Up the Gamma-Gompertz curve with gun and
camera
Alcuni approfondimenti della curva GammaGompertz
Gustavo De Santis and Giambattista Salinari
Abstract First, we propose a new procedure for the estimation of the parameters of
the Gamma-Gompertz model (GGM) of adult and senior mortality, so as to
circumvent the possible bias caused by period effects (e.g., a sudden reduction of
mortality in certain years, due to, for instance, medical progress).
Then, we check two of the assumptions of the model: the constancy of the rate of
ageing and the effect of the initial distribution of frailty on mortality deceleration.
On data taken from the Human Mortality Database (female cohorts, from nine
European countries, aged 75-99 years), the GGM failed both tests: ageing is not
constant, and the deceleration of mortality is stronger than the model predicts.
However, these deviations are relatively minor, especially at very old ages (90+).
Abstract Si tratta qui del modello Gamma-Gompertz (GG) per lo studio della
mortalità in età adulta e anziana. Prima ne stimiamo i parametri con un nuovo
metodo, che evita la distorsione causata dagli effetti di periodo.
Poi verifichiamo due assunti del modello: la costanza del tasso di senescenza e la
riduzione della variabilità della fragilità con l’età, con dati tratti dallo Human
Mortality Database, relativi a coorti di donne, in nove paesi Europei, osservate nella
fascia d’età 75-99 anni. Il tasso di senescenza non è però costante, e il processo di
decelerazione della variabilità della mortalità è più intenso di quello atteso. Tuttavia,
questi scostamenti sono di piccola entità e, a età molto elevate (90+), l’evoluzione
della mortalità sembra effettivamente convergere verso il modello GG.
Key words: adult mortality, Gamma-Gompertz model, period effect.
1
Gustavo De Santis, DISIA, Un. of Firenze; email: [email protected]
Giambattista Salinari, DISEA, Un. of Sassari; email: [email protected]
The research work has been financed by the P.O.R. Sardinia F.S.E. 2007–2013 in the context of research
project 13/D3-2 developed at the University of Sassari
PAGE 6
Gustavo De Santis and Giambattista Salinari
Introduction
In the demographic literature models abound that attempt to describe the evolution
of mortality with age (Yashin et al 2000). Among these, the most important is
probably the Gamma-Gompertz model, GGM (Vaupel et al. 1979), in part for its
mathematical tractability and elegance, but also, and more importantly, because the
theoretical and the empirical advances of the past few years have lent credit to the
conjecture that it may be the best possible mathematical description of mortality at
adult (and old) ages (Missov and Vaupel 2015).
Let us first recall two concepts that are important for this model: the force of
mortality x and frailty zi. The former is defined as the instantaneous rate of death.
The latter is the ratio between two forces of mortality: that of a specific individual i
and that of a standard individual, whose frailty is normalized to one (Vaupel et al.,
1979).
The GGM is based on four assumptions: 1) frailty is a constant, for every individual
i (i.e., it does not change as people get older); 2) among the members of a cohort,
frailty is Gamma distributed; 3) the individual force of mortality evolves as a
Gompertz (1825) curve 𝜇𝑖,𝑥 = 𝛼𝑖 𝑒 𝛽𝑥 ; 4) all the members of a cohort share the same
rate of ageing (same  in the Gompertz curve).
After analyzing the data collected by Kannisto on super-centenarians, Gampe (2010)
concluded that, at very old ages, somewhere between 110 and 120 years, there is a
sort of “mortality plateau”, when the annual probability of death levels off at a value
of about 0.5. This empirical finding had important theoretical consequences.
Finkelstein and Esaulova (2006), for instance, had already proved that if a mortality
plateau exists, then a whole family of models - the so-called accelerated life models
(where  is not the same for everybody) - cannot adequately describe the evolution
of mortality with age. Subsequently, Missov and Finkelstein (2011) proved that the
existence of a mortality plateau is compatible with just a handful of theoretical
distributions of frailty, among which the Gamma distribution. These and other
pieces of evidence led Missov and Vaupel (2015) to conclude that the GGM stands
out as the best, possibly the sole, model for the description of mortality at adult ages.
In this paper we initially propose a new, simple procedure for the estimation of the
parameters of the GGM which attenuates (and, possibly, eliminates) the risks of
period biases due to the reduction in mortality in certain years. This reduction may
be due to, for instance, the medical progress of the past one hundred years or so,
and, if it is not kept under control, it may cause an underestimate of the rate of
ageing .
With this procedure we obtained new estimates for , which we used to run two
different checks on the assumptions of the GGM. First, we compared different
groups of cohorts born in different periods and countries in order to test whether the
rate of ageing is truly a constant. If different cohorts exposed to different
geographical, environmental and historical conditions were to experience different
rates of ageing, then the rate of ageing could not be constant at the individual level,
thus contradicting one of the hypotheses underpinning the Gamma-Gompertz model.
Up the GG curve, with gun and camera
PAGE 7
It is probably for this reason that Vaupel (2010) advanced the hypothesis that all
human cohorts share the same rate of ageing at the individual level.
Secondly, we checked whether the initial heterogeneity of frailty (that is, the
parameter σ20 in the GGM) can adequately capture the process of mortality
deceleration for which it is responsible (through the selection of the fittest, who tend
to survive to older ages).
The GGM failed both our tests: the rate of ageing changed across countries and over
time, and mortality deceleration was stronger than the GGM predicted. However,
both deviations from the empirical reality were not very large, and progressively
vanished as generations aged. In short, Gamma-Gompertz seems to be a good
model, overall, and especially so at very old ages.
The Methodology
Vaupel et al. (1979) proved that, if the four assumptions that we recalled earlier
hold, then the average (cohort) force of mortality from age a onwards evolves as a
GGM, that is
(1)
2
ln(𝜇̅𝑗,𝑥 ) = 𝛼𝑗 + 𝛽𝑥 + 𝜎𝑗,0
ln(𝑠̅𝑗,𝑥 ) + 𝜀𝑗,𝑥
where x=age-a, 𝜇̅𝑗,𝑥 represents the average force of mortality of cohort j at age x,
2
𝜎𝑗,0
stands for the initial variance of frailty and 𝑠̅𝑗,𝑥 represents the average survival
function. This model can be estimated with weighted least squares (WLS), where the
weights are the number of deaths at age x in cohort j (Horiuchi and Wilmoth 1998),
or with maximum likelihood (ML).
In both cases, unfortunately, two types of bias may affect the estimates. The first of
these, the incidental parameter problem, stems from the fact that cohort-specific
2
parameters (𝛼𝑗 and 𝜎𝑗,0
) are estimated together with a parameter which, instead, is
assumed to be the same for all the cohorts, 𝛽. In cases like this, the maximum
likelihood estimation of the common parameter (𝛽) is not consistent (Neyman and
Scott 1948).
There are a few known ways of circumventing this problem (see, e.g., Salinari and
De Santis, 2014), but in this paper we suggest a new, and simpler, one: eq. (1) can
2
be estimated on a small number of contiguous cohorts so that 𝛼𝑗 and 𝜎𝑗,0
can be
approximated by their means, that is, respectively, 𝛼 (the mean value of the initial
mortality) and 𝜎02 (the mean value of the initial variance of frailty)
(2)
ln(𝜇̅𝑗,𝑥 ) = 𝛼 + 𝛽𝑥 + 𝜎02 ln(𝑠̅𝑗,𝑥 ) + 𝜀𝑗,𝑥
PAGE 6
Gustavo De Santis and Giambattista Salinari
This procedure can be repeated for several, adjacent (and, if so desired, partially
overlapping) groups of cohorts, yielding G different estimates of g, which, if 
were truly a constant, should not differ significantly from one another.
The second form of bias arises because of the historical process of mortality
reduction of the past two centuries. The simple version of eq. (2) [o (1), for that
matter] holds if mortality is constant over time. If, instead, mortality declines, its
force j,x will vary not only by age but also by period, a circumstance which, if not
controlled for, results in a downward bias in the estimate of the rate of ageing 𝛽.1 In
order to take this case into account, eq. (2) may be rewritten as follows:
ln(𝜇̅𝑗,𝑥 ) = 𝛼 + 𝛽𝑥 + 𝜎02 ln(𝑠̅𝑗,𝑥 ) + ∑𝐽𝑗=0 ∑𝜔
𝑥=0 𝛾𝑗,𝑥 𝜉𝑗,𝑥 + 𝜀𝑗,𝑥
(3)
where each 𝜉𝑡,𝑥 is a dummy variable whose value is 1 for cohort j at age x, and 0
otherwise, and where 𝛾𝑗,𝑥 represents the reduction in mortality due to period effects
(e.g., medical progress) in year t (= j+x) for cohort j at age x. In eq. (3) the very
general assumption is that period effects may affect the force of mortality in an agespecific way.
A convenient way of getting rid of period effects, which is frequently used in panel
data analysis, is to differentiate eq. (3) by age:
(4)
2
̅̅̅̅𝑗,𝑥 ) = 𝛽(∆𝑥) + 𝜎𝑗,0
̅̅̅𝑗,𝑥 ) + ∑𝐽𝑗=0 ∑𝜔
ln(∆𝜇
ln(∆𝑠
𝑥=0 ∆𝛾𝑗,𝑥 𝜉𝑗,𝑥 + 𝜈𝑗,𝑥
where
and where 𝜈𝑗,𝑥
̅̅̅̅𝑗,𝑥 = 𝜇̅𝑗,𝑥+1 − 𝜇̅𝑗,𝑥
∆𝜇
̅̅̅
∆𝑠𝑗,𝑥 = 𝑠̅𝑗,𝑥+1 − 𝑠̅𝑗,𝑥
∆𝛾𝑗,𝑥 = 𝛾𝑗,𝑥+1 − 𝛾𝑗,𝑥
is the new error term.
If we assume that period effects are virtually the same at two contiguous ages x and
x+1, then ∆𝛾𝑗,𝑥 ≅ 0, which leads to
(5)
1
2
̅̅̅̅𝑗,𝑥 ) = 𝛽(∆𝑥) + 𝜎𝑗,0
̅̅̅𝑗,𝑥 ) + 𝜈𝑗,𝑥
ln⁡(∆𝜇
ln(∆𝑠
Let us assume that, at the individual level, the mortality rate doubling time is seven
year (which, incidentally, is not far from reality). The (rate of) mortality of an
individual aged x+7 at time t+7 should therefore be twice as high as that of an
individual aged x at time t. Let us suppose, however, that the introduction of a
medical innovation between t and t+7 has suddenly shifted the mortality hazard
function downward, leaving the mortality rate doubling time unchanged. In this
case, mortality at age x+7 in year t+7 will be lower than expected, conveying the
wrong impression of a longer mortality rate doubling time, and, also, of a change
(reduction) in the rate of ageing.
Up the GG curve, with gun and camera
PAGE 7
Besides, if mortality is analyzed by single years of age, x=1 and eq. (5) further
simplifies to
̅̅̅̅𝑗,𝑥 ) = 𝛽 + 𝜎02 ln(∆𝑠
̅̅̅𝑗,𝑥 ) + 𝜈𝑗,𝑥
ln⁡(∆𝜇
(6)
Eq. (6) has several advantages: it is better protected from the two risks of bias
discussed earlier (the incidental parameter problem and period effects), it is very
simple, and it has very few parameters: this is why we used it for the empirical
estimates of this paper.
More precisely, in order to estimate the temporal evolution of the rate of ageing β,
we estimated eq. (6) repeatedly, on groups of adjacent (and partially overlapping)
cohorts, starting in year s and grouping cohorts as follows: [s, s+19], [s+1, s+1+19],
…, [1907, 1915]. The initial groups contain 20 cohorts, while the final groups
contain fewer, down to nine, because the cohort mortality of the cohorts born in
1916 or later is not known in its entirety, as yet. The estimates thus obtained have
been attributed to the starting year of each interval: s, s+1, ..., 1907. The estimates
were obtained with weighted least squares (WLS), with weights equal to half the
number of deaths dx at age x, for x ranging between 75 and 99. Note that we
̅̅̅̅𝑗,𝑥 ) is ( 1 + 1 ), which can be
implicitly assumed that the variance of ln⁡(∆𝜇
dx
2
dx+1
approximated by ( ) (Brillinger 1986; Horiuchi and Wilmoth 1998).
𝑑𝑥
Let us now turn to the other issue: the decline with age in the variability of frailty, as
predicted by the Gamma-Gompertz model. In order to investigate this aspect we
introduced several interaction terms in model 6, as follows:
(7)
2
̅̅̅̅𝑗,𝑥 ) = 𝛽 + 𝜎02 ln(∆𝑠
̅̅̅𝑗,𝑥 ) + ∑𝐾−1
̅̅̅
ln⁡(∆𝜇
𝑘=1 𝜁𝑥,𝑘 𝜎0,𝑘 ln(∆𝑠𝑗,𝑥 ) + 𝜈𝑗,𝑥
where 𝜁𝑥,𝑘 indicates a dummy variable whose value is one when x belongs to a given
age interval (for instance, the ages 75-79) and 0 otherwise. In practice, we
“segmented” the available age range in K segments, each one characterized by a
2
potentially specific value of 𝜎0,𝑘
. For this paper, in particular, we used 5 segments,
each of the length of 5 years, for the ages 75 to 99. If the GGM of eq. 6 correctly
predicts the reduction in the variance of frailty within a given group of cohorts, the
2
various 𝜎0,𝑘
′𝑠 should not differ significantly from 0.
2
If, instead, the 𝜎0,𝑘
′𝑠 turn out to be significantly different from 0, we can move a
step forward, and compare the evolution with age of the variance of frailty as
predicted by Gamma-Gompertz (eq. 6):
(8)
with that predicted by eq. (7):
𝜎𝑥2 =
𝜎02
[1+𝜎02 𝑀(𝑥)]2
PAGE 6
Gustavo De Santis and Giambattista Salinari
𝜎𝑥2 = ∑𝐾
𝑘=1 𝜁𝑥,𝑘
(9)
2
𝜎0,𝑘
2
[1+𝜎0,𝑘 𝑀(𝑥)]2
in order to assess whether the decline predicted by the GGM is faster or slower than
empirical data suggest.
The Data
We worked on the cohort life tables of the Human Mortality Database (HMD).
These are available for a sufficiently long period (roughly between 1850 and 1920 but it depends on the country), and for the age range that interested us (75-99 years),
only for a handful of countries: Denmark, England and Wales, Finland, France,
Italy, the Netherlands, Norway, Sweden, and Switzerland (Table 1).
In order to minimize the potential distortion (mainly, selection) caused by the two
world wars, we considered only females, and ignored males.
Table 1: Female cohorts analyzed in this paper
Country
First cohort Last cohort
Denmark
1835
1915
England-Wales
1841
1915
Finland
1878
1915
France
1820
1915
Italy
1872
1915
Netherlands
1850
1915
Norway
1846
1915
Sweden
1820
1915
Switzerland
1876
1915
Note: All cohorts analyzed only in the age range 75-99 years. The analysis is carried out on groups of
adjacent cohort, as in a moving average. E.g., for Denmark we analyze the following groups of cohorts
(1835-1854); (1836-1855), ..., (1906-1915), (1907-1915). Note that these groups are usually of 20
cohorts, but for the final groups which have better data but fewer cohorts (down to nine, in the worst
case).
Source: Human Mortality Database.
The Results
In Figure 1 we present a comparison of the estimates of the rate of ageing (the beta
parameter of the GGM) obtained by working on the levels (eq. 2) and, separately, on
Up the GG curve, with gun and camera
PAGE 7
the differences (eq. 6) of the average force of mortality of female cohorts. What
emerges is that the rate of ageing is not the same in the various countries: it is higher
when it is estimated on the differences than when it is estimated directly, on levels;
and the distance between the two sets of estimates is generally, although not always,
larger at the beginning of the period (for older cohorts). Apart from the geographical
differences, which had already been noted and discussed elsewhere (e.g., Salinari
and De Santis 2014), these findings are all in all consistent with the conjecture that,
by about mid-19th century, significant cross-sectional improvements in health and
survival occurred, basically due to medical progress.
The historical evolution of the rate of ageing is irregular, but its variability is not
negligible - especially in Denmark, Finland, France, Norway, and Sweden, where,
incidentally, the evolution of the rate of ageing follows a comparable pattern: it is
generally higher in older cohorts, then it declines strongly, and later still it increases
again. In the latest cohorts of our sample, in most countries, the rate of ageing seems
to converge to values that are in the vicinity of 0.12.
The process of cross-sectional change in the force of mortality, due in large part to
medical progress, if not controlled for, introduces a bias in the estimates of eq. 2.
This can be verified by looking at Figure 2 where we show the evolution of the
initial variance of frailty (𝜎02 ) estimated with models (2) and (6). In most cases the
estimates obtained with eq. 2 (which works on levels) were implausible: the 𝜎02
parameter appeared to be negative, and significantly so, which is simply impossible.
This never happened when the estimates were instead obtained with eq. 6 (which
works on differences): the variance of frailty was always positive - at worst, not
significantly different from zero.
In the GGM, the process of mortality deceleration stems from the progressive
elimination of the frailest individuals from the cohort. If the four assumptions of the
model hold, frailty is Gamma-distributed at every age x, but the variance and the
mean of such distribution decline with age, because the surviving members of the
cohort form a more and more homogeneous (and robust) set, as time goes by.
In order to test if the force of mortality evolves in line with the prediction of the
GGM, we compared the estimates of eq. (6) and eq. (7) on the female cohorts born
between 1890 and 1915 in each of the nine countries covered by the present
analysis, for the ages 75-99 (observed in the years 1965 to 2014, with no severe
mortality crisis, due, for instance, to wars or epidemics). Furthermore, since we
worked on mortality differences, in this application, the analysis should be sheltered
from the possible biases of period effects. In short, if selection works as predicted by
the GGM, the two sets of estimates should not differ significantly.
The results of our check are shown in Table 2. In all of the nine countries considered
here, model 7 (i.e., the more complicated one, with the interaction terms) always
offered a better fit to the data (see, in particular the results of the ANOVA test).
Furthermore, the results were very consistent, suggesting that the deviation from the
GGM was systematic. In particular, the rate of ageing and the initial variance of
frailty estimated with eq. (7) were always higher than that estimated with GammaGompertz, implying a stronger selection (i.e., elimination of the frailest). Actually,
Table 2 and Figure 3 show that eq. (7) may be a better description of reality, or, in
PAGE 6
Gustavo De Santis and Giambattista Salinari
other words, that the variance of frailty is initially higher, but then declines faster,
than the GGM predicts. Interestingly, however, both Table 2 and Figure 3 show that
at very old ages (90 and over, for instance) the two models tend to converge,
yielding basically the same estimates of the variance of frailty.
Figure 1: Evolution over time of the rate of ageing (β) according to the GGM
Denmark
England−Wales
Finland
France
Italy
Netherland
Norway
Sweden
Switzerland
0.20
0.15
0.10
0.05
Rate of Aging
0.20
0.15
0.10
0.05
0.20
0.15
0.10
0.05
1820
1840
1860
1880
1900
1820
1840
1860
1880
1900
1820
1840
1860
1880
1900
Year of Birth
Note: Black (continuous) lines indicate values estimated on the differences (eq. 6); red (dotted) lines
indicate values estimated on levels (eq. 2). In both cases, the dashed lines define the limits of the 5%
confidence interval.
Up the GG curve, with gun and camera
PAGE 7
Figure 2: Evolution over time of the initial variance of frailty (σ20 ) according to Gamma-Gompertz
England−Wales
Denmark
Finland
0.2
0.4
0.50
0.1
0.2
0.25
0.0
0.0
0.00
−0.2
−0.1
France
Italy
0.2
0.4
Initial Variance of Frailty
Netherland
0.2
0.1
0.2
0.1
0.0
0.0
0.0
−0.1
−0.1
Norway
Sweden
Switzerland
0.4
0.2
0.2
0.3
0.1
0.2
0.1
0.1
0.0
0.0
0.0
−0.1
−0.1
1850
1875
1900
1850
1875
1900
1850
1875
1900
Year of Birth
Note: Black lines indicate values estimated on the differences (eq. 6); red lines values estimated on levels
(eq. 2). In both cases, the dashed lines define the limits of the 5% confidence interval.
PAGE 6
Gustavo De Santis and Giambattista Salinari
Table 2: Testing the σ20 parameter of the Gamma-Gompertz model
Country
Model
Parameter
beta
Denmark
England-Wales
Finland
France
Italy
Netherlands
Norway
Sweden
Switzerland
𝜎02
ANOVA
2
𝜎0,1
2
𝜎0,2
2
𝜎0,3
2
𝜎0,4
F
Pr
eq. 6
0,121 0,135
-
-
-
-
-
-
eq. 7
0,211 2,334
-1,110
-1,549
-1,801
-1,947
7,360
0,000
eq. 6
0,102 0,081
-
-
-
-
-
-
eq. 7
0,110 0,317
-0,116
-0,175
-0,222
-0,200
4,442
0,001
eq. 6
0,112 0,140
-
-
-
-
-
-
eq. 7
0,217 1,839
-0,576
-1,035
-1,284
-1,418
eq. 6
0,120 0,152
-
-
-
-
-
-
eq. 7
0,148 1,014
-0,483
-0,649
-0,748
-0,767
7,354
0,000
eq. 6
0,110 0,104
-
-
-
-
-
-
eq. 7
0,138 0,802
-0,411
-0,484
-0,582
-0,613
3,745
0,005
eq. 6
0,119 0,122
-
-
-
-
-
-
eq. 7
0,172 1,435
-0,626
-0,929
-1,118
-1,148
eq. 6
0,125 0,168
-
-
-
-
-
-
eq. 7
0,186 1,591
-0,656
-0,999
-1,176
-1,246
6,807
0,000
eq. 6
0,113 0,075
-
-
-
-
-
-
eq. 7
0,166 1,358
-0,587
-0,895
-1,057
-1,138
6,222
0,000
eq. 6
0,127 0,180
-
-
-
-
-
-
eq. 7
0,198
2,22
13,096 0,000
11,237 0,000
-1,0236 -1,5683 -1,7615 -1,8293 13,012 0,000
Note: Pr, in the last column, indicates the p.value of the ANOVA test. F is the ratio of the “explained”
variances in the two models, which should approach one if there were no significant differences.
Up the GG curve, with gun and camera
Figure 3: The evolution of the variance of frailty according to eq. (6) and (7)
PAGE 7
England−Wales
Denmark
Finland
0.3
2.0
1.5
1.5
0.2
1.0
1.0
0.5
0.5
0.1
0.0
0.0
France
Italy
Variance of frailty
0.75
0.6
0.50
0.4
Netherland
1.5
0.8
1.00
1.0
0.5
0.25
0.2
0.0
Norway
Sweden
Switzerland
1.5
2.0
1.0
1.5
1.0
1.0
0.5
0.5
0.5
0.0
0.0
0.0
75
80
85
90
95
75
80
85
90
95
75
80
85
90
95
Age
Model
eq. (8)
eq. (9)
Discussion
The GGM, or Gamma-Gompertz model, for the analysis of adult mortality by age is
a very good one - the best we have available, thus far. However, at closer inspection,
PAGE 6
Gustavo De Santis and Giambattista Salinari
it reveals a few weak points. The weakest of these is probably its assumption of a
constant rate of ageing, which contradicts the evidence found on the differences
between males and females, between countries and epochs and, recently, between
causes of death (Barbi 2003; Barbi et al. 2003; Salinari and De Santis 2014).
Incidentally, also the experiments carried out on lab animals have shown that the
rate of ageing may be altered by, for instance, calorie restriction (Masoro 2005,
2009; Fontana et al 2010).
The violation of the hypothesis of a constant rate of ageing, which the analyses of
this paper have confirmed, especially with regard to its evolution over time, may be
responsible also for the second main result of our paper: as cohorts get older, the
reduction in the variability of frailty may be stronger than the GGM predicts.
In all cases, these deviations from the empirical reality, which are never a major
issue and emerge only with very detailed analyses, become smaller and smaller with
age: in other words, Gamma-Gompertz may be the model towards which the force
of mortality converges at very old ages.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Barbi, E.: Assessing the Rate of Ageing of the Human Population. MPIDR Working Paper WP
2003-008 (2003)
Barbi, E., Caselli, G., Vallin J.: Trajectories of Extreme Survival in Heterogeneous Populations.
Population 58(1): 43-65 (2003)
Brillinger, D. R.: The natural variability of vital rates and associated statistics. Biometrics, 42: 693734 (1986)
Finkelstein, M., Esaulova, V.: Asymptotic behavior of a general class of mixture failure rates.
Advances in Applied Probability 38: 242–262 (2006)
Fontana, L., Partridge, L., Longo, V.D.: Extending Healthy Life Span. From Yeast to Humans.
Science 328: 321-326 (2010)
Gampe, J.: Human mortality beyond age 110. In: Heiner, M., Gampe, J., Jeune, B. Robine, J.M.,
Vaupel J.W. (eds.) Supercentenarians. Demographic Research Monographs, No. 7. Springer,
Heidelberg, pp. 219–230 (2010)
Gompertz, B.: On the nature of the function expressive of the law of human mortality.
Phylosophical Transactions 27: 513-519 (1825)
Horiuchi, S., Wilmoth, J.R.: Deceleration in the age pattern of mortality at older ages. Demography
35(4): 391-412 (1998)
Masoro, E.J.: Overview of caloric restriction and ageing. Mechanisms of Ageing and Development,
126, 913 (2005)
Masoro, E.J.: Caloric restriction-induced life extension of rats and mice: A critique of proposed
mechanisms. Biochimica et Biophysica Acta, 1790, 1040 (2009)
Missov, T.I., Finkelstein, M.: Admissible mixing distributions for a general class of mixture
survival models with known asymptotics. Theoretical Population Biology 80(1): 64-70 (2011)
Missov, T.I., Vaupel, J.W.: Mortality Implications of Mortality Plateaus. SIAM Review 57(1): 6170 (2015)
Neyman, J., Scott, E.L.: Consistent estimation from partially consistent observation. Econometrica
16: 1-32 (1948)
Salinari, G., De Santis, G.: Comparing the rate of individual senescence across time and space.
Population-E, 69(2): 165-190 (2014)
Vaupel, J.W., Manton K.G., Stallard, E.: The impact of heterogeneity in individual frailty on the
dynamics of mortality. Demography 16(3): 439-454 (1979)
Vaupel, J. W.: Biodemography of human aging. Nature 464(7288): 536-542 (2010)
Yashin, A., Iachine, I.A., Begun A.S.: Mortality Modeling: a Review. Mathematical Population
Studies, 8(4): 305-332 (2000)