for
females. Across mortality conditions, for the smallest cohort
(la vs le), x,^ increased 9 years; 8 years for females. The
effect of changing mortality is largest for the large cohort (5a
vs 5e), i.e., 13 years for males and 17 years for females.
Discussion
We used 1960-1990 U.S. mortality data to test for life
span limit effects in the U.S. population. Five empirical and
simulation-based analyses were done.
First, extinct cohort life tables were calculated for U.S.
cohorts born 1870-1874, 1880-1884, and 1890-1894, attaining ages 70, 80, and 90 in 1960-1964. There were significant improvements in mortality, for both genders, at
Table 9. Maximum Expected Ages at Death for the Combination
of Five Life Tables and Five Cohort Sizes
Life Table Used in Calculations
a
b
c
d
e
Males
Females
109
113
115
118
109
113
115
118
118
121
Males
Females
113
116
118
121
113
117
118
121
121
127
Males
Females
114
117
120
122
110
118
115
123
125
131
Males
Females
115
117
121
124
116
119
121
124
127
133
Males
Females
117
119
122
126
118
122
123
126
130
136
Cohort Sizes
B373
comparable ages (e.g., 90) across cohorts. Mortality increases per year of age near 100 for the two older cohorts
were small for males (0.0-2.8%) and females (2.0-4.0%).
Second, we examined total and gender-specific changes in
the 1960-1990 U.S. empirical age at death distributions by
determining the age by which a given proportion of deaths
occurred. Changes were similar (2.5 to 4.0 years) at most
percentiles. We also compared changes in the ages of percentiles of the distributions from 1960-1962 to 1988-1990
to changes required to reach life span limits of 120 or 130
years. The percent changes for a 120-year limit increased
with age. For a 130-year limit, changes were fairly constant
above the 90th percentile. Thus, a more plausible life span
limit given this logic is 130 years.
Third, 1960 to 1990 U.S. age at death distributions were
adjusted for estimated differences in cohort size. This
showed that, if the average potential life span is no more than
10 years higher than the average age at death, and mortality
improvements continue at current rates, the age at death
distribution will not be constrained by the distribution of
individual life spans for 90 + years. If empirical and theoretical distribution differences are larger (e.g., potential life
spans for individuals range from 110 to 130 years), then,
starting with the current distribution of ages at death, the
ages by which specific proportions of deaths occur would
continue to increase.
Fourth, we examined changes in cause-specific age at
death distributions 1962-1964 to 1988-1990. Above 85, the
8 distributions showed increased means and standard deviations — with wide variation in the size of cause-specific
increases for total and gender-specific populations.
Finally, we examined the maximum expected age at death
(x^J in cohorts varying in size by 226 to 1 (within the range
of human experience, e.g., China vs Sweden) for different
life tables. In Table 8, x ^ is 6 years higher in the largest
cohort (using the same life table; lb vs 4b). Table 9 showed
differences of 6 + years in x^ between the largest and
smallest cohort by modifying mortality at late ages. For the
largest cohort with modified 1990 U.S. cohort life tables,
x^ was 130 for males and 136 for females, i.e., sufficient
for life expectancies of 95 + years to be achieved — even
with considerable individual life span variation (e.g., a
standard deviation of 4 years; Manton et al., 1994).
These analyses suggest life span limits are not yet manifest in U.S. mortality patterns. One analysis using parameters estimated from extinct cohorts suggests the maximum
individual longevity potential is over 130 years. This is
consistent with the highest documented age of 121.5 years,
and multiple reports of U.S. survivors to later ages. Given
the larger size of the U.S. population, and better late age
mortality than in other developed countries (Manton and
Vaupel, 1995), U.S. deaths reported above 120 may be
verified as SSA and Medicare registration matures.
The expectation of continuing U.S. mortality decreases is
based on jointly evaluating temporal changes in the mean
and variance of the empirical age at death distribution, the
potential distribution of individual life spans, and cohort size
changes. If the maximum life span is 130, the standard
deviation of the genetic potential for individual life spans has
to be large to prevent e0 from reaching 95 + years. The joint
B374
MANTON AND STALLARD
effects of the mean and variance of the potential life span
distribution, and changes in cohort size, are not considered
in estimates of human life expectancy limits of 85 years
(e.g., Olshansky et al., 1990). It would take extreme conditions (e.g., a large genetically determined variance for life
span) for life expectancy to be limited to 85 years.
Our analyses had two goals. One was to test if biological
limits constrained U.S. mortality reductions 1960 to 1990.
The data do not support this hypothesis. Second, we estimated fixed and stochastic limits. We found that current
estimates of life expectancy limits could be exceeded under
plausible scenarios (e.g., mortality reductions were not
proportional over age).
ACKNOWLEDGMENTS
The research reported in this article was supported by grant AG-01159
from the National Institute on Aging.
Address correspondence and requests for reprints to Dr. Kenneth G..
Manton, Duke University, Center for Demographic Studies, 2117 Campus
Drive, Box 90408, Durham, NC 27708-0408. E-mail: [email protected]
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Accepted February 1, 1996
Cerebrovascular Pathology in
Alzheimer's Disease
A New York Academy of Sciences Conference
November 12-15,1996 ~ East Rutherford, New Jersey
THIS CONFERENCE BRINGS TOGETHER EXPERTS at the forefront of new research
focusing on the critical role of the cerebrovasculature in the pathogenesis and
evolution of Alzheimer's disease. There are two central questions for discussion
at the conference. The first is whether cerebrovascular changes precede or follow the clinical onset of Alzheimer's disease. Second, what is the temporal relationship between amyloid deposition in blood vessels and cerebral tissue and
the emergence of cerebrovascular changes in Alzheimer's disease.
Research scientists and clinicians working in the fields of dementia, aging,
memory disorders, physiology and metabolism of cerebral blood flow, and
neurodegenerative disease, as well as primary care physicians, nurses and
health personnel who deal with Alzheimer's patients, will benefit from the
presentation of new findings and discussion of issues at this conference.
Investigators are invited to submit abstracts for poster sessions.
C O N F E R E N C E CHAIRS
Jack C. de la Torre, University of New Mexico School of Medicine, Albuquerque
Vladimir Hachinski, University of Western Ontario, London, Ontario, Canada
FOR MORE INFORMATION PLEASE CONTACT
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